Download POLITECNICO DI TORINO Invariance Principles and Extended

POLITECNICO DI TORINO
Dipartimento di Fisica
Invariance Principles and Extended Gravity:
Theories and Probes
Mariafelicia De Laurentis
Thesis submitted for the degree of
Doctor Philosophiae
Supervisors
Prof. Salvatore Capozziello
Prof. Angelo Tartaglia
Coordinator
Prof. Fausto Rossi
XXI CICLO
2006-2008
Soli Deo Gloria
J. S. Bach
I play the numbers as they are written but,
it is God who makes the physic!!!
-Sweet Nothings-John William Godward
Contents
Abstract
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Abstract
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List of Publications
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Notation
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Preface
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1 Introduction
1.1 General Relativity is the theory of gravity, isn’t it? . . . .
1.2 A high-energy theory of gravity? . . . . . . . . . . . . . .
1.2.1 Searching for the unknown . . . . . . . . . . . . .
1.2.2 Intrinsic limits in General Relativity and Quantum
1.2.3 A conceptual clash . . . . . . . . . . . . . . . . . .
1.2.4 The vision for unification . . . . . . . . . . . . . .
1.3 The Cosmological and Astrophysical riddles . . . . . . . .
1.3.1 Cosmology in a nutshell . . . . . . . . . . . . . . .
1.3.2 The first need for acceleration . . . . . . . . . . . .
1.3.3 Cosmological and Astronomical Observations . . .
1.3.4 The Cosmological Constant and its problems . . .
1.4 Is there a way out? . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Scalar fields as matter fields in Cosmology . . . . .
1.4.2 The dark energy problem . . . . . . . . . . . . . .
1.4.3 The dark matter problem . . . . . . . . . . . . . .
1.4.4 Towards Quantum Gravity, but how? . . . . . . .
1.4.5 Status of Gravity . . . . . . . . . . . . . . . . . . .
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Field Theory
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2 Extended Theories of Gravity
2.1 Theoretical motivations for Extended Theories of Gravity . . . . . . . .
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CONTENTS
2.2
2.3
2.4
What a good theory of Gravity has to do: General Relativity and
extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Structure of the Extended Theories of Gravity . . . . . . . . . . . .
2.3.1 Conformal transformations . . . . . . . . . . . . . . . . . . .
The Palatini Approach and the Intrinsic Conformal Structure . . . .
3 Gravity from Poincaré Gauge Invariance
3.1 What can generate the Gravity? . . . . .
3.2 Invariance Principle . . . . . . . . . . . .
3.3 Global Poincaré Invariance . . . . . . . .
3.4 Local Poincaré Invariance . . . . . . . . .
3.5 Spinors, Vectors and Tetrads . . . . . . .
3.6 Curvature, Torsion and Metric . . . . . .
3.7 Field Equations for Gravity . . . . . . . .
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4 Space-time deformations and conformal transformations towards extended theories of gravity
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4.1 Deformation and conformal transformations, how? . . . . . . . . . . . .
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4.2 Generalities on space-time deformations . . . . . . . . . . . . . . . . . .
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4.3 Properties of deforming matrices . . . . . . . . . . . . . . . . . . . . . .
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4.4 Metric deformations as perturbations and gravitational waves . . . . . . 102
4.5 Approximate Killing vectors . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.6 Deformations in f (R) -Theories . . . . . . . . . . . . . . . . . . . . . . . 106
5 Probing the Minkowskian limit: Gravitational waves in f (R)-Theories107
5.1 Why the gravitational waves in f (R)-Theories? . . . . . . . . . . . . . . 107
5.2 Stochastic background of gravitational waves ”tuned” by f (R) gravity . 108
5.3 Massive gravitational waves from f (R) theories of gravity: Potential detection with LISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4 Stochastic background of relic scalar gravitational waves from scalartensor gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6 Further probe: Parametrized Post Newtonian limit
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6.1 f (R) gravity constrained by PPN parameters and stochastic background
of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2 f (R) gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.3 f (R) viable models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.4 Constraining f (R)-models by PPN parameters . . . . . . . . . . . . . . 146
6.5 Stochastic backgrounds of gravitational waves to constrain f (R)-gravity 150
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Future perspectives and conclusions
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7.1 Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.2 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
CONTENTS
Bibliography
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Abstract
This thesis is devoted to the study of gravitational theories which can be seen as modifications or generalisations of General Relativity. The motivation for considering such
theories, stemming from Cosmology, High Enery Physics and Astrophysics in throrougly
discussed (cosmological problems, dark energy and dark matter problems, the lack of
succes so far in obtaining a successful formulation for Quantum Gravity). The basic
principles which a gravitational theory should follow, and their geometrical interpretation, are analysed in a broad perspective which highlights the basic assumptions of
General Relativity and suggests possible modifications which might be made. A number of such modifications are presented, focusing on certain specific classes of theories:
scalar-tensor theories, metric f (R) theories, Palatini f (R) theories. The caracteristics
of these theories are fully explored and attention is payed to issues of dynamical equivalence between them. Also, cosmological phenomenology within the realm of each of the
theories is discussed and it is shown that they can potentially address the well-known
cosmological problems. A number of viability cirteria are presented: cosmolgical tests,
Solar System tests....etc...
Finally, future perspectives in the field of modified gravity are discussed and the
possibility for going beyond a trial and error approach to modified gravity is explored.
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ABSTRACT
List of Publications
The research presented in this thesis was mainly conducted in Politecnico di Torino and
in Universit degli Studi di Napoli ”Federico II” between January 2006 and December
2008. This thesis is the result of the authors own work, as well as the outcome of
scientific collaborations stated below, except where explicit reference is made to the
results of others.
The content of this thesis is based on the following research papers pubblished in
refereed Journals or refereed conference proceedings:
Refereed papers
1. S. Capozziello, M. De Laurentis, M. Francaviglia: Stochastic background of
Gravitational waves as a Benchmark for Extended Theories of Gravity, Special
issue ” Problem in Modern Cosmology” Ed. Lavronov, 2009.
2. S. Capozziello, M. De Laurentis, F. Garufi, L. Milano:Relativistic orbits with
gravitomagnetic effects accepted for Physica Scripta 2009.
3. S. Capozziello, M. De Laurentis, S. Nojiri, S.D. Odintsov: f (R) gravity constrained by PPN parameters and stochastic background of gravitational waves ,
accepted for General Relativity and Gravitation 2009.
4. S. Capozziello, M. De Laurentis: Gravity from Local Poincare Gauge Invariance,
accettato su International Journal of Geometric Methods in Modern Physics to
appear in vol. 6, N 1 February (2009).
5. S. Capozziello, M. De Laurentis, C. Corda: Massive gravitational waves from
f (R) theories of gravity potential detecton with LISA, Physics Letter B 699, 255259 (2008).
6. S.Capozziello, M. De Laurentis: Gravitational waves from stellar encounters
Astroparticle Physics Volume 30,p. 105-112 (2008) .
7. S. Capozziello, M. De Laurentis, M. Francaviglia: Higher-order gravity and the
cosmological background of gravitational waves, Astroparticle Physics Volume 29,
Issue 2, p. 125-129 (2008).
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LIST OF PUBLICATION
8. S. Capozziello, M. De Laurentis, F. De Paolis, G. Ingrosso, A. Nucita: Gravitational waves from hyperbolic encounters, Modern Physics Letters A, Volume 23,
Issue 02, pp. 99-107 (2008).
9. S.Capozziello,C. Corda, M. De Laurentis: Stochastic background of relic scalar
gravitational waves from scalar-tensor gravity ,Modern Physics Letters A, Volume
22, Issue 35,(2007).
10. S. Capozziello, C. Corda, M. De Laurentis: Stochastic background of gravitational waves ”tuned” by f (R) gravity, Modern Physics Letters A vol.22 n15 pp.
1097-1104 (2007).
Proceeding
1. S. Capozziello, C.Corda,M. De Laurentis: Tuning the stochastic background
of gravitational waves with theory and observations Proc. of Societ Italiana di
Relativit Generale e Fisica della Gravitazione XVII Congresso SIGRAV General
Relativity and Gravitational Physics Torino, 4-7 Settembre 2006.
2. C. Corda, M. De Laurentis:Gravitational waves from the R−1 high-order theory
of gravity, To appear in Proc. of the 10th ICATPP International Conference on
Advanced Technology and Particle Physics (Villa Olmo Como, 8-12 october 2007)
published on World Scientific.
3. A. Tartaglia, A. Nagar, N. Radicella, M. De Laurentis and 26 coauthors: Summary of session B3: Analytic approximations, perturbation methods and their
applications, Classical and Quantum Gravity, Volume 25, Issue 11, pp. 114020
(2008).
4. S. Capozziello, C.Corda, M. De Laurentis: Tuning the stochastic background of
gravitational waves with theory and observations, To appear in the proceedings of
4th Italian-Sino Workshop on Relativistic Astrophysics, Pescara, Italy, 20-30 Jul
2007. Published in AIP Conf.Proc.966:257-263, (2008).
5. S. Capozziello, M. De Laurentis, L. Izzo: Stochastic Background of gravitational
waves tuned by f(R) gravityProc. of 3 Sueckelberg Workshop on Relativistic Field
Theories Pescara, Italy, 19-21 Jul 2008, Published in AIP Conf.Proc.(2009).
6. S. Capozziello,M. De Laurentis, L. Izzo: Detection of Cosmological Stochastic
Background of Gravitational waves in f(R) gravity with FASTICA Proc. of 3
Sueckelberg Workshop on Relativistic Field Theories Pescara, Italy, 19-21 Jul 2008,
Published in AIP Conf.Proc.(2009).
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Submitted papers
1. S. Capozziello, M. De Laurentis, M. Francaviglia, S. Mercadante: From Dark
Energy and Dark Matter to Dark Metric, submitted to Foundations of Physics
2009.
2. S. Capozziello, M. De Laurentis: Deformations in f(R) -theories, submitted to
Physics Letters B (2008).
3. S. Capozziello, M. De Laurentis, N. Radicella: Solutions to LQC with f(R)
submitted to Physics Letters B (2008)
4. S. Bellucci, S. Capozziello, M. De Laurentis, V. Faraoni:Position and frequency
shifts induced by massive modes of the gravitational wave background in alternative
gravity submitted to Physical Review D.
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LIST OF PUBLICATION
Notation
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14
NOTATION
Preface
The terms “modified gravity” and “alternative theory of gravity” have become standard
terminology for theories proposed for describing the gravitational interaction which differ
from the most conventional one, General Relativity. Modified or alternative theories of
gravity have a long history. The first attempts date back to the 1920s, soon after the
introduction of Einstein’s theory. Interest in this research field, which was initially
driven by curiosity or a desire to challenge the then newly introduced General Theory
of Relativity, has subsequently varied depending on circumstances, responding to the
appearance of new motivations. However, there has been more or less continuous activity
in this subject over the last 85 years.
When the research presented in this thesis began, interest in modified gravity was
already at a high point and it has continued increasing further until the present day.
This recent stimulus has mostly been due to combined motivation coming from the wellknown cosmological problems related to the accelerated expansion of the universe and
the feedback from High Energy Physics.
Due to the above, and even though the main scope of this thesis is to present the
research conducted by the author during the period january 2006 December 2008, a
significant efforts has been made so that this thesis can also serve as a guide for readers
who have recentetly developed an interest in this field. To this end, special attention
has been paid to giving a very general analysis of the foundations of gravitation theory.
Also, an effort has been made to present the theories discussed thorougly, so that readers
less familiar with this subject can be introduced to them before gradually moving on to
their more complicate characteristics and applications.
The outline of this thesis is as follows: In the Introduction, several open issues
related to gravity are discussed, including the cosmological problems related to dark
matter and dark energy, and the search for a theory of Quantum Gravity. Through the
presentation of a historical timeline of the passage from Newtonian gravity to General
Relativity, and a comparison with the current status of the latter in the light of the
problems just mentioned, the motivations for considering alternative theories of gravity
are introduced.
Chapter 2 is devoted a survey of what is intended for Extended Theories of Gravity
in the so called ”metric” and ”Palatini” approaches. In the Chapter 3 a compact,
self-contained approach to gravitation, based on the local Poincare gauge invariance, is
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PREFACE
proposed. Starting from the general invariance principle, we discuss the global and the
local Poincare invariance developing the spinor, vector and tetrad formalisms. These
tools allow to construct the curvature, torsion and metric tensors by the Fock-Ivanenko
covariant derivative. The resulting Einstein-Cartan theory describes a space endowed
with non-vanishing curvature and torsion while the gravitational field equations are
similar to the Yang-Mills equations of motion with the torsion tensor playing the role
of the Yang-Mills field strength. A definition of space-time metric deformations on an
n-dimensional manifold is given in the Chapter 4. We show that such deformations can
be regarded as extended conformal transformations. In particular, their features can
be related to the perturbation theory giving a natural picture by which gravitational
waves are described by small deformations of the metric. As further result, deformations
can be related to approximate Killing vectors (approximate symmetries) by which it is
possible to parameterize the deformed region of a given manifold. Space-time metric
deformations can be immediately recast in terms of perturbation theory allowing a
completely covariant approach to the problem of gravitational waves (GW) and then in
the Chapter 5 we show that the stochastic background of gravitational waves, produced
in the early cosmological epochs, strictly depends on the assumed theory of gravity. In
particular, the specific form of the function f (R), where R is the Ricci scalar, is related
to the evolution and the production mechanism of gravitational waves.
Then we given a generalization of previous results on gravitational waves (GWs)
from f (R) theories of gravity where the process is further generalized, showing that
every f (R) theory can be linearized producting a third massive mode of gravitational
radiation. The potential detectability of such massive GWs with LISA is also discussed
with the auxilium of longitudinal response functions.
Afetr we provide a distinctive spectrum of relic gravitational waves. In the framework
of scalar-tensor gravity, we discuss the scalar modes of gravitational waves and the
primordial production of this scalar component which is generated beside tensorial one.
We discuss also the upper limit for such a relic scalar component with respect to the
WMAP constraints. On the other hand, detecting the stochastic background by the
forthcoming interferometric experiments (VIRGO, LIGO, LISA) could be a further tool
to select the effective theory of gravity.
Finally in the Chapter 6 viable f (R)-gravity models are discussed toward Solar
System tests and stochastic background of gravitational waves. The aim is to achieve
experimental bounds for the theory at local and cosmological scales in order to select
models capable of addressing the accelerating cosmological expansion without cosmological constant but evading the weak field constraints. Beside large scale structure and
galactic dynamics, these bounds can be considered complementary in order to select
self-consistent theories of gravity working at the infrared limit.
A number of people have contributed in this thesis in various ways. First and
foremost, I would thank my PhD advisors, Salvatore Capozziello an Angelo Tartaglia,
for their constant support during the course of this work. It is difficult for me to imagine
having better advisors than Salvatore Capozziello and Angelo Tartaglia, to whom I am
17
truly grateful, not only for their guidance but also for standing by me in all my choices
and for the impressive amount of patience they have exhibited during the course of this
years. Special thanks to Capozziello for his untiring correction of my spelling, grammar
and (ab)use of the English language.
I cannot thank enough Mauro Francaviglia and Silvio Mercadante,not only for their
hard work on our common projects, but also for numerous hours of conversation and
debate moslty, but definitely not exclusively, on scientific issues. It has really been a
pleasure for me to collaborate with them. I am also very grateful to ......referee, my thesis
examiners, for undertaking the task of reviewing this manuscript and for their invaluable
suggestions. During the course of this research I have benefited from systematic or
occasional but always stimulating discussions with a number of people, besides those
already mentioned. Aware of the fact that I am running the risk of forgetting a few —
and apologising in advance for that — I will attempt to name them here and express
my gratitude:
Shin’ichi Nojiri, Sergei Odintsov, Fernando Atrio Barandela,Valerio Faraoni, J. Michael
Alim, Cosimo Stornaiolo, Roberto Cianci, Stefano Vignolo, Francesco de Paolis, G. Ingrosso, Achille Nucita, Fabio Garufi, Leopoldo Milano and Virgo Laboratory, Giampiero
Esposito, Christian Corda, Luca Izzo, Giovanni Covone, Piero Quarati, Ninfa Radicella,
Monica Capone, Matteo Ruggiero, Alessandro Nagar and Guido Rizzi.
The Politecnico has provided an ideal environment for conducting my research over
the last three years. I would like to thank all of my colleagues for contributing to that.
Special thanks to my office mate Marco Zamparo, Katarzyna Szymanska and Simone
Musso for all the fun we had while sharing a room in Departiment.
Last, but definitely not least, I would like to thank my family for their love and
their help in finding my way through life. If it was not for their continuous and untiring
support it would have been impossible for me to start, let alone finish, this PhD.
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PREFACE
Chapter 1
Introduction
1.1
General Relativity is the theory of gravity, isn’t it?
It is remarkable that gravity is probably the fundamental interaction which still remains
the most enigmatic, even though it is so related with phenomena experienced in everyday
life and is the one most easily conceived of without any sophisticated knowledge. As
a matter of fact, the gravitational interaction was the first one to be put under the
microscope of experimental investigation, obviously due to exactly the simplicity of
constructing a suitable experimental apparatus.
Galileo Galilei was the first to introduce pendulums and inclined planes to the study
of terrestrial gravity at the end of the 16th century. It seems that gravity played an
important role in the development of Galileo’s ideas about the necessity of experiment in
the study of science, which had a great impact on modern scientific thinking. However, it
was not until 1665, when Sir Isaac Newton introduced the now renowned “inverse-square
gravitational force law”, that terrestrial gravity was actually united with celestial gravity
in a single theory. Newton’s theory made correct predictions for a variety of phenomena
at different scales, including both terrestrial experiments and planetary motion.
Obviously, Newton’s contribution to gravity — quite apart from his enormous contribution to physics overall — is not restricted to the expression of the inverse square
law. Much attention should be paid to the conceptual basis of his gravitational theory,
which incorporates two key ideas:
1. The idea of absolute space, i.e. the view of space as a fixed, unaffected structure;
a rigid arena in which physical phenomena take place.
2. The idea of what was later called the Weak Equivalence Principle which, expressed
in the language of Newtonian theory, states that the inertial and the gravitational
mass coincide.
Asking whether Newton’s theory, or any other physical theory for that matter, is
right or wrong, would be ill-posed to begin with, since any consistent theory is apparently
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1. INTRODUCTION
“right”. A more appropriate way to pose the question would be to ask how suitable is
this theory for describing the physical world or, even better, how large a portion of the
physical world is sufficiently described by this theory. Also, one could ask how unique
the specific theory is for the description of the relevant phenomena. It was obvious in
the first 20 years after the introduction of Newtonian gravity that it did manage to
explain all of the aspects of gravity known at that time. However, all of the questions
above were posed sooner or later.
In 1855, Urbain Le Verrier observed a 35 arc-second excess precession of Mercury’s
orbit and later on, in 1882, Simon Newcomb measured this precession more accurately
to be 43 arc-seconds. This experimental fact was not predicted by Newton’s theory.
It should be noted that Le Verrier initially tried to explain the precession within the
context of Newtonian gravity, attributing it to the existence of another, yet unobserved,
planet whose orbit lies within that of Mercury. He was apparently influenced by the
fact that examining the distortion of the planetary orbit of Uranus in 1846 had led him,
and, independently, John Couch Adams, to the discovery of Neptune and the accurate
prediction of its position and momenta. However, this innermost planet was never found.
On the other hand, in 1893 Ernst Mach stated what was later called by Albert
Einstein “Mach’s principle”. This is the first constructive attack on Newton’s idea of
absolute space after the 17th century debate between Gottfried Wilhelm Leibniz and
Samuel Clarke (Clarke was acting as Newton’s spokesman) on the same subject, known
as the Leibniz–Clarke Correspondence. Mach’s idea can be considered as rather vague
in its initial formulation and it was essentially brought into mainstream physics later on
by Einstein along the following lines:
” ...inertia originates in a kind of interaction between bodies...”.
This is obviously in contradiction with Newton’s ideas, according to which inertia was
always relative to the absolute frame of space. There exists also a later, probably clearer
interpretation of Mach’s Principle, which, however, also differs in substance. This was
given by Dicke:
” The gravitational constant should be a function of the mass distribution in the universe”.
This is different from Newton’s idea of the gravitational constant as being universal and
unchanging. Now Newton’s basic axioms were being reconsidered.
But it was not until 1905, when Albert Einstein completed Special Relativity, that
Newtonian gravity would have to face a serious challenge. Einstein’s new theory, which
managed to explain a series of phenomena related to non-gravitational physics, appeared
1.2. A HIGH-ENERGY THEORY OF GRAVITY?
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to be incompatible with Newtonian gravity. Relative motion and all the linked concepts
had gone well beyond the ideas of Galileo and Newton and it seemed that Special Relativity should somehow be generalised to include non-inertial frames. In 1907, Einstein
introduced the equivalence between gravitation and inertia and successfully used it to
predict the gravitational redshift. Finally, in 1915, he completed the theory of General
Relativity, a generalisation of Special Relativity which included gravity. Remarkably,
the theory matched perfectly the experimental result for the precession of Mercury’s
orbit, as well as other experimental findings like the Lense-Thirring gravitomagnetic
precession (1918) and the gravitational deflection of light by the Sun, as measured in
1919 during a Solar eclipse by Arthur Eddington.
General Relativity overthrew Newtonian gravity and continues to be up to now an
extremely successful and well-accepted theory for gravitational phenomena. As mentioned before, and as often happens with physical theories, Newtonian gravity did not
lose its appeal to scientists. It was realised, of course, that it is of limited validity
compared to General Relativity, but it is still sufficient for most applications related
to gravity. What is more, at a certain limit of gravitational field strength and velocities, General Relativity inevitably reduces to Newtonian gravity. Newton’s equations
for gravity might have been generalised and some of the axioms of his theory may have
been abandoned, like the notion of an absolute frame, but some of the cornerstones of his
theory still exist in the foundations of General Relativity, the most prominent example
being the Equivalence Principle, in a more suitable formulation of course.
This brief chronological review, besides its historical interest, is outlined here also for
a practical reason. General Relativity is bound to face the same questions as were faced
by Newtonian gravity and many would agree that it is actually facing them now. In
the forthcoming sections, experimental facts and theoretical problems will be presented
which justify that this is indeed the case. Remarkably, there exists a striking similarity
to the problems which Newtonian gravity faced, i.e. difficulty in explaining particular
observations, incompatibility with other well established theories and lack of uniqueness.
This is the reason behind the question mark in the title of this section.
1.2
A high-energy theory of gravity?
Many will agree that modern physics is based on two great pillars: General Relativity
and Quantum Field Theory. Each of these two theories has been very successful in
its own arena of physical phenomena: General Relativity in describing gravitating systems and non-inertial frames from a classical viewpoint or on large enough scales, and
Quantum Field Theory in revealing the mysteries of high energy or small scale regimes
where a classical description breaks down. However, Quantum Field Theory assumes
that spacetime is flat and even its extensions, such as Quantum Field Theory in curved
space time, consider spacetime as a rigid arena inhabited by quantum fields. General
Relativity, on the other hand, does not take into account the quantum nature of mat-
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1. INTRODUCTION
ter. Therefore, it comes naturally to ask what happens if a strong gravitational field
is present at small, essentially quantum, scales? How do quantum fields behave in the
presence of gravity? To what extent are these amazing theories compatible?
Let us try to pose the problem more rigorously. Firstly, what needs to be clarified is
that there is no precise proof that gravity should have some quantum representation at
high energies or small scales, or even that it will retain its nature as an interaction. The
gravitational interaction is so weak compared with other interactions that the characteristic scale under which one would expect to experience non-classical effects relevant
to gravity, the Planck scale, is 10−33 cm. Such a scale is not of course accessible by
any current experiment and it is doubtful whether it will ever be accessible to future
experiments either1 . However, there are a number of reasons for which one would prefer
to fit together General Relativity and Quantum Field Theory [1, 2]. Let us list some of
the most prominent ones here and leave the discussion about how to address them for
the next section.
1.2.1
Searching for the unknown
Curiosity is probably the motivation leading scientific research. From this perspective
it would be at least unusual if the gravity research community was so easily willing to
abandon any attempt to describe the regime where both quantum and gravitational
effects are important. The fact that the Planck scale seems currently experimentally
inaccessible does not, in any way, imply that it is physically irrelevant. On the contrary,
one can easily name some very important open issues of contemporary physics that are
related to the Planck scale.
A particular example is the Big Bang scenario in which the universe inevitably goes
though an era in which its dimensions are smaller than the Planck scale (Planck era).
On the other hand, spacetime in General Relativity is a continuum and so in principle
all scales are relevant. From this perspective, in order to derive conclusions about the
nature of spacetime one has to answer the question of what happens on very small scales.
1.2.2
Intrinsic limits in General Relativity and Quantum Field Theory
The predictions of a theory can place limits on the extent of its ability to describe
the physical world. General Relativity is believed by some to be no exception to this
rule. Surprisingly, this problem is related to one of the most standard processes in a
gravitational theory: gravitational collapse. Studying gravitational collapse is not easy
since generating solutions to Einstein’s field equations can be a tedious procedure. We
only have a few exact solutions to hand and numerical or approximate solutions are often
the only resort. However, fortunately, this does not prevent one from making general
arguments about the ultimate fate of a collapsing object.
1
This does not imply, of course, that imprints of Quantum Gravity phenomenology cannot be found
in lower energy experiments.
1.2. A HIGH-ENERGY THEORY OF GRAVITY?
23
This was made possible after the proof of the Penrose–Hawking singularity theorems
[3, 4]. These theorems state that a generic spacetime cannot remain regular beyond a finite proper time, since gravitational collapse (or time reversal of cosmological expansion)
will inevitably lead to spacetime singularities. In a strict interpretation, the presence
of a singularity is inferred by geodesic incompleteness, i.e. the inability of an observer
travelling along a geodesic to extend this geodesic for an infinite time as measured by
his clock. In practical terms this can be loosely interpreted to mean that an observer
free-falling in a gravitational field will “hit” a singularity in a finite time and Einstein’s
equation cannot then predict what happens next. Such singularities seem to be present
in the centre of black holes. In the Big Bang scenario, the universe itself emerges out of
such a singularity.
Wheeler has compared the problem of gravitational collapse in General Relativity
with the collapse of the classical Rutherford atom due to radiation [5]. This raises hopes
that principles of quantum mechanics may resolve the problem of singularities in General
Relativity, as happened for the Rutherford model. In a more general perspective, it
is reasonable to hope that quantization can help to overcome these intrinsic limits of
General Relativity.
On the other hand, it is not only General Relativity that has an intrinsic limit. Quantum Field Theory presents some disturbing ultraviolet divergences. Such divergences,
caused by the fact that integrals corresponding to the Feynman diagrams diverge due to
very high energy contributions — hence the name ultraviolet — are discretely removed
by a process called renormalization. These divergences are attributed to the perturbative nature of the quantization process and the renormalization procedure is somehow
unappealing and probably not so fundamental, since it appears to cure them in a way
that can easily be considered as non-rigorous from a mathematical viewpoint. A nonperturbative approach is believed to be free of such divergences and there is hope that
Quantum Gravity may allow that (for early results see [6, 7, 8, 9, 10]).
1.2.3
A conceptual clash
Every theory is based on a series of conceptual assumption and General Relativity and
Quantum Field Theory are no exceptions. On the other hand, for two theories to work in
a complementary way to each other and fit well together, one would expect an agreement
between their conceptual bases. This is not necessarily the case here.
There are two main points of tension between General Relativity and Quantum Field
Theory. The first has to do with the concept of time: Time is given and not dynamical in
Quantum Field Theory and this is closely related to the fact that spacetime is considered
as a fixed arena where phenomena take place, much like Newtonian mechanics. On the
other hand, General Relativity considers spacetime as being dynamical, with time alone
not being such a relevant concept. It is more of a theory describing relations between
different events in spacetime than a theory that describes evolution over some running
parameter. One could go further and seek for the connection between what is mentioned
24
1. INTRODUCTION
here and the differences between gauge invariance as a symmetry of Quantum Field
Theory and diffeomorphism invariance as a symmetry of General Relativity.
The second conceptual issue has to do with Heisenberg’s uncertainty principle in
Quantum Theory which is absent in General Relativity as a classical theory. It is
interesting to note that General Relativity, a theory in which background independence
is a key concept, actually introduces spacetime as an exact and fully detailed record of
the past, the present and the future. Everything would be fixed for a super-observer
that could look at this 4-dimensional space from a fifth dimension. On the other hand,
Quantum Field Theory, a background dependent theory, manages to include a degree
of uncertainty for the position of any event in spacetime.
Having a precise mathematical structure for a physical theory is always important,
but getting answers to conceptual issues is always the main motivation for studying
physics in the first place. Trying to attain a quantum theory of gravity could lead to
such answers.
1.2.4
The vision for unification
Apart from strictly scientific reasons for trying to make a match between Quantum Field
Theory and General Relativity, there is also a long-standing intellectual desire, maybe
of a philosophical nature or stemming from physical intuition, to bring the fundamental
interactions to a unification. This was the vision of Einstein himself in his late years. His
perspective was that a geometric description might be the solution. Nowdays most of the
scientists active in this field would disagree with this approach to unification and there is
much debate about whether the geometric interpretation or a field theory interpretation
of General Relativity is actually preferable — Steven Weinberg for example even claimed
in [11] that “no-one” takes a geometric viewpoint of gravity “seriously”. However, very
few would argue that such a unification should not be one of the major goals of modern
physics. An elegant theory leading to a much deeper understanding of both gravity and
the quantum world could be the reward for achieving this.
1.3
1.3.1
The Cosmological and Astrophysical riddles
Cosmology in a nutshell
Taking things in chronological order, we started by discussing the possible shortcomings
of General Relativity on very small scales, as those were the first to appear in the
literature. However, if there is one scale for which gravity is by far of the utmost
importance, this is surely the cosmic scale. Given the fact that other interactions are
short-range and that at cosmological scales we expect matter characteristics related to
them to have “averaged out” — for example we do not expect that the universe has
an overall charge — gravity should be the force which rules cosmic evolution. Let us
1.3. THE COSMOLOGICAL AND ASTROPHYSICAL RIDDLES
25
see briefly how this comes about by considering Einstein’s equations combined with our
more obvious assumptions about the main characteristics of the observable universe.
Even though matter is not equally distributed through space and by simple browsing
through the sky one can observe distinct structures such as stars and galaxies, if attention
is focused on larger scales the universe appears as if it was made by patching together
multiple copies of the same pattern, i.e. a suitably large elementary volume around
the Earth and another elementary volume of the same size elsewhere will have little
difference. This suitable scale is actually ≈ 108 light years, slightly larger than the
typical size of a cluster of galaxies. In Cosmology one wants to deal with scales larger
than that and to describe the universe as a whole. Therefore, as far as Cosmology is
concerned the universe can be very well described as homogeneous and isotropic.
To make the above statement useful from a quantitative point of view, we have to
turn it into an idealized assumption about the matter and geometry of the Universe.
Note that the universe is assumed to be spatially homogeneous and isotropic at each
instant of cosmic time. In more rigorous terms, we are talking about homogeneity on
each one of a set of 3-dimensional space-like hypersurfaces. For the matter, we assume
a perfect fluid description and these spacelike hypersurfaces are defined in terms of a
family of fundamental observers who are comoving with this perfect fluid and who can
synchronise their comoving clocks so as to measure the universal cosmic time. The
matter content of the universe is then just described by two parameters, a uniform
density ρ and a uniform pressure p, as if the matter in stars and atoms is scattered
through space. For the geometry we idealize the curvature of space to be everywhere
the same.
Let us proceed by imposing these assumption on the equation describing gravity
and very briefly review the derivation of the equations governing the dynamics of the
universe, namely the Friedmann equations. We refer the reader to standard textbooks
for a more detailed discussion of the precise geometric definitions of homogeneity and
isotropy and their implications for the form of the metric (e.g. [11]). Additionally, for
what comes next, the reader is assumed to be acquainted with the basics of General
Relativity, some of which will also be reviewed in the next chapter.
Einstein’s equation has the following form
Gµν = 8 π G Tµν ,
where
(1.1)
1
R gµν
(1.2)
2
is the Einstein tensor and Rµν and R are the Ricci tensor and Ricci scalar of the metric
gµν . G is the gravitational constant and Tµν is the matter stress-energy tensor. Under
the assumptions of homogeneity and isotropy, the metric can take the form
dr 2
2 2
2
2
2
2
2
2
+ r dθ + r sin (θ)dφ ,
(1.3)
ds = −dt + a (t)
1 − kr 2
Gµν ≡ Rµν −
26
1. INTRODUCTION
known as the Friedmann-Lemaı̂tre-Robertson-Walker metric (FLRW). k = −1, 0, 1 according to whether the universe is hyperspherical (“closed”), spatially flat, or hyperbolic
(“open”) and a(t) is called the scale factor2 . Inserting this metric into eq. (1.1) and taking into account that for a perfect fluid
T µν = (ρ + p)uµ uν + p gµν ,
(1.4)
where uµ denotes the four-velocity of an observer comoving with the fluid and ρ and p
are the energy density and the pressure of the fluid, one gets the following equations
2
8π Gρ
k
ȧ
=
− 2,
a
3
a
ä
4π G
= −
(ρ + 3p) ,
a
3
(1.5)
(1.6)
where an overdot denotes differentiation with respect to coordinate time t.
Eqs. (1.5) and (1.6) are called the Friedmann equations. By imposing homogeneity
and isotropy as characteristics of the universe that remain unchanged with time on suitably large scales we have implicitly restricted any evolution to affect only one remaining
characteristic: its size. This is the reason why the Friedmann equations are equations
for the scale factor, a(t), which is a measure of the evolution of the size of any length
scale in the universe. Eq. (1.5), being an equation in ȧ, tells us about the velocity of
the expansion or contraction, whereas eq. (1.6), which involves ä, tells us about the acceleration of the expansion or the contraction. According to the Big Bang scenario, the
universe starts expanding with some initial velocity. Setting aside the contribution of
the k-term for the moment, eq. (1.5) implies that the universe will continue to expand
as long as there is matter in it. Let us also take into consideration the contribution
of the k-term, which measures the spatial curvature and in which k takes the values
−1, 0, 1. If k = 0 the spatial part of the metric (1.3) reduces to a flat metric expressed
in spherical coordinates. Therefore, the universe is spatially flat and eq. (1.5) implies
that it has to become infinite, with ρ approaching zero, in order for the expansion to
halt. On the other hand, if k = 1 the expansion can halt at a finite density at which the
matter contribution is balanced by the k-term. Therefore, at a finite time the universe
will stop expanding and will re-collapse. Finally for k = −1 one can see that even if
matter is completely dissolved, the k-term will continue to “pump” the expansion which
means that the latter can never halt and the universe will expand forever.
2
The traditional cosmological language of “closed”/“flat”/“open” is inaccurate and quite misleading
and, therefore, should be avoided. Even if one ignores the possibility of nonstandard topologies, the
k = 0 spatially flat 3-manifold is, in any sensible use of the word, “open”. If one allows nonstandard
topologies (by modding out by a suitable symmetry group) then there are, in any sensible use of the
word, “closed” k = 0 spatially flat 3-manifolds (tori), and also “closed” k = −1 hyperbolic 3-manifolds.
Finally the distinction between flat and spatially flat is important, and obscuring this distinction is
dangerous.
1.3. THE COSMOLOGICAL AND ASTROPHYSICAL RIDDLES
27
Let us now focus on eq. (1.6) which, as already mentioned, governs the acceleration
of the expansion. Notice that k does not appear in this equation, i.e. the acceleration
does not depend on the characteristics of the spatial curvature. Eq. (1.6) reveals what
would be expected by simple intuition: that gravity is always an attractive force. Let
us see this in detail. The Newtonian analogue of eq. (1.6) would be
ä
4πG
=−
ρ,
a
3
(1.7)
where ρ denotes the matter density. Due to the minus sign on the right hand side and
the positivity of the density, this equation implies that the expansion will always be
slowed by gravity.
The presence of the pressure term in eq. (1.6) is simply due to the fact that in General
Relativity, it is not simply matter that gravitates but actually energy and therefore the
pressure should be included. For what could be called ordinary matter (e.g. radiation,
dust, perfect fluids, etc.) the pressure can be expected to be positive, as with the density.
More precisely, one could ask that the matter satisfies the four energy conditions [263]:
1. Null Energy Condition: ρ + p ≥ 0,
2. Weak Energy Condition: ρ ≥ 0,
ρ + p ≥ 0,
3. Strong Energy Condition: ρ + p ≥ 0,
ρ + 3p ≥ 0,
4. Dominant Energy Condition: ρ ≥ |p|.
We give these conditions here in terms of the components of the stress-energy tensor of
a perfect fluid but they can be found in a more generic form in [263]. Therefore, once
positivity of the pressure or the validity of the Strong Energy Condition is assumed,
gravity remains always an attractive force also in General Relativity 3 .
To sum up, even without attempting to solve the Friedmann equations, we have
already arrived at a well-established conclusion: Once we assume, according to the Big
Bang scenario, that the universe is expanding, then, according to General Relativity
and with ordinary matter considerations, this expansion should always be decelerated.
Is this what actually happens though?
1.3.2
The first need for acceleration
We derived the Friedmann equations using two assumptions: homogeneity and isotropy
of the universe. Both assumptions seem very reasonable considering how the universe
appears to be today. However, there are always the questions of why does the universe
appear to be this way and how did it arrive at its present form through its evolution.
3
When quantum effects are taken into account, one or more of the energy conditions can be violated,
even though a suitably averaged version may still be satisfied. However, there are even classical fields
that can violate the energy conditions, as we will see latter on.
28
1. INTRODUCTION
More importantly though, one has to consider whether the description of the universe
by the Big Bang model and the Friedmann equations is self-consistent and agrees not
only with a rough picture of the universe but also with the more precise current picture
of it.
Let us put the problem in more rigorous terms. First of all one needs to clarify what
is meant by “universe”. Given that the speed of light (and consequently of any signal
carrying information) is finite and adopting the Big Bang scenario, not every region
of spacetime is accessible to us. The age of the universe sets an upper limit for the
largest distance from which a point in space may have received information. This is
what is called a “particle horizon” and its size changes with time. What we refer to
as the universe is the part of the universe causally connected to us — the part inside
our particle horizon. What happens outside this region is inaccessible to us but more
importantly it does not affect us, at least not directly. However, it is possible to have
two regions that are both accessible and causally connected to us, or to some other
observer, but are not causally connected with each other. They just have to be inside
our particle horizon without being inside each other’s particle horizons. It is intuitive
that regions that are causally connected can be homogeneous — they have had the time
to interact. However, homogeneity of regions which are not causally connected would
have to be attributed to some initial homogeneity of the universe since local interactions
cannot be effective for producing this.
The picture of the universe that we observe is indeed homogeneous and isotropic
on scales larger than we would expect based on our calculation regarding its age and
causality. This problem was first posed in the late 1960s and has been known as the
horizon problem [11, 13]. One could look to solve it by assuming that the universe
is perhaps much older and this is why in the past the horizon problem has also been
reformulated in the form of a question: how did the universe grow to be so old? However,
this would require the age of the universe to differ by orders of magnitude from the value
estimated by observations. So the homogeneity of the universe, at least at first sight
and as long as we believe in the cosmological model at hand, appears to be built into
the initial conditions.
Another problem, which is similar and appeared at the same time, is the flatness
problem. To pose it rigorously let us return to the Friedmann equations and more
specifically to eq. (1.5). The Hubble parameter H is defined as H = ȧ/a. We can use it
to define what is called the critical density
ρc =
3 H2
,
8π G
(1.8)
which is the density which would make the 3-geometry flat. Finally, we can use the
1.3. THE COSMOLOGICAL AND ASTROPHYSICAL RIDDLES
29
critical density in order to create the dimensionless fractions
Ω =
ρ
,
ρc
Ωk = −
(1.9)
k
a2 H 2
.
(1.10)
It is easy to verify from eq. (1.5) that
Ω + Ωk = 1.
(1.11)
As dimensionless quantities, Ω and Ωk are measurable, and by the 1970s it was already
known that the current value of Ω appears to be very close to 1 (see for example [14]).
Extrapolating into the past reveals that Ω would have had to be even closer to 1, making
the contribution of Ωk , and consequently of the k-term in eq. (1.5), exponentially small.
The name “flatness problem” can be slightly misleading and therefore it needs to
be clarified that the value of k obviously remains unaffected by the evolution. To avoid
misconceptions it is therefore better to formulate the flatness problems in terms of Ω
itself. The fact that Ω seems to be taking a value so close to the critical one at early
times is not a consequence of the evolution and once more, as happened with the horizon
problem, it appears as a strange coincidence which can only be attributed to some fine
tuning of the initial conditions.
But is it reasonable to assume that the universe started in such a homogeneous state,
even at scales that where not causally connected, or that its density was dramatically
close to its critical value without any apparent reason? Even if the universe started with
extremely small inhomogeneities it would still not present such a homogeneous picture
currently. Even if shortcomings like the horizon and flatness problems do not constitute
logical inconsistencies of the standard cosmological model but rather indicate that the
present state of the universe depends critically on some initial state, this is definitely a
feature that many consider undesirable.
So, by the 1970s Cosmology was facing new challenges. Early attempts to address
these problems involved implementing a recurring or oscillatory behaviour for the universe and therefore were departing from the standard ideas of cosmological evolution
[15, 16, 17]. This problem also triggered Charles W. Misner to propose the “Mixmaster Universe” (Bianchi type IX metric), in which a chaotic behaviour was supposed to
ultimately lead to statistical homogeneity and isotropy [18]. However, all of these ideas
have proved to be non-viable descriptions of the observed universe.
A possible solution came in the early 1980s when Alan Guth proposed that a period
of exponential expansion could be the answer [19]. The main idea is quite simple:
an exponential increase of the scale factor a(t) implies that the Hubble parameter H
remains constant. On the other hand, one can define the Hubble radius c/H(t) which,
roughly speaking, is a measure of the radius of the observable universe at a certain
time t. Then, when a(t) increases exponentially, the Hubble radius remains constant,
whereas any physical length scale increases exponentially in size. This implies that in a
30
1. INTRODUCTION
short period of time, any lengthscale which could, for example, be the distance between
two initially causally connected observers, can become larger than the Hubble radius.
So, if the universe passed through a phase of very rapid expansion, then the part of
it that we can observe today may have been significantly smaller at early times than
what one would naively calculate using the Friedmann equations. If this period lasted
long enough, then the observed universe could have been small enough to be causally
connected at the very early stage of its evolution. This rapid expansion would also drive
Ωk to zero and consequently Ω to 1 today, due to the very large value that the scale factor
a(t) would currently have, compared to its initial value. Additionally, such a procedure
is very efficient in smoothing out inhomogeneities, since the physical wavelength of a
perturbation can rapidly grow to be larger than the Hubble radius. Thus, both of the
problems mentioned above seem to be effectively addressed.
Guth was not the only person who proposed the idea of an accelerated phase and
some will argue he was not even the first. Contemporaneously with him, Alexei Starobinski had proposed that an exponential expansion could be triggered by quantum corrections to gravity and provide a mechanism to replace the initial singularity [20]. There
are also earlier proposals whose spirit is very similar to that of Guth, such as those by
Demosthenes Kazanas [21], Katsuhiko Sato [22] and Robert Brout et al. [23]. However,
Guth’s name is the one most related with these idea since he was the first to provide
a coherent and complete picture on how an exponential expansion could address the
cosmological problems mentioned above. This period of accelerated expansion is known
as inflation, a terminology borrowed from economics due to the apparent similarity between the growth of the scale factor in Cosmology and the growth of prices during an
inflationary period. To be more precise, one defines as inflation any period in the cosmic
evolution for which
ä > 0.
(1.12)
However, a more detailed discussion reveals that an exponential expansion, or at
least quasi-exponential since what is really needed is that the physical scales increase
much more rapidly than the Hubble radius increases, is not something trivial to achieve.
As discussed in the previous section, it does not appear to be easy to trigger such an era
in the evolution of the universe, since accelerated expansion seems impossible according
to eq. (1.6), as long as both the density and the pressure remain positive. In other
words, satisfying eq. (1.12) requires
(ρ + 3p) < 0 ⇒ ρ < −3p,
(1.13)
and assuming that the energy density cannot be negative, inflation can only be achieved
if the overall pressure of the ideal fluid which we are using to describe the universe
becomes negative. In more technical terms, eq. (1.13) implies the violation of the Strong
Energy Condition [263].
It does not seem possible for any kind of baryonic matter to satisfy eq. (1.13), which
directly implies that a period of accelerated expansion in the universe evolution can
1.3. THE COSMOLOGICAL AND ASTROPHYSICAL RIDDLES
31
only be achieved within the framework of General Relativity if some new form of matter
field with special characteristics is introduced. Before presenting any scenario of this
sort though, let us resort to observations to convince ourselves about whether such a
cosmological era is indeed necessary.
1.3.3
Cosmological and Astronomical Observations
In reviewing the early theoretical shortcomings of the Big Bang evolutionary model of
the universe we have seen indications for an inflationary era. The best way to confirm
those indications is probably to resort to the observational data at hand for having a
verification. Fortunately, there are currently very powerful and precise observations that
allow us to look back to very early times.
A typical example of such observations is the Cosmic Microwave Background Radiation (CMBR). In the early universe, baryons, photons and electrons formed a hot
plasma, in which the mean free path of a photon was very short due to constant interactions of the photons with the plasma through Thomson scattering. However, due
to the expansion of the universe and the subsequent decrease of temperature, it subsequently became energetically favourable for electrons to combine with protons to form
hydrogen atoms (recombination). This allowed photons to travel freely through space.
This decoupling of photons from matter is believed to have taken place at a redshift of
z ∼ 1088, when the age of the universe was about 380, 000 years old or approximately
13.7 billion years ago. The photons which left the last scattering surface at that time,
then travelled freely through space and have continued cooling since then. In 1965 Penzias and Wilson noticed that a Dicke radiometer which they were intending to use for
radio astronomy observations and satellite communication experiments had an excess
3.5K antenna temperature which they could not account for. They had, in fact, detected
the CMBR, which actually had already been theoretically predicted in 1948 by George
Gamow. The measurement of the CMBR, apart from giving Penzias and Wilson a Nobel
prize publication [24], was also to become the number one verification of the Big Bang
model.
Later measurements showed that the CMBR has a black body spectrum corresponding to approximately 2.7 K and verifies the high degree of isotropy of the universe. However, it was soon realized that attention should be focused not on the overall isotropy,
but on the small anisotropies present in the CMBR, which reveal density fluctuations
[25, 26]. This triggered a numbered of experiments, such as COBE, Toco, BOOMERanG
and MAXIMA [27, 28, 29, 30, 31, 32]. The most recent one is the Wilkinson Microwave
Anisotropy Probe (WMAP) [33] and there are also new experiments planned for the
near future, such as the Planck mission [34].
The density fluctuations indicated by the small anisotropies in the temperature of
CMBR are believed to act as seeds for gravitational collapse, leading to gravitationally
bound objects which constitute the large scale matter structures currently present in
the universe [35]. This allows us to build a coherent scenario about how these structures
32
1. INTRODUCTION
were formed and to explain the current small scale inhomogeneities and anisotropies.
Besides the CMBR, which gives information about the initial anisotropies, one can
resort to galaxy surveys for complementary information. Current surveys determining
the distribution of galaxies include the 2 degree Field Galaxy Redshift Survey (2dF
GRS) [36] and the ongoing Sloan Digital Sky Survey (SDSS) [37]. There are also other
methods used to measure the density variations such as gravitational lensing [38] and
X-ray measurements [39].
Besides the CMBR and Large Scale Structure surveys, another class of observations
that appears to be of special interest in Cosmology are those of type Ia supernovae. These
exploding stellar objects are believed to be approximately standard candles, i.e. astronomical objects with known luminosity and absolute magnitude. Therefore, they can be
used to reveal distances, leading to the possibility of forming a redshift-distance relation
and thereby measuring the expansion of the universe at different redshifts. For this
purpose, there are a number of supernova surveys [40, 41, 42].
But let us return to how we can use the outcome of the experimental measurements
mentioned above in order to infer whether a period of accelerated expansion has occurred. The most recent CMBR dataset is that of the Three-Year WMAP Observations
[43] and results are derived using combined WMAP data and data from supernova and
galaxy surveys in many cases. To begin with, let us focus on the value of Ωk . The
WMAP data (combined with Supernova Legacy Survey data [41]) indicates that
Ωk = −0.015+0.020
−0.016 ,
(1.14)
i.e. that Ω is very close to unity and the universe appears to be spatially flat, while the
power spectrum of the CMBR appears to be consistent with gaussianity and adiabaticity [44, 45]. Both of these facts are in perfect agreement with the predictions of the
inflationary paradigm.
In fact, even though the theoretical issues mentioned in the previous paragraph
(i.e. the horizon and the flatness problem) were the motivations for introducing the
inflationary paradigm, it is the possibility of relating large scale structure formation with
initial quantum fluctuations that appears today as the major advantage of inflation [46].
Even if one would choose to dismiss, or find another way to address, problems related
to the initial conditions, it is very difficult to construct any other theory which could
successfully explain the presence of over-densities with values suitable for leading to the
present picture of our universe at smaller scales [35]. Therefore, even though it might
be premature to claim that the inflationary paradigm has been experimentally verified,
it seems that the evidence for there having been a period of accelerated expansion of
the universe in the past is very compelling.
However, observational data hold more surprises. Even though Ω is measured to be
very close to unity, the contribution of matter to it, Ωm , is only of the order of 24%.
Therefore, there seems to be some unknown form of energy in the universe, often called
dark energy. What is more, observations indicate that, if one tries to model dark energy
1.3. THE COSMOLOGICAL AND ASTROPHYSICAL RIDDLES
33
as a perfect fluid with equation of state p = wρ then
wde = −1.06+0.13
−0.08 ,
(1.15)
so that dark energy appears to satisfy eq. (1.13). Since it is the dominant energy
component today, this implies that the universe should be undergoing an accelerated
expansion currently as well. This is also what was found earlier using supernova surveys
[40].
As is well known, between the two periods of acceleration (inflation and the current
era) the other conventional eras of evolutionary Cosmology should take place. This
means that inflation should be followed by Big Bang Nucleosynthesis (BBN), referring
to the production of nuclei other than hydrogen. There are very strict bounds on
the abundances of primordial light elements, such as deuterium, helium and lithium,
coming from observations [47] which do not seem to allow significant deviations from
the standard cosmological model [48]. This implies that BBN almost certainly took
place during an era of radiation domination, i.e. a period in which radiation was the
most important contribution to the energy density. On the other hand, the formation
of matter structures requires that the radiation dominated era is followed by a matter
dominated era. The transition, from radiation domination to matter domination, comes
naturally since the matter energy density is inversely proportional to the volume and,
therefore, proportional to a−3 , whereas the radiation energy density is proportional to
a−4 and so it decreases faster than the matter energy density as the universe expands.
To sum up, our current picture of the evolution of the universe as inferred from
observations comprises a pre-inflationary (probably quantum gravitational) era followed
by an inflationary era, a radiation dominated era, a matter dominated era and then a
second era of accelerated expansion which is currently taking place. Such an evolution
departs seriously from the one expected if one just takes into account General Relativity
and conventional matter and therefore appears to be quite unorthodox.
But puzzling observations do not seem to stop here. As mentioned before, Ωm
accounts for approximately 24% of the energy density of the universe. However, one also
has to ask how much of this 24% is actually ordinary baryonic matter. Observations
indicate that the contribution of baryons to that, Ωb , is of the order of Ωb ∼ 0.04 leaving
some 20% of the total energy content of the universe and some 83% of the matter
content to be accounted for by some unknown unobserved form of matter, called dark
matter. Differently from dark energy, dark matter has the gravitational characteristics
of ordinary matter (hence the name) and does not violate the Strong Energy Condition.
However, it is not directly observed since it appears to interact very weakly if at all.
The first indications for the existence of dark matter did not come from Cosmology.
Historically, it was Fritz Zwicky who first posed the “missing mass” question for the
Coma cluster of galaxies [49, 50] in 1933. After applying the virial theorem in order to
compute the mass of the cluster needed to account for the motion of the galaxies near
to its edges, he compared this with the mass obtained from galaxy counts and the total
34
1. INTRODUCTION
brightness of the cluster. The virial mass turned out to be larger by a factor of almost
400.
Later, in 1959, Kahn and Waljter were the first to propose the presence of dark
matter in individual galaxies [51]. However, it was in the 1970s that the first compelling
evidence for the existence of dark matter came about: the rotation curves of galaxies,
i.e. the velocity curves of stars as functions of the radius, did not appear to have the
expected shapes. The velocities, instead of decreasing at large distances as expected from
Keplerian dynamics and the fact that most of the visible mass of a galaxy is located
near to its centre, appeared to be flat [52, 53, 54]. As long as Keplerian dynamics are
considered correct, this implies that there should be more matter than just the luminous
matter, and this additional matter should have a different distribution within the galaxy
(dark matter halo).
Much work has been done in the last 35 years to analyse the problem of dark matter in
astrophysical environments (for recent reviews see [55, 56, 57]) and there are also recent
findings, such as the observations related to the Bullet Cluster, that deserve a special
mention4 . The main conclusion that can be drawn is that some form of dark matter
is present in galaxies and clusters of galaxies. What is more, taking also into account
the fact that dark matter appears to greatly dominate over ordinary baryonic matter
at cosmological scales, it is not surprising that current models of structure formation
consider it as a main ingredient (e.g. [59]).
1.3.4
The Cosmological Constant and its problems
We have just seen some of the main characteristics of the universe as inferred from
observations. Let us now set aside for the moment the discussion of the earlier epochs
of the universe and inflation and concentrate on the characteristic of the universe as
it appears today: it is probably spatially flat (Ωk ∼ 0), expanding in a accelerated
manner as confirmed both from supernova surveys and WMAP, and its matter energy
composition consists of approximately 76% dark energy, 20% dark matter and only 4%
ordinary baryonic matter. One has to admit that this picture is not only surprising but
maybe even embarrassing, since it is not at all close to what one would have expected
based on the standard cosmological model and what is more it reveals that almost 96%
of the energy content of the universe has a composition which is unknown to us.
In any case, let us see which is the simplest model that agrees with the observational
data. To begin with, we need to find a simple explanation for the accelerated expansion.
The first physicist to consider a universe which exhibits an accelerated expansion was
probably Willem de Sitter [60]. A de Sitter space is the maximally symmetric, simplyconnected, Lorentzian manifold with constant positive curvature. It may be regarded
as the Lorentzian analogue of an n-sphere in n dimensions. However, the de Sitter
4
Weak lensing observations of the Bullet cluster (1E0657-558), which is actually a unique cluster
merger, appear to provide direct evidence for the existence of dark matter [58].
1.3. THE COSMOLOGICAL AND ASTROPHYSICAL RIDDLES
35
spacetime is not a solution of the Einstein equations, unless one adds a cosmological
constant Λ to them, i.e. adds on the left hand side of eq. (1.1) the term Λgµν .
Such a term was not included initially by Einstein, even though this is technically
possible since, according to the reasoning which he gave for arriving at the gravitational
field equations, the left hand side has to be a second rank tensor constructed from
the Ricci tensor and the metric, which is divergence free. Clearly, the presence of a
cosmological constant does not contradict these requirements. In fact, Einstein was
the first to introduce the cosmological constant, thinking that it would allow him to
derive a solution of the field equations describing a static universe [61]. The idea of
a static universe was then rapidly abandoned however when Hubble discovered that
the universe is expanding and Einstein appears to have changed his mind about the
cosmological constant: Gamow quotes in his autobiography, My World Line (1970):
“Much later, when I was discussing cosmological problems with Einstein, he remarked
that the introduction of the cosmological term was the biggest blunder of his life” and
Pais quotes a 1923 letter of Einstein to Weyl with his reaction to the discovery of
the expansion of the universe: “If thereis no quasi-static world, then away with the
cosmological term!” [62].
In any case, once the cosmological term is included in the Einstein equations, de
Sitter space becomes a solution. Actually, the de Sitter metric can be brought into the
form of the FLRW metric in eq. (1.3) with the scale factor and the Hubble parameter
given by5 .
a(t) = eH t ,
8π G
Λ.
H2 =
3
(1.16)
(1.17)
This is sometimes referred to as the de Sitter universe and it can be seen that it is
expanding exponentially.
The de Sitter solution is a vacuum solution. However, if we allow the cosmological
term to be present in the field equations, the Friedmann equations (1.5) and (1.6) will
be modified so as to include the de Sitter spacetime as a solution:
2
ȧ
=
a
ä
=
a
k
8π Gρ + Λ
− 2,
3
a
Λ 4π G
−
(ρ + 3p) .
3
3
(1.18)
(1.19)
From eq. (1.19) one infers that the universe can now enter a phase of accelerated expansion once the cosmological constant term dominates over the matter term on the right
hand side. This is bound to happen since the value of the cosmological constant stays
5
Note that de Sitter space is an example of a manifold that can be sliced in 3 ways — k = +1, k = 0,
k = −1 — with each coordinate patch covering a different portions of spacetime. We are referring here
just to the k = 0 slicing for simplicity.
36
1. INTRODUCTION
unchanged during the evolution, whereas the matter density decreases like a3 . In other
words, the universe is bound to approach a de Sitter space asymptotically in time.
On the other hand Ω in eq. (1.11) can now be split in two different contributions,
ΩΛ = Λ/(3 H 2 ) and Ωm , so that eq. (1.11) takes the form
Ωm + ΩΛ + Ωk = 1.
(1.20)
In this sense, the observations presented previously can be interpreted to mean that
ΩΛ ∼ 0.72 and the cosmological constant can account for the mysterious dark energy
responsible for the current accelerated expansion. One should not fail to notice that
Ωm does not only refer to baryons. As mentioned before, it also includes dark matter,
which is actually the dominant contribution. Currently, dark matter is mostly treated as
being cold and not baryonic, since these characteristics appear to be in good accordance
with the data. This implies that, apart from the gravitational interaction, it does not
have other interactions — or at least that it interacts extremely weakly — and can be
regarded as collisionless dust, with an effective equation of state p = 0 (we will return
to the distinction between cold and hot dark matter shortly).
We have sketched our way to what is referred to as the Λ Cold Dark Matter or
ΛCDM model. This is a phenomenological model which is sometimes also called the
concordance model of Big Bang Cosmology, since it is more of an empirical fit to the data.
It is the simplest model that can fit the cosmic microwave background observations as
well as large scale structure observations and supernova observations of the accelerating
expansion of the universe with a remarkable agreement (see for instance [43]). As a
phenomenological model, however, it gives no insight about the nature of dark matter,
or the reason for the presence of the cosmological constant, neither does it justify the
value of the latter.
While it seems easy to convince someone that an answer is indeed required to the
question “what exactly is dark matter and why is it almost 9 times more abundant than
ordinary matter”, the presence of the cosmological constant in the field equations might
not be so disturbing for some. Therefore, let us for the moment put aside the dark
matter problem — we will return to it shortly — and consider how natural it is to try
to explain the dark energy problem by a cosmological constant (see [63, 64, 65, 66] for
reviews).
It has already been mentioned that there is absolutely no reason to discard the presence of a cosmological constant in the field equations from a gravitational and mathematical perspective. Nonetheless, it is also reasonable to assume that there should be
a theoretical motivation for including it — after all there are numerous modifications
that could be made to the left hand side of the gravitational field equation and still lead
to a consistent theory from a mathematical perspective and we are not aware of any
other theory that includes more than one fundamental constant. On the other hand, it
is easy to see that the cosmological term can be moved to the right hand side of the
field equations with the opposite sign and be regarded as some sort of matter term.
It can then be put into the form of a stress-energy tensor Tνµ = diag(Λ, −Λ, −Λ, −Λ),
1.3. THE COSMOLOGICAL AND ASTROPHYSICAL RIDDLES
37
i.e. resembling a perfect fluid with equation of state p = −ρ or w = −1. Notice the
very good agreement with the value of wde inferred from observations (eq. (1.15)), which
explains the success of the ΛCDM model.
Once the cosmological constant term is considered to be a matter term, a natural
explanation for it seems to arise: the cosmological constant can represent the vacuum
energy associated with the matter fields. One should not be surprised that empty space
has a non-zero energy density once, apart from General Relativity, field theory is also
taken into consideration. Actually, Local Lorentz Invariance implies that the expectation
value of the stress energy tensor in vacuum is
hTµν i = −hρigµν ,
(1.21)
and hρi is generically non-zero. To demonstrate this, we can take the simple example of
a scalar field [67]. Its energy density will be
1
1
ρφ = φ̇2 + (∇sp φ)2 + V (φ),
2
2
(1.22)
where ∇sp denotes the spatial gradient and V (φ) is the potential. The energy density
will become constant for any constant value φ = φ0 and there is no reason to believe
that for φ = φ0 , V (φ0 ) should be zero. One could in general assume that there is some
principle or symmetry that dictates it, but nothing like this has been found up to now.
So in general one should expect that matter fields have a non-vanishing vacuum energy,
i.e. that hρi is non-zero.
Within this perspective, effectively there should be a cosmological constant in the
field equations, given by
Λ = 8 π Ghρi.
(1.23)
One could, therefore, think to use the Standard Model of particle physics in order to
estimate its value. Unfortunately, however, hρi actually diverges due to the contribution
of very high-frequency modes. No reliable exact calculation can be made but it is easy
to make a rough estimate once a cutoff is considered (see for instance [11, 67]). Taking
the cutoff to be the Planck scale (MPlank = 1018 GeV), which is a typical scale at which
the validity of classical gravity is becoming questionable, the outcome is
ρΛ ∼ (1027 eV)4 .
(1.24)
On the other hand, observations indicate that
ρΛ ∼ (10−3 eV)4 .
(1.25)
Obviously the discrepancy between these two estimates is very large for being attributed
to any rough approximation. There is a difference of 120-orders-of-magnitude, which is
large enough to be considered embarrassing. One could validly claim that we should not
be comparing energy densities but mass scales by considering a mass scale for the vacuum
38
1. INTRODUCTION
implicitly defined through ρΛ = MΛ4 . However, this will not really make a difference,
since a 30-orders-of-magnitude discrepancy in mass scale hardly makes a good estimate.
This constitutes the so-called cosmological constant problem.
Unfortunately, this is not the only problem related to the cosmological constant. The
other known problem goes under the name of the coincidence problem. It is apparent
from the data that ΩΛ ∼ 0.72 and Ωm ∼ 0.28 have comparable values today. However,
as the universe expands their fractional contributions change rapidly since
ρΛ
ΩΛ
=
∝ a3 .
Ωm
ρm
(1.26)
Since Λ is a constant, ρΛ should once have been negligible compared to the energy
densities of both matter and radiation and, as dictated by eq. (1.26), it will come to
dominate completely at some point in the late time universe. However, the striking fact
is that the period of transition between matter domination and cosmological constant
domination is very short compared to cosmological time scales6 . The puzzle is, therefore, why we live precisely in this very special era [67]. Obviously, the transition from
matter domination to cosmological constant domination, or, alternatively stated, from
deceleration to acceleration, would happen eventually. The question is, why now?
To sum up, including a cosmological constant in the field equations appears as an
easy way to address issues like the late time accelerated expansion but unfortunately
it comes with a price: the cosmological constant and coincidence problems. We will
return to this discussion from this point later on but for the moment let us close the
present section with an overall comment about the ΛCDM model. Its value should
definitely not be underestimated. In spite of any potential problems that it may have,
it is still a remarkable fit to observational data while at the same time being elegantly
simple. One should always bear in mind how useful a simple empirical fit to the data
may be. On the other hand, the ΛCDM model should also not be over-estimated. Being
a phenomenological model, with poor theoretical motivation at the moment, one should
not necessarily expect to discover in it some fundamental secrets of nature.
1.4
Is there a way out?
In the previous sections, some of the most prominent problems of contemporary physics
were presented. As one would expect, since these questions were initially posed, many
attempts to address one or more of them have been pursued. These problems may be
viewed as being unrelated to each other, or grouped in different categories at will. For
instance, one could follow a broad research field grouping, much like the one attempted in
the previous section, dividing them into problems related with Cosmology and problems
related with high energy physics, or group them according to whether they refer to
6
Note that in the presence of a positive cosmological constant there is an infinite future in which Λ
is dominating.
1.4. IS THERE A WAY OUT?
39
unexplained observations or theoretical shortcomings. In any case there is one common
denominator in all of these problems. They are all somehow related to gravity.
The way in which one chooses to group or divide these problems proposes a natural
path to follow for their solution. In this section let us very briefly review some of the
most well-known and conventional solutions proposed in the literature, which mainly
assume that all or at least most of these issues are unrelated. Then we can proceed to
argue why and how the appearance of so many yet to be explained puzzles related to
gravity and General Relativity may imply that there is something wrong with our current
understanding of the gravitational interaction even at a classical level, resembling the
historically recorded transition from Newtonian gravity to General Relativity described
in section 1.1. With that we will conclude this introductory chapter.
1.4.1
Scalar fields as matter fields in Cosmology
We have already discussed the need for an inflationary period in the early universe.
However, we have not yet attempted to trace the cause of such an accelerated expansion.
Since the presence of a cosmological constant could in principle account for that, one is
tempted to explore this possibility, as in the case of late time acceleration. Unfortunately,
this simple solution is bound not to work for a very simple reason: once the cosmological
constant dominates over matter there is no way for matter to dominate again. Inflation
has to end at some point, as already mentioned, so that Big Bang Nucleosynthesis and
structure formation can take place. Our presence in the universe is all the evidence
one needs for that. Therefore, one is forced to seek other, dynamical solutions to this
problem.
As long as one is convinced that gravity is well described by General Relativity, the
only option left is to assume that it is a matter field that is responsible for inflation.
However, this matter field should have a rather unusual property: its effective equation
of state should satisfy eq. (1.13), i.e. it should have a negative pressure and actually
violate the Strong Energy Condition. Fortunately, matter fields with this property do
exist. A typical simple example is a scalar field φ.
A scalar field minimally coupled to gravity, satisfies the Klein–Gordon equation
∇2 φ + V ′ (φ) = 0,
(1.27)
where ∇µ denotes the covariant derivative, ∇2 ≡ ∇µ ∇µ , V (φ) is the potential and the
prime denotes partial differentiation with respect to the argument. Assuming that the
scalar field is homogeneous and therefore φ ≡ φ(t) we can write its energy density and
pressure as
ρφ =
pφ =
1 2
φ̇ + V (φ),
2
1 2
φ̇ − V (φ),
2
(1.28)
(1.29)
40
1. INTRODUCTION
while, in a FLRW spacetime, eq. (1.27) takes the following form:
φ̈ + 3H φ̇ + V ′ (φ) = 0.
(1.30)
It is now apparent that if φ̇2 < V (φ) then the pressure is indeed negative. In fact
wφ = pφ /ρφ approaches −1 when φ̇2 ≪ V (φ).
In general a scalar field that leads to inflation is referred to as the inflaton. Since
we invoked such a field instead of a cosmological constant, claiming that in this way we
can successfully end inflation, let us see how this is achieved. Assuming that the scalar
dominates over both matter and radiation and neglecting for the moment the spatial
curvature term for simplicity, eq. (1.5) takes the form
8π G 1 2
2
φ̇ + V (φ) .
(1.31)
H ≈
3
2
If, together with the condition φ̇2 < V (φ), we require that φ̈ is negligible in eq. (1.30)
then eqs. (1.31) and (1.30) reduce to
8π G
V (φ),
3
3H φ̇ ≈ −V ′ (φ).
H2 ≈
(1.32)
(1.33)
This constitutes the slow-roll approximation since the potential terms are dominant with
respect to the kinetic terms, causing the scalar to roll slowly from one value to another.
To be more rigorous, one can define two slow-roll parameters
′ 2
V
,
(1.34)
ε(φ) = 4 π G
V
V ′′
η(φ) = 8 π G ,
(1.35)
V
for which the conditions ε(φ) ≪ 1 and η(φ) ≪ 1 are necessary in order for the slow-roll
approximation to hold [68, 69]. Note that these are not sufficient conditions since they
only restrict the form of the potential. One also has to make sure that eq. (1.33) is
satisfied. In any case, what we want to focus on at this point is that one can start
with a scalar that initially satisfies the slow-roll conditions but, after some period, φ
can be driven to such a value so as to violate them. A typical example is that of
V (φ) = m2 φ2 /2, where these conditions are satisfied as long as φ2 > 16 π G but, as φ
approaches the minimum of the potential, a point will be reached where φ2 > 16 π G
will cease to hold. Once the slow-roll conditions are violated, inflation can be naturally
driven to an end since φ̇2 can begin to dominate again in eq. (1.29).
However, just ending inflation is not enough. After such an era the universe would
be a cold and empty place unable to evolve dynamically to anything close to the picture
which we observe today. A viable model for inflation should include a mechanism that
1.4. IS THERE A WAY OUT?
41
will allow the universe to return to the standard Big Bang scenario. This mechanism
is called reheating and consists mainly of three processes: a period of non-inflationary
scalar field dynamics, after the slow-roll approximation has ceased to be valid, the
creation and decay of inflaton particles and the thermalization of the products of this
decay [35]. Reheating is an extensive and intricate subject and analyzing it goes beyond
the scope of this introduction. We refer the reader to [70, 71, 72, 73, 74, 75, 76] for more
information.
On the same grounds, we will refrain here from mentioning specific models for inflation and from discussing subtleties with using inflation in order to address problems
of initial conditions such as those stated in paragraph 1.3.2. We refer the reader to the
literature for further reading [75, 76, 77, 340, 79].
Before closing this paragraph, it should be mentioned that scalar fields can be used
to account for the late-time accelerated expansion of the universe in the same way as
the inflaton is used in inflationary models. Since, however, this subject overlaps with
the subject of dark energy, we will discuss it in the next sub-section which is dedicated
to the dark energy problem.
1.4.2
The dark energy problem
We have already seen that there seems to be compelling observational evidence that the
universe is currently undergoing an accelerated expansion and we have also discussed
the problems that arise if a cosmological constant is considered to be responsible for this
acceleration within the framework of the ΛCDM model. Based on that, one can classify
the attempts to address the problem of finding a mechanism that will account for the
late-time accelerated expansion in two categories: those that try to find direct solutions
to the cosmological constant and the coincidence problems and consequently attempt
to provide an appealing theoretical explanation for the presence and the value of the
cosmological constant, and those that abandon the idea of the cosmological constant
altogether and attempt to find alternative ways to explain the acceleration.
Let us state two of the main approaches followed to solve the cosmological constant
problem directly:
The first approach resorts to High Energy Physics. The general idea is simple and
can be summed up in the question: Are we counting properly? This refers to the
quite naive calculation mentioned previously, according to which the energy density
of the cosmological constant as calculated theoretically should be 10120 times larger
than its observed value. Even though the question is simple and reasonable, giving
a precise answer to it is actually very complicated since, as mentioned already, little
is known about how to make an exact calculation of the vacuum energy of quantum
fields. There are indications coming from contemporary particle physics theories, such
as supersymmetry (SUSY), which imply that one can be led to different values for the
energy density of vacuum from the one mentioned before (eq. (1.24)). For instance,
since no superpartners of known particles have been discovered in accelerators, one can
42
1. INTRODUCTION
assume that supersymmetry was broken at some scale of the order of 103 GeV or higher.
If this is the case, one would expect that
4
ρΛ ∼ MSUSY
≥ (1012 eV)4 .
(1.36)
This calculation gives an estimate for the energy density of the vacuum which is 60
orders of magnitude smaller than the one presented previously in eq. (1.24). However,
the value estimated here is still 60 orders of magnitude larger than the one inferred from
observations (eq. (1.25)). Other estimates with or without a reference to supersymmetry
or based on string theory or loop quantum gravity exist. One example is the approach
of Ref. [80] where an attempt is made to use our knowledge from condensed matter
systems in order to explain the value of the cosmological constant. We will not, however,
list further examples here but refer the reader to [63, 65] and references therein for
more details. In any case, the general flavour is that it is very difficult to avoid the
cosmological constant problem by following such approaches without making some fine
tuning within the fundamental theory used to perform the calculation for the energy
density of vacuum. Also, such approaches mostly fail to address the second problem
related to the cosmological constant: the coincidence problem.
he second direct approach for solving problems related to the cosmological constant
has a long history and was given the name “anthropic principle” by Brandon Carter
[81, 82, 83]. Unfortunately, the anthropic principle leaves a lot of room for different
formulations or even misinterpretations. Following [63] we can identify at least three
versions, starting from a very mild one, that probably no one really disagrees with but
is not very useful for answering questions, stating essentially that our mere existence
can potentially serve as an experimental tool. The second version on the other hand is a
rather strong one, stating that the laws of nature are by themselves incomplete and become complete only if one adds the requirement that conditions should allow intelligent
life to arise, for only in the presence of intelligent life does science become meaningful.
It is apparent that such a formulation places intelligent life or science at the centre of
attention as far as the universe is concerned. From this perspective one cannot help but
notice that the anthropic principle becomes reminiscent of the Ptolemaic model. Additionally, to quote Weinberg: “...although science is clearly impossible without scientists,
it is not clear that the universe is impossible without science”. The third and most
moderate version of the anthropic principle, known as the “weak anthropic principle”
states essentially that observers will only observe conditions which allow for observers.
This version is the one mostly discussed from a scientific perspective and even though
it might seem tautological, it acquires a meaning if one invokes probability theory.
To be more concrete, as opposed to the second stronger formulation, the weak anthropic principle does not assume some sort of conspiracy of nature aimed at creating
intelligent life. It merely states that, since the existence of intelligent observes requires
certain conditions, it is not possible for them in practice to observe any other conditions, something that introduces a bias in any probabilistic analysis. This, of course,
requires one extra assumption: that parts of the universe, either in space or time, might
1.4. IS THERE A WAY OUT?
43
indeed be in alternative conditions. Unfortunately we cannot conclude at this point
whether this last statement is true. Assuming that it is, one could put constrains on
the value of the cosmological constant by requiring that it should be small enough for
galaxies to form as in [84] and arrive at the conclusion that the currently observed value
of the cosmological constant is by no means unlikely. Some modern theories do allow
such alternative states of the universe to co-exist (multiverse), and for this reason it has
recently been argued that the anthropic principle could even be placed on firm ground
by using the ideas of string theory for the “anthropic or string landscape”, consisting
of a large number of different false vacua [85]. However, admitting that there are limits
on our ability to unambiguously and directly explain the observable universe inevitably
comes with a disappointment. It is for this reason that many physicists would refrain
from using the anthropic principle or at least they would consider it only as a last resort,
when all other possibilities have failed.
Let us now proceed to the indirect ways of solving problems related with the cosmological constant. As already mentioned, the main approach of this kind is to dismiss the
cosmological constant completely and assume that there is some form of dynamical dark
energy. In this sense, dark energy and vacuum energy are not related and therefore the
cosmological constant problem ceases to exist, at least in the strict formulation given
above. However, this comes with a cost: as mentioned previously, observational data
seem to be in very good agreement with having a cosmological constant, therefore implying that any form of dynamical dark energy should be able to mimic a cosmological
constant very precisely at present times. This is not something easy to achieve. In order
to be clearer and also to have the possibility to discuss how well dynamical forms of
dark energy can address the cosmological constant and coincidence problems, let us use
an example.
Given the discussion presented earlier about inflation, it should be clear by now
that if a matter field is to account for accelerated expansion, it should have a special
characteristic: negative pressure or more precisely p ≤ −ρ/3. Once again, as in the
inflationary paradigm, the obvious candidate is a scalar field. When such a field is used
to represent dark energy it is usually called quintessence [86, 87, 88, 89, 90, 91, 92, 93, 94].
Quintessence is one of the simplest and probably the most common alternative to the
cosmological constant.
If the scalar field is taken to be spatially homogeneous, its equation of motion in
an FLRW spacetime will be given by eq. (1.30) and its energy density and pressure
will be given by eqs. (1.28) and (1.29) respectively, just like the inflaton. As dictated
by observations through eq. (1.15), a viable candidate for dark energy should have an
effective equation of state with w very close to minus one. In the previous section it
was mentioned that this can be achieved for a scalar field if the condition φ̇2 ≪ V (φ)
holds. This should not be confused with the slow-roll condition for inflation, which
just requires that φ̇2 < V (φ) and also places a constraint for φ̈. However, there is a
similarity in the spirit of the two conditions, namely that in both cases the scalar field
is required, roughly speaking, to be slowly-varying. It is worth mentioning that the
44
1. INTRODUCTION
condition φ̇2 ≪ V (φ) effectively restricts the form of the potential V .
Let us see how well quintessence can address the cosmological constant problem.
One has to bear in mind that the value given in eq. (1.25) for the energy density of
the cosmological constant now becomes the current value of the energy density of the
scalar ρφ . Since we have asked that the potential terms should be very dominant with
respect to the kinetic terms, this value for the energy density effectively constrains the
current value of the potential. What is more, the equation of motion for the scalar field,
eq. (1.30) is that of a damped oscillator, 3H φ̇ being the friction term. This implies
that, p
for φ to be rolling slowly enough so that φ̇2 ≪ V (φ) could be satisfied, then
be
H ∼ V ′′ (φ). Consequently, this means that the current value of V ′′ (φ) should
p
that of the observed cosmological constant or, taking also into account that V ′′ (φ)
represents the effective mass of the scalar mφ , that
mφ ∼ 10−33 eV.
(1.37)
Such a small value for the mass of the scalar field raises doubts about whether
quintessence really solves the cosmological constant problem or actually just transfers it
from the domain of Cosmology to the domain of particle physics. The reason for this is
that the scalar fields usually present in quantum field theory have masses many orders
of magnitude larger than that given in eq. (1.37) and, hence, this poses a naturalness
question (see [65] for more details). For instance, one of the well-known problems in
particle physics, the hierarchy problem, concerns explaining why the Higgs field appears
to have a mass of 1011 eV which is much smaller that the grand unification/Planck
scale, 1025 -1028 eV. As commented in [67], one can then imagine how hard it could
be to explain the existence of a field with a mass equal to 10−33 eV. In all fairness to
quintessence, however, it should be stated that the current value of the energy density
of dark energy (or vacuum, depending on the approach) is an observational fact, and so
it does not seem possible to completely dismiss this number in some way. All that is
left to do, therefore, is to put the cosmological constant problem on new grounds that
will hopefully be more suitable for explaining it.
One should not forget, however, also the coincidence problem. There are attempts
to address it within the context of quintessence mainly based on what is referred to as
tracker models [95, 96, 97, 98, 99, 100, 101]. These are specific models of quintessence
whose special characteristic is that the energy density of the scalar parallels that of
matter or radiation for a part of the evolution which is significant enough so as to
remove the coincidence problem. What is interesting is that these models do not in
general require specific initial conditions, which means that the coincidence problem is
not just turned into an initial conditions fine-tuning problem. Of course, the dependence
of such approaches on the parameters of the potential remains inevitable.
It is also worth mentioning that φ should give rise to some force, which judging
from its mass should be long-range, if the scalar couples to ordinary matter. From a
particle physics point of view, one could expect that this is indeed the case, even if
those interactions would have to be seriously suppressed by powers of the Planck scale
1.4. IS THERE A WAY OUT?
45
[102, 103]. However, current limits based on experiments concerning a fifth-force or time
dependence of coupling constants, appear to be several orders of magnitude lower than
this expectation [102, 103]. This implies that, if quintessence really exists, then there
should be a mechanism — probably a symmetry — that suppresses these couplings.
Yet another possibility for addressing the cosmological constant problems, or more
precisely for dismissing them, comes when one adopts the approach that the accelerated
expansion as inferred by observations is not due to some new physics but is actually
due to a misinterpretation or an abuse of the underlying model being used. The Big
Bang model is based on certain assumptions, such as homogeneity and isotropy, and
apparently all calculations made rely on these assumptions. Even though at present one
cannot claim that there is compelling evidence for this, it could be, for example, that
the role of inhomogeneities is underestimated in the standard cosmological model and
a more detailed model may provide a natural solution to the problem of dark energy,
even by changing our interpretation of current observations (for instance see [104] and
references therein).
1.4.3
The dark matter problem
As we have already seen, the presence of dark matter is indirectly inferred from observations through its gravitational interaction. Therefore, if one accepts that General
Relativity describes gravity correctly, then an explanation for the nature of dark matter
as some form of matter yet to be observed in the universe or in the laboratory should
be given. Note that dark matter is used here generically to mean matter that does not
emit light. So, to begin with, its nature could be either baryonic and non-baryonic. The
candidates for baryonic dark matter are mostly quite conventional astrophysical objects
such as brown dwarfs, massive black holes and cold diffuse gas. However, there is precise
evidence from observations that only a small fraction of dark matter can be baryonic
(see for example [43] and [105, 106] for reviews). Therefore, the real puzzle regards the
nature of non-baryonic dark matter.
One can separate the candidates into two major categories: hot dark matter, i.e. nonbaryonic particles which move (ultra-)relativistically, and cold dark matter i.e. nonbaryonic particles which move non-relativistically. The most prominent candidate for
hot dark matter is the the neutrino. However, studies of the cosmic microwave background, numerical simulations and other astrophysical observations indicate that dark
matter has clumped to form some structures on rather small scales and therefore it cannot consist mainly of particles with high velocities, since this clumping would then have
been suppressed (see for example [107, 108] and references in [106]). For this reason,
and because of its simplicity, cold dark matter currently gives the favoured picture.
There are many cold dark matter candidates and so we will refrain from listing them
all or discussing their properties in detail here and refer the reader to the literature
[106]. The most commonly considered ones are the axion and a number of weakly
interacted massive particles (WIMPs) naturally predicted in supersymmetry theories,
46
1. INTRODUCTION
such as the neutralino, the sneutrino, the gravitino, the axino etc. There are a number
of experiments aiming for direct and indirect detection of dark matter and some of
them, such as the DAMA/NaI experiment [109], even claim to have already achieved
that (see [110] for a full list of dark matter detection experiments and [105] for a review
of experimental searches for dark matter). Great hope is also being placed on the
Large Hadron Collider (LHC) [111], which is due to start operating shortly, to constrain
the parameter space of particles arising from supersymmetric theories. Finally, the
improvement of cosmological and astrophysical observations obviously plays a crucial
role. Let us close by saying that the general flavour or expectation seems to be that
one of the proposed candidates will soon be detected and that the relevant dark matter
scenario will be verified. Of course expectations are not always fulfilled and it is best to
be prepared for surprises.
1.4.4
Towards Quantum Gravity, but how?
In Section 1.2 we discussed some of the more prominent motivations for seeking a high
energy theory of gravity which would allow a matching between General Relativity and
Quantum Field Theory. These triggered research in this direction at a very early stage
and already in the 1950s serious efforts were being made towards what is referred to
as Quantum Gravity. Early attempts followed the conventional approach of trying to
quantize the gravitational field in ways similar to the quantization of Electromagnetism,
which had resulted in Quantum Electrodynamics (QED). This led to influential papers
about the canonical formulation of General Relativity [112, 113]. However, it was soon
realized that the obvious quantization techniques could not work, since General Relativity is not renormalizable as is the case with Quantum Electrodynamics [114]. In simple
terms, this means that if one attempts to treat gravity as another particle field and to
assign a gravity particle to it (graviton) then the sum of the interactions of the graviton diverges. This would not be a problem if these divergences were few enough to be
removable via the technique called renormalization and this is indeed what happens in
Quantum Electrodynamics, as also mentioned in Section 1.2. Unfortunately, this is not
the case for General Relativity and renormalization cannot lead to sensible and finite
results.
It was later shown that a renormalizable gravitation theory — although not a unitary
one — could be constructed, but only at the price of admitting corrections to General
Relativity [114, 188]. Views on renormalization have changed since then and more
modern ideas have been introduced such as the concept of effective field theories. These
are approximate theories with the following characteristic: according to the length-scale,
they take into account only the relevant degrees of freedom. Degrees of freedom which are
only relevant to shorter length-scales and higher energies and are, therefore, responsible
for divergences, are ignored. A systematic way to integrate out short-distance degrees
of freedom is given by the renormalization group (see [116] for an introduction to these
concepts).
1.4. IS THERE A WAY OUT?
47
In any case, quantizing gravity has proved to be a more difficult task than initially
expected and quantum corrections seem to appear, introducing deviations away from
General Relativity [117, 118, 119]. Contemporary research is mainly focused on two
directions: String Theory and Loop Quantum Gravity. Analysing the basis of either
of these two approaches would go beyond the scope of this introduction and so we
will only make a short mention of them. We refer the reader to [120, 121, 122] and
[123, 124, 125, 126, 127] for text books and topical reviews in String Theory and Loop
Quantum Gravity respectively.
String Theory attempts to explain fundamental physics and unify all interactions
under the key assumption that the building blocks are not point particles but one dimensional objects called strings. There are five different versions of String Theory, namely
Type I, Type IIA, Type IIB and two types of Heterotic String Theory. M-Theory is
a proposed theory under development that attempts to unify all of the above types.
A simplified version of the idea behind String Theory would be that its fundamental
constituents, strings, vibrate at resonant frequencies. Different strings have different
resonances and this is what determines their nature and results in the discrimination
between different forces.
Loop Quantum Gravity follows a more direct approach to the quantization of gravity.
It is close to the picture of canonical quantization and relies on a non-perturbative
method called loop quantization. One of its main disadvantages is that it is not yet
clear whether it can become a theory that can include the description of matter as well
or whether it is just a quantum theory of gravitation.
It is worth mentioning that a common problem with these two approaches is that,
at the moment, they do not make any experimentally testable predictions which are
different from those already know from the standard model of particle physics. As
far as gravity is concerned, String Theory appears to introduce deviations from General
Relativity (see for example [128, 129, 130]), whereas, the classical limit of Loop Quantum
gravity is still under investigation.
1.4.5
Status of Gravity
In this introductory chapter, an attempt has been made to pose clearly a series of
open questions related, in one way or the other, to gravity and to discuss some of the
most common approaches currently being pursued for their solution. This brings us
to the main question motivating the research presented in this thesis: could all or at
least some of the problems mentioned earlier be somehow related and is the fact that
General Relativity is now facing so many challenges indicative of a need for some new
gravitational physics, even at a classical level?
Let us be more analytic. In Section 1.1 we presented a brief chronological review
of some landmarks in the passage from Newtonian Gravity to General Relativity. One
could find striking similarities with what has happened in the last decades with General
Relativity itself. For instance, the cosmological and astrophysical observations which
48
1. INTRODUCTION
are interpreted as indicating the existence of dark matter and/or dark energy could
be compared with Le Verrier’s observation of the excess precession of Mercury’s orbit.
Remarkably, the first attempt to explain this phenomenon, was exactly the suggestion
that an extra unseen — and therefore dark, in a way — planet orbited the Sun inside
Mercury’s orbit. The basic motivation behind this attempt, much like the contemporary
proposals for matter fields to describe dark matter and dark energy, was to solve the
problem within the context of an otherwise successful theory, instead of questioning
the theory itself. Another example one could give, is the theoretical problems faced
by Newtonian gravity once Special Relativity was established. The desire for a unified
description of coordinate frames, inertial or not, and the need for a gravitational theory
that is in good accordance with the conceptual basis of Special Relativity (e.g. Lorentz
invariance) does not seem to be very far from the current desire for a unified description
of forces and the need to resolve the conceptual clash between General Relativity and
Quantum Field Theory.
The idea of looking for an alternative theory to describe the gravitational interaction
is obviously not new. We already mentioned previously that attempts to unify gravity
with quantum theory have included such considerations in the form of making quantum
corrections to the gravitational field equations (or to the action, from a field theory
perspective). Such corrections became effective at small scales or high energies. Additionally, many attempts have been made to modify General Relativity on both small
and large scales, in order to address specific problems, such as those discussed earlier.
Since we will refer to such modification extensively in the forthcoming chapters we will
refrain from listing them here to avoid repetition. At present we will confine ourselves
to giving two very early examples of such attempts which were not triggered so much
by a theoretical or observational need for a new theory, but by another important issue
in our opinion: the desire to test the uniqueness of General Relativity as the only viable
gravitational theory and the need to verify its conceptual basis.
Sir Arthur Stanley Eddington, the very man who performed the deflection of light
experiment during the Solar eclipse of 1919 which was one of the early experimental
verifications of General Relativity, was one of first people to question whether Einstein’s
theory was the unique theory that could describe gravity [131]. Eddington tried to
develop alternative theories sharing the same conceptual basis with General Relativity, most probably for the sake of theoretical completeness, since at the time there was
no apparent reason coming from observations. Robert Dicke was also one of the pioneers in exploring the conceptual basis of General Relativity and questioning Einstein’s
equivalence principle. He reformulated Mach’s principle and together with Carl Brans
developed an alternative theory, known as Brans–Dicke theory [199, 133]. Part of the
value of Dicke’s work lies on the fact that it helped people to understand that we do
not know as much as we thought about the basic assumptions of General Relativity, a
subject that we will discuss shortly.
Even though the idea of an alternative theory for gravitation is not new, a new
perspective about it has emerged quite recently. The quantum corrections predicted in
1.4. IS THERE A WAY OUT?
49
the 1960s were expected to appear only at small scales. On the other hand, Eddington’s
modification or Brans–Dicke theory were initially pursued as a conceptual alternative
of General Relativity and had phenomenological effects on large scales as well. Now,
due to both the shortcomings of Quantum Gravity and the puzzling cosmological and
astrophysical observations, these ideas have stopped being considered unrelated. It seem
worthwhile to consider the possibility of developing a gravitation theory that will be in
agreement with observations and at the same time will be closer to the theories that
emerge as a classical limit of our current approaches to Quantum Gravity, especially
since it has been understood that quantum corrections might have an effect on large
scale phenomenology as well.
Unfortunately, constructing a viable alternative to General Relativity with the above
characteristics is far from being an easy task since there are numerous theoretical and
observational restrictions. Two main paths have been followed towards achieving this
goal: proposing phenomenological models tailored to fit observations, with the hope
that they will soon gain some theoretical motivation from high energy physics and current Quantum Gravity candidates, and developing ideas for Quantum Gravity, with the
hope that they will eventually give the answer in the form of an effective gravitational
theory through their classical limit which will account for unexplained observations. In
this thesis a different approach will be followed in an attempt to combine and complement these two. At least according to the author’s opinion, we seem to be still at too
early a stage in the development of our ideas about Quantum Gravity to be able to
give precise answers about the type and form of the expected quantum corrections to
General Relativity. Current observations still leave scope for a wide range of different
phenomenological models and so it seems a good idea to attempt exploring the limits
of classical gravity by combining theory and observations. In a sense, this approach lies
somewhere in the middle between the more conventional approaches mentioned before.
Instead of starting from something known in order to extrapolate to the unknown, we
attempt here to jump directly into the unknown, hoping that we will find an answer.
To this end, we will examine theories of gravity, trying to determine how far one
can go from General Relativity. These theories have been chosen in such a way as to
present a resemblance with the low energy effective actions of contemporary candidates
for Quantum Gravity in a quest to study the phenomenology of the induced corrections.
Their choice has also been motivated by a desire to fit recent unexplained observations.
However, it should be stressed that both of these criteria have been used in a loose
manner, since the main scope of this study is to explore the limits of alternative theories
of gravity and hopefully shed some light on the strength and validity of the several
assumptions underlying General Relativity. The main motivation comes from the fear
that we may not know as much as we think or as much as needed to be known before
making the key steps pursued in the last 50 years in gravitational physics; and from the
hope that a better understanding of classical gravity might have a lot to offer in this
direction.
As a conclusion to this introduction it is worth saying the following: it is probably too
50
1. INTRODUCTION
early to conclude whether it is General Relativity that needs to be modified or replaced
by some other gravitational theory or whether other solutions to the problems presented
in this chapter, such as those mentioned earlier, will eventually give the required answers.
However, in scientific research, pursuing more than one possible solution to a problem
has always been the wisest and most rewarding choice; not only because there is an
already explored alternative when one of the proposed solutions fails, but also due to the
fact that trial and error is one of the most efficient ways to get a deeper understanding
of a physical theory. Exploring alternative theories of gravity, although having some
disadvantages such as complexity, also presents a serious advantage: it is bound to be
fruitful even if it leads to the conclusion that General Relativity is the only correct
theory for gravitation, as it will have helped us both to understand General Relativity
better and to secure our faith in it.
Chapter 2
Extended Theories of Gravity
2.1
Theoretical motivations for Extended Theories of Gravity
Due to the problems of Standard Cosmological Model and to the problems of the solution
found to solve them, and, first of all, to the lack of a denitive quantum gravity theory,
alternative theories have been considered in order to attempt, at least, a semi-classical
scheme where Gen- eral Relativity and its positive results could be recovered. One of the
most fruitful approaches has been that of Extended Theories of Gravity (ETG) which
have become a sort of paradigm in the study of gravitational interaction. They are
based on corrections and enlargements of the Einstein theory. The paradigm consists,
essentially, in adding higher-order curvature invariants and minimally or non-minimally
coupled scalar fields into dynamics which come out from the effective action of quantum
gravity [404].
Other motivations to modify GR come from the issue of a full recovering of the
Mach principle which leads to assume a varying gravitational coupling. The principle
states that the local inertial frame is determined by some average of the motion of
distant astronomical objects [135]. This fact implies that the gravitational coupling can
be scale-dependent and related to some scalar field. As a consequence, the concept of
“inertia” and the Equivalence Principle have to be revised. For example, the BransDicke theory [136] is a serious attempt to define an alternative theory to the Einstein
gravity: it takes into account a variable Newton gravitational coupling, whose dynamics
is governed by a scalar field non-minimally coupled to the geometry. In such a way,
Mach’s principle is better implemented [136, 137, 138].
Besides, every unification scheme as Superstrings, Supergravity or Grand Unified
Theories, takes into account effective actions where non-minimal couplings to the geometry or higher-order terms in the curvature invariants are present. Such contributions
are due to one-loop or higher-loop corrections in the high-curvature regimes near the full
(not yet available) quantum gravity regime [404]. Specifically, this scheme was adopted
51
52
2. EXTENDED THEORIES OF GRAVITY
in order to deal with the quantization on curved spacetimes and the result was that the
interactions among quantum scalar fields and background geometry or the gravitational
self-interactions yield corrective terms in the Hilbert-Einstein Lagrangian [139]. Moreover, it has been realized that such corrective terms are inescapable in order to obtain
the effective action of quantum gravity at scales closed to the Planck one [140]. All
these approaches are not the “full quantum gravity” but are needed as working schemes
toward it.
In summary, higher-order terms in curvature invariants (such as R2 , Rµν Rµν , Rµναβ Rµναβ ,
R R, or R k R) or non-minimally coupled terms between scalar fields and geometry
(such as φ2 R) have to be added to the effective Lagrangian of gravitational field when
quantum corrections are considered. For instance, one can notice that such terms occur
in the effective Lagrangian of strings or in Kaluza-Klein theories, when the mechanism
of dimensional reduction is used [141].
On the other hand, from a conceptual viewpoint, there are no a priori reason to
restrict the gravitational Lagrangian to a linear function of the Ricci scalar R, minimally
coupled with matter [142]. Furthermore, the idea that there are no “exact” laws of
physics could be taken into serious account: in such a case, the effective Lagrangians
of physical interactions are “stochastic” functions. This feature means that the local
gauge invariances (i.e. conservation laws) are well approximated only in the low energy
limit and the fundamental physical constants can vary [381].
Besides fundamental physics motivations, all these theories have acquired a huge
interest in cosmology due to the fact that they “naturally” exhibit inflationary behaviors
able to overcome the shortcomings of Cosmological Standard Model (based on GR). The
related cosmological models seem realistic and capable of matching with the CMBR
observations [144, 145, 146]. Furthermore, it is possible to show that, via conformal
transformations, the higher-order and non-minimally coupled terms always correspond
to the Einstein gravity plus one or more than one minimally coupled scalar fields [147,
148, 149, 150, 151].
More precisely, higher-order terms appear always as contributions of order two in
the field equations. For example, a term like R2 gives fourth order equations [152],
R R gives sixth order equations [151, 153], R 2 R gives eighth order equations [154]
and so on. By a conformal transformation, any 2nd-order derivative term corresponds
to a scalar field1 : for example, fourth-order gravity gives Einstein plus one scalar field,
sixth-order gravity gives Einstein plus two scalar fields and so on [151, 155].
Furthermore, it is possible to show that the f (R)-gravity is equivalent not only to a
scalar-tensor one but also to the Einstein theory plus an ideal fluid [156]. This feature
results very interesting if we want to obtain multiple inflationary events since an early
stage could select “very” large-scale structures (clusters of galaxies today), while a late
stage could select “small” large-scale structures (galaxies today) [153]. The philosophy
1
The dynamics of such scalar fields is usually given by the corresponding Klein-Gordon Equation,
which is second order.
2.1. THEORETICAL MOTIVATIONS FOR EXTENDED THEORIES OF GRAVITY
53
is that each inflationary era is related to the dynamics of a scalar field. Finally, these
extended schemes could naturally solve the problem of “graceful exit” bypassing the
shortcomings of former inflationary models [146, 157].
In addition to the revision of Standard Cosmology at early epochs (leading to the Inflation), a new approach is necessary also at late epochs. ETGs could play a fundamental
role also in this context. In fact, the increasing bulk of data that have been accumulated
in the last few years have paved the way to the emergence of a new cosmological model
usually referred to as the Concordance Model.
The Hubble diagram of Type Ia Supernovae (hereafter SNeIa), measured by both
the Supernova Cosmology Project [158] and the High - z Team [159] up to redshift z ∼ 1,
has been the first evidence that the Universe is undergoing a phase of accelerated expansion. On the other hand, balloon born experiments, such as BOOMERanG [160] and
MAXIMA [161], determined the location of the first and second peak in the anisotropy
spectrum of the cosmic microwave background radiation (CMBR) strongly pointing out
that the geometry of the Universe is spatially flat. If combined with constraints coming from galaxy clusters on the matter density parameter ΩM , these data indicate that
the Universe is dominated by a non-clustered fluid with negative pressure, generically
dubbed dark energy, which is able to drive the accelerated expansion. This picture has
been further strengthened by the recent precise measurements of the CMBR spectrum,
due to the WMAP experiment [162, 163, 164], and by the extension of the SNeIa Hubble
diagram to redshifts higher than 1 [165]. After these observational evidences, an overwhelming flood of papers has appeared: they present a great variety of models trying to
explain this phenomenon. In any case, the simplest explanation is claiming for the well
known cosmological constant Λ [166]. Although it is the best fit to most of the available
astrophysical data [162], the ΛCDM model fails in explaining why the inferred value of Λ
is so tiny (120 orders of magnitude lower!) if compared with the typical vacuum energy
values predicted by particle physics and why its energy density is today comparable to
the matter density (the so called coincidence problem).
As a tentative solution, many authors have replaced the cosmological constant with
a scalar field rolling down its potential and giving rise to the model now referred to as
quintessence [167, 168]. Even if successful in fitting the data, the quintessence approach
to dark energy is still plagued by the coincidence problem since the dark energy and
matter densities evolve differently and reach comparable values for a very limited portion
of the Universe evolution coinciding at present era. To be more precise, the quintessence
dark energy is tracking matter and evolves in the same way for a long time. But then, at
late time, somehow it has to change its behavior into no longer tracking the dark matter
but starting to dominate as a cosmological constant. This is the coincidence problem of
quintessence.
Moreover, it is not clear where this scalar field originates from, thus leaving a great
uncertainty on the choice of the scalar field potential. The subtle and elusive nature
of dark energy has led many authors to look for completely different scenarios able to
give a quintessential behavior without the need of exotic components. To this aim, it is
54
2. EXTENDED THEORIES OF GRAVITY
worth stressing that the acceleration of the Universe only claims for a negative pressure
dominant component, but does not tell anything about the nature and the number of
cosmic fluids filling the Universe.
This consideration suggests that it could be possible to explain the accelerated expansion by introducing a single cosmic fluid with an equation of state causing it to act like
dark matter at high densities and dark energy at low densities. An attractive feature of
these models, usually referred to as Unified Dark Energy (UDE) or Unified Dark Matter
(UDM) models, is that such an approach naturally solves, al least phenomenologically,
the coincidence problem. Some interesting examples are the generalized Chaplygin gas
[169], the tachyon field [170] and the condensate cosmology [171]. A different class of
UDE models has been proposed [172] where a single fluid is considered: its energy density scales with the redshift in such a way that the radiation dominated era, the matter
era and the accelerating phase can be naturally achieved. It is worth noticing that such
class of models are extremely versatile since they can be interpreted both in the framework of UDE models and as a two-fluid scenario with dark matter and scalar field dark
energy. The main ingredient of the approach is that a generalized equation of state can
be always obtained and observational data can be fitted.
Actually, there is still a different way to face the problem of cosmic acceleration. As
stressed in [173], it is possible that the observed acceleration is not the manifestation of
another ingredient in the cosmic pie, but rather the first signal of a breakdown of our
understanding of the laws of gravitation (in the infra-red limit).
From this point of view, it is thus tempting to modify the Friedmann equations
to see whether it is possible to fit the astrophysical data with models comprising only
the standard matter. Interesting examples of this kind are the Cardassian expansion
[174] and the DGP gravity [175]. Moving in this same framework, it is possible to find
alternative schemes where a quintessential behavior is obtained by taking into account
effective models coming from fundamental physics giving rise to generalized or higherorder gravity actions [176] (for a comprehensive review see [177]).
For instance, a cosmological constant term may be recovered as a consequence of a
non - vanishing torsion field thus leading to a model which is consistent with both SNeIa
Hubble diagram and Sunyaev - Zel’dovich data coming from clusters of galaxies [178].
SNeIa data could also be efficiently fitted including higher-order curvature invariants in
the gravity Lagrangian [179, 181, 182, 183]. It is worth noticing that these alternative
models provide naturally a cosmological component with negative pressure whose origin
is related to the geometry of the Universe thus overcoming the problems linked to the
physical significance of the scalar field.
It is evident, from this short overview, the high number of cosmological models which
are viable candidates to explain the observed accelerated expansion. This abundance
of models is, from one hand, the signal of the fact that we have a limited number
of cosmological tests to discriminate among rival theories, and, from the other hand,
that a urgent degeneracy problem has to be faced. To this aim, it is useful to remark
that both the SNeIa Hubble diagram and the angular size - redshift relation of compact
2.1. THEORETICAL MOTIVATIONS FOR EXTENDED THEORIES OF GRAVITY
55
radio sources [184] are distance based methods to probe cosmological models so then
systematic errors and biases could be iterated. From this point of view, it is interesting
to search for tests based on time-dependent observables.
For example, one can take into account the lookback time to distant objects since this
quantity can discriminate among different cosmological models. The lookback time is
observationally estimated as the difference between the present day age of the Universe
and the age of a given object at redshift z. Such an estimate is possible if the object
is a galaxy observed in more than one photometric band since its color is determined
by its age as a consequence of stellar evolution. It is thus possible to get an estimate
of the galaxy age by measuring its magnitude in different bands and then using stellar
evolutionary codes to choose the model that reproduces the observed colors at best.
Coming to the weak-field-limit approximation, which essentially means considering
Solar System scales, ETGs are expected to reproduce GR which, in any case, is firmly
tested only in this limit [185]. This fact is matter of debate since several relativistic
theories do not reproduce exactly the Einstein results in the Newtonian approximation
but, in some sense, generalize them. As it was firstly noticed by Stelle [188], a R2 -theory
gives rise to Yukawa-like corrections in the Newtonian potential. Such a feature could
have interesting physical consequences. For example, some authors claim to explain
the flat rotation curves of galaxies by using such terms [189]. Others [190] have shown
that a conformal theory of gravity is nothing else but a fourth-order theory containing
such terms in the Newtonian limit. Besides, indications of an apparent, anomalous,
long-range acceleration revealed from the data analysis of Pioneer 10/11, Galileo, and
Ulysses spacecrafts could be framed in a general theoretical scheme by taking corrections
to the Newtonian potential into account [191, 371].
In general, any relativistic theory of gravitation yields corrections to the Newton
potential (see for example [193]) which, in the post-Newtonian (PPN) formalism, could
be a test for the same theory [185]. Furthermore the newborn gravitational lensing
astronomy [194] is giving rise to additional tests of gravity over small, large, and very
large scales which soon will provide direct measurements for the variation of the Newton
coupling [195], the potential of galaxies, clusters of galaxies and several other features
of self-gravitating systems.
Such data will be, very likely, capable of confirming or ruling out the physical consistency of GR or of any ETG. In summary, the general features of ETGs are that the
Einstein field equations result to be modified in two senses: i) geometry can be nonminimally coupled to some scalar field, and/or ii) higher than second order derivative
terms in the metric come out. In the former case, we generically deal with scalar-tensor
theories of gravity; in the latter, we deal with higher-order theories. However combinations of non-minimally coupled and higher-order terms can emerge as contributions in
effective Lagrangians. In this case, we deal with higher-order-scalar-tensor theories of
gravity.
Considering a mathematical viewpoint, the problem of reducing more general theories to Einstein standard form has been extensively treated; one can see that, through
56
2. EXTENDED THEORIES OF GRAVITY
a “Legendre” transformation on the metric, higher-order theories, under suitable regularity conditions on the Lagrangian, take the form of the Einstein one in which a
scalar field (or more than one) is the source of the gravitational field (see for example
[142, 197, 196, 198]); on the other side, as discussed above, it has been studied the
mathematical equivalence between models with variable gravitational coupling with the
Einstein standard gravity through suitable conformal transformations (see [199, 200]).
In any case, the debate on the physical meaning of conformal transformations is far to
be solved [see [347] and references therein for a comprehensive review]. Several authors
claim for a true physical difference between Jordan frame (higher-order theories and/or
variable gravitational coupling) since there are experimental and observational evidences
which point out that the Jordan frame could be suitable to better match solutions with
data. Others state that the true physical frame is the Einstein one according to the
energy theorems [198]. However, the discussion is open and no definitive statement has
been formulated up to now.
The problem should be faced from a more general viewpoint and the Palatini approach to gravity could be useful to this goal. The Palatini approach in gravitational
theories was firstly introduced and analyzed by Einstein himself [203]. It was, however, called the Palatini approach as a consequence of an historical misunderstanding
[204, 205].
The fundamental idea of the Palatini formalism is to consider the (usually torsionless) connection Γ, entering the definition of the Ricci tensor, to be independent of the
metric g defined on the spacetime M. The Palatini formulation for the standard HilbertEinstein theory results to be equivalent to the purely metric theory: this follows from the
fact that the field equations for the connection Γ, firstly considered to be independent of
the metric, give the Levi-Civita connection of the metric g. As a consequence, there is
no reason to impose the Palatini variational principle in the standard Hilbert-Einstein
theory instead of the metric variational principle.
However, the situation completely changes if we consider the ETGs, depending on
functions of curvature invariants, as f (R), or non-minimally coupled to some scalar
field. In these cases, the Palatini and the metric variational principle provide different
field equations and the theories thus derived differ [198, 206]. The relevance of Palatini approach, in this framework, has been recently proven in relation to cosmological
applications [176, 177, 207, 208, 209].
It has also been studied the crucial problem of the Newtonian potential in alternative
theories of Gravity and its relations with the conformal factor [211]. From a physical
viewpoint, considering the metric g and the connection Γ as independent fields means to
decouple the metric structure of spacetime and its geodesic structure (being, in general,
the connection Γ not the Levi-Civita connection of g). The chronological structure of
spacetime is governed by g while the trajectories of particles, moving in the spacetime,
are governed by Γ.
This decoupling enriches the geometric structure of spacetime and generalizes the
purely metric formalism. This metric-affine structure of spacetime is naturally trans-
2.2. WHAT A GOOD THEORY OF GRAVITY HAS TO DO: GENERAL RELATIVITY AND ITS
EXTENSIONS
57
lated, by means of the same (Palatini) field equations, into a bi-metric structure of
spacetime. Beside the physical metric g, another metric h is involved. This new metric
is related, in the case of f (R)-gravity, to the connection. As a matter of fact, the connection Γ results to be the Levi-Civita connection of h and thus provides the geodesic
structure of spacetime.
If we consider the case of non-minimally coupled interaction in the gravitational
Lagrangian (scalar-tensor theories), the new metric h is related to the non-minimal coupling. The new metric h can be thus related to a different geometric and physical aspect
of the gravitational theory. Thanks to the Palatini formalism, the non-minimal coupling
and the scalar field, entering the evolution of the gravitational fields, are separated from
the metric structure of spacetime. The situation mixes when we consider the case of
higher-order-scalar-tensor theories. Due to these features, the Palatini approach could
greatly contribute to clarify the physical meaning of conformal transformation [210].
2.2
What a good theory of Gravity has to do: General
Relativity and its extensions
From a phenomenological point of view, there are some minimal requirements that any
relativistic theory of gravity has to match. First of all, it has to explain the astrophysical
observations (e.g. the orbits of planets, the potential of self-gravitating structures).
This means that it has to reproduce the Newtonian dynamics in the weak-energy
limit. Besides, it has to pass the classical Solar System tests which are all experimentally
well founded [185].
As second step, it should reproduce galactic dynamics considering the observed
baryonic constituents (e.g. luminous components as stars, sub-luminous components
as planets, dust and gas), radiation and Newtonian potential which is, by assumption,
extrapolated to galactic scales.
Thirdly, it should address the problem of large scale structure (e.g. clustering
of galaxies) and finally cosmological dynamics, which means to reproduce, in a selfconsistent way, the cosmological parameters as the expansion rate, the Hubble constant,
the density parameter and so on. Observations and experiments, essentially, probe the
standard baryonic matter, the radiation and an attractive overall interaction, acting at
all scales and depending on distance: the gravity.
The simplest theory which try to satisfies the above requirements was formulated by
Albert Einstein in the years 1915-1916 [212] and it is known as the Theory of General
Relativity. It is firstly based on the assumption that space and time have to be entangled
into a single spacetime structure, which, in the limit of no gravitational forces, has to
reproduce the Minkowski spacetime structure. Einstein profitted also of ideas earlier
put forward by Riemann, who stated that the Universe should be a curved manifold
and that its curvature should be established on the basis of astronomical observations
[213].
58
2. EXTENDED THEORIES OF GRAVITY
In other words, the distribution of matter has to influence point by point the local
curvature of the spacetime structure. The theory, eventually formulated by Einstein in
1915, was strongly based on three assumptions that the Physics of Gravitation has to
satisfy.
The ”Principle of Relativity”, that amounts to require all frames to be good frames
for Physics, so that no preferred inertial frame should be chosen a priori (if any exist).
The ”Principle of Equivalence”, that amounts to require inertial effects to be locally
indistinguishable from gravitational effects (in a sense, the equivalence between the
inertial and the gravitational mass).
The ”Principle of General Covariance”, that requires field equations to be ”generally
covariant” (today, we would better say to be invariant under the action of the group of
all spacetime diffeomorphisms) [214].
And - on the top of these three principles - the requirement that causality has to be
preserved (the ”Principle of Causality”, i.e. that each point of spacetime should admit
a universally valid notion of past, present and future).
Let us also recall that the older Newtonian theory of spacetime and gravitation that Einstein wanted to reproduce at least in the limit of small gravitational forces
(what is called today the ”post-Newtonian approximation”) - required space and time
to be absolute entities, particles moving in a preferred inertial frame following curved
trajectories, the curvature of which (i.e., the acceleration) had to be determined as a
function of the sources (i.e., the ”forces”).
On these bases, Einstein was led to postulate that the gravitational forces have to
be expressed by the curvature of a metric tensor field ds2 = gµν dxµ dxν on a fourdimensional spacetime manifold, having the same signature of Minkowski metric, i.e.,
the so-called ”Lorentzian signature”, herewith assumed to be (+, −, −, −). He also
postulated that spacetime is curved in itself and that its curvature is locally determined
by the distribution of the sources, i.e. - being spacetime a continuum - by the fourdimensional generalization of what in Continuum Mechanics is called the ”matter stressm.
energy tensor”, i.e. a rank-two (symmetric) tensor Tµν
Once a metric gµν is given, its curvature is expressed by the Riemann (curvature)
tensor
Rα βµν = Γαβν ,µ − Γαβµ ,ν + Γσβν Γασµ − Γσβµ Γασν
(2.1)
where the comas are partial derivatives. Its contraction
Rα µαν = Rµν ,
(2.2)
R = Rµ µ = gµν Rµν
(2.3)
is the ”Ricci tensor” and the scalar
is called the ”scalar curvature” of gµν . Einstein was led to postulate the following
2.2. WHAT A GOOD THEORY OF GRAVITY HAS TO DO: GENERAL RELATIVITY AND ITS
EXTENSIONS
59
equations for the dynamics of gravitational forces
Rµν =
κ m
T
2 µν
(2.4)
where κ = 8πG, with c = 1 is a coupling constant. These equations turned out to be
physically and mathematically unsatisfactory.
As Hilbert pointed out [214], they were not of a variational origin, i.e. there was
no Lagrangian able to reproduce them exactly (this is slightly wrong, but this remark
is unessential here). Einstein replied that he knew that the equations were physically
unsatisfactory, since they were contrasting with the continuity equation of any reasonable
kind of matter. Assuming that matter is given as a perfect fluid, that is
m
Tµν
= (p + ρ)uµ uν − pgµν
(2.5)
where uµ uν is a comoving observer, p is the pressure and ρ the density of the fluid,
m to be covariantly constant, i.e. to satisfy the
then the continuity equation requires Tµν
conservation law
m
∇µ Tµν
= 0,
(2.6)
where ∇µ denotes the covariant derivative with respect to the metric.
In fact, it is not true that ∇µ Rµν vanishes (unless R = 0). Einstein and Hilbert
reached independently the conclusion that the wrong field equations (2.4) had to be
replaced by the correct ones
m
Gµν = κTµν
(2.7)
where
1
(2.8)
Gµν = Rµν − gµν R
2
that is currently called the ”Einstein tensor” of gµν . These equations are both variational
and satisfy the conservation laws (2.6) since the following relation holds
∇µ Gµν = 0 ,
(2.9)
as a byproduct of the so-called ”Bianchi identities” that the curvature tensor of gµν has
to satisfy [11].
The Lagrangian that allows to obtain the field equations (2.7) is the sum of a ”matter
m , i.e.
Lagrangian” Lm , the variational derivative of which is exactly Tµν
m
Tµν
=
δLm
δgµν
(2.10)
and of a ”gravitational Lagrangian”, currently called the Hilbert-Einstein Lagrangian
√
√
(2.11)
LHE = gµν Rµν −g = R −g ,
√
where −g denotes the square root of the value of the determinant of the metric gµν .
60
2. EXTENDED THEORIES OF GRAVITY
The choice of Hilbert and Einstein was completely arbitrary (as it became clear a
few years later), but it was certainly the simplest one both from the mathematical and
the physical viewpoint. As it was later clarified by Levi-Civita in 1919, curvature is
not a ”purely metric notion” but, rather, a notion related to the ”linear connection” to
which ”parallel transport” and ”covariant derivation” refer [215].
In a sense, this is the precursor idea of what in the sequel would be called a ”gauge
theoretical framework” [216], after the pioneering work by Cartan in 1925 [217]. But at
the time of Einstein, only metric concepts were at hands and his solution was the only
viable.
It was later clarified that the three principles of relativity, equivalence and covariance,
together with causality, just require that the spacetime structure has to be determined
by either one or both of two fields, a Lorentzian metric g and a linear connection Γ,
assumed to be torsionless for the sake of simplicity.
The metric g fixes the causal structure of spacetime (the light cones) as well as its
metric relations (clocks and rods); the connection Γ fixes the free-fall, i.e. the locally
inertial observers. They have, of course, to satisfy a number of compatibility relations
which amount to require that photons follow null geodesics of Γ, so that Γ and g can
be independent, a priori, but constrained, a posteriori, by some physical restrictions.
These, however, do not impose that Γ has necessarily to be the Levi-Civita connection
of g [218].
This justifies - at least on a purely theoretical basis - the fact that one can envisage
the so-called ”alternative theories of gravitation”, that we prefer to call ”Extended Theories of Gravitation” since their starting points are exactly those considered by Einstein
and Hilbert: theories in which gravitation is described by either a metric (the so-called
”purely metric theories”), or by a linear connection (the so-called ”purely affine theories”) or by both fields (the so-called ”metric-affine theories”, also known as ”first
order formalism theories”). In these theories, the Lagrangian is a scalar density of the
curvature invariants constructed out of both g and Γ.
The choice (2.11) is by no means unique and it turns out that the Hilbert-Einstein
Lagrangian is in fact the only choice that produces an invariant that is linear in second
derivatives of the metric (or first derivatives of the connection). A Lagrangian that,
unfortunately, is rather singular from the Hamiltonian viewpoint, in much than same way
as Lagrangians, linear in canonical momenta, are rather singular in Classical Mechanics
(see e.g. [219]).
A number of attempts to generalize GR (and unify it to Electromagnetism) along
these lines were followed by Einstein himself and many others (Eddington, Weyl, Schrodinger,
just to quote the main contributors; see, e.g., [220]) but they were eventually given up
in the fifties of XX Century, mainly because of a number of difficulties related to the
definitely more complicated structure of a non-linear theory (where by ”non-linear” we
mean here a theory that is based on non-linear invariants of the curvature tensor), and
also because of the new understanding of Physics that is currently based on four fundamental forces and requires the more general ”gauge framework” to be adopted (see
2.2. WHAT A GOOD THEORY OF GRAVITY HAS TO DO: GENERAL RELATIVITY AND ITS
EXTENSIONS
61
[221]).
Still a number of sporadic investigations about ”alternative theories” continued even
after 1960 (see [185] and refs. quoted therein for a short history). The search of a
coherent quantum theory of gravitation or the belief that gravity has to be considered
as a sort of low-energy limit of string theories (see, e.g., [222]) - something that we are
not willing to enter here in detail - has more or less recently revitalized the idea that
there is no reason to follow the simple prescription of Einstein and Hilbert and to assume
that gravity should be classically governed by a Lagrangian linear in the curvature.
Further curvature invariants or non-linear functions of them should be also considered, especially in view of the fact that they have to be included in both the semi-classical
expansion of a quantum Lagrangian or in the low-energy limit of a string Lagrangian.
Moreover, it is clear from the recent astrophysical observations and from the current
cosmological hypotheses that Einstein equations are no longer a good test for gravitation
at Solar System, galactic, extra-galactic and cosmic scale, unless one does not admit
that the matter side of Eqs.(2.7) contains some kind of exotic matter-energy which is
the ”dark matter” and ”dark energy” side of the Universe.
The idea which we propose here is much simpler. Instead of changing the matter
side of Einstein Equations (2.7) in order to fit the ”missing matter-energy” content
of the currently observed Universe (up to the 95% of the total amount!), by adding
any sort of inexplicable and strangely behaving matter and energy, we claim that it is
simpler and more convenient to change the gravitational side of the equations, admitting
corrections coming from non-linearities in the Lagrangian. However, this is nothing else
but a matter of taste and, since it is possible, such an approach should be explored. Of
course, provided that the Lagrangian can be conveniently tuned up (i.e., chosen in a huge
family of allowed Lagrangians) on the basis of its best fit with all possible observational
tests, at all scales (solar, galactic, extragalactic and cosmic).
Something that - in spite of some commonly accepted but disguised opinion - can
and should be done before rejecting a priori a non-linear theory of gravitation (based
on a non-singular Lagrangian) and insisting that the Universe has to be necessarily described by a rather singular gravitational Lagrangian (one that does not allow a coherent
perturbation theory from a good Hamiltonian viewpoint) accompanied by matter that
does not follow the behavior that standard baryonic matter, probed in our laboratories,
usually satisfies.
62
2. EXTENDED THEORIES OF GRAVITY
2.3
Structure of the Extended Theories of Gravity
With the above considerations in mind, let us start with a general class of higher-orderscalar-tensor theories in four dimensions 2 given by the action
Z
i
√ h
ε
(2.12)
A = d4 x −g F (R, R, 2 R, ..k R, φ) − gµν φ;µ φ;ν + Lm ,
2
where F is an unspecified function of curvature invariants and of a scalar field φ. The
term Lm , as above, is the minimally coupled ordinary matter contribution. We shall use
physical units 8πG = c = ~ = 1; ε is a constant which specifies the theory. Actually its
values can be ε = ±1, 0 fixing the nature and the dynamics of the scalar field which can
be a standard scalar field, a phantom field or a field without dynamics (see [223, 224]
for details).
In the metric approach, the field equations are obtained by varying (4.22) with
respect to gµν . We get
1
1
Gµν =
T µν + gµν (F − GR) + (gµλ gνσ − gµν gλσ )G;λσ
G
2
i
k
1 X X µν λσ
µλ νσ
j−i
i−j ∂F
(g g + g g )( );σ +
2
∂i R ;λ
i=1 j=1
#
∂F
,
(2.13)
−gµν gλσ (j−1 R);σ i−j
∂i R ;λ
where Gµν is the above Einstein tensor and
n
X
∂F
j
G≡
.
∂j R
(2.14)
j=0
The differential Eqs.(4.23) are of order (2k + 4). The stress-energy tensor is due to the
kinetic part of the scalar field and to the ordinary matter:
ε
1
m
Tµν = Tµν
+ [φ;µ φ;ν − φα; φ;α ] .
2
2
(2.15)
The (eventual) contribution of a potential V (φ) is contained in the definition of F .
From now on, we shall indicate by a capital F a Lagrangian density containing also the
contribution of a potential V (φ) and by F (φ), f (R), or f (R, R) a function of such
fields without potential.
By varying with respect to the scalar field φ, we obtain the Klein-Gordon equation
εφ = −
2
∂F
.
∂φ
(2.16)
For the aims of this review, we do not need more complicated invariants like Rµν Rµν , Rµναβ Rµναβ ,
Cµναβ C µναβ which are also possible.
2.3. STRUCTURE OF THE EXTENDED THEORIES OF GRAVITY
63
Several approaches can be used to deal with such equations. For example, as we said, by
a conformal transformation, it is possible to reduce an ETG to a (multi) scalar-tensor
theory of gravity [193, 149, 150, 151, ?].
The simplest extension of GR is achieved assuming
F = f (R) ,
ε = 0,
(2.17)
in the action (4.22); f (R) is an arbitrary (analytic) function of the Ricci curvature
scalar R. We are considering here the simplest case of fourth-order gravity but we could
α . The standard
construct such kind of theories also using other invariants in Rµν or Rβµν
Hilbert-Einstein action is, of course, recovered for f (R) = R. Varying with respect to
gαβ , we get the field equations
µν
1
f ′ (R)Rαβ − f (R)gαβ = f ′ (R); (gαµ gβν − gαβ gµν ) ,
2
(2.18)
which are fourth-order equations due to the term f ′ (R);µν ; the prime indicates the
derivative with respect to R. Eq.(2.18) is also the equation for Tµν = 0 when the matter
term is absent.
By a suitable manipulation, the above equation can be rewritten as:
1
1
(2.19)
gαβ f (R) − Rf ′ (R) + f ′ (R);αβ − gαβ f ′ (R) ,
Gαβ = ′
f (R) 2
where the gravitational contribution due to higher-order terms can be simply reinterpreted as a stress-energy tensor contribution. This means that additional and higherorder terms in the gravitational action act, in principle, as a stress-energy tensor, related
to the form of f (R). Considering also the standard perfect-fluid matter contribution,
we have
m
m
Tαβ
Tαβ
1
1
′
′
′
curv
gαβ f (R) − Rf (R) + f (R);αβ − gαβ f (R) + ′
= Tαβ + ′
,
Gαβ = ′
f (R) 2
f (R)
f (R)
(2.20)
curv is an effective stress-energy tensor constructed by the extra curvature terms.
where Tαβ
curv identically vanishes while the standard, minimal coupling is
In the case of GR, Tαβ
recovered for the matter contribution. The peculiar behavior of f (R) = R is due to
the particular form of the Lagrangian itself which, even though it is a second order
Lagrangian, can be non-covariantly rewritten as the sum of a first order Lagrangian
plus a pure divergence term. The Hilbert-Einstein Lagrangian can be in fact recast as
follows:
h
i
√
(2.21)
LHE = LHE −g = pαβ (Γρασ Γσρβ − Γρρσ Γσαβ ) + ∇σ (pαβ uσ αβ )
where:
pαβ =
√
−ggαβ =
∂L
∂Rαβ
(2.22)
64
2. EXTENDED THEORIES OF GRAVITY
Γ is the Levi-Civita connection of g and uσαβ is a quantity constructed out with the
variation of Γ [11]. Since uσαβ is not a tensor, the above expression is not covariant;
however a standard procedure has been studied to recast covariance in the first order
theories [225]. This clearly shows that the field equations should consequently be second
order and the Hilbert-Einstein Lagrangian is thus degenerate.
From the action (4.22), it is possible to obtain another interesting case by choosing
F = F (φ)R − V (φ) ,
ε = −1 .
(2.23)
In this case, we get
A=
Z
4
d x
√
1 µν
−g F (φ)R + g φ;µ φ;ν − V (φ)
2
(2.24)
V (φ) and F (φ) are generic functions describing respectively the potential and the coupling of a scalar field φ. The Brans-Dicke theory of gravity is a particular case of the
action (2.24) for V (φ)=0 [226]. The variation with respect to gµν gives the second-order
field equations
1
1 φ
− gµν g F (φ) + F (φ);µν ,
(2.25)
F (φ)Gµν = F (φ) Rµν − Rgµν = − Tµν
2
2
here g is the d’Alembert operator with respect to the metric g The energy-momentum
tensor relative to the scalar field is
1
φ
Tµν
= φ;µ φ;ν − gµν φ;α φα; + gµν V (φ)
2
(2.26)
The variation with respect to φ provides the Klein - Gordon equation, i.e. the field
equation for the scalar field:
g φ − RFφ (φ) + Vφ (φ) = 0
(2.27)
where Fφ = dF (φ)/dφ, Vφ = dV (φ)/dφ. This last equation is equivalent to the Bianchi
contracted identity [227]. Standard fluid matter can be treated as above.
2.3.1
Conformal transformations
Let us now introduce conformal transformations to show that any higher-order or scalartensor theory, in absence of ordinary matter, e.g. a perfect fluid, is conformally equivalent to an Einstein theory plus minimally coupled scalar fields. If standard matter is
present, conformal transformations allow to transfer non-minimal coupling to the matter
component [198]. The conformal transformation on the metric gµν is
g̃µν = e2ω gµν
(2.28)
2.3. STRUCTURE OF THE EXTENDED THEORIES OF GRAVITY
65
in which e2ω is the conformal factor. Under this transformation, the Lagrangian in
(2.24) becomes
√
√
1 µν
= −g̃e−2ω F R̃ − 6F g̃ ω+
−g F R + g φ;µ φ;ν − V
2
(2.29)
1
−6F ω;α ω;α + g̃µν φ;µ φ;ν − e−2ω V
2
in which R̃ and g̃ are the Ricci scalar and the d’Alembert operator relative to the
metric g̃. Requiring the theory in the metric g̃µν to appear as a standard Einstein
theory [201], the conformal factor has to be related to F , that is
e2ω = −2F.
(2.30)
where F must be negative in order to restore physical coupling. Using this relation and
introducing a new scalar field φ̃ and a new potential Ṽ , defined respectively by
s
3Fφ 2 − F
V (φ)
,
(2.31)
φ;α , Ṽ (φ̃(φ)) =
φ̃;α =
2
2F
4F 2 (φ)
we see that the Lagrangian (2.29) becomes
p
√
1 µν
1
1
α
−g F R + g φ;µ φ;ν − V = −g̃ − R̃ + φ̃;α φ̃; − Ṽ
2
2
2
which is the usual Hilbert-Einstein Lagrangian plus the standard Lagrangian relative
to the scalar field φ̃. Therefore, every non-minimally coupled scalar-tensor theory, in
absence of ordinary matter, e.g. perfect fluid, is conformally equivalent to an Einstein
theory, being the conformal transformation and the potential suitably defined by (2.30)
and (2.31). The converse is also true: for a given F (φ), such that 3Fφ 2 − F > 0, we
can transform a standard Einstein theory into a non-minimally coupled scalar-tensor
theory. This means that, in principle, if we are able to solve the field equations in the
framework of the Einstein theory in presence of a scalar field with a given potential,
we should be able to get the solutions for the scalar-tensor theories, assigned by the
coupling F (φ), via the conformal transformation (2.30) with the constraints given by
(2.31). Following the standard terminology, the “Einstein frame” is the framework of
the Einstein theory with the minimal coupling and the “Jordan frame” is the framework
of the non-minimally coupled theory [198].
In the context of alternative theories of gravity, as previously discussed, the gravitational
contribution to the stress-energy tensor of the theory can be reinterpreted by means of
a conformal transformation as the stress-energy tensor of a suitable scalar field and then
as “matter” like terms. Performing the conformal transformation (2.28) in the field
equations (2.19), we get:
1
1
′
′
′
(2.32)
gαβ f (R) − Rf (R) + f (R);αβ − gαβ f (R) +
G̃αβ = ′
f (R) 2
66
2. EXTENDED THEORIES OF GRAVITY
1
+2 ω;α;β + gαβ ω − ω;αω;β + gαβ ω;γ ω ;γ
2
.
We can then choose the conformal factor to be
ω=
1
ln |f ′ (R)| ,
2
(2.33)
which has now to be substituted into (2.20). Rescaling ω in such a way that
kφ = ω ,
and k =
p
1/6, we obtain the Lagrangian equivalence
p
√
1
1
α
−gf (R) = −g̃ − R̃ + φ̃;α φ̃; − Ṽ
2
2
(2.34)
(2.35)
and the Einstein equations in standard form
1
G̃αβ = φ;α φ;β − g̃αβ φ;γ φ;γ + g̃αβ V (φ) ,
2
(2.36)
with the potential
i 1 f (R) − Rf ′ (R)
e−4kφ h
P(φ) − N e2kφ e2kφ =
.
(2.37)
2
2
f ′ (R)2
R
Here N is the inverse function of P ′ (φ) and P(φ) = exp(2kφ)dN . However, the
problem is completely solved if P ′ (φ) can be analytically inverted. In summary, a fourthorder theory is conformally equivalent to the standard second-order Einstein theory plus
a scalar field (see also [142, 196]).
This procedure can be extended to more general theories. If the theory is assumed to
be higher than fourth order, we may have Lagrangian densities of the form [204, 151],
V (φ) =
L = L(R, R, ...k R) .
(2.38)
Every operator introduces two further terms of derivation into the field equations.
For example a theory like
L = RR ,
(2.39)
is a sixth-order theory and the above approach can be pursued by considering a conformal
factor of the form
∂L 1 ∂L
.
(2.40)
+
ω = ln 2
∂R
∂R In general, increasing two orders of derivation in the field equations (i.e. for every term
R), corresponds to adding a scalar field in the conformally transformed frame [151].
A sixth-order theory can be reduced to an Einstein theory with two minimally coupled
2.4. THE PALATINI APPROACH AND THE INTRINSIC CONFORMAL STRUCTURE
67
scalar fields; a 2n-order theory can be, in principle, reduced to an Einstein theory plus
(n − 1)-scalar fields. On the other hand, these considerations can be directly generalized
to higher-order-scalar-tensor theories in any number of dimensions as shown in [148].
As concluding remarks, we can say that conformal transformations work at three
levels: i) on the Lagrangian of the given theory; ii) on the field equations; iii) on the solutions. The table below summarizes the situation for fourth-order gravity (FOG), nonminimally coupled scalar-tensor theories (NMC) and standard Hilbert-Einstein (HE)
theory. Clearly, direct and inverse transformations correlate all the steps of the table
but no absolute criterion, at this point of the discussion, is able to select which is the
“physical” framework since, at least from a mathematical point of view, all the frames
are equivalent [198]. This point is up to now unsolved even if wide discussions are
present in literature [347].
LF OG
l
FOG Eqs.
l
FOG Solutions
2.4
←→
←→
←→
LN M C
l
NMC Eqs.
l
NMC Solutions
←→
←→
←→
LHE
l
Einstein Eqs.
l
Einstein Solutions
The Palatini Approach and the Intrinsic Conformal
Structure
As we said, the Palatini approach, considering g and Γ as independent fields, is “intrinsically” bi-metric and capable of disentangling the geodesic structure from the chronological structure of a given manifold. Starting from these considerations, conformal
transformations assume a fundamental role in defining the affine connection which is
merely “Levi-Civita” only for the Hilbert-Einstein theory.
In this section, we work out examples showing how conformal transformations assume
a fundamental physical role in relation to the Palatini approach in ETGs [210].
Let us start from the case of fourth-order gravity where Palatini variational principle
is straightforward in showing the differences with Hilbert-Einstein variational principle,
involving only metric. Besides, cosmological applications of f (R) gravity have shown the
relevance of Palatini formalism, giving physically interesting results with singularity free solutions [207]. This last nice feature is not present in the standard metric approach.
An important remark is in order at this point. The Ricci scalar entering in f (R)
is R ≡ R(g, Γ) = gαβ Rαβ (Γ) that is a generalized Ricci scalar and Rµν (Γ) is the Ricci
tensor of a torsion-less connection Γ, which, a priori, has no relations with the metric
g of spacetime. The gravitational part of the Lagrangian is controlled by a given real
√
analytical function of one real variable f (R), while −g denotes a related scalar density
of weight 1. Field equations, deriving from the Palatini variational principle are:
1
m
f ′ (R)R(µν) (Γ) − f (R)gµν = Tµν
2
(2.41)
68
2. EXTENDED THEORIES OF GRAVITY
√
∇Γα ( −gf ′ (R)gµν ) = 0
(2.42)
where ∇Γ is the covariant derivative with respect to Γ. As above, we assume 8πG = 1.
We shall use the standard notation denoting by R(µν) the symmetric part of Rµν , i.e.
R(µν) ≡ 21 (Rµν + Rνµ ).
In order to get (2.42), one has to additionally assume that Lm is functionally ing
dependent of Γ; however it may contain metric covariant derivatives ∇ of fields. This
m = T m (g, Ψ) depends on the metric g
means that the matter stress-energy tensor Tµν
µν
and some matter fields denoted here by Ψ, together with their derivatives (covariant
derivatives with respect to the Levi-Civita connection of g). From (2.42) one sees that
√
−gf ′ (R)gµν is a symmetric twice contravariant tensor density of weight 1. As previously discussed in [206, 210], this naturally leads to define a new metric hµν , such that
the following relation holds:
√
√
(2.43)
−gf ′ (R)gµν = −hhµν .
This ansatz is suitably made in order to impose Γ to be the Levi-Civita connection of
√
h and the only restriction is that −gf ′ (R)gµν should be non-degenerate. In the case
of Hilbert-Einstein Lagrangian, it is f ′ (R) = 1 and the statement is trivial.
The above Eq.(2.43) imposes that the two metrics h and g are conformally equivalent.
The corresponding conformal factor can be easily found to be f ′ (R) (in dim M = 4)
and the conformal transformation results to be ruled by:
hµν = f ′ (R)gµν
(2.44)
Therefore, as it is well known, Eq.(2.42) implies that Γ = ΓLC (h) and R(µν) (Γ) =
Rµν (h) ≡ Rµν . Field equations can be supplemented by the scalar-valued equation
obtained by taking the trace of (2.41), (we define τ = trT̂ )
m
≡ τm
f ′ (R)R − 2f (R) = gαβ Tαβ
(2.45)
which controls solutions of (2.42). We shall refer to this scalar-valued equation as the
structural equation of the spacetime. In the vacuum case (and spacetimes filled with
radiation, such that τ m = 0) this scalar-valued equation admits constant solutions,
which are different from zero only if one add a cosmological constant. This means that
the universality of Einstein field equations holds [206], corresponding to a theory with
cosmological constant [228].
In the case of interaction with matter fields, the structural equation (2.44), if explicitly solvable, provides an expression of R = F (τ ), where F is a generic function,
and consequently both f (R) and f ′ (R) can be expressed in terms of τ . The matter
content of spacetime thus rules the bi-metric structure of spacetime and, consequently,
both the geodesic and metric structures which are intrinsically different. This behavior
generalizes the vacuum case and corresponds to the case of a time-varying cosmological
constant. In other words, due to these features, conformal transformations, which allow
2.4. THE PALATINI APPROACH AND THE INTRINSIC CONFORMAL STRUCTURE
69
to pass from a metric structure to another one, acquire an intrinsic physical meaning
since “select” metric and geodesic structures which, for a given ETG, in principle, do
not coincide.
Let us now try to extend the above formalism to the case of non-minimally coupled
scalar-tensor theories. The effort is to understand if and how the bi-metric structure of
spacetime behaves in this cases and which could be its geometric and physical interpretation.
We start by considering scalar-tensor theories in the Palatini formalism, calling A1
the action functional. After, we take into account the case of decoupled non-minimal
interaction between a scalar-tensor theory and a f (R) theory, calling A2 this action
functional. We finally consider the case of non-minimal-coupled interaction between
the scalar field φ and the gravitational fields (g, Γ), calling A3 the corresponding action
functional. Particularly significant is, in this case, the limit of low curvature R. This
resembles the physical relevant case of present values of curvatures of the Universe and
it is important for cosmological applications.
The action (2.24) for scalar-tensor gravity can be generalized, in order to better
develop the Palatini approach, as:
Z
gµ
g
√
ε g
−g [F (φ)R + ∇µ φ ∇ φ − V (φ) + Lm (Ψ, ∇ Ψ)]d4 x .
(2.46)
A1 =
2
As above, the values of ε = ±1 selects between standard scalar field theories and
quintessence (phantom) field theories. The relative “signature” can be selected by conformal transformations. Field equations for the gravitational part of the action are,
respectively for the metric g and the connection Γ:
1
φ
m Γ √
+ Tµν
∇α ( −gF (φ)gµν ) = 0
F (φ)[R(µν) − Rgµν ] = Tµν
2
(2.47)
R(µν) is the same defined in (2.41). For matter fields we have the following field equations:
δLm
εφ = −Vφ (φ) + Fφ (φ)R
= 0.
(2.48)
δΨ
In this case, the structural equation of spacetime implies that:
R=−
τφ + τm
F (φ)
(2.49)
which expresses the value of the Ricci scalar curvature in terms of the traces of the
stress-energy tensors of standard matter and scalar field (we have to require F (φ) 6= 0).
The bi-metric structure of spacetime is thus defined by the ansatz:
√
√
−gF (φ)gµν = −hhµν
(2.50)
such that g and h result to be conformally related
hµν = F (φ)gµν .
(2.51)
70
2. EXTENDED THEORIES OF GRAVITY
The conformal factor is exactly the interaction factor. From (2.49), it follows that in
the vacuum case τ φ = 0 and τ m = 0: this theory is equivalent to the standard Einstein
one without matter. On the other hand, for F (φ) = F0 we recover the Einstein theory
plus a minimally coupled scalar field: this means that the Palatini approach intrinsically
gives rise to the conformal structure (2.51) of the theory which is trivial in the Einstein,
minimally coupled case. Beside fundamental physics motivations, these theories have
acquired a huge interest in Cosmology due to the fact that they naturally exhibit inflationary behaviors able to overcome the shortcomings of Cosmological Standard Model
(based on GR). The related cosmological models seem realistic and capable of matching with the Cosmic Microwave Background Radiation (CMBR) observations [?] As a
further step, let us generalize the previous results considering the case of a non-minimal
coupling in the framework of f (R) theories. The action functional can be written as:
Z
gµ
g
√
ε g
(2.52)
−g [F (φ)f (R) + ∇µ φ ∇ φ − V (φ) + Lm (Ψ, ∇ Ψ)]d4 x
A2 =
2
where f (R) is, as usual, any analytical function of the Ricci scalar R. Field equations
(in the Palatini formalism) for the gravitational part of the action are:
1
φ
m Γ √
+ Tµν
∇α ( −gF (φ)f ′ (R)gµν ) = 0 .
F (φ)[f ′ (R)R(µν) − f (R)gµν ] = Tµν
2
(2.53)
For scalar and matter fields we have, otherwise, the following field equations:
εφ = −Vφ (φ) +
√
−gFφ (φ)f (R)
δLm
=0
δΨ
(2.54)
where the non-minimal interaction term enters into the modified Klein-Gordon equations. In this case the structural equation of spacetime implies that:
f ′ (R)R − 2f (R) =
τφ + τm
.
F (φ)
(2.55)
We remark again that this equation, if solved, expresses the value of the Ricci scalar
curvature in terms of traces of the stress-energy tensors of standard matter and scalar
field (we have to require again that F (φ) 6= 0). The bi-metric structure of spacetime is
thus defined by the ansatz:
√
√
(2.56)
−gF (φ)f ′ (R)gµν = −hhµν
such that g and h result to be conformally related by:
hµν = F (φ)f ′ (R)gµν .
(2.57)
Once the structural equation is solved, the conformal factor depends on the values of
the matter fields (φ, Ψ) or, more precisely, on the traces of the stress-energy tensors
2.4. THE PALATINI APPROACH AND THE INTRINSIC CONFORMAL STRUCTURE
71
and the value of φ. From equation (2.55), it follows that in the vacuum case, i.e. both
τ φ = 0 and τ m = 0, the universality of Einstein field equations still holds as in the case
of minimally interacting f (R) theories [206]. The validity of this property is related to
the decoupling of the scalar field and the gravitational field.
Let us finally consider the case where the gravitational Lagrangian is a general
function of φ and R. The action functional can thus be written as:
Z
gµ
g
√
ε g
−g [K(φ, R) + ∇µ φ ∇ φ − V (φ) + Lm (Ψ, ∇ Ψ)]d4 x
A3 =
(2.58)
2
Field equations for the gravitational part of the action are:
1
∂ K(φ, R) µν
∂ K(φ, R)
φ
m Γ √
R(µν) − K(φ, R)gµν = Tµν + Tµν ∇α
g
= 0.
−g
∂R
2
∂R
(2.59)
For matter fields, we have:
∂ K(φ, R) δLmat
εφ = −Vφ (φ) +
= 0.
(2.60)
∂φ
δΨ
The structural equation of spacetime can be expressed as:
∂K(φ, R)
R − 2K(φ, R) = τ φ + τ m
∂R
(2.61)
This equation, if solved, expresses again the form of the Ricci scalar curvature in terms of
traces of the stress-energy tensors of matter and scalar field (we have to impose regularity
conditions and, for example, K(φ, R) 6= 0). The bi-metric structure of spacetime is thus
defined by the ansatz:
√ ∂K(φ, R) µν √
−g
(2.62)
g = −hhµν
∂R
such that g and h result to be conformally related by
hµν =
∂K(φ, R)
gµν
∂R
(2.63)
Again, once the structural equation is solved, the conformal factor depends just on the
values of the matter fields and (the trace of) their stress energy tensors. In other words,
the evolution, the definition of the conformal factor and the bi-metric structure is ruled
by the values of traces of the stress-energy tensors and by the value of the scalar field
φ. In this case, the universality of Einstein field equations does not hold anymore in
general. This is evident from (2.61) where the strong coupling between R and φ avoids
the possibility, also in the vacuum case, to achieve simple constant solutions.
We consider, furthermore, the case of small values of R, corresponding to small
curvature spacetimes. This limit represents, as a good approximation, the present epoch
72
2. EXTENDED THEORIES OF GRAVITY
of the observed Universe under suitably regularity conditions. A Taylor expansion of
the analytical function K(φ, R) can be performed:
K(φ, R) = K0 (φ) + K1 (φ)R + o(R2 )
where only the first leading term in R is considered and we have defined:
∂K(φ, R)
.
K0 (φ) = K(φ, R)R=0 K1 (φ) =
∂R
R=0
(2.64)
(2.65)
Substituting this expression in (2.61) and (2.63) we get (neglecting higher order approximations in R) the structural equation and the bi-metric structure in this particular
case. From the structural equation, we get:
R=
1
[−(τ φ + τ m ) − 2K0 (φ)]
K1 (φ)
(2.66)
such that the value of the Ricci scalar is always determined, in this first order approximation, in terms of τ φ , τ m , φ. The bi-metric structure is, otherwise, simply defined by
means of the first term of the Taylor expansion, which is
hµν = K1 (φ)gµν .
(2.67)
It reproduces, as expected, the scalar-tensor case (2.51). In other words, scalar-tensor
theories can be recovered in a first order approximation of a general theory where gravity
and non-minimal couplings are any (compare (2.66) with (2.55)). This fact agrees with
the above considerations where Lagrangians of physical interactions can be considered
as stochastic functions with local gauge invariance properties [381].
Finally we have to say that there are also bi-metric theories which cannot be conformally related (see for example the summary of alternative theories given in [185]) and
torsion field should be taken into account, if one wants to consider the most general
viewpoint [186, 187]. We will not take into account these general theories in this review.
After this short review of ETGs in metric and Palatini approach, we are going to
face some remarkable applications to cosmology and astrophysics. In particular, we
deal with the straightforward generalization of GR, the f (R) gravity, showing that,
in principle, no further ingredient, a part a generalized gravity, could be necessary to
address issues as missing matter (dark matter) and cosmic acceleration (dark energy).
However what we are going to consider here are nothing else but toy models which are
not able to fit the whole expansion history, the structure growth law and the CMB
anisotropy and polarization. These issues require more detailed theories which, up to
now, are not available but what we are discussing could be a useful working paradigm as
soon as refined experimental tests to probe such theories will be proposed and pursued.
In particular, we will outline an independent test, based on the stochastic background of
gravitational waves, which could be extremely useful to discriminate between ETGs and
2.4. THE PALATINI APPROACH AND THE INTRINSIC CONFORMAL STRUCTURE
73
GR or among the ETGs themselves. In this latter case, the data delivered from groundbased interferometers, like VIRGO and LIGO, or the forthcoming space interferometer
LISA, could be of extreme relevance in such a discrimination.
Finally, we do not take into account the well known inflationary models based on
ETGs (e.g. [144, 146]) since we want to show that also the last cosmological epochs,
directly related to the so called Precision Cosmology, can be framed in such a new
”economic” scheme.
74
2. EXTENDED THEORIES OF GRAVITY
Chapter 3
Gravity from Poincaré Gauge
Invariance
3.1
What can generate the Gravity?
Following the prescriptions of General Relativity, the physical spacetime is assumed to
be a four-dimensional differential manifold (see [229] for a general discussion on gravity
theories and their prescriptions). In Special Relativity, this manifold is the Minkwoski
flat-spacetime M4 while, in General Relativity, the underlying spacetime is assumed to
be curved in order to describe the effects of gravitation.
Utiyama [230] was the first to propose that General Relativity can be seen as a
gauge theory based on the local Lorentz group SO(3, 1) in much the same way that
the Yang-Mills gauge theory [231] was developed on the basis of the internal iso-spin
gauge group SU (2). In this formulation the Riemannian connection is the gravitational
counterpart of the Yang-Mills gauge fields. While SU (2), in the Yang-Mills theory, is an
internal symmetry group, the Lorentz symmetry represents the local nature of spacetime
rather than internal degrees of freedom. The Einstein Equivalence Principle, asserted
for General Relativity, requires that the local spacetime structure can be identified with
the Minkowski spacetime possessing Lorentz symmetry. In order to relate local Lorentz
symmetry to the external spacetime, we need to solder the local space to the external
space. The soldering tools are the tetrad fields. Utiyama regarded the tetrads as objects
given a priori. Soon after, Sciama [232] recognized that spacetime should necessarily
be endowed with torsion in order to accommodate spinor fields. In other words, the
gravitational interaction of spinning particles requires the modification of the Riemann
spacetime of General Relativity to be a (non-Riemannian) curved spacetime with torsion.
Although Sciama used the tetrad formalism for his gauge-like handling of gravitation,
his theory fell shortcomings in treating tetrad fields as gauge fields. Kibble [233] made
a comprehensive extension of the Utiyama gauge theory of gravitation by showing that
the local Poincaré symmetry SO(3, 1) ⋊ T (3, 1) (⋊ represents the semi-direct product)
75
76
3. GRAVITY FROM POINCARÉ GAUGE INVARIANCE
can generate a spacetime with torsion as well as curvature. The gauge fields introduced
by Kibble include the tetrads as well as the local affine connection. There have been a
variety of gauge theories of gravitation based on different local symmetries which gave
rise to several interesting applications in theoretical physics [234, 235, 236, 237, 238,
239, 240, 241, 242, 243].
Following the Kibble approach, it can be demonstrated how gravitation can be formulated starting from a pure gauge viewpoint. In particular, the aim of this chapter is
to show, in details, how a theory of gravitation is a gauge theory which can be obtained
starting from the local Poincaré symmetry.
A gauge theory of gravity based on a nonlinear realization of the local conformalaffine group of symmetry transformations has been formulated [244]. The coframe fields
and gauge connections of the theory have been obtained. The tetrads and Lorentz group
metric have been used to induce a spacetime metric. The inhomogenously transforming
(under the Lorentz group) connection coefficients gave rise to gravitational gauge potentials used to define covariant derivatives accommodating minimal coupling of matter
and gauge fields. On the other hand, the tensor valued connection forms have been
used as auxiliary dynamical fields associated with the dilation, special conformal and
deformation (shear) degrees of freedom inherent to the bundle manifold. This allowed
to define the bundle curvature of the theory. Then boundary topological invariants have
been constructed. They served as a prototype (source free) gravitational Lagrangian.
Finally the Bianchi identities, covariant field equations and gauge currents have been
obtained.
Here, starting from the Invariance Principle, we consider first the Global Poincaré
Invariance and then the Local Poincaré Invariance. This approach lead to construct a
given theory of gravity as a gauge theory. This viewpoint, if considered in detail, can
avoid many shortcomings and could be useful to formulate self-consistent schemes for
quantum gravity.
3.2
Invariance Principle
As it is well-known, the field equations and conservation laws can be obtained from a
least action principle. The same principle is the basis of any gauge theory so we start
from it to develop our considerations. Let us start from a least action principle and the
Noether theorem.
Let χ(x) be a multiplet field defined at a spacetime point x and L{χ(x), ∂j χ(x); x}
be the Lagrangian density of the system. The action integral of the system over a given
spacetime volume Ω is defined by
I(Ω) =
Z
Ω
L{χ(x), ∂j χ(x); x} d4 x.
(3.1)
3.2. INVARIANCE PRINCIPLE
77
Now let us consider the infinitesimal variations of the coordinates
xi → x′i = xi + δxi ,
(3.2)
χ(x) → χ′ (x′ ) = χ(x) + δχ(x).
(3.3)
and the field variables
Correspondingly, the variation of the action is given by
Z
Z
Z
′ ′
′ ′
4 ′
4
L (x )||∂j x′j || − L(x) d4 x.
L (x ) d x −
L(x) d x =
δI =
Ω′
Ω
(3.4)
Ω
Since the Jacobian for the infinitesimal variation of coordinates becomes
||∂j x′j || = 1 + ∂j (δxj ),
the variation of the action takes the form,
Z
δL(x) + L(x) ∂j (δxj ) d4 x
δI =
(3.5)
(3.6)
Ω
where
δL(x) = L′ (x′ ) − L(x).
(3.7)
For any function Φ(x) of x, it is convenient to define the fixed point variation δ0 by,
δ0 Φ(x) := Φ′ (x) − Φ(x) = Φ′ (x′ ) − Φ(x′ ).
(3.8)
Expanding the function to first order in δxj as
Φ(x′ ) = Φ(x) + δxj ∂j Φ(x),
(3.9)
we obtain
δΦ(x) = Φ′ (x′ ) − Φ(x) = Φ′ (x′ ) − Φ(x′ ) + Φ(x′ ) − Φ(x) = δ0 Φ(x) + δxj ∂j Φ(x), (3.10)
or
δ0 Φ(x) = δΦ(x) − δxj ∂j Φ(x).
(3.11)
The advantage to have the fixed point variation is that δ0 commutes with ∂j :
δ0 ∂j Φ(x) = ∂j δ0 Φ(x).
(3.12)
δχ = δ0 χ + δxi ∂i χ,
(3.13)
δ∂i χ = ∂i (δ0 χ) − ∂(δxj )∂i χ.
(3.14)
For Φ(x) = χ(x), we have
and
78
3. GRAVITY FROM POINCARÉ GAUGE INVARIANCE
Using the fixed point variation in the integrand of (3.6) gives
Z
δ0 L(x) + ∂j (δxj L(x)) d4 x.
δI =
(3.15)
δL + L∂j (δxj ) = δ0 L + ∂j (Lδxj ) = 0.
(3.16)
Ω
If we require the action integral defined over any arbitrary region Ω be invariant, that
is, δI = 0, then we must have
If ∂j (δxj ) = 0, then δL = 0, that is, the Lagrangian density L is invariant. In general,
however, ∂j (δxj ) 6= 0, and L transforms like a scalar density. In other words, L is a
Lagrangian density unless ∂j (δxj ) = 0.
For convenience, let us introduce a function h(x) that behaves like a scalar density,
namely
δh + h∂j (δxj ) = 0.
(3.17)
We further assume L(χ, ∂j χ; x) = h(x)L(χ, ∂j χ; x). Then we see that
δL + L∂j (δxj ) = hδL.
(3.18)
Hence the action integral remains invariant if
δL = 0.
(3.19)
The newly introduced function L(χ, ∂j χ; x) is the scalar Lagrangian of the system.
Let us calculate the integrand of (3.15) explicitly. The fixed point variation of L(x)
is a consequence of a fixed point variation of the field χ(x),
δ0 L =
∂L
∂L
δ0 χ +
δ0 (∂j χ)
∂χ
∂(∂j χ)
(3.20)
which can be cast into the form,
δ0 L = [L]χ δ0 χ + ∂j
where
∂L
− ∂j
[L]χ ≡
∂χ
∂L
δ0 χ
∂(∂j χ)
∂L
∂(∂j χ)
.
Consequently, we have the action integral in the form
Z ∂L
j
k
δI =
[L]χ δ0 χ + ∂j
δχ − Tk δx
d4 x,
∂(∂
χ)
j
Ω
where
T j k :=
∂L
∂k χ − δkj L
∂(∂j χ)
(3.21)
(3.22)
(3.23)
(3.24)
3.3. GLOBAL POINCARÉ INVARIANCE
79
is the canonical energy-momentum tensor density. If the variations are chosen in such
a way that δxj = 0 over Ω and δ0 χ vanishes on the boundary of Ω, then δI = 0 gives us
the Euler-Lagrange equation,
∂L
∂L
[L]χ =
− ∂j
= 0.
(3.25)
∂χ
∂(∂j χ)
On the other hand, if the field variables obey the Euler-Lagrange equation, [L]χ = 0,
then we have
∂L
j
k
δχ − T k δx = 0,
(3.26)
∂j
∂(∂j χ)
which gives rise, considering also the Noether theorem, to conservation laws. These very
straightforward considerations are at the basis of our following discussion.
3.3
Global Poincaré Invariance
As standard, we assert that our spacetime in the absence of gravitation is a Minkowski
space M4 . The isometry group of M4 is the group of Poincaré transformations (PT)
which consists of the Lorentz group SO(3, 1) and the translation group T (3, 1). The
Poincaré transformations of coordinates are
PT
xi → x′i = ai j xj + bi ,
(3.27)
where aij and bi are real constants, and aij satisfy the orthogonality conditions aik akj = δji .
For infinitesimal variations,
δx′i = χ′ (x′ ) − χ(x) = εi j xj + εi
(3.28)
where εij + εji = 0. While the Lorentz transformation forms a six parameter group,
the Poincaré group has ten parameters. The Lie algebra for the ten generators of the
Poincaré group is
[Ξij , Ξkl ] = ηik Ξjl + ηjl Ξik − ηjk Ξil − ηil Ξjk ,
(3.29)
[Ξij , Tk ] = ηjk Ti − ηik Tj , [Ti , Tj ] = 0,
where Ξij are the generators of Lorentz transformations, and Ti are the generators of
four-dimensional translations. Obviously, ∂i (δxi ) = 0 for the Poincaré transformations
(3.27). Therefore, our Lagrangian density L, which is the same as L with h(x) = 1 in
this case, is invariant; namely, δL = δL = 0 for δI = 0.
Suppose that the field χ(x) transforms under the infinitesimal Poincaré transformations as
1
(3.30)
δχ = εij Sij χ,
2
80
3. GRAVITY FROM POINCARÉ GAUGE INVARIANCE
where the tensors Sij are the generators of the Lorentz group, satisfying
Sij = −Sji , [Sij , Skl ] = ηik Sjl + ηjl Sik − ηjk Sil − ηil Sjk .
(3.31)
Correspondingly, the derivative of χ transforms as
1
δ(∂k χ) = εij Sij ∂k χ − εi k ∂i χ.
2
(3.32)
Since the choice of infinitesimal parameters εi and εij is arbitrary, the vanishing variation
of the Lagrangian density δL = 0 leads to the identities,
∂L
∂L
Sij χ +
(Sij ∂k χ + ηki ∂j χ − ηkj ∂i χ) = 0.
∂χ
∂(∂k χ)
We also obtain the following conservation laws
∂j Tkj = 0, ∂k S k ij − xi T k j + xj T k i = 0,
where
S k ij := −
∂L
Sij χ.
∂(∂k χ)
(3.33)
(3.34)
(3.35)
These conservation laws imply that the energy-momentum and angular momentum,
respectively
Z
Z
0
0 3
S ij − xi T 0 j − xj T 0 i d3 x,
Pl = Tl d x, Jij =
(3.36)
are conserved. This means that the system invariant under the ten parameter symmetry
group has ten conserved quantities. This is an example of Noether symmetry. The first
term of the angular momentum integral corresponds to the spin angular momentum
while the second term gives the orbital angular momentum. The global Poincaré invariance of a system means that, for the system, the spacetime is homogeneous (all spacetime
points are equivalent) as dictated by the translational invariance and is isotropic (all directions about a spacetime point are equivalent) as indicated by the Lorentz invariance.
It is interesting to observe that the fixed point variation of the field variables takes the
form
1
(3.37)
δ0 χ = εj k Ξj k χ + εj Tj χ,
2
where
(3.38)
Ξj k = Sj k + xj ∂k − xk ∂j , Tj = −∂j .
We remark that Ξj k are the generators of the Lorentz transformation and Tj are those
of the translations.
3.4. LOCAL POINCARÉ INVARIANCE
3.4
81
Local Poincaré Invariance
As next step, let us consider a modification of the infinitesimal Poincaré transformations
(3.28) by assuming that the parameters εjk and εj are functions of the coordinates and
by writing them altogether as
δxµ = εµ ν (x) xν + εµ (x) = ξ µ ,
(3.39)
which we call the local Poincaré transformations (or the general coordinate transformations). In order to make a distinction between the global transformation and the local
transformation, we use the Latin indices (j, k = 0, 1, 2, 3) for the former and the Greek
indices (µ, ν = 0, 1, 2, 3) for the latter. The variation of the field variables χ(x) defined
at a point x is still the same as that of the global Poincaré transformations,
1
δχ = εij S ij χ.
2
(3.40)
The corresponding fixed point variation of χ takes the form,
δ0 χ =
1
εij S ij χ − ξ ν ∂ν χ.
2
(3.41)
Differentiating both sides of (3.41) with respect to xµ , we have
1
1
δ0 ∂µ χ = εij Sij ∂µ χ + (∂µ εij ) S ij χ − ∂µ (ξ ν ∂ν χ).
2
2
(3.42)
By using these variations, we obtain the variation of the Lagrangian L,
1
δL + ∂µ (δxµ )L = hδL = δ0 L + ∂ν (Lδxν ) = − (∂µ εij ) S µ ij − ∂µ ξ ν T µν ,
2
(3.43)
which is no longer zero unless the parameters εij and ξ ν become constants. Accordingly,
the action integral for the given Lagrangian density L is not invariant under the local
Poincaré transformations. We notice that while ∂j (δxj ) = 0 for the local Poincaré
transformations, ∂µ ξ µ does not vanish under local Poincaré transformations. Hence, as
expected L is not a Lagrangian scalar but a Lagrangian density. As mentioned earlier,
in order to define the Lagrangian L, we have to select an appropriate non-trivial scalar
function h(x) satisfying
δh + h∂µ ξ µ = 0.
(3.44)
Now we consider a minimal modification of the Lagrangian so as to make the action
integral invariant under the local Poincaré transformations. It is rather obvious that if
there is a covariant derivative ∇k χ which transforms as
δ(∇k χ) =
1 ij
ε Sij ∇k χ − εi k ∇i χ,
2
(3.45)
82
3. GRAVITY FROM POINCARÉ GAUGE INVARIANCE
then a modified Lagrangian L′ (χ, ∂k χ, x) = L(χ, ∇k χ, x), obtained by replacing ∂k χ of
L(χ, ∂k χ, x) by ∇k χ, remains invariant under the local Poincaré transformations, that
is
∂L′
∂L′
δχ +
δ(∇k χ) = 0.
(3.46)
δL′ =
∂χ
∂(∇k χ)
To find such a k-covariant derivative, we introduce the gauge fields Aij µ = −Aji µ and
define the µ-covariant derivative
1
∇µ χ := ∂µ χ + Aij µ Sij χ,
2
(3.47)
in such a way that the covariant derivative transforms as
1
δ0 ∇µ χ = Sij ∇µ χ − ∂µ (ξ ν ∇ν χ).
2
(3.48)
The transformation properties of Aabµ are determined by ∇µ χ and δ∇µ χ. Making use
of
δ∇µ χ =
1 ij
1
1
1
ε ,µ Sij χ + εij Sij ∂µ χ − (∂µ ξ ν ) ∂ν ψ + δAij µ Sij χ + Aij µ Sij εkl Skl χ (3.49)
2
2
2
4
and comparing with (3.47) we obtain,
1 ij kl
ij
ij
ij kl
δA µ Sij χ + ε ,µ Sij χ +
A µ ε − ε A µ Sij Skl χ + (∂µ ξ ν ) Aij ν Sij χ = 0. (3.50)
2
Using the antisymmetry in ij and kl to rewrite the term in parentheses on the RHS of
(3.50) as [Sij , Skl ] Aij µ εkl χ, we see the explicit appearance of the commutator [Sij , Skl ].
Using the expression for the commutator of Lie algebra generators
[Sij , Skl ] =
1 [ef ]
c [ij][kl] Sef ,
2
(3.51)
[ef ]
where c [ij][kl] (the square brackets denote anti-symmetrization) is the structure constants of the Lorentz group (deduced below), we have
[Sij , Skl ] Aij µ εkl =
1 ic j
i
Aµ εc − Acj
µ εc Sij .
2
(3.52)
The substitution of this equation and the consideration of the antisymmetry of εcb = −εbc
enables us to write
δAij µ = εi k Akj µ + εj k Aik µ − (∂µ ξ ν )Aij ν − ∂µ εij .
(3.53)
We require the k-derivative and µ-derivative of χ to be linearly related as
∇k χ = Vk µ (x)∇µ χ,
(3.54)
3.5. SPINORS, VECTORS AND TETRADS
83
where the coefficients Vk µ (x) are position-dependent and behave like a new set of field
variables. From (3.54) it is evident that ∇k χ varies as
δ∇k χ = δVkµ ∇µ χ + Vkµ δ∇µ χ.
(3.55)
Comparing with δ∇k χ = 12 εab Sab ∇k χ − εjk ∇j χ we obtain,
Vαk δVkµ ∇µ χ − ξ ν ,α ∇ν χ + Vαk εjk ∇j χ = 0.
(3.56)
δVk µ = Vk ν ∂ν ξ µ − Vi µ εi k .
(3.57)
Exploiting δ Vαk Vkµ = 0 we find the quantity Vk µ transforms according to
It is also important to recognize that the inverse of det(Vk µ ) transforms like a scalar
density as h(x) does. For our minimal modification of the Lagrangian density, we utilize
this available quantity for the scalar density h; namely, we let
h(x) = [det(Vk µ )]−1 .
(3.58)
In the limiting case, when we consider Poincaré transformations, that are not spacetime
dependent, Vk µ → δkµ so that h(x) → 1. This is a desirable property. Then we replace
the Lagrangian density L(χ, ∂k χ, x), invariant under the global Poincaré transformations, by a Lagrangian density
L(χ, ∂µ χ; x) → h(x)L(χ, ∇k χ).
(3.59)
The action integral with this modified Lagrangian density remains invariant under the
local Poincaré transformations. Since the local Poincaré transformations δxµ = ξ µ (x)
are nothing else but generalized coordinate transformations, the newly introduced gauge
fields Viλ and Aij µ can be interpreted, respectively, as the tetrad (vierbein) fields which
set the local coordinate frame and as a local affine connection with respect to the tetrad
frame (see also [245]).
3.5
Spinors, Vectors and Tetrads
Let us consider first the case where the multiplet field χ is the Dirac field ψ(x) which
behaves like a four-component spinor under the Lorentz transformations and transforms
as
ψ(x) → ψ ′ (x′ ) = S(Λ)ψ(x),
(3.60)
where S(Λ) is an irreducible unitary representation of the Lorentz group. Since the
bilinear form v k = iψγ k ψ is a vector, it transforms according to
v j = Λjk v k ,
(3.61)
84
3. GRAVITY FROM POINCARÉ GAUGE INVARIANCE
where Λji is a Lorentz transformation matrix satisfying
Λij + Λji = 0.
(3.62)
The invariance of v i (or the covariance of the Dirac equation) under the transformation
ψ(x) → ψ ′ (x′ ) leads to
S −1 (Λ)γ µ S(Λ) = Λµν γ ν ,
(3.63)
where the γ ′ s are the Dirac γ-matrices satisfying the anticommutator,
γi γj + γj γi = ηij 1.
Furthermore, we notice that the γ-matrices have the following properties:

2

(γ0 )† = −γ0 , γ 0 = (γ0 )2 = −1, γ0 = −γ 0 and γ0 γ 0 = 1





2
(γk )† = γk , γ k = (γk )2 = 1; (k = 1, 2, 3) and γk = γ k






(γ5 )† = −γ5 , (γ5 )2 = −1 and γ 5 = γ5 .
(3.64)
(3.65)
µν
We assume the transformation S(Λ) can be put into the form S(Λ) = eΛµν γ . Expanding S(Λ) about the identity and only retaining terms up to the first order in the
infinitesimals and expanding Λµν to the first order in εµν
Λµν = δµν + εµν , εij + εji = 0,
(3.66)
we get
1
(3.67)
S(Λ) = 1 + εij γij .
2
In order to determine the form of γij , we substitute (3.66) and (3.67) into (3.63) to
obtain
1 h ij k i
(3.68)
εij γ , γ = η ki εji γ j .
2
Rewriting the RHS of (3.68) using the antisymmetry of εij as
yields
1 η ki εji γ j = εij η ki γ j − η kj γ i ,
2
(3.69)
i
γ k , γ ij = η ki γ j − η kj γ i .
(3.70)
h
Assuming the solution to have the form of an antisymmetric product of two matrices,
we obtain the solution
1 i j
γ,γ .
(3.71)
γ ij :=
2
3.5. SPINORS, VECTORS AND TETRADS
85
If χ = ψ, the group generator Sij appearing in (3.31) is identified with
1
Sij ≡ γij = (γi γj − γj γi ).
2
(3.72)
To be explicit, the Dirac field transforms under Lorentz transformations (LT) as
δψ(x) =
1 ij
ε γij ψ(x).
2
(3.73)
The Pauli conjugate of the Dirac field is denoted ψ and defined by
ψ(x) := iψ † (x) γ0 , i
C.
(3.74)
The conjugate field ψ transforms under LTs as,
1
δψ = −ψ εij ψγij .
2
(3.75)
Under local LTs, εab (x) becomes a function of spacetime. Now, unlike ∂µ ψ(x),
the derivative of ψ ′ (x′ ) is no longer homogenous due to the occurrence of the term
γ ab [∂µ εab (x)] ψ(x) in ∂µ ψ ′ (x′ ), which is non-vanishing unless εab is constant. When
going from locally flat to curved spacetime, we must generalize ∂µ to the covariant
derivative ∇µ to compensate for this extra term, allowing to gauge the group of LTs.
Thus, by using ∇µ , we can preserve the invariance of the Lagrangian for arbitrary local
LTs at each spacetime point
∇µ ψ ′ (x′ ) = S(Λ(x))∇µ ψ(x).
(3.76)
To determine the explicit form of the connection belonging to ∇µ , we study the derivative
of S(Λ(x)). The transformation S(Λ(x)) is given by
1
S(Λ(x)) = 1 + εab (x)γ ab .
2
(3.77)
Since εab (x) is only a function of spacetime for local Lorentz coordinates, we express this
infinitesimal LT in terms of general coordinates only by shifting all spacetime dependence
of the local coordinates into tetrad fields as
εab (x) = Va λ (x)V νb (x)ελν .
Substituting this expression for εab (x), we obtain
i
h
∂µ εab (x) = ∂µ Va λ (x)V νb (x)ελν .
(3.78)
(3.79)
However, since ελν has no spacetime dependence, this reduces to
∂µ εab (x) = Va λ (x)∂µ Vbλ (x) − Vb ν (x)∂µ Vaν (x).
(3.80)
86
3. GRAVITY FROM POINCARÉ GAUGE INVARIANCE
Letting
ωµba := Vb ν (x)∂µ Vaν (x),
(3.81)
the first and second terms in Eq.(3.80) become Vaλ (x)∂µ Vbλ (x) = 12 ωµab and Vbν (x)∂µ Vaν (x) =
1
2 ωµba respectively. Using the identification
∂µ εab (x) = ωµab ,
(3.82)
we write
1
∂µ S(Λ(x)) = − γ ab ωµab .
2
According to (3.47), the covariant derivative of the Dirac spinor is
1
∇µ ψ = ∂µ ψ + Aij µ γij ψ.
2
(3.83)
(3.84)
Correspondingly, the covariant derivative of ψ̄ is given by
1
∇µ ψ = ∂µ ψ − Aij µ ψ̄γij .
2
(3.85)
Using the covariant derivatives of ψ and ψ̄, we can show that
∇µ vj = ∂µ vj − Ai jµ vi .
(3.86)
The same covariant derivative should be used for any covariant vector vk under the
Lorentz transformation. Since ∇µ (vi v i ) = ∂µ (vi v i ), the covariant derivative for a contravariant vector v i must be
∇µ v i = ∂µ v i + Ai jµ v j .
(3.87)
Since the tetrad Vi µ is a covariant vector under Lorentz transformations, its covariant derivative must transform according to the same rule. Using ∇a = Vaµ (x)∇µ , the
covariant derivatives of a tetrad in local Lorentz coordinates read
∇ν Vi µ = ∂ν Vi µ − Ak iν Vk µ , ∇ν V i µ = ∂ν V i µ + Ai kν V k µ .
(3.88)
The inverse of Vi µ is denoted by V i µ and satisfies
V i µ Vi ν = δµ ν ,
V i µ Vj µ = δi j .
(3.89)
To allow the transition to curved spacetime, we take account of the general coordinates
of objects that are covariant under local Poincaré transformations. Here we define the
covariant derivative of a quantity v λ which behaves like a contravariant vector under
the local Poincaré transformation. Namely
Dν v λ ≡ Vi λ ∇ν v i = ∂ν v λ + Γλ µν v µ , Dν vµ ≡ V i µ ∇ν vi = ∂ν vµ − Γλ µν vλ ,
(3.90)
3.5. SPINORS, VECTORS AND TETRADS
87
where
Γλ µν := Vi λ ∇ν V i µ ≡ −V i µ ∇ν Vi λ .
(3.91)
The definition of Γλ µν implies
Dν Vi λ = ∇ν Vi λ + Γλ µν Vi µ = ∂ν Vi λ − Ak iν Vk λ + Γλ µν Vi µ = 0,
(3.92)
Dν V i µ = ∇ν V i µ − Γλ µν V i λ = ∂ν V i µ + Ai kν V k µ − Γλ µν V i λ = 0.
From (3.92) we find,
Ai kν = V i λ ∂ν Vk λ + Γλ µν V i λ Vk µ = −Vk λ ∂ν V i λ + Γλ µν V i λ Vk µ .
(3.93)
or, equivalently, in terms of ω defined in (3.81),
Ai kν = ω iνk + Γλ µν V i λ Vk µ = −ωkν i + Γλ µν V i λ Vk µ .
(3.94)
Using this in (3.84), we may write
∇µ ψ = (∂µ − Γµ )ψ,
(3.95)
where
1 i
(3.96)
ω jµ − Γλ µν V i λ Vj ν γi j ,
4
which is known as the Fock-Ivanenko connection.
We now study the transformation properties of Aµab . Recall ωµab = Va λ (x)∂µ Vβλ (x)
and since ∂µ ηab = 0, we write
Γµ =
Λa a ηab ∂µ Λb b = Λa a ∂µ Λab .
(3.97)
Note that barred indices are equivalent to the primed indices used above. Hence, the
spin connection transforms as
Aabc = Λa a Λb b Λc c Aabc + Λa a Λc c V µa (x)∂µ Λbc .
(3.98)
To determine the transformation properties of
Γabc = Aabc − [V µa (x)∂µ V νb (x)] Vνc (x),
we consider the local LT of [Va µ (x)∂µ V νb (x)] Vνc (x) which is,
h
i
V µa (x)∂µ V νb Vνc (x) = Λa a Λb b Λc c [Aνab Vνc (x)] + Λa a Λc c V µa (x)∂µ Λcb .
(3.99)
(3.100)
From this result, we obtain the following transformation law,
Γabc = Λa a Λb b Λc c Γabc .
(3.101)
88
3. GRAVITY FROM POINCARÉ GAUGE INVARIANCE
We now explore the consequence of the antisymmetry of ωabc in bc. Recalling the
equation for Γabc , exchanging b and c and adding the two equations, we obtain
Γabc + Γacb = −V µa (x) [(∂µ V νb (x)) Vνc (x) + (∂µ V νc (x)) Vνb (x)] .
(3.102)
We know however, that
∂µ [V νb (x)Vνc (x)] = Vνc (x)∂µ V νb (x) + Vλb (x)∂µ V λc (x) + Vb ν (x)V λc (x)∂µ gλν . (3.103)
Letting λ → ν and exchanging b and c, we obtain
∂µ [V νb (x)Vνc (x)] = −Vb λ (x)V νc (x)∂µ gνλ ,
(3.104)
Γabc + Γacb = Va µ (x)Vb λ (x)Vc ν (x)∂µ gνλ .
(3.105)
so that, finally,
This, however, is equivalent to
Γabc + Γa cb = V µa (x)V λb (x)V νc (x)∂µ gνλ ,
(3.106)
Γµλν + Γµνλ = ∂µ gνλ ,
(3.107)
and then
which we recognize as the general coordinate connection. It is known that the covariant
derivative for general coordinates is
∇µ Aν λ = ∂µ Aν λ + Γλ µσ Aν σ − Γσ µν Aσ λ .
(3.108)
In a Riemannian manifold, the connection is symmetric under the exchange of µν, that
is, Γλ µν = Γλ νµ . Using the fact that the metric is a symmetric tensor we can now
determine the form of the Christoffel connection by cyclically permuting the indices of
the general coordinate connection equation (3.107) yielding
Γµνλ =
1
(∂µ gνλ + ∂ν gλµ − ∂λ gµν ) .
2
(3.109)
Since Γµνλ = Γνµλ is valid for general coordinate systems, it follows that a similar
constraint must hold for local Lorentz transforming coordinates as well, so we expect
Γabc = Γbac . Recalling the equation for Γabc and exchanging a and b, we obtain
ωabc − ωbac = Vνc (x) V µa (x)∂µ V νb (x) − V µb (x)∂µ V νa (x) .
(3.110)
We now define the objects of anholonomicity as
Ωcab := Vνc (x) V µa (x)∂µ V νb (x) − V µb (x)∂µ V νa (x) .
(3.111)
3.6. CURVATURE, TORSION AND METRIC
89
Using Ωcab = −Ωcba , we permute indices in a similar manner as was done for the derivation of the Christoffel connection above yielding,
1
[Ωcab + Ωbca − Ωabc ] V cµ ≡ ∆abµ .
(3.112)
2
For completeness, we determine the transformation law of the Christoffel connection.
Making use of Γλµν eλ = ∂µ eν where
ωabµ =
we can show
∂µ eν = X µµ X ν ν ∂µ eν + X µµ (∂µ X ν ν ) eν ,
(3.113)
Γλ µ ν = X µµ X ν ν Xλ λ Γλ µν + X µµ Xν λ X ν µν ,
(3.114)
X ν µν ≡ ∂µ ∂ν xν .
(3.115)
where
In the light of the above considerations, we may regard infinitesimal local gauge
transformations as local rotations of basis vectors belonging to the tangent space [243,
248] of the manifold. For this reason, given a local frame on a tangent plane to the
point x on the base manifold, we can obtain all other frames on the same tangent plane
by means of local rotations of the original basis vectors. Reversing this argument, we
observe that by knowing all frames residing in the horizontal tangent space to a point x
on the base manifold enables us to deduce the corresponding gauge group of symmetry
transformations.
3.6
Curvature, Torsion and Metric
From the definition of the Fock-Ivanenko covariant derivative, we can find the second
order covariant derivative
1
1
Dν Dµ ψ = ∂ν ∂µ ψ + Scd ψ∂ν Aµ cd + Aµ cd ∂ν ψ + Γρ µν Dρ ψ + Sef Aν ef ∂µ ψ
2
2
1
ef
cd
+ Sef Scd Aν Aµ ψ.
(3.116)
4
Recalling Dν V cµ = 0, we can solve for the spin connection in terms of the Christoffel
connection
Aµ cd = −V dλ ∂µ V cλ − Γµ cd .
(3.117)
The derivative of the spin connection is then
∂µ Acd ν = −V dλ ∂µ ∂ν V cλ − ∂ν V cλ ∂µ Vλ d − ∂µ Γcdν .
(3.118)
Noting that the Christoffel connection is symmetric and partial derivatives commute,
we find
i 1
i
h
h
1
[Dµ , Dν ] ψ = Scd ∂ν Acd µ − ∂µ Acd ν ψ + Sef Scd Aef ν Acd µ − Aef µ Acd ν ψ ,
2
4
(3.119)
90
3. GRAVITY FROM POINCARÉ GAUGE INVARIANCE
where
∂ν Acd µ − ∂µ Acd ν = ∂µ Γcd ν − ∂ν Γcdµ .
Relabeling running indices, we can write
1
1
Sef Scd Aef ν Acdµ − Aef µ Acd ν ψ = [Scd , Sef ] Aefµ Acdν ψ.
4
4
(3.120)
(3.121)
Using {γa , γb } = 2ηab to deduce
{γa , γb } γc γd = 2ηab γc γd ,
we find that the commutator of bi-spinors is given by
i
1h
ηce δda δfb − ηde δca δfb + ηcf δea δdb − ηdf δea δcb Sab .
[Scd , Sef ] =
2
(3.122)
(3.123)
Clearly the terms in brackets on the RHS of (3.123) are antisymmetric in cd and ef
and also antisymmetric under the exchange of pairs of indices cd and ef . Since the
alternating spinor is antisymmetric in ab, so it must be the terms in brackets: this
means that the commutator does not vanish. Hence, the term in brackets is totally
antisymmetric under interchange of indices ab, cd and ef and exchange of these pairs of
indices. We identify this as the structure constant of the Lorentz group [249]
i
h
(3.124)
ηce δda δfb − ηde δca δfb + ηcf δea δdb − ηdf δea δcb = c[cd][ef ] [ab] = c[ab] [cd][ef ],
with the aid of which we can write
where
i
h
1
1
[Scd , Sef ] Aef µ Acd ν ψ = Sab Aa eν Aeb µ − Ab eν Aaeµ ψ,
4
2
(3.125)
Aa eν Aeb µ − Ab eν Aaeµ = Γaνe Γeb µ − Γbνe Γeaµ .
(3.126)
Combining these results, the commutator of two µ-covariant differentiations gives
1
[∇µ , ∇ν ]χ = − Rij µν Sij χ,
2
(3.127)
Ri jµν = ∂ν Ai jµ − ∂µ Ai jν + Ai kν Ak jµ − Ai kµ Ak jν .
(3.128)
where
Using the Jacobi identities for the commutator of covariant derivatives, it follows that
the field strength Ri jµν satisfies the Bianchi identity
∇λ Ri jµν + ∇µ Ri jνλ + ∇ν Ri jλµ = 0.
(3.129)
Permuting indices, this can be put into the cyclic form
εαβρσ ∇β Rijρσ = 0,
(3.130)
3.6. CURVATURE, TORSION AND METRIC
91
where εαβρσ is the Levi-Civita alternating symbol. Furthermore, Rij µν = η jk Ri kµν is
antisymmetric with respect to both pairs of indices,
Rij µν = −Rji µν = Rji νµ = −Rij νµ .
(3.131)
This condition is known as the first curvature tensor identity.
To determine the analogue of [∇µ , ∇ν ]χ in local coordinates, we start from ∇k ψ =
V µk ∇µ ψ. From ∇k ψ we obtain,
∇l ∇k ψ = V νl ∇ν V µk ∇µ ψ + V νl V µk ∇ν ∇µ ψ.
(3.132)
Permuting indices and recognizing
Vµ a ∇ν V µk = −Vk µ ∇ν V aµ ,
Vµa Vkµ = 0), we arrive at
(which follows from ∇ν
V νl ∇ν V µk ∇µ ψ − V µk (∇µ V νl ) ∇ν ψ = V µl V νk − V µk V νl ∇ν Vµ a ∇a ψ.
Defining
C akl := V µk V νl − V µl V νk ∇ν Vµ a ,
(3.133)
(3.134)
(3.135)
the commutator of the k-covariant differentiations takes the final form [233]
1
[∇k , ∇l ]χ = − Rij kl Sij χ + C i kl ∇i χ,
2
(3.136)
Rij kl = Vk µ Vl ν Rij µν .
(3.137)
where
As done for Ri jµν using the Jacobi identities for the commutator of covariant derivatives,
we find the Bianchi identity in Einstein-Cartan spacetime [255, 247]
εαβρσ ∇β Rijρσ = εαβρσ Cβρ λ Rijσλ .
(3.138)
The second curvature identity
leads to,
Rk [ρσλ] = 2∇[ρ Cσλ] k − 4C[ρσ b Cλ]b k
(3.139)
εαβρσ ∇β Cρσ k = εαβρσ Rk jρσ V jβ .
(3.140)
Γλµν = Vi λ ∇ν V i µ = −Vµ i ∇ν V λi ,
(3.141)
Notice that if
then
Γλµν − Γλνµ = Viλ ∇ν V i µ − ∇µ V i ν .
(3.142)
92
3. GRAVITY FROM POINCARÉ GAUGE INVARIANCE
Contracting by Vkµ Vlν , we obtain [233],
C akl = Vk µ Vl ν Vλ a
Γλµν − Γλνµ .
(3.143)
We therefore conclude that C akl is related to the antisymmetric part of the affine connection
(3.144)
Γλ[µν] = Vµ k Vν l Va λ C a kl ≡ T λµν ,
which is usually interpreted as spacetime torsion T λµν . Considering ∆abµ defined in
(3.112), we see that the most general connection in the Poincaré gauge approach to
gravitation is
(3.145)
Aabµ = ∆abµ − Kabµ + Γλ νµ Vaλ Vb ν ,
where
Kabc = − T λ νµ − Tνµλ + Tµ λν Vaλ Vb ν Vc µ ,
(3.146)
Rρ σµν = ∂ν Γρ σµ − ∂µ Γρ σν + Γρ λν Γλ σµ − Γρ λµ Γλ σν .
(3.147)
ρ
is the contorsion tensor. Now, the quantity Rσµν
= Vi ρ Ri σµν may be expressed as
Therefore, we can regard Rρ σµν as the curvature tensor with respect the affine connection Γλ µν . By using the inverse of the tetrad, we define the metric of the spacetime
manifold by
gµν = V i µ V j ν ηij .
(3.148)
From (3.92) and the fact that the Minkowski metric is constant, it is obvious that the
metric so defined is covariantly constant, that is,
Dλ gµν = 0.
(3.149)
The spacetime thus specified by the local Poincaré transformation is said to be metric.
It is not difficult to show that
√
−g = [det V i µ ] = [det Vi µ ]−1 ,
(3.150)
√
where g = det gµν . Hence we may take −g for the density function h(x).
3.7
Field Equations for Gravity
Finally, we are able to deduce the field equations for the gravitational field. From the
curvature tensor Rρ σµν , given in (3.147), the Ricci tensor follows
Rσν = Rµ σµν .
(3.151)
and the scalar curvature
L
R = Rν ν = R + ∂i Ka ia − Ta bc Kbc a
(3.152)
3.7. FIELD EQUATIONS FOR GRAVITY
93
L
where R denotes the usual Ricci scalar of General Relativity. Using this scalar curvature
R, we choose the Lagrangian density for free Einstein-Cartan gravity
L
1 √
ia
bc
a
(3.153)
LG =
−g R + ∂i Ka − Ta Kbc − 2Λ ,
2κ
where κ is a gravitational coupling constant, and Λ is the cosmological constant. These
L
considerations can be easily extended to any function of R as in [246]. Observe that the
second term is a divergence and may be ignored. The field equation can be obtained
from the total action,
Z
(3.154)
Lfield (χ, ∂µ χ, Vi µ , Aij µ ) + LG d4 x,
S=
where the matter Lagrangian density is taken to be
Lfield =
1 a
ψγ Da ψ − Da ψ γ a ψ .
2
(3.155)
Modifying the connection to include Christoffel, spin connection and contorsion contributions so as to operate on general, spinoral arguments, we have
L
1
σ
σ
σ
Γµ = gλσ ∆ µρ − Γ ρµ − K ρµ γ λρ .
(3.156)
4
It is important to keep in mind that ∆σ µρ act only on multi-component spinor fields,
L
while Γ σ ρµ act on vectors and arbitrary tensors. The gauge covariant derivative for a
spinor and adjoint spinor is then given by
Dµ ψ = (∂µ − Γµ ) ψ, Dµ ψ = ∂µ ψ − ψΓµ .
(3.157)
The variation of the field Lagrangian is
δLfield = ψ (δγ µ Dµ + γ µ δΓµ ) ψ.
We know that the Dirac gamma matrices are covariantly vanishing, so
i
h
b κ = 0.
Dκ γι = ∂κ γι − Γµικ γµ + γι , Γ
(3.158)
(3.159)
b κ are real matrices used to induce similarity transformations on
The 4 × 4 matrices Γ
quantities with spinor transformation [254] properties according to
b κ leads to,
Solving for Γ
b −1 γi Γ.
b
γi′ = Γ
b κ = 1 [(∂κ γι ) γ ι − Γµ ικ γµ γ ι ] .
Γ
8
(3.160)
(3.161)
94
3. GRAVITY FROM POINCARÉ GAUGE INVARIANCE
bκ,
Taking the variation of Γ
bκ =
δΓ
=
1 (∂κ δγι ) γ ι + (∂κ γι ) δγ ι − (δΓµ ικ ) γµ γ ι
−Γµ ικ ((δγµ ) γ ι + γµ δγ ι )
8
1
[(∂κ δγι ) γ ι − (δΓµ ικ ) γµ γ ι ] .
8
(3.162)
Since we require the anticommutator condition on the gamma matrices {γ µ , γ ν } = 2gµν
to hold, the variation of the metric gives
2δgµν = {δγ µ , γ ν } + {γ µ δγ ν }.
(3.163)
One solution to this equation is,
δγ ν =
1
γσ δγ σν .
2
(3.164)
With the aid of this result, we can write
1
(∂κ δγι ) γ ι = ∂κ (γ ν δgνι ) γ ι .
2
(3.165)
Finally, exploiting the anti-symmetry in γµν we obtain
bκ =
δΓ
1
gνσ δΓµκσ − gµσ δΓνκσ γ µν .
8
(3.166)
The field Lagrangian defined in the Einstein-Cartan spacetime can be written [250, 251,
255, 388, 247] explicitly in terms of its Lorentzian and contorsion components as
o
n
L
L
1
~c
µ
µ
µ
αβ
Lfield =
ψ.
(3.167)
D µ ψ γ ψ − ψγ Dµ ψ − Kµαβ ψ γ , γ
2
8
Using the following relations
 1
βα γ µ ψ − 1 K
µ αβ
 − 4 Kµαβ ψ γ µ , γ αβ ψ =14 Kµαβ ψγ
4 µαβ ψγ γ ψ,
µ , γ νλ ε
γ µ γ ν γ λ εµνλσ
γ
µνλσ = 3!γσ γ5 ,
µ= νλ

γ ,γ
= γ [µ γ ν γ λ] ,
(3.168)
we obtain
o
n
1
Kµαβ ψ γ µ , γ αβ ψ = Kµαβ εαβµν ψγ5 γν ψ .
2i
Here we define the contorsion axial vector
Kν :=
1 αβµν
ε
Kαβµ .
3!
Multiplying through by the axial current jν5 = ψγ5 γν ψ, we obtain
ψγ5 γν ψ εαβµν Kµαβ = −6ijν5 K ν .
(3.169)
(3.170)
(3.171)
3.7. FIELD EQUATIONS FOR GRAVITY
95
Thus, the field Lagrangian density becomes
L
L
3i~c
1
µ
µ
D µ ψ γ ψ − ψγ Dµ ψ +
Kµ j5µ .
Lfield =
2
8
The total action reads
δI
Z
Z
√
= δ LG −gd x + δ Lfield −gd4 x
Z
√
=
(δLG + δLfield ) −gd4 x.
√
4
(3.172)
(3.173)
Writing the metric in terms of the tetrads gµν = V µi V νi , we observe
√
1√
−g δV µi Vµi + Vνi δV νi .
δ −g = −
2
By using
(3.174)
δV νi = δ η ij V νj = η ij δV jν ,
we are able to deduce
(3.175)
√
√
δ −g = − −gVµi δVi µ .
(3.176)
µ
For the variation of the Ricci tensor Riν = Vi Rµν we have
L
L
L
δRiν = δVi µ Rµν + Vi µ δRµν .
(3.177)
In an inertial frame, the Ricci tensor reduces to
L
L
Rµν = ∂ν Γ
so that
L
µ
L
δRiν = δVi Rµν + Vi
µ
β
βµ
L
− ∂β Γ
L
∂ν δΓ
β
νµ ,
β
βµ
(3.178)
L
− ∂β δΓ
β
νµ
.
(3.179)
The second term can be converted into a surface term, so it may be ignored. Collecting
our results, we have


δgµν = −gµρ gνσ δgρσ ,

√
√
√



δ −g = − 21 −ggµν δgµν = − −gVµi δVi µ ,


L
L
ρ
µ
λρ
λρ
λ
(3.180)
∇
δΓ
−
∇
δΓ
δR
=
g
ν
ν
µν
ρµ
λ
λ + Tλµ δΓ ρν , δ Riν = δVi Rµν



L
L
L

 δR = R
µν δg
µν ∇ δ Γ λ
λ
bc
a


µν + g
λ
µν − ∇ν δ Γ µλ − Ta δKbc .
From the above results, we obtain


Z
Ri µ − 12 Vi µ R − Vi µ Λ δV iµ + 2gρλ Tµλσ δΓµ ρσ
√
1
L
L

 −gd4 x.
δIG =
µν
λ
λ
+g
∇λ δΓ µν − ∇ν δΓ µλ
16π
(3.181)
96
3. GRAVITY FROM POINCARÉ GAUGE INVARIANCE
The last term in the action can be converted into a surface term, so it may be ignored.
Using the four-current v µ introduced earlier, the action for the matter fields read [254]
Z h
i
bµ ψ √−gd4 x
ψδγ µ ∇µ ψ + ψγ µ δΓ
(3.182)
δIfield =

 Z  21 gµν ψγi (∇ν ψ) + T µρσ Ti ρσ − δiµ Tλρσ T λρσ δV iµ 
√
L
L
=
−gd4 x.
1
σ
ρν
µ
ρµ
ν
σ

 + 8 (g v − g v ) gµσ δΓ νρ − gνσ δΓ µρ
Removing the derivatives of variations of the metric appearing in δΓσ νρ via partial
integration, and equating to zero the coefficients of δgµν and δT σνρ in the variation of
the action integral, we obtain
1
1
1
1
Rµν − gµν R − gµν Λ +
(3.183)
ψγν ∇µ ψ − ∇µ vν
0 =
16π
2
2
4
+∇σ Tµν σ + Tµρσ Tνρσ − gµν Tλρσ T λρσ
and
Tρσλ = 8πτρσλ .
(3.184)
Eqs.(3.183) have the form of Einstein equations
Gµν − gµν Λ = 8πΣµν ,
(3.185)
where the Einstein tensor and non-symmetric energy-momentum tensors are
1
Gµν = Rµν − gµν R,
2
(3.186)
Σµν = Θµν + Tµν ,
(3.187)
respectively. Here we identify Θµν as the canonical energy-momentum
Θµ ν =
∂Lfield
∇ν χ − δµν Lfield ,
∂(∇µ χ)
(3.188)
while Tµν is the stress-tensor form of the non-Riemannian manifold. For the case of
spinor fields being considered here the explicit form of the energy-momentum components [253] are (after symmetrization of corresponding canonical source terms in the
Einstein equation),
(3.189)
Θµν = − ψγµ ∇ν ψ − ∇ν ψ γµ ψ + ψγν ∇µ ψ − ∇µ ψ γν ψ
and by using the second field equation (3.184), we determine
Tµν = ∇σ Tµν σ + Tµρσ τνρσ − gµν Tλρσ τ λρσ ,
(3.190)
3.7. FIELD EQUATIONS FOR GRAVITY
97
where τµν σ is the so-called spin - energy potential [251, 255]
τµν σ :=
∂Lfield
γµν χ.
∂(∇σ χ)
(3.191)
Explicitly, the spin energy potential reads τ µνσ = ψγ [µ γ ν γ σ] ψ. The equation of motion
obtained from the variation of the action with respect to ψ reads [251, 255]
3
γ µ ∇µ ψ + Tµνσ γ [µ γ ν γ σ] ψ = 0.
8
(3.192)
It is interesting to observe that this generalized curved spacetime Dirac equation can be
recast into the nonlinear equation of the Heisenberg-Pauli type
3
γ µ ∇µ ψ + ε ψγ µ γ5 ψ γµ γ5 ψ = 0.
8
(3.193)
Although the gravitational field equation is similar in form to the Einstein field
equation, it differs from the original Einstein equations because the curvature tensor,
containing spacetime torsion, is non-Riemannian. Assuming that the Euler-Lagrange
equations for the matter fields are satisfied, we obtain the following conservation laws
for the angular - momentum and energy - momentum
V µi V νj Σ[µν] = ∇ν τij ν ,
Vµ k ∇ν Σν κ = Σν κ T kµν + τ νij Rijµν .
(3.194)
98
3. GRAVITY FROM POINCARÉ GAUGE INVARIANCE
Chapter 4
Space-time deformations and
conformal transformations
towards extended theories of
gravity
4.1
Deformation and conformal transformations, how?
The issue to consider a general way to deform the space-time metrics is not new. It
has been posed in different ways and is related to several physical problems ranging
from the spontaneous symmetry breaking of unification theories up to gravitational
waves, considered as space-time perturbations. In cosmology, for example, one faces the
problem to describe an observationally lumpy universe at small scales which becomes
isotropic and homogeneous at very large scales according to the Cosmological Principle.
In this context, it is crucial to find a way to connect background and locally perturbed
metrics [256]. For example, McVittie [257] considered a metric which behaves as a
Schwarzschild one at short ranges and as a Friedman-Lemaitre-Robertson-Walker metric
at very large scales. Gautreau [258] calculated the metric generated by a Schwarzschild
mass embedded in a Friedman cosmological fluid trying to address the same problem. On
the other hand, the post-newtonian parameterization, as a standard, can be considered
as a deformation of a background, asymptotically flat Minkowski metric.
In general, the deformation problem has been explicitly posed by Coll and collaborators [259, 260, 261] who conjectured the possibility to obtain any metric from the
deformation of a space-time with constant curvature. The problem was solved only
for 3-dimensional spaces but a straightforward extension should be to achieve the same
result for space-times of any dimension.
In principle, new exact solutions of the Einstein field equations can be obtained by
studying perturbations. In particular, dealing with perturbations as Lorentz matrices
99
4. SPACE-TIME DEFORMATIONS AND CONFORMAL TRANSFORMATIONS TOWARDS EXTENDED
100
THEORIES OF GRAVITY
of scalar fields ΦAC reveals particularly useful. Firstly they transform as scalars with
respect the coordinate transformations. Secondly, they are dimensionless and, in each
point, the matrix ΦAC behaves as the element of a group. As we shall see below, such an
approach can be related to the conformal transformations giving an ”extended” interpretation and a straightforward physical meaning of them (see [347, 210] and references
therein for a comprehensive review). Furthermore scalar fields related to space-time
deformations have a straightforward physical interpretation which could contribute to
explain several fundamental issues as the Higgs mechanism in unification theories, the
inflation in cosmology and other pictures where scalar fields play a fundamental role in
dynamics. In this chapter, we are going to discuss the properties of the deforming matrices ΦAC and we will derive, from the Einstein equations, the field equations for them,
showing how them can parameterize the deformed metrics, according to the boundary
and initial conditions and to the energy-momentum tensor. The layout is the following,
we define the space-time perturbations in the framework of the metric formalism giving
the notion of first and second deformation matrices. When are devoted to the main
properties of deformations. particular, we discuss how deformation matrices can be
split in their trace, traceless and skew parts. We derive the contributions of deformation
to the geodesic equation and, starting from the curvature Riemann tensor, the general
equation of deformations. We discuss the notion of linear perturbations under the standard of deformations. In particular, we recast the equation of gravitational waves and
the transverse traceless gauge under the standard of deformations. After we discuss the
action of deformations on the Killing vectors. The result consists in achieving a notion
of approximate symmetry.
4.2
Generalities on space-time deformations
In order to start our considerations, let us take into account a metric g on a space-time
manifold M. Such a metric is assumed to be an exact solution of the Einstein field
equations. We can decompose it by a co-tetrad field ω A (x)
g = ηAB ω A ω B .
(4.1)
Let us define now a new tetrad field ω
e = ΦAC (x) ω C , with ΦAC (x) a matrix of scalar
f with the metric e
fields. Finally we introduce a space-time M
g defined in the following
way
e = ηAB ΦAC ΦB D ω C ω D = γCD (x)ω C ω D ,
g
(4.2)
where also γCD (x) is a matrix of fields which are scalars with respect to the coordinate
transformations.
If ΦAC (x) is a Lorentz matrix in any point of M, then
ge ≡ g
(4.3)
4.3. PROPERTIES OF DEFORMING MATRICES
101
f is a deformed M. If all the functions
otherwise we say that ge is a deformation of g and M
A
of Φ C (x) are continuous, then there is a one - to - one correspondence between the
f
points of M and the points of M.
f
In particular, if ξ is a Killing vector for g on M, the corresponding vector ξe on M
could not necessarily be a Killing vector.
A particular subset of these deformation matrices is given by
A
ΦA
C (x) = Ω(x) δ C .
(4.4)
which define conformal transformations of the metric,
ge = Ω2 (x)g .
(4.5)
γCD (x) = ηAB ΦAC (x)ΦB D (x).
(4.6)
In this sense, the deformations defined by Eq. (4.2) can be regarded as a generalization of the conformal transformations.
We call the matrices ΦAC (x) first deformation matrices, while we can refer to
as the second deformation matrices, which, as seen above, are also matrices of scalar
fields. They generalize the Minkowski matrix ηAB with constant elements in the definition of the metric. A further restriction on the matrices ΦAC comes from the theorem
proved by Riemann by which an n-dimensional metric has n(n − 1)/2 degrees of freedom
(see [260] for details). With this definitions in mind, let us consider the main properties
of deforming matrices.
4.3
Properties of deforming matrices
Let us take into account a four dimensional space-time with Lorentzian signature. A
family of matrices ΦAC (x) such that
ΦAC (x)
GL(4) ∀x,
(4.7)
are defined on such a space-time.
These functions are not necessarily continuous and can connect space-times with
f with a
different topologies. A singular scalar field introduces a deformed manifold M
space-time singularity.
As it is well known, the Lorentz matrices ΛAC leave the Minkowski metric invariant
and then
g = ηEF ΛE A ΛF B ΦAC ΦB D ω C ω D = ηAB ΦAC ΦB D ω C ω D .
(4.8)
It follows that ΦAC give rise to right cosets of the Lorentz group, i.e. they are the elements of the quotient group GL(4, R)/SO(3, 1). On the other hand, a right-multiplication
of ΦAC by a Lorentz matrix induces a different deformation matrix.
4. SPACE-TIME DEFORMATIONS AND CONFORMAL TRANSFORMATIONS TOWARDS EXTENDED
102
THEORIES OF GRAVITY
The inverse deformed metric is
A
geab = η CD Φ−1
A
CΦ
−1 B
a b
D eA eB
(4.9)
C
A.
where Φ−1 C ΦC B = ΦAC Φ−1 B = δB
Let us decompose now the matrix ΦAB = ηAC ΦC B in its symmetric and antisymmetric parts
ΦAB = Φ(AB) + Φ[AB] = Ω ηAB + ΘAB + ϕAB
(4.10)
where Ω = ΦAA , ΘAB is the traceless symmetric part and ϕAB is the skew symmetric
part of the first deformation matrix respectively. Then standard conformal transformations are nothing else but deformations with ΘAB = ϕAB = 0 [263].
A
Finding the inverse matrix Φ−1 C in terms of Ω, ΘAB and ϕAB is not immediate,
but as above, it can be split in the three terms
Φ−1
A
C
= αδAC + ΨAC + ΣAC
(4.11)
where α, ΨAC and ΣAC are respectively the trace, the traceless symmetric part and the
antisymmetric part of the inverse deformation matrix. The second deformation matrix,
from the above decomposition, takes the form
C
D
γAB = ηCD (Ω δA
+ ΘC A + ϕC A )(Ω δB
+ ΘDB + ϕDB )
(4.12)
and then
γAB = Ω2 ηAB + 2Ω ΘAB + ηCD ΘC A ΘDB + ηCD (ΘC A ϕDB
+ϕC A ΘDB ) + ηCD ϕC A ϕDB .
(4.13)
In general, the deformed metric can be split as
g̃ ab = Ω2 gab + γ ab
(4.14)
γ ab = 2Ω ΘAB + ηCD ΘC A ΘDB + ηCD (ΘC A ϕD B + ϕC A ΘDB )
B
+ηCD ϕC A ϕD B ω A
a ωb
(4.15)
where
In particular, if ΘAB = 0, the deformed metric simplifies to
geab = Ω2 gab + ηCD ϕ CA ϕ DB ω Aa ω Bb
(4.16)
and, if Ω = 1, the deformation of a metric consists in adding to the background metric
a tensor γab . We have to remember that all these quantities are not independent as, by
the theorem mentioned in [260], they have to form at most six independent functions in
a four dimensional space-time.
4.3. PROPERTIES OF DEFORMING MATRICES
103
Similarly the controvariant deformed metric can be always decomposed in the following way
gab = α2 gab + λab
e
(4.17)
f
c
bc
Let us find the relation between γ ab and λab . By using gf
ab g = δa , we obtain
α2 Ω2 δac + α2 γac + Ω2 λca + γ ab λbc = δac
(4.18)
if the deformations are conformal transformations, we have α = Ω−1 , so assuming such
a condition, one obtain the following matrix equation
α2 γac + Ω2 λca + γ ab λbc = 0 ,
(4.19)
(δab + Ω−2 γab )λcb = −Ω−4 γac
(4.20)
λcb = −Ω−4 (δ + Ω−2 γ)−1ab γac
(4.21)
φab = ΦAB ωbB eaA
(4.22)
gab = gcd φca φdb
e
(4.23)
φAB = φab ωaA ebB .
(4.24)
and
and finally
where (δ + Ω−2 γ)−1 is the inverse tensor of (δab + Ω−2 γab ).
To each matrix ΦAB , we can associate a (1,1)-tensor φab defined by
such that
which can be decomposed as in Eq.(4.16). Vice-versa from a (1,1)-tensor φab , we can
define a matrix of scalar fields as
The Levi Civita connection corresponding to the metric (4.14) is related to the
original connection by the relation (see the NOTE for details)
(see [263]), where
e c = Γc + C c
Γ
ab
ab
ab
1
gcd gd(a ∇b) Ω − gab gecd ∇d Ω + gecd (∇a γdb + ∇b γad − ∇d γab ) .
C cab = 2e
2
(4.25)
(4.26)
Therefore, in a deformed space-time, the connection deformation acts like a force that
deviates the test particles from the geodesic motion in the unperturbed space-time. As
a matter of fact the geodesic equation for the deformed space-time
b
a
d 2 xc
c dx dx
+
Γ̃
=0
ab
dλ2
dλ dλ
(4.27)
4. SPACE-TIME DEFORMATIONS AND CONFORMAL TRANSFORMATIONS TOWARDS EXTENDED
104
THEORIES OF GRAVITY
becomes
d 2 xc
dxa dxb
dxa dxb
= −C cab
.
+ Γcab
2
dλ
dλ dλ
dλ dλ
The deformed Riemann curvature tensor is then
(4.28)
e d = R d + ∇b C d − ∇a C d + C eac C d − C e C d ,
R
ae
bc
be
bc
ac
abc
abc
(4.29)
eab = Rab + ∇d C d − ∇a C d + C e C d − C e C d
R
ae
db
de
ab
db
ab
(4.30)
while the deformed Ricci tensor obtained by contraction is
and the curvature scalar
i
h
e = geab R
eab = e
R
gab Rab + geab ∇d C dab − ∇a C ddb + C eab C dde − C edb C dae
(4.31)
From the above curvature quantities, we obtain finally the equations for the deformations. In the vacuum case, we simply have
eab = Rab + ∇d C d − ∇a C d + C e C d − C e C d = 0
R
ab
db
ab
de
db
ae
(4.32)
where Rab must be regarded as a known function. In presence of matter, we consider
the equation
1
Rab + ∇d C dab − ∇a C ddb + C eab C dde − C edb C dae = Teab − geab Te
2
(4.33)
we are assuming, for the sake of simplicity 8πG = c = 1. This last equation can be
improved by considering the Einstein field equations
1
Rab = Tab − gab T
2
(4.34)
and then
∇d C dab
−
∇a C ddb
+ C eab C dde
−
C edb C dae
1
1
= Teab − geab Te − Tab − gab T
2
2
(4.35)
is the most general equation for deformations.
4.4
Metric deformations as perturbations and gravitational
waves
Metric deformations can be used to describe perturbations. To this aim we can simply
consider the deformations
4.4. METRIC DEFORMATIONS AS PERTURBATIONS AND GRAVITATIONAL WAVES
105
ΦAB = δAB + ϕAB
(4.36)
| ϕAB | ≪ 1,
(4.37)
| ∂ϕAB | ≪ 1 .
(4.38)
with
together with their derivatives
With this approximation, immediately we find the inverse relation
(Φ−1 )AB ≃ δAB − ϕAB .
(4.39)
As a remarkable example, we have that gravitational waves are generally described, in
linear approximation, as perturbations of the Minkowski metric
gab = ηab + γab .
(4.40)
In our case, we can extend in a covariant way such an approximation. If ϕAB is an
antisymmetric matrix, we have
gab = gab + γab
e
(4.41)
where the first order terms in ϕAB vanish and γab is of second order
γab = ηAB ϕAC ϕB D ω Ca ω Db .
(4.42)
geab = gab + γ ab
(4.43)
Consequently
where
γ ab = η AB (ϕ−1 )CA (ϕ−1 )DB eC a eDb .
(4.44)
Let us consider the background metric gab , solution of the Einstein equations in the
vacuum
Rab = 0.
(4.45)
We obtain the equation of perturbations considering only the linear terms in Eq.(4.32)
and neglecting the contributions of quadratic terms. We find
eab = ∇d C d − ∇a C d = 0 ,
R
ab
db
and, by the explicit form of C dab , this equation becomes
∇d ∇a γ db + ∇d ∇b γ da − ∇d ∇d γ ab − ∇a ∇d γ db + ∇a ∇b γ dd − ∇a ∇d γ db = 0 .
(4.46)
(4.47)
Imposing the transverse traceless gauge on γab , i.e. the standard gauge conditions
∇a γab = 0
(4.48)
4. SPACE-TIME DEFORMATIONS AND CONFORMAL TRANSFORMATIONS TOWARDS EXTENDED
106
THEORIES OF GRAVITY
and
γ = γ aa = 0
(4.49)
Eq.(4.47) reduces to
d
b
γbd = 0 ,
∇b ∇b γac − 2Rac
(4.50)
see also [263]. In our context, this equation is a linearized equation for deformations and
it is straightforward to consider perturbations and, in particular, gravitational waves,
as small deformations of the metric. This result can be immediately translated into the
above scalar field matrix equations. Note that such an equation can be applied to the
conformal part of the deformation, when the general decomposition is considered.
As an example, let us take into account the deformation matrix equations applied
to the Minkowski metric, when the deformation matrix assumes the form (4.36). In this
case, the equations (4.47), become ordinary wave equations for γab . Considering the
deformation matrices, these equations become, for a tetrad field of constant vectors,
∂ d ∂d ϕC A ϕCB + 2 ∂d ϕC A ∂ d ϕCB + ϕC A ∂ d ∂d ϕCB = 0 .
(4.51)
The above gauge conditions are now
ϕAB ϕBA = 0
and
C
edD ∂d ϕCA ϕC
B + ϕCA ∂d ϕB = 0 .
(4.52)
(4.53)
This result shows that the gravitational waves can be fully recovered starting from the
scalar fields which describe the deformations of the metric. In other words, such scalar
fields can assume the meaning of gravitational wave modes.
4.5
Approximate Killing vectors
Another important issue which can be addressed starting from space-time deformations
is related to the symmetries. In particular, they assume a fundamental role in describing
when a symmetry is preserved or broken under the action of a given field. In General
Relativity, the Killing vectors are always related to the presence of given space-time
symmetries [263].
Let us take an exact solution of the Einstein equations, which satisfies the Killing
equation
(Lξ g)ab = 0
(4.54)
where ξ, being the generator of an infinitesimal coordinate transformation, is a Killing
vector. If we take a deformation of the metric with the scalar matrix
ΦAB = δAB + ϕAB
(4.55)
4.5. APPROXIMATE KILLING VECTORS
107
with
| ϕAB | ≪ 1 ,
(4.56)
(Lξ e
g)ab 6= 0 ,
(4.57)
(Lξ eA )a = 0 ,
(4.58)
(Lξ ϕ)AB = ξ a ∂a ϕAB 6= 0 .
(4.59)
and
being
we have
If there is some region D of the deformed space-time Mdef ormed where
| (Lξ ϕ)AB | ≪ 1
(4.60)
we say that ξ is an approximate Killing vector on D. In other words, these approximate
Killing vectors allow to ”control” the space-time symmetries under the action of a given
deformation. 1
We can calculate the modified connection Γˆcab in many alternative ways. Let us introduce the tetrad
eA and cotetrad ω B satisfying the orthogonality relation
1
B
ieA ω B = δA
(4.61)
and the non-integrability condition (anholonomy)
1 A B
Ω ω ∧ ωC .
2 BC
(4.62)
”
′
′
′
1“ A
AA′
ηCC ′ ΩC
ΩBC − η AA ηBB′ ΩB
A′ B
A′ C − η
2
(4.63)
dω A =
The corresponding connection is
ΓA
BC =
If we deform the metric as in (4.2), we have two alternative ways to write this expression: either writing
the “deformation” of the metric in the space of tetrads or “deforming” the tetrad field as in the following
expression
ĝ = ηAB ΦAC ΦB D ω C ω D = γAB ω A ω B = ηAB ω̂ A ω̂ B .
(4.64)
In the first case, the contribution of the Christoffel symbols, constructed by the metric γAB , appears
”
′
′
′
1“ A
AA′
Γ̂A
γCC ′ ΩC
ΩBC − γ AA γBB′ ΩB
BC =
A′ B
A′ C − γ
2
´
′ `
1
(4.65)
+ γ AA ieC dγBA′ − ieB dγCA′ − ieA′ dγBC
2
A
In the second case, using (4.62), we can define the new anholonomy objects ĈBC
.
dω̂ A =
1 A B
Ω̂ ω̂ ∧ ω̂ C .
2 BC
(4.66)
After some calculations, we have
A
−1
Ω̂A
BC = Φ E Φ
D
−1 F
BΦ
C
“
”
A
a
−1 G
−1 F
ΩE
DF + 2Φ F eG Φ
[B ∂a Φ
C]
(4.67)
4. SPACE-TIME DEFORMATIONS AND CONFORMAL TRANSFORMATIONS TOWARDS EXTENDED
108
THEORIES OF GRAVITY
4.6
Deformations in f (R) -Theories
Resuming the equation 4.71, i.e
g = Ω2 (x)g .
e
(4.71)
we can recast the conformal factor with f ′ (R)
As we are assuming a constant metric in tetradic space, the deformed connection is
”
′
′
′
1“ A
AA′
Ω̂BC − η AA ηBB′ Ω̂B
ηCC ′ Ω̂C
Γ̂A
BC =
A′ C − η
A′ B .
2
(4.68)
A
A
−1
Substituting (4.67) in (4.68), the final expression of Γ̂A
BC , as a function of ΩBC , Φ B , Φ
D
C
and eaG
is
Γ̂ABC = ∆DEF
ABC
»
′
1
E′
F′
K
a −1 G
−1 H
ηDG ΦGG′ Φ−1 E Φ−1 F ΩG
[E ∂|a| Φ
F]
E ′ F ′ + ηDK Φ H eG Φ
2
–
(4.69)
where
D E F
D E F
D E F
∆DEF
ABC = δA δC δB − δB δC δA + δC δA δB .
(4.70)
Chapter 5
Probing the Minkowskian limit:
Gravitational waves in
f (R)-Theories
5.1
Why the gravitational waves in f (R)-Theories?
Recently, the data analysis of interferometric GWs detectors has been started (for the
current status of GWs interferometers see [312, 313, 314, 315, 316, 317, 318, 319]) and the
scientific community aims at a first direct detection of GWs in next years. The design
and the construction of a number of sensitive detectors for gravitational waves (GWs)
is underway today. There are some laser interferometers like the VIRGO detector,
built in Cascina, near Pisa, Italy, by a joint Italian-French collaboration, the GEO
600 detector built in Hanover, Germany, by a joint Anglo-German collaboration, the
two LIGO detectors built in the United States (one in Hanford, Washington and the
other in Livingston, Louisiana) by a joint Caltech-Mit collaboration, and the TAMA
300 detector, in Tokyo, Japan. Many bar detectors are currently in operation too, and
several interferometers and bars are in a phase of planning and proposal stages (for the
current status of gravitational waves experiments see [402, 403]). The results of these
detectors will have a fundamental impact on astrophysics and gravitation physics and
will be important for a better knowledge of the Universe and either to confirm or ruling
out the physical consistency of General Relativity or any other theory of gravitation
[320, 321, 322, 323, 324, 325].
Moreover as we write in the above Chapters the emergence of issues as dark matter
and dark energy cloud be related to the need of revising the theory of gravitation at
astrophysical and cosmological scales and/or at strong field regimes. However, also considering the recent flurry of papers on the argument, a comprehensive effective theory of
gravity, acting consistently at any scale, is far, up to now, to be found out, and demands
an improvement of observational datasets and the search for experimentally testable
109
110
5. PROBING THE MINKOWSKIAN LIMIT: GRAVITATIONAL WAVES IN F (R)-THEORIES
theories. A more pragmatic point of view could be to “reconstruct” the suitable theory
of gravity starting from data. The main issues of this “inverse scattering” approach
is matching consistently observations at different scales and taking into account wide
classes of gravitational theories where “ad hoc” hypotheses are avoided. In principle, the
most popular dark energy models can be achieved considering f (R) theories of gravity
[320] and the same track can be followed, at completely different scales, to match galactic
dynamics [316]. Here, f (R) is a generic analytic function of the Ricci curvature scalar
R. As we show in the above Chapter 4 the deformations can be described as extended
conformal transformations and this fact gives a straightforward physical interpretation
of conformal transformations because conformally related metrics can be seen as the
”background” and the ”perturbed” metrics. Then space-time metric deformations can
be immediately recast in terms of perturbation theory allowing a completely covariant
approach to the problem of gravitational waves (GW). In this Chapter, we want to face
the problem of how the GW stochastic background and f (R) gravity can be related
showing, vice-versa, that a revealed stochastic GW signal could be a powerful probe for
a given effective theory of gravity. Our goal is to show that the conformal treatment of
GWs can be used to parameterize in a natural way f (R) theories.
5.2
Stochastic background of gravitational waves ”tuned”
by f (R) gravity
GWs are the perturbations hµν of the metric gµν which transform as 3-tensors. Following
[332], the GW-equations in the transverse-traceless gauge are
hji = 0
(5.1)
where ≡ (−g)−1/2 ∂µ (−g)1/2 gµν ∂ν is the usual d’Alembert operator and these equations are derived from the Einstein field equations deduced from the Hilbert-Lagrangian
density L = R. Clearly matter perturbations do not appear in (5.1) since scalar and
vector perturbations do not couple with tensor perturbations in Einstein equations. The
Latin indexes run from 1 to 3 the Greek ones from 0 to 3. Our task is now to derive the
analog of Eqs. (5.1) assuming a generic theory of gravity given by the action
Z
√
1
d4 x −gf (R)
(5.2)
A=
2k
where, for the sake of simplicity, we have discarded matter contributions. A conformal
analysis will help to this goal. In fact, assuming the conformal transformation
geµν = e2Φ gµν
with
e2Φ = f ′ (R)
(5.3)
where the prime indicates the derivative with respect to the Ricci scalar R and Φ is the
“conformal scalar field”, we obtain the conformally equivalent Hilbert-Einstein action
Z p
h
i
1
e + L Φ,Φ;µ
A=
(5.4)
−e
g d4 x R
2k
5.2. STOCHASTIC BACKGROUND OF GRAVITATIONAL WAVES ”TUNED” BY F (R) GRAVITY
where L Φ,Φ;µ is the conformal scalar field contribution derived from
1
;δ
;δ
e
Rµν = Rµν + 2 Φ;µ Φ;ν − gµν Φ;δ Φ − Φ;µν − gµν Φ ;δ
2
111
(5.5)
and
e = e−2Φ R − 6Φ − 6Φ;δ Φ;δ
(5.6)
R
In any case, as we will see, the L Φ,Φ;µ -term does not affect the GW-tensor equations
so it will not be considered any longer1 .
Starting from the action (5.4) and deriving the Einstein-like conformal equations,
the GW-equations are
ee
hji = 0
(5.7)
expressed in the conformal metric geµν . Since no scalar perturbation couples to the tensor
part of gravitational waves, we have
e
hji = e
glj δe
gil = e−2Φ glj e2Φ δgil = hji
(5.8)
which means that hji is a conformal invariant.
As a consequence, the plane-wave amplitude hji = h(t)eji exp(iki xi ), where eji is the
polarization tensor, are the same in both metrics. In any case, the d’Alembert operator
transforms as
e = e−2Φ + 2Φ;λ ∂;λ
(5.9)
and this means that the background is changing while the tensor wave amplitude not.
In order to study the cosmological stochastic background, the operator (5.9) can be
specified for a Friedmann-Robertson-Walker metric and then Eq. (5.7) becomes
ḧ + 3H + 2Φ̇ ḣ + k2 a−2 h = 0
(5.10)
∂
∂
+ 3H , a(t) the scale factor and k the wave number.
2
∂t
∂t
It is worth stressing that Eq. (5.10) applies to any f (R) theory whose conformal
transformation can be defined as e2Φ = f ′ (R). The solution, i.e. the GW amplitude,
depends on the specific cosmological background (i.e. a(t)) and the specific theory
of gravity (i.e. Φ(t)). For example, if we assume power law behaviors for a(t) and
Φ(t) = 21 ln f ′ (R(t)), that is
being =
Φ(t) = f ′ (R) = f ′ 0 (t/t0 )m , a(t) = a0 (t/t0 )n
(5.11)
it is easy show that general relativity is recovered for m = 0 while
n=
1
m2 + m − 2
m+2
(5.12)
Actually a scalar component in gravitational radiation is often considered [322, 324] but here we are
taking into account only the genuine tensor part of stochastic background.
112
5. PROBING THE MINKOWSKIAN LIMIT: GRAVITATIONAL WAVES IN F (R)-THEORIES
is the relation between the parameters for a generic f (R) = f0 Rs where s = 1 − m
2 with
s 6= 1 [325]. Eq. (5.10) can be recast in the form
ḧ + (3n + m) t−1 ḣ + k2 a0 (t0 /t))2n h = 0
(5.13)
whose general solution is
h(t) =
t
t0
β
[C1 Jα (x) + C2 J−α (x)]
(5.14)
Jα ’s are Bessel functions and
α=
1 − 3n − m
2(n − 1)
, β=
1 − 3n − m
2
, x=
kt1−n
1−n
(5.15)
while t0 , C1 , C2 are constants related to the specific values of n and m. In Fig. (5.1),
some examples are given. The plots are labelled by the set of parameters {m, n, s}
which assign the time evolution of Φ(t) and a(t) with respect to a given power-law
theory f (R) = f0 Rs .
The time units are in terms of the Hubble radius H −1 ; n = 1/2 is a radiation-like
evolution; n = 2/3 is a dust-like evolution, n = 2 labels power-law inflationary phases
and n = −5 is a pole-like inflation. From Eq. (5.12), a singular case is for m = −2 and
s = 2. It is clear that the conformally invariant plane-wave amplitude evolution of the
tensor GW strictly depends on the background.
Let us now take into account the issue of the production of GWs contributing to
the stochastic background. Several mechanism can be considered as cosmological populations of astrophysical sources [319], vacuum fluctuations, phase transitions [322] and
so on. In principle, we could seek for contributions due to every high-energy physical
process in the early phases of the Universe evolution.
It is important to distinguish processes coming from transitions like inflation, where
the Hubble flow emerges in the radiation dominated phase and process, like the early
star formation rates, where the production takes place during the dust dominated era.
In the first case, stochastic GW background is strictly related to the cosmological model.
This is the case we are considering here which is, furthermore, also connected to the
specific theory of gravity. In particular, one can assume that the main contribution to
the stochastic background comes from the amplification of vacuum fluctuations at the
transition between an inflationary phase and the radiation dominated era. However, in
any inflationary model, we can assume that the GWs generated as zero-point fluctuation
during the inflation undergo adiabatically damped oscillations (∼ 1/a) until they reach
the Hubble radius H −1 . This is the particle horizon for the growth of perturbations.
On the other hand, any other previous fluctuation is smoothed away by the inflationary
expansion. The GWs freeze aut for a/k ≫ H −1 and reenter the H −1 radius after
the reheating in the Friedmann era (see also [285, 286]). The reenter in the radiationdominated or in the dust-dominated era depends on the scale of the GW. After the
5.2. STOCHASTIC BACKGROUND OF GRAVITATIONAL WAVES ”TUNED” BY F (R) GRAVITY
hHtL
hHtL
n=12 , m=0 , s=1
0.3
113
n=23 , m=0 , s=1
0.15
0.2
0.1
0.1
0.05
100
200
300
400
500
t
100
-0.1
200
300
400
500
t
-0.05
-0.2
-0.1
-0.3
hHtL
hHtL
n=23 , m=-6 , s=4
n=2 , m=2 , s=-12
13
4·10
10000
5000
13
2·10
t
1000 2000 3000 4000 5000
13
-2·10
hHtL
0.15
0.2
0.25
t
-5000
hHtL
n=-2 , m=-1 , s=32
n=-5 , m=-4 , s=3
40
10
20
1
-20
0.1
-10000
20
-10
0.05
2
3
4
5
t
1.2 1.4 1.6 1.8
2
2.2 2.4
t
-20
-40
Figure 5.1: Evolution of the GW amplitude for some power-law behaviors of a(t), Φ(t)
and f (R). The scales of time and amplitude strictly depend on the cosmological background giving a ”signature” for the model.
114
5. PROBING THE MINKOWSKIAN LIMIT: GRAVITATIONAL WAVES IN F (R)-THEORIES
reenter, Gws can be detected by their Sachs-Wolfe effect on the temperature anisotropy
△T /T at the decoupling [328]. When Φ acts as the inflaton [329] we have Φ̇ ≪ H during
the inflation. Considering also the conformal time dη = dt/a, Eq. (5.10) reads
h′′ + 2
χ′ ′
h + k2 h = 0
χ
(5.16)
where χ = aeΦ and derivation
is with respect to η. Inflation means that a(t) =
R
a0 exp(Ht) and then η = dt/a = (aH)−1 and χ′ /χ = −η −1 . The exact solution
of (5.16) is
p
(5.17)
h(η) = k−3/2 2/k [C1 (sin kη − cos kη) + C2 (sin kη + cos kη)]
Inside the H −1 radius we have kη ≫ 1. Furthermore considering the absence of gravitons in the initial vacuum state, we have only negative-frequency modes and then the
adiabatic behavior is
p
1
h = k1/2 2/π
C exp(−ikη) .
(5.18)
aH
At the first horizon crossing (aH = k), the averaged amplitude Ah = (k/2π)3/2 |h|
of the perturbation is
1
Ah = 2 C
(5.19)
2π
when the scale a/k grows larger than the Hubble radius H −1 , the growing mode of evolution is constant, that is it is frozen. This situation corresponds to the limit −kη ≪ 1
in Eq. (5.17). Since Φ acts as the inflaton field, it is Φ ∼ 0 at reenter (after the end
of inflation). Then the amplitude Ah of the wave is preserved until the second horizon
crossing after which it can be observed, in principle, as an anisotropy perturbation on
the CMBR. It can be shown that △T /T . Ah as an upper limit to Ah since other effects
can contribute to the background anisotropy [330]. From this consideration, it is clear
that the only relevant quantity is the initial amplitude C in Eq. (5.18) which is conserved
until the reenter. Such an amplitude directly depends on the fundamental mechanism
generating perturbations. Inflation gives rise to processes capable of producing perturbations as zero-point energy fluctuations. Such a mechanism depends on the adopted
theory of gravitation and then (△T /T ) could constitute a further constraint to select
a suitable f (R)-theory. Considering a single graviton in the form of a monochromatic
wave, its zero-point amplitude is derived through the commutation relations:
[h(t, x), πh (t, y)] = iδ3 (x − y)
(5.20)
calculated at a fixed time t, where the amplitude h is the field and πh is the conjugate
momentum operator. Writing the Lagrangian for h
1p
−e
ggeµν h;µ h;ν
Le =
2
(5.21)
5.2. STOCHASTIC BACKGROUND OF GRAVITATIONAL WAVES ”TUNED” BY F (R) GRAVITY
in the conformal FRW metric geµν (h is conformally invariant), we obtain
πh =
The Eq. (5.20) becomes
h
∂ Le
= e2Φ a3 ḣ
∂ ḣ
i
δ3 (x − y)
h(t, x), ḣ(y, y) = i 3 2Φ
a e
115
(5.22)
(5.23)
and the fields h and ḣ can be expanded in terms of creation and annihilation operators
Z
i
h
1
3
−ikx
∗
+ikx
,
(5.24)
h(t, x) =
d
k
h(t)e
+
h
(t)e
(2π)3/2
Z
i
h
1
3
−ikx
∗
+ikx
ḣ(t, x) =
.
(5.25)
d
k
ḣ(t)e
+
ḣ
(t)e
(2π)3/2
The commutation relations is conformal time are then
i(2π)3
(5.26)
hh′∗ − h∗ h′ = 3 2Φ
a e
√
Inserting (5.18) and (5.19), we obtain C = 2π 2 He−Φ where H and Φ are calculated
at the first horizon-crossing and then
√
2 −Φ
He
(5.27)
Ah =
2
which means that the amplitude of GWs produced during inflation directly depends on
the given f (R) theory being Φ = 12 ln f ′ (R). Explicitly, it is
Ah = p
H
2f ′ (R)
.
(5.28)
This result deserves some discussion and can be read in two ways. From one side the
amplitude of GWs produced during inflation depends on the given theory of gravity that,
if different from general relativity, gives extra degrees of freedom which assume the role
of inflaton field in the cosmological dynamics [329]. On the other hand, the Sachs-Wolfe
effect related to the CMBR temperature anisotropy could constitute a powerful tool to
test the true theory of gravity at early epochs, i.e. at very high redshift. This probe,
related with data at medium [320] and low redshift [331], could strongly contribute i) to
reconstruct cosmological dynamics at every scale; ii) to further test general relativity or
to rule out it against alternative theories, iii) to give constraints on the GW-stochastic
background, if f (R) theories are independently probed at other scales.
In summary, we have shown that amplitudes of tensor GWs are conformally invariant
and their evolution depends on the cosmological background. Such a background is
5. PROBING THE MINKOWSKIAN LIMIT: GRAVITATIONAL WAVES IN F (R)-THEORIES
116
tuned by conformal scalar field which is not present in the standard general relativity.
Assuming that primordial vacuum fluctuations produce stochastic GWS, beside scalar
perturbations, kinematical distortions and so on, the initial amplitude of these ones is a
function of the f (R)-theory of gravity and then the stochastic background can be, in a
certain sense “tuned” by the theory. Vice versa, data coming from the Sachs-Wolfe effect
could contribute to select a suitable f (R) theory which can be consistently matched with
other observations. However, further and accurate studies are needed in order to test
the relation between Sachs-Wolfe effect and f (R) gravity. This goal could be achieved
very soon through the forthcoming space (LISA) and ground-based (VIRGO, LIGO)
interferometers.
5.3
Massive gravitational waves from f (R) theories of gravity: Potential detection with LISA
Now, we will analyse the general case, i.e.
Z
√
S = d4 x −gf (R) + Lm ,
(5.29)
where f (R) is a generic high order theory of gravity.
Of course, the cases which have been analysed in [327] and in [330] are particular
cases of the more general case that we are going to analyse now.
As we will interact with gravitational waves, i.e. the linearized theory in vacuum,
Lm = 0 will be put and the pure curvature action
Z
√
(5.30)
S = d4 x −gf (R)
will be considered.
By varying the action (5.30) in respect to gµν (see refs. [327, 328, 330] for a parallel
computation) the field equations are obtained (note that in this thesis we work with
G = 1, c = 1 and ~ = 1):
1
f ′ (R)Rµν − f (R)gµν − f ′ (R);µ;ν + gµν f ′ (R) = 0
(5.31)
2
which are the modified Einstein field equations. f ′ (R) is the derivative of f in respect
to the Ricci scalar. Writing down, exlplicitly, the Einstein tensor, eqs. (5.31) become
Gµν =
1
1
{ gµν [f (R) − f ′ (R)R] + f ′ (R);µ;ν − gµν f ′ (R)}.
f ′ (R) 2
(5.32)
Taking the trace of the field equations (5.32) one gets
3f ′ (R) + Rf ′ (R) − 2f (R) = 0,
(5.33)
5.3. MASSIVE GRAVITATIONAL WAVES FROM F (R) THEORIES OF GRAVITY: POTENTIAL
DETECTION WITH LISA
117
and, with the identifications [334]
Φ → f ′ (R)
dV
dΦ
and
→
2f (R)−Rf ′ (R)
3
(5.34)
a Klein - Gordon equation for the effective Φ scalar field is obtained:
dV
.
(5.35)
dΦ
To study gravitational waves, the linearized theory has to be analyzed, with a little
perturbation of the background, which is assumed given by a a Minkowskian background
plus Φ = Φ0 ,i.e. we are linearizing into a background with constant curvature [330, 335].
We also assume Φ0 to be a minimum for V :
Φ =
1
dV
V ≃ αδΦ2 ⇒
≃ m2 δΦ,
2
dΦ
and the constant m has mass dimension.
Putting
(5.36)
gµν = ηµν + hµν
(5.37)
Φ = Φ0 + δΦ.
eµνρσ , R
eµν and R
e the linearized quantity
to first order in hµν and δΦ, calling R
which correspond to Rµνρσ , Rµν and R, the linearized field equations are obtained
[327, 328, 330]:
eµν −
R
e
R
2 ηµν
= (∂µ ∂ν hf − ηµν hf )
hf =
(5.38)
m2 hf ,
where
hf ≡
δΦ
.
Φ0
eµνρσ and eqs. (5.38) are invariants for gauge transformations [327, 328, 330]
R
hµν → h′µν = hµν − ∂(µ εν)
δΦ →
δΦ′
(5.39)
(5.40)
= δΦ;
then
h
ηµν + ηµν hf
2
can be defined, and, considering the transformation for the parameter εµ
h̄µν ≡ hµν −
(5.41)
5. PROBING THE MINKOWSKIAN LIMIT: GRAVITATIONAL WAVES IN F (R)-THEORIES
118
εν = ∂ µ h̄µν ,
(5.42)
a gauge parallel to the Lorenz one of electromagnetic waves can be choosen:
∂ µ h̄µν = 0.
(5.43)
In this way, field equations read like
h̄µν = 0
(5.44)
hf = m2 hf
(5.45)
Solutions of eqs. (5.44) and (5.45) are plan waves [327, 328, 330]:
→
h̄µν = Aµν (−
p ) exp(ipα xα ) + c.c.
(5.46)
→
hf = a(−
p ) exp(iq α xα ) + c.c.
(5.47)
where
→
kα ≡ (ω, −
p)
qα
→
≡ (ωm , −
p)
→
ω = p ≡ |−
p|
ωm =
p
m2
+
(5.48)
p2 .
In eqs. (5.44) and (5.46) the equation and the solution for the standard waves
of General Relativity [332, 333] have been obtained, while eqs. (5.45) and (5.47) are
respectively the equation and the solution for the massive mode (see also [327, 328, 330]).
The fact that the dispersion law for the modes of the massive field hf is not linear
has to be emphatized. The velocity of every “ordinary” (i.e. which arises from General
Relativity) mode h̄µν is the light speed c, but the dispersion law (the second of eq.
(5.48)) for the modes of hf is that of a massive field which can be discussed like a wave→
packet [327, 328, 330]. Also, the group-velocity of a wave-packet of hf centered in −
p
is
−
→
p
−
,
v→
G =
ω
(5.49)
→
which is exactly the velocity of a massive particle with mass m and momentum −
p.
From the second of eqs. (??) and eq. (5.49) it is simple to obtain:
√
ω 2 − m2
.
(5.50)
vG =
ω
Then, wanting a constant speed of the wave-packet, it has to be [327, 328, 330]
5.3. MASSIVE GRAVITATIONAL WAVES FROM F (R) THEORIES OF GRAVITY: POTENTIAL
DETECTION WITH LISA
q
m=
2 )ω.
(1 − vG
119
(5.51)
Now, the analysis can remain in the Lorenz gauge with trasformations of the type
εν = 0; this gauge gives a condition of transversality for the ordinary part of the field:
kµ Aµν = 0, but does not give the transversality for the total field hµν . From eq. (5.41)
it is
h̄
ηµν + ηµν hf .
(5.52)
2
At this point, if being in the massless case [327, 328, 330], it could been put
hµν = h̄µν −
εµ = 0
(5.53)
∂µ
εµ
=
− h̄2
+ hf ,
which gives the total transversality of the field. But in the massive case this is
impossible. In fact, applying the Dalembertian operator to the second of eqs. (5.53)
and using the field equations (5.44) and (5.45) it results
εµ = m2 hf ,
(5.54)
which is in contrast with the first of eqs. (5.53). In the same way, it is possible to
show that it does not exist any linear relation between the tensorial field h̄µν and the
massive field hf . Thus a gauge in wich hµν is purely spatial cannot be chosen (i.e. it
cannot be put hµ0 = 0, see eq. (5.52)) . But the traceless condition to the field h̄µν can
be put :
εµ = 0
(5.55)
∂µ
εµ
=
− h̄2 .
These equations imply
∂ µ h̄µν = 0.
(5.56)
To save the conditions ∂µ h̄µν and h̄ = 0 transformations like
εµ = 0
(5.57)
∂µ
εµ
=0
→
can be used and, taking −
p in the z direction, a gauge in which only A11 , A22 , and
A12 = A21 are different to zero can be chosen. The condition h̄ = 0 gives A11 = −A22 .
Now, putting these equations in eq. (5.52), it results
120
5. PROBING THE MINKOWSKIAN LIMIT: GRAVITATIONAL WAVES IN F (R)-THEORIES
×
(×)
hµν (t, z) = A+ (t − z)e(+)
µν + A (t − z)eµν + hf (t − vG z)ηµν .
(+)
(5.58)
(×)
The term A+ (t − z)eµν + A× (t − z)eµν describes the two standard polarizations of
gravitational waves which arise from General Relativity, while the term hf (t − vG z)ηµν
is the massive field arising from the generic high order f (R) theory. In other words, the
function f ′ (R) of the Ricci scalar generates a third massive polarization for gravitational
waves which is not present in standard General Relativity. Note that the line element
(5.58) has been obtained in both of references [327] and [330] . Here we have shown that
such a line element is characteristic of every f (R) theory of gravity.
The analysis of the two standard polarization is well known in the literature [313,
314, 332, 333]. For a the pure polarization arising from the f (R) theory eq. (5.101) can
be rewritten as
hµν (t − vG z) = hf (t − vG z)ηµν
(5.59)
and the corrispondent line element is the conformally flat one
ds2 = [1 + hf (t − vG z)](−dt2 + dz 2 + dx2 + dy 2 ).
(5.60)
In [330] it has been shown that in this kind of line element the effect of the mass
is the generation of a longitudinal force (in addition to the transverse one) while in the
limit m → 0 the longitudinal force vanishes.
Now, before starting the analysis, it has to be discussed if there are fenomenogical
limitations to the mass of the GW [330, 335]. A strong limitation arises from the fact that
the GW needs a frequency which falls in the frequency-range for both of earth based
and space based gravitational antennas, that is the interval 10−4 Hz ≤ f ≤ 10KHz
[312, 315, 316, 317, 318, 319, 337, 338]. For a massive GW, from [323, 325, 327, 330] it
is:
2πf = ω =
p
m2 + p 2 ,
(5.61)
were p is the momentum. Thus, it needs
0eV ≤ m ≤ 10−11 eV.
(5.62)
A stronger limitation is given by requirements of cosmology and Solar System tests
on extended theories of gravity. In this case it is [335]
0eV ≤ m ≤ 10−33 eV.
(5.63)
For these light scalars, their effect can be still discussed as a coherent GW.
The frequency-dependent response function, for a massive mode of gravitational
radiation, has been obtained in [330] for the particular case f (R) = R + R−1 . Here the
5.3. MASSIVE GRAVITATIONAL WAVES FROM F (R) THEORIES OF GRAVITY: POTENTIAL
DETECTION WITH LISA
121
computation will be performed with another treatment and the results will be appllied
to LISA, following the advice in [335].
Eq. (5.60) can be rewritten as
(
dx
dy
dz
1
dt 2
) − ( )2 − ( )2 − ( )2 =
,
dτ
dτ
dτ
dτ
(1 + hf )
(5.64)
where τ is the proper time of the test masses.
From eqs. (5.60) and (5.64) the geodesic equations of motion for test masses (i.e.
the beam-splitter and the mirrors of the interferometer), can be obtained
d2 x
dτ 2
=
0
d2 y
dτ 2
=
0
d2 t
dτ 2
=
1 ∂t (1+hf )
2 (1+hf )2
d2 z
dτ 2
= − 21
(5.65)
∂z (1+hf )
.
(1+hf )2
The first and the second of eqs. (5.65) can be immediately integrated obtaining
dx
= C1 = const.
dτ
(5.66)
dy
= C2 = const.
dτ
(5.67)
dt 2
dz
1
) − ( )2 =
.
dτ
dτ
(1 + hf )
(5.68)
In this way eq. (5.64) becomes
(
If we assume that test masses are at rest initially we get C1 = C2 = 0. Thus we see
that, even if the GW arrives at test masses, we do not have motion of test masses within
the x − y plane in this gauge. We could understand this directly from eq. (5.60) because
the absence of the x and of the y dependences in the metric implies that test masses
momentum in these directions (i.e. C1 and C2 respectively) is conserved. This results,
for example, from the fact that in this case the x and y coordinates do not esplicitly
enter in the Hamilton-Jacobi equation for a test mass in a gravitational field [313].
Now we will see that, in presence of the GW, we have motion of test masses in the
z direction which is the direction of the propagating wave. An analysis of eqs. (5.65)
shows that, to simplify equations, we can introduce the retarded and advanced time
coordinates (u, v):
122
5. PROBING THE MINKOWSKIAN LIMIT: GRAVITATIONAL WAVES IN F (R)-THEORIES
u = t − vG z
(5.69)
v = t + vG z.
From the third and the fourth of eqs. (5.65) we have
∂v [1 + hf (u)]
d du
=
= 0.
dτ dτ
(1 + hf (u))2
(5.70)
This equation can be integrated obtaining
du
= α,
dτ
(5.71)
where α is an integration constant. From eqs. (5.68) and (5.71), we also get
dv
β
=
dτ
1 + hf
(5.72)
τ = βu + γ,
(5.73)
where β ≡ α1 , and
where the integration constant γ correspondes simply to the retarded time coordinate
translation u. Thus, without loss of generality, we can put it equal to zero. Now let us
see what is the meaning of the other integration constant β. We can write the equation
for z from eqs. (5.71) and (5.72):
1
β2
dz
=
(
− 1).
dτ
2β 1 + hf
(5.74)
When it is hf = 0 (i.e. before the GW arrives at the test masses) eq. (5.74) becomes
1 2
dz
=
(β − 1).
dτ
2β
(5.75)
But this is exactly the initial velocity of the test mass, thus we have to choose β = 1
because we suppose that test masses are at rest initially. This also imply α = 1.
To find the motion of a test mass in the z direction we see that from eq. (5.73) we
dτ
have dτ = du, while from eq. (5.72) we have dv = 1+h
. Because it is vG z = v−u
2 we
f
obtain
dz =
which can be integrated as
dτ
1
− du),
(
2vG 1 + hf
(5.76)
5.3. MASSIVE GRAVITATIONAL WAVES FROM F (R) THEORIES OF GRAVITY: POTENTIAL
DETECTION WITH LISA
z = z0 +
= z0 −
R
1
2vG
1
2vG
du
− du) =
( 1+h
f
R t−vG z
123
(5.77)
hf (u)
1+hf (u) du,
−∞
where z0 is the initial position of the test mass. Now the displacement of the test
mass in the z direction can be written as
∆z = z − z0 = − 2v1G
≃
− 2v1G
R t−vG z0 −vG ∆z
−∞
R t−vG z0
−∞
hf (u)
1+hf (u) du
(5.78)
hf (u)
1+hf (u) du.
We can also rewrite the results in function of the time coordinate t:
x(t) =
x0
y(t) =
y0
z(t) = z0 −
1
2vG
R t−vG z0
−∞
hf (u)
1+hf (u) d(u)
(5.79)
t − vG z(t),
τ (t) =
Calling l and L + l the unperturbed positions of the beam-splitter and of the mirror
and using the third of eqs. (5.79) the varying position of the beam-splitter and of the
mirror are given by
zBS (t) = l −
1
2vG
zM (t) = L + l −
1
2vG
R t−vG l
−∞
hf (u)
1+hf (u) d(u)
R t−vG (L+l)
−∞
(5.80)
hf (u)
1+hf (u) d(u)
But we are interested in variations in the proper distance (time) of test masses, thus,
in correspondence of eqs. (5.80), using the fourth of eqs. (5.79) we get
τBS (t) = t − vG l −
1
2
τM (t) = t − vG L − vG l −
R t−vG l
1
2
−∞
hf (u)
1+hf (u) d(u)
R t−vG (L+l)
−∞
(5.81)
hf (u)
1+hf (u) d(u).
Then the total variation of the proper time is given by
1
△τ (t) = τM (t) − τBS (t) = vG L −
2
Z
t−vG (L+l)
t−vG l
hf (u)
d(u).
1 + hf (u)
(5.82)
124
5. PROBING THE MINKOWSKIAN LIMIT: GRAVITATIONAL WAVES IN F (R)-THEORIES
In this way, recalling that in the used units the unperturbed proper distance (time)
is T = L, the difference between the total variation of the proper time in presence and
the total variation of the proper time in absence of the GW is
1
δτ (t) ≡ △τ (t) − L = −L(vG + 1) −
2
Z
t−vG (L+l)
t−vG l
hf (u)
d(u).
1 + hf (u)
(5.83)
This quantity can be computed in the frequency domain, defining the Fourier transform of hf as
Z ∞
e
dt hf (t) exp(iωt).
(5.84)
hf (ω) =
−∞
and using the translation and derivation Fourier theorems, obtaining
2 ) exp[iωL(1 + v )] +
δe
τ (ω) = L(1 − vG
G
L
2 −1)2
2ωL(vG
[exp[2iωL](vG + 1)3 (−2i + ωL(vG − 1) + 2L exp[iωL(1 + vG )]
(5.85)
3 − ωL + ωLv 4 ) + L(v + 1)3 (−2i + ωL(v + 1))]e
(6ivG + 2ivG
hR .
G
G
G
A “signal” can be also defined:
e
S(ω)
≡
δe
τ (ω)
L
2 ) exp[iωL(1 + v )] +
= (1 − vG
G
1
2 −1)2
2ωL(vG
[exp[2iωL](vG + 1)3 (−2i + ωL(vG − 1) + 2 exp[iωL(1 + vG )]
(5.86)
3 − ωL + ωLv 4 ) + (v + 1)3 (−2i + ωL(v + 1))]e
(6ivG + 2ivG
hR .
G
G
G
Then the function
2 ) exp[iωL(1 + v )] +
Υl (ω) ≡ (1 − vG
G
1
2 −1)2
2ωL(vG
[exp[2iωL](vG + 1)3 (−2i + ωL(vG − 1) + 2 exp[iωL(1 + vG )]
(5.87)
3 − ωL + ωLv 4 ) + (v + 1)3 (−2i + ωL(v + 1))],
(6ivG + 2ivG
G
G
G
is the response function of an arm of the interferometer located in the z-axis, due
to the longitudinal component of the massive gravitational wave arising from the high
order gravity theory and propagating in the same direction of the axis.
For vG → 1 it is Υl (ω) → 0. Such a response function has been obtained in [330]
too, but with a different kind of analysis.
In figures 1 and 2 are shown the response functions (5.87) for an arm of LISA
(L = 5 ∗ 106 Km) [337, 338] for vG = 0.1 (non-relativistic case) and vG = 0.9 (relativistic
5.3. MASSIVE GRAVITATIONAL WAVES FROM F (R) THEORIES OF GRAVITY: POTENTIAL
DETECTION WITH LISA
125
1.125 ÈHHfLÈ
1.1
1.075
1.05
1.025
Hz
0.002
0.004
0.006
0.008
f
0.01
Figure 5.2: the longitudinal response function (5.87) of an arm of LISA for vG = 0.1
(non-relativistic case)
0.215
ÈHHfLÈ
0.21
0.205
0.002
0.004
0.006
0.008
Hz
0.01
f
0.195
0.19
Figure 5.3: the longitudinal response function (5.87) of an arm of LISA for vG = 0.9
(relativistic case)
5. PROBING THE MINKOWSKIAN LIMIT: GRAVITATIONAL WAVES IN F (R)-THEORIES
126
case). We see that in the non-relativistic case the signal is stronger as it could be
expected (for m → 0 we expectΥl (ω) → 0).
It is very important to emphasize that, differently from the response functions of
massless gravitational waves, this longitudinal response function increases with frequency, .i.e , the presence of the mass prevents signal to drop off the regime in the
high-frequency portion of the sensitivity band. Thus, considering such a high-frequency
portion of the sensitivity band becomes fundamental if LISA would detect massive GWs
arising from f (R) theories of gravity which are not banned by requirements of Cosmology
and Solar System tests [335, 336].
5.4
Stochastic background of relic scalar gravitational waves
from scalar-tensor gravity
In this section consider the framework of scalar-tensor gravity and we discuss the scalar
modes of gravitational waves and the primordial production of this scalar component
which is generated beside tensorial one. Then the Scalar-tensor gravity theories are a
particular case of Extended Theories of Gravity which are revealing a useful paradigm
to deal with several problems in cosmology, astrophysics and fundamental physics (for
a comprehensive discussion see, for example [304, 305]). In the most general case,
considering only the Ricci scalar among the curvature invariants, they arises from the
action
Z
i
√ h
ε
(5.88)
S = d4 x −g F (R, R, 2 R, k R, φ) − gµν φ;µ φ;ν + Lm ,
2
where F is an unspecified function of curvature invariants and of a scalar field φ and
is the D’Alembert operator. The term Lm is the minimally coupled ordinary matter
contribution. Scalar-tensor gravity, is recovered from (5.88) through the choice
F (R, φ) = f (φ)R − V (φ),
ε = −1 .
(5.89)
Considering (5.89), a general action for scalar-tensor gravity in four dimensions is
Z
√
1
(5.90)
S = d4 x −g f (φ)R + gµν φ;µ φ;ν − V (φ) + Lm ,
2
which can be recast in a Brans-Dicke-like form [306] by
ϕ = f (φ) , ω(ϕ) =
,,
and then
S=
Z
f (φ)
2′ f (φ)
, W (ϕ) = V (φ(ϕ))
√
ω(ϕ) µν
d4 x −g ϕR −
g ϕ;µ ϕ;ν − W (ϕ) + Lm .
ϕ
(5.91)
(5.92)
5.4. STOCHASTIC BACKGROUND OF RELIC SCALAR GRAVITATIONAL WAVES FROM
SCALAR-TENSOR GRAVITY
127
By varying the action (5.92) with respect to gµν , we obtain the field equations
(m)
Gµν = − 4πϕG̃ Tµν +
ω(ϕ)
(ϕ;µ ϕ;ν
ϕ2
+ ϕ1 (ϕ;µν − gµν ϕ) +
− 21 gµν gαβ ϕ;α ϕ;β )+
(5.93)
1
2ϕ gµν W (ϕ)
while the variation with respect to ϕ gives the Klein - Gordon equation
1
dω(ϕ) µν
(m)
′
g ϕ;µ ϕ;ν .
ϕ =
−4π G̃T
+ 2W (ϕ) + ϕW (ϕ) +
2ω(ϕ) + 3
dϕ
(5.94)
(m)
We are assuming physical units G = 1, c = 1 and ~ = 1. Tµν is the matter stressenergy tensor and G̃ is a dimensional, strictly positive, gravitational coupling constant
[294, 295]. The Newton constant is replaced by the effective coupling
Gef f = −
1
,
2ϕ
(5.95)
which is, in general, different from G. General Relativity is recovered for
1
ϕ = ϕ0 = − .
2
(5.96)
In order to study gravitational waves, we assume first-order, small perturbations in
(m)
vacuum (Tµν = 0). This means
gµν = ηµν + hµν ,
ϕ = ϕ0 + δϕ
(5.97)
and
1
(5.98)
W ≃ αδϕ2 ⇒ W ′ ≃ αδϕ
2
for the self-interacting, scalar-field potential. These assumptions allow to derive the
eµνρσ , R
eµν and R
e which correspond to Rµνρσ , Rµν
”linearized” curvature invariants R
and R, and then the linearized field equations [295, 392]
eµν −
R
e
R
2 ηµν
where
Φ≡−
= −∂µ ∂ν Φ + ηµν Φ
Φ =
δϕ
,
ϕ0
(5.99)
m2 Φ,
m2 ≡
αϕ0
.
2ω + 3
(5.100)
The case ω = const and W = 0 has been analyzed in [295] considering the so-called
“canonical” linearization [392]. In particular, the transverse-traceless (TT) gauge (see
[392]) can be generalized to scalar-tensor gravity obtaining the total perturbation of a
128
5. PROBING THE MINKOWSKIAN LIMIT: GRAVITATIONAL WAVES IN F (R)-THEORIES
GW incoming in the z+ direction in this gauge as we have see in the above section 5.3
as
×
(×)
(s)
hµν (t − z) = A+ (t − z)e(+)
(5.101)
µν + A (t − z)eµν + Φ(t − z)eµν .
(+)
(×)
The term A+ (t − z)eµν + A× (t − z)eµν describes the two standard (i.e. tensorial)
polarizations of a gravitational wave arising from General Relativity in the TT gauge
(s)
[392], while the term Φ(t − z)eµν is the extension of the TT gauge to the scalar case.
This means that, in scalar-tensor gravity, the scalar field generates a third component
for the tensor polarization of GWs. This is because three different degrees of freedom are
present (see Eq.(32) of [295]), while only two are present in standard General Relativity.
Let us now take into account the primordial physical process which gave rise to a
characteristic spectrum Ωsgw for the early stochastic background of relic scalar GWs.
The production physical process has been analyzed, for example, in [393, 394, 395]
but only for the first two tensorial components of eq. (5.101) due to standard General
Relativity. Actually the process can be improved considering also the third scalar-tensor
component.
Before starting with the analysis, it has to be emphasized that, considering a stochastic background of scalar GWs, it can be described in terms of the scalar field Φ and
characterized by a dimensionless spectrum (see the analogous definition for tensorial
waves in [393, 396, 384, 394])
Ωsgw (f ) =
where
1 dρsgw
,
ρc d ln f
(5.102)
3H02
(5.103)
8πG
is the (actual) critical energy density of the Universe, H0 the today observed Hubble
expansion rate, and dρsgw is the energy density of the scalar part of the gravitational
radiation contained in the frequency range f to f +df . We are considering now standard
units.
The existence of a relic stochastic background of scalar GWs is a consequence of
general assumptions. Essentially it derives from basic principles of Quantum Field Theory and General Relativity. The strong variations of gravitational field in the early
Universe amplifies the zero-point quantum fluctuations and produces relic GWs. It is
well known that the detection of relic GWs is the only way to learn about the evolution of the very early universe, up to the bounds of the Planck epoch and the initial
singularity [393, 396, 384, 394, 301]. It is very important to stress the unavoidable and
fundamental character of such a mechanism. It directly derives from the inflationary
scenario [397, 398], which well fit the WMAP data in particular good agreement with
almost exponential inflation and spectral index ≈ 1, [399, 400].
A remarkable fact about the inflationary scenario is that it contains a natural mechanism which gives rise to perturbations for any field. It is important for our aims that such
ρc ≡
5.4. STOCHASTIC BACKGROUND OF RELIC SCALAR GRAVITATIONAL WAVES FROM
SCALAR-TENSOR GRAVITY
129
a mechanism provides also a distinctive spectrum for relic scalar GWs. These perturbations in inflationary cosmology arise from the most basic quantum mechanical effect:
the uncertainty principle. In this way, the spectrum of relic GWs that we could detect
today is nothing else but the adiabatically-amplified zero-point fluctuations [393, 394].
The calculation for a simple inflationary model can be performed for the scalar field
component of eq. (5.101). Let us assume that the early Universe is described an inflationary de Sitter phase emerging in a radiation dominated phase [393, 396, 394]. The
conformal metric element is
→
→
ds2 = a2 (η)[−dη 2 + d−
x 2 + hµν (η, −
x )dxµ dxν ],
(5.104)
where, for a purely scalar GW the metric perturbation (5.101) reduces to
hµν = Φe(s)
µν ,
(5.105)
Following [393, 394], in the de Sitter phase, we have:
η < η1
P = −ρ
η12 η0−1 (2η − η)−1
Hds = cη0 /η12
conformal time
equation of state
scale factor
Hubble constant
while, in the radiation dominated phase we have, respectively,
η > η1
P = ρ/3
η/η0
H = cη0 /η 2
conformal time
equation of state
scale factor
Hubble constant
η1 is the inflation-radiation transition conformal time and η0 is the value of conformal
time today. If we express the scale factor in terms of comoving time cdt = a(t)dη, we
have
√
a(t) ∝ exp(Hds t),
a(t) ∝ t
(5.106)
for the de Sitter and radiation phases respectively. In order to solve the horizon and
a(η0 )
> 1027 has to be satisfied. The relic scalar-tensor
flatness problems, the condition
a(η1 )
→
GWs are the weak perturbations hµν (η, −
x ) of the metric (5.105) which can be written
in the form
−
→ −
→
hµν = e(s)
µν (k̂)X(η) exp( k · x ),
(5.107)
130
5. PROBING THE MINKOWSKIAN LIMIT: GRAVITATIONAL WAVES IN F (R)-THEORIES
−
→
in terms of the conformal time η where k is a constant wavevector. From Eq.(5.107),
the scalar component is
−
→ →
−
→ →
Φ(η, k , −
x ) = X(η) exp( k · −
x ).
(5.108)
Assuming Y (η) = a(η)X(η), from the Klein-Gordon equation in the FRW metric, one
gets
−
→
a′′
(5.109)
Y ′′ + (| k |2 − )Y = 0
a
where the prime ′ denotes derivative with respect to the conformal time. The solutions
of Eq. (5.109) can be expressed in terms of Hankel functions in both the inflationary
and radiation dominated eras, that is:
For η < η1
a(η1 )
X(η) =
[1 + Hds ω −1 ] exp −ik(η − η1 ),
(5.110)
a(η)
for η > η1
X(η) =
a(η1 )
[α exp −ik(η − η1 ) + β exp ik(η − η1 ),
a(η)
(5.111)
where ω = ck/a is the angular frequency of the wave (which is function of the time
−
→
being k = | k | constant), α and β are time-independent constants which we can obtain
demanding that both X and dX/dη are continuous at the boundary η = η1 between the
inflationary and the radiation dominated eras. By this constraint, we obtain
α = 1+i
√
Hds H0 Hds H0
−
,
ω
2ω 2
β=
Hds H0
2ω 2
(5.112)
In Eqs. (5.112), ω = ck/a(η0 ) is the angular frequency as observed today, H0 = c/η0 is
the Hubble expansion rate as observed today. Such calculations are referred in literature
as the Bogoliubov coefficient methods [393, 394].
In an inflationary scenario, every classical or macroscopic perturbation is damped
out by the inflation, i.e. the minimum allowed level of fluctuations is that required
by the uncertainty principle. The solution (5.110) corresponds to a de Sitter vacuum
state. If the period of inflation is long enough, the today observable properties of
the Universe should be indistinguishable from the properties of a Universe started in
the de Sitter vacuum state. In the radiation dominated phase, the eigenmodes which
describe particles are the coefficients of α, and these which describe antiparticles are
the coefficients of β (see also [401]). Thus, the number of particles created at angular
frequency ω in the radiation dominated phase is given by
Nω = |βω |2 =
Hds H0
2ω 2
2
.
(5.113)
5.4. STOCHASTIC BACKGROUND OF RELIC SCALAR GRAVITATIONAL WAVES FROM
SCALAR-TENSOR GRAVITY
131
Now it is possible to write an expression for the energy density of the stochastic scalar
relic gravitons background in the frequency interval (ω, ω + dω) as
dρsgw = 2~ω
ω 2 dω
2π 2 c3
Nω =
2 H2
2 H2
~Hds
~Hds
0 df
0 dω
=
,
2
3
2
3
4π c ω
4π c f
(5.114)
where f , as above, is the frequency in standard comoving time. Eq. (5.114) can be
rewritten in terms of the today and de Sitter value of energy density being
H0 =
8πGρc
,
3c2
Introducing the Planck density ρP lanck =
Ωsgw (f ) =
Hds =
8πGρds
.
3c2
(5.115)
c7
the spectrum is given by
~G2
1 dρsgw
f dρsgw
16 ρds
=
=
.
ρc d ln f
ρc df
9 ρP lanck
(5.116)
At this point, some comments are in order. First of all, such a calculation works for a
simplified model that does not include the matter dominated era. If also such an era
is also included, the redshift at equivalence epoch has to be considered. Taking into
account also results in [395], we get
Ωsgw (f ) =
16 ρds
(1 + zeq )−1 ,
9 ρP lanck
(5.117)
for the waves which, at the epoch in which the Universe becomes matter dominated,
have a frequency higher than Heq , the Hubble parameter at equivalence. This situation
corresponds to frequencies f > (1 + zeq )1/2 H0 . The redshift correction in Eq.(5.117) is
needed since the today observed Hubble parameter H0 would result different without a
matter dominated contribution. At lower frequencies, the spectrum is given by [393, 394]
Ωsgw (f ) ∝ f −2 .
(5.118)
As a further consideration, let us note that the results (5.116) and (5.117), which are not
frequency dependent, does not work correctly in all the range of physical frequencies.
For waves with frequencies less than today observed H0 , the notion of energy density has
no sense, since the wavelength becomes longer than the Hubble scale of the Universe.
In analogous way, at high frequencies, there is a maximal frequency above which the
spectrum rapidly drops to zero. In the above calculation, the simple assumption that the
phase transition from the inflationary to the radiation dominated epoch is instantaneous
has been made. In the physical Universe, this process occurs over some time scale ∆τ ,
being
a(t1 ) 1
,
(5.119)
fmax =
a(t0 ) ∆τ
132
5. PROBING THE MINKOWSKIAN LIMIT: GRAVITATIONAL WAVES IN F (R)-THEORIES
which is the redshifted rate of the transition. In any case, Ωsgw drops rapidly. The two
cutoffs at low and high frequencies for the spectrum guarantee that the total energy
density of the relic scalar gravitons is finite. For GUT energy-scale inflation it is of the
order [393]
ρds
≈ 10−12 .
(5.120)
ρP lanck
These results can be quantitatively constrained considering the recent WMAP release.
In fact, it is well known that WMAP observations put strongly severe restrictions on the
spectrum. In Fig. 5.4 the spectrum Ωsgw is mapped : considering the ratio ρds /ρP lanck ,
the relic scalar GW spectrum seems consistent with the WMAP constraints on scalar
perturbations. Nevertheless, since the spectrum falls off ∝ f −2 at low frequencies, this
means that today, at LIGO-VIRGO and LISA frequencies (indicated in fig. 5.4), one
gets
Ωsgw (f )h2100 < 2.3 × 10−12 .
(5.121)
It is interesting to calculate the corresponding strain at ≈ 100Hz, where interferometers
like VIRGO and LIGO reach a maximum in sensitivity. The well known equation for
the characteristic amplitude [393, 394, 301] adapted to the scalar component of GWs
can be used:
q
−18 1Hz
(5.122)
) h2100 Ωsgw (f ),
Φc (f ) ≃ 1.26 × 10 (
f
and then we obtain
Φc (100Hz) < 2 × 10−26 .
(5.123)
Φc (100Hz) < 2 × 10−25 .
(5.124)
Then, since we expect a sensitivity of the order of 10−22 for the above interferometers
at ≈ 100Hz, we need to gain four order of magnitude. Let us analyze the situation also
at smaller frequencies. The sensitivity of the VIRGO interferometer is of the order of
10−21 at ≈ 10Hz and in that case it is
The sensitivity of the LISA interferometer will be of the order of 10−22 at ≈ 10−3 Hz
and in that case it is
Φc (100Hz) < 2 × 10−21 .
(5.125)
This means that a stochastic background of relic scalar GWs could be, in principle,
detected by the LISA interferometer.
5.4. STOCHASTIC BACKGROUND OF RELIC SCALAR GRAVITATIONAL WAVES FROM
SCALAR-TENSOR GRAVITY
133
Energy
LISA
-15
-10
-6
-5
LIGO
5
10
Hz H… Graphics …L
-10
-12
-14
-16
-18
Figure 5.4: The spectrum of relic scalar GWs in inflationary models is flat over a wide
range of frequencies. The horizontal axis is log10 of frequency, in Hz. The vertical
axis is log10 Ωgsw . The inflationary spectrum rises quickly at low frequencies (wave
which re-entered in the Hubble sphere after the Universe became matter dominated) and
falls off above the (appropriately redshifted) frequency scale fmax associated with the
fastest characteristic time of the phase transition at the end of inflation. The amplitude
of the flat region depends only on the energy density during the inflationary stage;
we have chosen the largest amplitude consistent with the WMAP constrains on scalar
perturbations. This means that at LIGO and LISA frequencies, Ωsgw < 2.3 ∗ 10−12
134
5. PROBING THE MINKOWSKIAN LIMIT: GRAVITATIONAL WAVES IN F (R)-THEORIES
Chapter 6
Further probe: Parametrized
Post Newtonian limit
6.1
f (R) gravity constrained by PPN parameters and stochastic background of gravitational waves
The idea that Einstein gravity should be extended or corrected at large scales (infrared
limit) or at high energies (ultraviolet limit) is suggested by several theoretical and observational issues. Quantum field theory in curved spacetimes, as well as the low-energy
limit of String/M theory, both imply semi-classical effective actions containing higherorder curvature invariants or scalar-tensor terms. In addition, GR has been definitely
tested only at Solar System scales while it may show several shortcomings if checked at
higher energies or larger scales. Besides, the Solar System experiments are, up to now,
not so conclusive to state that the only viable theory of gravity is GR: for example,
the limits on PPN parameters should be greatly improved to fully remove degeneracies
[342].
Of course, modifying the gravitational action asks for several fundamental challenges. These models can exhibit instabilities [343] or ghost - like behavior [344], while,
on the other hand, they have to be matched with observations and experiments in the
appropriate low energy limit.
Despite of all these issues, in the last years, some interesting results have been
achieved in the framework of the so called f (R)-gravity at cosmological, Galactic and
Solar System scales. Here f (R) is a general (analytic) function of the Ricci scalar R (see
Refs. [345, 346, 347] for review).
For example, there exist cosmological solutions that give the accelerated expansion
of the universe at late times [348, 349, 350, 351]. In addition, it has been discovered
that some stability conditions can lead to avoid ghost and tachyon solutions. Furthermore there exist viable f (R) models which satisfy both background cosmological constraints and stability conditions [353, 355, 352, 356, 357, 358, 359, 360] and results have
135
136
6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT
been achieved in order to place constraints on f (R) cosmological models by CMBR
anisotropies and galaxy power spectrum [361, 362, 363]. Moreover, some of such viable models lead to the unification of early-time inflation with late-time acceleration
[358, 359, 360].
On the other hand, by considering f (R)-gravity in the low energy limit, it is possible
to obtain corrected gravitational potentials capable of explaining the flat rotation curves
of spiral galaxies or the dynamics of galaxy clusters without considering huge amounts
of dark matter [364, 365, 366, 367, 368, 369].
Furthermore, several authors have dealt with the weak field limit of fourth order
gravity, in particular considering the PPN limit [371, 372, 373, 374, 375, 376, 377, 383]
and the spherically symmetric solutions [378, 379, 380, 382].
This great deal of work needs an essential issue to be pursued: we need to compare
experiments and probes at local scales (e.g. Solar System) with experiments and probes
at large scales (Galaxy, extragalactic scales, cosmology) in order to achieve self-consistent
f (R) models. Some work has been done in this direction (see e.g. [355]) but the
large part of efforts has been devoted to address single data sets (observations at a
given redshift) by a single model which, several time, is not working at other scales
than the one considered. In particular, a given f (R) model, evading Solar System
tests, should be not simply extrapolated at extragalactic and cosmological scales only
requiring accelerated cosmological solutions but it should be confronted with data and
probes coming from cosmological observations. Reliable models are then those matching
data at very different scales (and redshifts).
In order to constrain further viable f (R)-models, one could take into account also
the stochastic background of gravitational waves (GW) which, together with cosmic
microwave background radiation (CMBR), would carry a huge amount of information on
the early stages of the Universe evolution. In fact, if detected, such a background could
constitute a further probe for these theories at very high red-shift [401]. On the other
hand, a key role for the production and the detection of the relic gravitational radiation
background is played by the adopted theory of gravity [384, 385]. This means that the
effective theory of gravity should be probed at zero, intermediate and high redshifts to
be consistent at all scales and not simply extrapolated up to the last scattering surface,
as in the case of GR.
The aim of this chapter is to discuss the PPN Solar-System constraints and the
GW stochastic background considering some recently proposed f (R) gravity models
[352, 353, 355, 358, 359, 360] which satisfy both cosmological and stability conditions
mentioned above. Using the definition of PPN-parameters γ and β in terms of f (R)models [377] and the definition of scalar GWs [386], we compare and discuss if it is
possible to search for parameter ranges of f (R)-models working at Solar System and
GW stochastic background scale. This phenomenological approach is complementary
to the one proposed, e.g. in [355, 363] where also galactic and cosmological scales have
been considered to constraint the models.
6.2. F (R) GRAVITY
6.2
137
f (R) gravity
Let us start from the following action
S = Sg + Sm
1
= 2
k
Z
√
d4 x −g [R + f (R) + Lm ] ,
(6.1)
where we have considered the gravitational and matter contributions and k2 ≡ 16πG.
The non-linear f (R) term has been put in evidence with respect to the standard HilbertEinstein term R and Lm is the perfect-fluid matter Lagrangian. The field equations are
1
k2 (m)
gµν F (R) − Rµν F ′ (R) − gµν F ′ (R) + ∇µ ∇ν F ′ (R) = − Tµν
.
2
2
(6.2)
(m)
Here F (R) = R+f (R) and Tµν is the matter energy - momentum tensor. By introducing
the auxiliary field A, one can rewrite the gravitational part in the Action (6.1) as
Z
√ 1
Sg = 2 d4 x −g 1 + f ′ (A) (R − A) + A + f (A) .
(6.3)
k
2
′
2
As it is clear from Eq.(6.3), if F ′ (R) = 1 + f ′ (R) < 0, the coupling kef
f = k /F (A)
becomes negative and the theory enters the anti-gravity regime. Note that it is not the
case for the standard GR.
Action (6.3) can be recast in a scalar-tensor form. By using the conformal scale
transformation gµν → eσ gµν with σ = − ln (1 + f ′ (A)), the action can be written in the
Einstein frame as follows [345]:
Z
3 ρσ
1
4 √
(6.4)
SE = 2 d x −g R − g ∂ρ σ∂σ σ − V (σ) ,
k
2
where
V (σ) = eσ g e−σ − e2σ f g e−σ =
A
F ′ (A)
−
F (A)
.
F ′ (A)2
(6.5)
The form of g (e−σ ) is given by solving σ = − ln (1 + f ′ (A)) = ln F ′ (A) as A =
The transformation gµν → eσ gµν induces a coupling of the scalar field σ with
matter.
In general, an effective mass for σ is defined as [360]
1 d2 V (σ)
A
1
4F (A)
1
2
mσ ≡
=
−
+
,
(6.6)
2 dσ 2
2 F ′ (A) (F ′ (A))2 F ′′ (A)
g (e−σ ).
which, in the weak field limit, could induce corrections to the Newton law. This allows, as
it is well known, to deal with the extra degrees of freedom of f (R)-gravity as an effective
scalar field which reveals particularly useful in considering ”chameleon” models [354].
This ”parameterization” will be particularly useful to deal with the scalar component
of GWs.
138
6.3
6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT
f (R) viable models
Let us consider now a class of f (R) models which do not contain cosmological constant
and are explicitly designed to satisfy cosmological and Solar-System constraints in given
limits of the parameter space. In practice, we choose a class of functional forms of
f (R) capable of matching, in principle, observational data (see [350] for the general
approach). Firstly, the cosmological model should reproduce the CMBR constraints
in the high-redshift regime (which agree with the presence of an effective cosmological
constant). Secondly, it should give rise to an accelerated expansion, at low redshift,
according to the ΛCDM model. Thirdly, there should be sufficient degrees of freedom
in the parameterization to encompass low redshift phenomena (e.g. the large scale
structure) according to the observations [363]. Finally, small deviations from GR should
be consistent with Solar System tests. All these requirements suggest that we can assume
the limits
lim f (R) = constant,
(6.7)
R→∞
lim f (R) = 0,
(6.8)
R→0
which are satisfied by a general class of broken power law models, proposed in [355],
which are
R n
c
1
2
m
n
(6.9)
fI (R) = −m2
c2 mR2 + 1
or otherwise written as
FI (R) = R − λRc 2n
R
Rc
2n
R
Rc
(6.10)
+1
where m is a mass scale and c1,2 are dimensionless parameters.
Besides, another viable class of models was proposed in [352]
#
"
−p
R2
−1 .
FII (R) = R + λRc
1+ 2
Rc
(6.11)
Since F (R = 0) = 0, the cosmological constant has to disappear in a flat spacetime.
The parameters {n, p, λ, Rc } are constants which should be determined by experimental
bounds.
Other interesting models with similar features have been studied in [360, 358, 356,
357, 359]. In all these models, a de-Sitter stability point, responsible for the late-time
acceleration, exists for R = R1 (> 0), where R1 is derived by solving the equation
R1 f,R (R1 ) = 2f (R1 ) [381]. For example, in the model (6.11), we have R1 /Rc = 3.38
for λ = 2 and p = 1. If λ is of the unit order, R1 is of the same order of Rc . The
6.3. F (R) VIABLE MODELS
139
stability conditions, f,R > 0 and f,RR > 0, are fulfilled for R > R1 [352, 357]. Moreover
the models satisfy the conditions for the cosmological viability that gives rise to the
sequence of radiation, matter and accelerated epochs [357].
In the region R ≫ Rc both classes of models (6.9) and (6.11) behave as
i
h
FIII (R) ≃ R − λRc 1 − (Rc /R)2s ,
(6.12)
where s is a positive constant. The model approaches ΛCDM in the limit R/Rc → ∞.
Finally, let also consider the class of models [353, 370, 362]
FIV (R) = R − λRc
R
Rc
q
.
(6.13)
Also in this case λ, q and Rc are positive constants (note that n, p, s and q have to
converge toward the same values to match the observations). We do not consider the
models whit negative q, because they suffer for instability problems associated with
negative F,RR [387, 361]. In Fig.(6.1), we have plotted some of the selected models as
R
for suitable values of {p, n, q, s, λ} .
function of
Rc
Let us now estimate mσ for the models discussed above. For Model I [355], when
the curvature is large, we find
fI (R) ∼ −
m2 c1 m2+2n c1
+ ··· ,
+
c2
c22 Rn
(6.14)
and obtain the following expression:
m2σ
m2 c22
∼
2n(n + 1)c1
R
m2
n+2
.
(6.15)
Here the order of the mass-dimensional parameter m2 should be m2 ∼ 10−64 eV2 . Then
in Solar System, where R ∼ 10−61 eV2 , the mass is given by m2σ ∼ 10−58+3n eV2 while
on the Earth atmosphere, where R ∼ 10−50 eV2 , it has to be m2σ ∼ 10−36+14n eV2 . The
−1
order of the radius of the Earth is 107 m ∼ 10−14 eV
. Therefore the scalar field σ
is enough heavy if n ≫ 1 and the correction to the Newton law is not observed, being
extremely small. In fact, if we choose n = 10, the order of the Compton length of
the scalar field σ becomes that of the Earth radius. On the other hand, in the Earth
atmosphere, if we choose n = 10, for example, we find that the mass is extremely large:
mσ ∼ 1043 GeV ∼ 1029 × MPlanck .
(6.16)
Here MPlanck is the Planck mass. Hence, the Newton law correction should be extremely
small.
140
6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT
In Model II
fII (R) = −λR0
"
R2
1− 1+ 2
R0
−p #
,
(6.17)
if R is large compared with R0 , whose order of magnitude is that of the curvature in
the present universe, we find
fII (R) = −λR0 + λ
R02p+1
+ ··· .
R2p
(6.18)
By comparing Eq.(6.18) with Eq. (6.14), if the curvature is large enough when compared with R0 or m2 , as in the Solar System or on the Earth, we can set the following
identifications:
λR0 ↔
m2 c1
,
c2
λR02p+1 ↔
m2+2n c1
,
c22
2p ↔ n .
(6.19)
We have 41m2 ∼ R0 . Then, if p is large enough, there is no correction to the Newton
law as in Model I given by Eq.(6.10).
Let us now discuss the instability of fluid matter proposed in [387], which may appear
if the matter-energy density (or the scalar curvature) is large enough when compared
with the average density the Universe, as it is inside the Earth. Considering the trace
of the above field equations and with a little algebra, one obtains
R +
F ′ (R)R
2F (R)
κ2
F (3) (R)
ρ
∇
R∇
R
+
−
=
T.
ρ
F (2) (R)
3F (2) (R) 3F (2) (R)
6F (2) (R)
(6.20)
(m)ρ
Here T is the trace of the matter energy-momentum tensor: T ≡ Tρ . We also denote
the derivative dn F (R)/dRn by F (n) (R). Let us now consider the perturbation of the
Einstein gravity solutions. We denote the scalar curvature, given by the matter density
in the Einstein gravity, by Rb ∼ (κ2 /2)ρ > 0 and separate the scalar curvature R into
the sum of Rb (background) and the perturbed part Rp as R = Rb + Rp (|Rp | ≪ |Rb |).
Then Eq.(6.20) leads to the perturbed equation:
0 = Rb +
−
F (3) (Rb )
F ′ (Rb )Rb
ρ
∇
R
∇
R
+
ρ
b
b
F (2) (Rb )
3F (2) (Rb )
Rb
F (3) (Rb )
2F (Rb )
−
+
R
+
2
∇ρ Rb ∇ρ Rp + U (Rb )Rp .(6.21)
p
(2)
(2)
(2)
3F (Rb ) 3F (Rb )
F (Rb )
Here the potential U (Rb ) is given by
F (4) (Rb ) F (3) (Rb )2
−
F (2) (Rb ) F (2) (Rb )2
U (Rb ) ≡
−
!
∇ ρ Rb ∇ ρ Rb +
Rb
3
F (1) (Rb )F (3) (Rb )Rb
F (1) (Rb )
2F (Rb )F (3) (Rb ) F (3) (Rb )Rb
−
+
−
(6.22)
3F (2) (Rb )2
3F (2) (Rb )
3F (2) (Rb )2
3F (2) (Rb )2
6.3. F (R) VIABLE MODELS
141
It is convenient to consider the case where Rb and Rp are uniform and do not depend on
the spatial coordinates. Hence, the d’Alembert operator can be replaced by the second
derivative with respect to the time, that is: Rp → −∂t2 Rp . Eq.(6.22) assumes the
following structure:
0 = −∂t2 Rp + U (Rb )Rp + const .
(6.23)
√
If U (Rb ) > 0, Rp becomes exponentially large with time, i.e. Rp ∼ e U (Rb )t , and the
system becomes unstable.
In the 1/R-model, considering the background values, we find
3
−2
Rb3
ρm
R03
−26
U (Rb ) = −Rb + 4 ∼ 4 ∼ 10 sec
,
6µ
µ
g cm−3
−2
ρm
3
.
(6.24)
Rb ∼ 10 sec
g cm−3
Here the mass parameter µ is of the order
−1
µ−1 ∼ 1018 sec ∼ 10−33 eV
.
(6.25)
Eq.(6.24) tells us that the model is unstable and it would decay in 10−26 sec (considering
the Earth size). In Model I, however, U (Rb ) is negative:
U (R0 ) ∼ −
(n + 2)m2 c22
<0.
c1 n(n + 1)
(6.26)
Therefore, there is no matter instability.
For Model (6.17), as it is clear from the identifications (6.19), there is no matter
instability too.
In order to study the stability of the de Sitter solution, let us proceed as follows.
From the field equations (6.2), we obtain the trace
f ′ (R) =
1
R − f ′ (R)R + 2f (R) + κ2 T .
3
(6.27)
(m)
Here, as above, F (R) is F (R) = R + f (R) and T ≡ gµν Tµν .
Now we consider the (in)stability around the de Sitter solution, where R = R0 , and
therefore f (R0 ) and f ′ (R0 ), are constants. Then since the l.h.s. in Eq.(6.27) vanishes
for R = R0 , we find
R0 − f ′ (R0 )R0 + 2f (R0 ) + κ2 T0 = 0 .
(6.28)
Let us expand both sides of (6.28) around R = R0 as
R = R0 + δR .
One obtains
f ′′ (R0 )δR =
1
1 − f ′′ (R0 )R0 + f ′ (R0 ) δR .
3
(6.29)
(6.30)
142
6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT
Since
δR = −
dδR
d2 δR
− 3H0
,
2
dt
dt
(6.31)
in the de Sitter background, if
C(R0 ) ≡ lim
R→R0
1 − f ′′ (R)R + f ′ (R)
>0,
f ′′ (R)
(6.32)
the de Sitter background is stable but, if C(R0 ) < 0, the de Sitter background is unstable.
The expression for C(R0 ) could be valid even if f ′′ (R0 ) = 0. More precisely, the solution
of (6.30) is given by
δR = A+ eλ+ t + A− eλ− t .
(6.33)
Here A± are constants and
λ± =
−3H0 ±
p
9H02 − C(R0 )
.
2
(6.34)
Then, if C(R0 ) < 0, λ+ is always positive and the perturbation grows up. This leads to
the instability. We have also to note that, when C(R0 ) is positive, if C(R0 ) > 9H02 , δR
oscillates and the amplitude becomes exponentially small being:
p
C(R0 ) − 9H02
−3H0 t/2
δR = (A cos ω0 t + B sin ω0 t) e
, ω≡
.
(6.35)
2
Here A and B are constant. On the other hand, if C(R0 ) < 9H02 , there is no oscillation
in δR.
Let us now consider the case where the matter contribution T can be neglected in the
de Sitter background and assume f ′ (R) = 0 in the same background. We can assume
that there are two de Sitter background solutions satisfying f ′ (R) = 0, for R = R1
and R = R2 as it could be the physical case if one asks for an inflationary and a dark
energy epoch. We also assume f ′ (R) 6= 0 if R1 < R < R2 or R2 < R < R1 . In the
case C(R1 ) < 0 and C(R2 ) > 0, the de Sitter solution, corresponding to R = R1 , is
unstable but the solution corresponding to R = R2 is stable. Then there should be
a solution where the (nearly) de Sitter solution corresponding to R1 transits to the
(nearly) de Sitter solution R2 . Since the solution corresponding to R2 is stable, the
universe remains in the de Sitter solution corresponding to R2 and there is no more
transition to any other de Sitter solution.
As an example, we consider Model I. For large curvature values, we find
fI (R) = −Λ +
α
R2n+1
.
(6.36)
Here Λ and α are positive constants and n is a positive integer. Then we find
C(R) ∼
1
R2n+2
∼
> 0.
f ′′ (R)
2n(2n + 1)α
(6.37)
6.3. F (R) VIABLE MODELS
143
This means that the de Sitter solution in Model I can be stable. We have also to
note that C(R0 ) ∼ H04n+4 /m4n+2 . Here m2 is the mass scale introduced in [355] and
m2 ≪ H02 : this means that C(R0 ) ≫ 9H02 and therefore there could be no oscillation.
We may also consider the model proposed in [358](here Model V):
fV (R) =
αR2n − βRn
.
1 + γRn
(6.38)
Here α, β, and γ are positive constants and n is a positive integer. In Fig.6.2, we
show the behavior of Model V and of its first derivative. When the curvature is large
(R → ∞), f (R) behaves as a power law. Since the derivative of f (R) is given by
nRn−1 αγR2n − 2αRn − β
′
,
(6.39)
fV (R) =
(1 + γRn )2
we find that the curvature R0 in the present universe, which satisfies the condition
f ′ (R0 ) = 0, is given by
!#1/n
"
r
βγ
1
1+ 1+
,
(6.40)
R0 =
γ
α
and
α
f (R0 ) ∼ −2R̃0 = 2
γ
!
p
(1 − βγ/α) 1 + βγ/α
p
.
1+
2 + 1 + βγ/α
(6.41)
As shown in [358], the magnitudes of the parameters is given by
α ∼ 2R̃0 R0−2n , β ∼ 4R̃02 R0−2n RIn−1 , γ ∼ 2R̃0 R0−2n RIn−1 .
(6.42)
Here RI is the curvature in the inflationary epoch and we have assumed f (RI ) ∼
(α/γ)RIn ∼ RI .
C(R0 ) in (6.32) is given by
C(R0 ) ∼
1
1 + γR0n
.
=
f ′′ (R0 )
2n2 αR02n−2 (γR0n − 1)
(6.43)
By using the relations (6.42), we find
C(R0 ) ∼
R02
,
4n2 R̃0
(6.44)
which is positive and therefore the de Sitter solution is stable. We notice that C(R0 ) <
9H02 and therefore, there could occur oscillations as in (6.35).
Furthermore, we can take into account the following model [359] (Model VI):
#
"
b (R − R0 )
bR0
eb(R−R0 ) − 1 ebR0 − 1
fV I (R) = −α tanh
+ bR0
+ tanh
= −α b(R−R )
0 + 1
2
2
e
+1
e
(6.45)
144
6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT
where α and b are positive constants. When R → 0, we find that
fV I (R) → −
αbR
,
2 cosh2 bR2 0
and thus f (0) = 0. On the other hand, when R → +∞,
bR0
.
fV I (R) → −2Λeff ≡ −α 1 + tanh
2
(6.46)
(6.47)
If R ≫ R0 , in the present universe, Λeff plays the role of the effective cosmological
constant. We also obtain
fV′ I (R) = −
2 cosh
which has a minimum when R = R0 , that is:
2
αb
fV′ I (R0 ) = −
b(R−R0 )
2
,
αb
.
2
(6.48)
(6.49)
Then in order to avoid anti-gravity, we find
0 < 1 + fV′ I (R0 ) = 1 −
αb
.
2
(6.50)
Beside the above model, we can consider a model which is able to describe, in
principle, both the early inflation and the late acceleration epochs. The following twostep model [359] (Model VII):
b0 R0
bI RI
bI (R − RI )
b0 (R − R0 )
+ tanh
−αI tanh
+ tanh
,
fV II (R) = −α0 tanh
2
2
2
2
(6.51)
could be useful to this goal. Let us assume
RI ≫ R0 ,
αI ≫ α0 ,
bI ≪ b0 ,
(6.52)
and
bI RI ≫ 1 .
When R → 0 or R ≪ R0 ≪ RI , fV II (R) behaves as


α0 b0
αI bI
+
 R ,
fV II (R) → − 
2 b0 R0
2 bI RI
2 cosh
2
cosh
2
2
(6.53)
(6.54)
6.3. F (R) VIABLE MODELS
145
and we find again fV II (0) = 0. When R ≫ RI , we find
b0 R0
bI RI
bI RI
f (R)V II → −2ΛI ≡ −α0 1 + tanh
−αI 1 + tanh
∼ −αI 1 + tanh
.
2
2
2
(6.55)
On the other hand, when R0 ≪ R ≪ RI , we find
αI bI R
b0 R0
b0 R0
∼ −2Λ0 ≡ −α0 1 + tanh
−
.
fV II (R) → −α0 1 + tanh
2
2
2 cosh2 bI2RI
(6.56)
Here, we have assumed the condition (6.53). We also find
fV′ II (R) = −
2 cosh
2
α0 b0
b0 (R−R0 )
2
−
2 cosh
2
αI bI
bI (R−RI )
2
,
(6.57)
which has two minima for R ∼ R0 and R ∼ RI . When R = R0 , we obtain
fV′ II (R0 ) = −α0 b0 −
2 cosh2
On the other hand, when R = RI , we get
fV′ II (RI ) = −αI bI −
2 cosh2
α b
I I
> −αI bI − α0 b0 .
(6.58)
α b
0 0
> −αI bI − α0 b0 .
(6.59)
bI (R0 −RI )
2
b0 (R0 −RI )
2
Then, in order to avoid the anti-gravity behavior, we find
αI bI + α0 b0 < 1 .
(6.60)
Let us now investigate the correction to the Newton potential and the matter instability issue related to Models VI and VII. In the Solar System domain, on or inside the
Earth, where R ≫ R0 , f (R) in Eq.(6.45) can be approximated by
fV I (R) ∼ −2Λeff + 2αe−b(R−R0 ) .
(6.61)
On the other hand, since R0 ≪ R ≪ RI , by assuming Eq. (6.53), f (R) in (6.51) can be
also approximated by
fV II (R) ∼ −2Λ0 + 2αe−b0 (R−R0 ) ,
(6.62)
which has the same expression, after having identified Λ0 = Λeff and b0 = b. Then, we
may check the case of (6.61) only. In this case, the effective mass has the following form
m2σ ∼
eb(R−R0 )
,
4αb2
(6.63)
146
6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT
which could be again very large. In fact, in the
Solar System, we find R ∼ 10−61 eV2 .
2
Even if we choose α ∼ 1/b ∼ R0 ∼ 10−33 eV , we find that m2σ ∼ 101,000 eV2 , which
is, ultimately, extremely heavy. Then, there will be no appreciable correction to the
Newton law. In the Earth atmosphere, R ∼ 10−50 eV2 , and even if we choose α ∼ 1/b ∼
2
R0 ∼ 10−33 eV again, we find that m2σ ∼ 1010,000,000,000 eV2 . Then, a correction to
the Newton law is never observed in such models. In this case, we find that the effective
potential U (Rb ) has the form
1 −b(Re −R0 )
1
2Λ +
e
,
(6.64)
U (Re ) = −
2αb
b
which could be negative, what would suppress any instability.
In order that a de Sitter solution exists in f (R)-gravity, the following condition has
to be satisfied:
R = Rf ′ (R) − 2f (R) .
(6.65)
For the model (6.45), the r.h.s of (6.65) has the following form:
bR0
b (R − R0 )
bαR
+ 2α tanh
+ tanh
.
R=−
0)
2
2
2 cosh2 b(R−R
2
For large R, the r.h.s. behaves as
b (R − R0 )
bαR
bR0
+ 2α tanh
−
+ tanh
→ 2α ,
0)
2
2
2 cosh2 b(R−R
2
(6.66)
(6.67)
although the l.h.s. goes to infinity. On the other hand, when R is small, the r.h.s.
behaves as
bR0
bαR
b (R − R0 )
bαR
.
+ tanh
→
−
+ 2α tanh
2 b(R−R0 )
2 bR0
2
2
2 cosh
2
cosh
2
2
(6.68)
Then if
bα
>1,
(6.69)
2 cosh2 bR2 0
there is a de Sitter solution. Combining Eq.(6.69) with Eq.(6.50), we find
2 > αb >
1
2 cosh2
bR0
2
.
(6.70)
The stability, as above, is given by C(RdS ), where RdS is the solution of (6.66). The
expression is given by
2 cosh3 b(RdS2−R0 )
1
−
.
C(RdS ) = −RdS +
(6.71)
b(RdS −R0 )
b(RdS −R0 )
2
αb sinh
b
tanh
2
2
6.3. F (R) VIABLE MODELS
147
Let us now rewrite Eq.(6.66) as follows,
RdS = 2α tanh
b (RdS − R0 )
2
+ tanh
bR0
2

1 +
2 cosh
2
αb
b(RdS −R0 )
2
Then by using (6.72), we may rewrite (6.71) in the following form:
i
h
−α2 b2 1 − x2 (x − x0 )2 + 1 − x20 + 4
,
C(RdS ) =
αb2 x (1 − x2 ) [2 + αb (1 − x2 )]
where
x = tanh
and therefore we have
b (RdS − R0 )
2
,
−1 < x0 ≤ x < 1 ,
x0 = − tanh
bR0
2
x0 < 0 .
,
−1

.
(6.72)
(6.73)
(6.74)
(6.75)
Let us now consider (6.66) in order to find a de Sitter solution. Since Eq.(6.66) is
difficult to solve in general, we assume 0 < RdS ≪ R0 . Then we find
2 bR0
2
cosh
2
2
ε
.
=1−
, ε≡1−
(6.76)
RdS =
bx0
αb
αb 1 − x20
Eq.(6.69) tells that the parameter ε is positive and, by assumption, very small: 0 < ε ≪
1. Since ε is small, by using Eqs.(6.74), we find
1 − x20
ε + O ε2 .
(6.77)
x = x0 +
2x0
Then by using the expression (6.73) for C(RdS ), we find
2
−α2 b2 1 − x20 + 4
.
C(RdS ) ∼ 2
αb x0 1 − x20 2 + αb 1 − x20
From the definition of ε in (6.76), we find
αb 1 − x20 = 2 + 2ε + O ε2 ,
(6.78)
(6.79)
and then, from Eq.(6.79), Eq.(6.78) can be written as follows;
C(RdS ) ∼ −
ε
.
bx0
(6.80)
Since x0 < 0 in the condition (6.75), we find C(RdS ) > 0 and therefore the de Sitter
solution is stable.
148
6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT
Mercury perihelion Shift
Lunar Laser Ranging
Very Long Baseline Interferometer
Cassini Spacecraft
|2γ − β − 1| < 3 × 10−3
4β − γ − 3 = (0.7 ± 1) × 10−3
|γ − 1| < 4 × 10−4
γ − 1 = (2.1 ± 2.3) × 10−5
Table 6.1: Solar System experimental constraints on the PPN parameters.
In Fig. 6.3, we have plotted the two models (6.45) and (6.51) written in the
form F (R) = R + f (R). We have used the inequalities (6.52) assuming, RI ∼ ρg ∼
10−24 g/cm3 for the Galactic density in the Solar vicinity and R0 ∼ ρg ∼ 10−29 g/cm3
for the present cosmological density. .
Our task is now to find reliable experimental bounds for such models working at
small and large scales. To this goal, we shall take into account constraints coming
from Solar System experiments (which, at present, are capable of giving upper limits
on the PPN parameters) and constraints coming from interferometers, in particular
those giving limits on the (eventual) scalar components of GWs. If constraints (and in
particular the ranges of model parameters given by them) are comparable, this could
constitute, besides other experimental and observational probes, a good hint to achieve
a self-consistent f (R) theory at very different scales.
6.4
Constraining f (R)-models by PPN parameters
The above models can be constrained at Solar System level by considering the PPN
formalism. This approach is extremely important in order to test gravitational theories
and to compare them with GR. As it is shown in [372, 377], one can derive the PPNparameters γ and β in terms of a generic analytic function F (R) and its derivative
F ′′ (R)2
,
F ′ (R) + 2F ′′ (R)2
dγ
F ′ (R) · F ′′ (R)
1
.
β−1=
4 2F ′ (R) + 3F ′′ (R)2 dR
γ−1=−
(6.81)
(6.82)
These quantities have to fulfill the constraints coming from the Solar System experimental tests summarized in Table I. They are the perihelion shift of Mercury [388], the
Lunar Laser Ranging [389], the upper limits coming from the Very Long Baseline Interferometry (VLBI) [390] and the results obtained from the Cassini spacecraft mission in
the delay of the radio waves transmission near the Solar conjunction [391].
6.4. CONSTRAINING F (R)-MODELS BY PPN PARAMETERS
149
Let us take into account before the f (R)-models (6.10)-(6.13). Specifically, we want
to investigate the values or the ranges of parameters in which they match the SolarSystem experimental constraints in Table 6.1. In other words, we use these models
to search under what circumstances it is possible to significantly address cosmological
observations by f (R)-gravity and, simultaneously, evade the local tests of gravity.
By integrating Eqs. (6.81)-(6.82), one obtains f (R) solutions depending on β and
γ which has to be confronted with βexp and γexp [377]. If we plug into such equations
the models (6.10)-(6.13) and the experimental values of PPN parameters, we will obtain
algebraic constraints for the phenomenological parameters {n, p, q, λ, s}. This is the
issue which we want to take into account in this section.
1−γ ′
′′
2
, we obtain
From Eq.(6.81), assuming F (R)+2F (R) 6= 0 and defining A = 2γ − 1 ′′
2
F (R) − AF ′ (R) = 0 .
(6.83)
The general solution of such an equation is a polynomial function [377].
Considering Model II given by (6.11), we obtain


1 −
2pR
R2
R2c
2
−p−1 
−2p
2
+1
+
1
λ  γ − 1 4p2 R
Rc2 Rc2 − (2p + 1)R2 λ2
R2c
−
= 0.
 Rc
2γ − 1 (R2 + Rc2 )4
(6.84)
Our issue is now to find the values of λ, p, and R/Rc for which the Solar System
experimental constraints are satisfied. Some preliminary considerations are in order at
this point. Considering the de Sitter solution achieved from (6.11), we have R = const =
R1 = x1 Rc , and x1 > 0. It is straightforward to obtain
p+1
x1 1 + x21
i.
λ= h
p+1
2 1 + x21
− 1 − (p + 1) x21
(6.85)
On the other hand, the stability conditions F,R > 0 and F,RR > 0 give the inequality
1 + x21
p+2
> 1 + (p + 2) x21 + (p + 1) (2p + 1) x41 ,
√
(6.86)
8
3 and then λ > √ = 1.5396.
3 3
In addition, the value of x1 satisfying the relation (6.86) is also the point where λ(x1 ),
in Eq.(6.85), reaches its minimum.
To determine values of R compatible with PPN constraints, let us consider the trace
of the field equations (6.2) and explicit solutions, given the density profile ρ(r), in the
Solar vicinity. One can set the boundary condition considering F,R∞ = FRg
which has to be satisfied. In particular, for p = 1, it is x1 >
150
6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT
F,Rg = F,R (R = k2 ρg )
(6.87)
where ρg ∼ 10−24 g/cm3 is the observed Galactic density in the Solar neighborhoods.
At this point, we can see when the relation (6.84) satisfies the constraints for very Long
Baseline Interferometer (γ − 1 = 4 × 10−4 ) and Cassini Spacecraft (γ − 1 = 2.1 × 10− 5).
This allows to find out suitable values for p.
An important remark is in order at this point. These constraint equations work if
stability conditions hold. In the range
0<
1
R
<√
Rc
2p + 1
(6.88)
F,RR is negative for the model (6.11) and then stability conditions are violated. To
avoid this range, we need, at least, RRc > 1. For example, we can choose RRc = 3.38,
corresponding to de Sitter behavior. Then we have √
p = 1 and λ = 2. On the other
R
hand, for 0.944 < λ < 0.966, we have p = 2 and Rc = 3; finally, for R >> Rc , we have
λ = 2 and p = 1.5. For these values of parameters, the Solar System tests are evaded.
Let us consider now Model I, given by (6.9). Inserting it into the relation (6.83), we
get
R3
R
Rc
2n
#
4 " 2
2 2n
2n
2n
4n
γ−1 R
R
R
2 (2n + 1) R
R
−
2n
+1
+
1
R
λ
−
2n
+
1
−
4n
c
Rc
Rc
2γ−1
Rc
Rc
R4
2n
R
Rc
6
+1
(6.89)
Using the same procedure as above, λ is related to the de Sitter behavior. This means
λ=
1 + x2n
1
2
x12n−1 2 + 2x2n
1 − 2n
while, from the stability conditions, we get
,
2x41 − (2n − 1) (2n + 4) x2n
1 + (2n − 1) (2n − 2) ≥ 0 .
For n = 1, one obtains x1 >
√
3,λ>
8
√
.
3 3
R
<
0<
Rc
(6.90)
(6.91)
In this model, F,RR is negative for
2n − 1
2n + 1
1
2n
.
(6.92)
The VLBI constraint is satisfied for n = 1 and λ = 2, while, for n = 1 and λ = 1.5,
Cassini constraint holds.
6.4. CONSTRAINING F (R)-MODELS BY PPN PARAMETERS
151
By inserting Model III, given by Eq.(6.12), into the relation (6.83), we obtain
h
2s i γ−1 4s 2
2
λ 2γ−1 − 4 2s2 + s Rc2 Rrc
λ
R3 R − 2sRc RRc
= 0.
(6.93)
R4
The de-Sitter point corresponds to
λ=
x2s+1
1
.
2(x2s
1 − s − 1)
(6.94)
2
are
while the stability condition is x2s
1 > 2s + 3s + 1. VLBI and Cassini constraints √
R
R
satisfied by the sets of values: s = 1, λ = 1.53, for Rc ∼ 1; s = 2, λ = 0.95, for Rc = 3,
; s = 1, λ = 2, for RRc = 3.38.
Finally let us consider Model VI, given by Eq.(6.45), and Model VII, given by
Eq.(6.51). Using Eq.(6.83) for (6.45), we get
γ−1 1
3
2 1
2 1
= 0.
b(R − R0 ) b αsech
b(R − R0 ) tanh
b(R − R0 ) − 2 2
2
2
2γ − 1 (6.95)
As above, considering the stability conditions and the de Sitter behavior, we get the
parameter ranges 0 < b < 2 and 0 < α ≤ 2 which satisfy both VLBI and Cassini
constraints. Inserting now Model VII in (6.83), we have
1
− bαsech2
4
1 γ − 1 2 1
2 1
bαsech
b(R − R0 ) − bI αI sech
bI (R − RI ) + 2
2 2γ − 1 2
2
2
1 2
1
1
2
2 1
2 1
−
= 0.
b(R − R0 ) tanh
b(R − R0 ) − bI αI sech
bI (R − RI ) tanh
bI (R − RI )
b αsech
4
2
2
2
2
(6.96)
From the stability condition, we have that F,R > 0 for R > 0, (see Fig.6.6) and
F,RR < 0 for 0 < R < 2.35 in suitable units (see Fig.6.7). Observational constraints
from VLBI and Cassini experiments are fulfilled for
RI ≫ R0 ,
αI ≫ α ,
bI ≪ b .
(6.97)
Plots for b = 2, bI = 0.5, α = 1.5 and αI = 2, verifying the constraints, are reported in
Figs. 6.6 and 6.7.
Considering now the relation for β given by Eq. (6.82), one can easily verify that it
is
d
F ′′ (R)2
dγ
=−
= 0,
(6.98)
dR
dR F ′ (R) + 2F ′′ (R)2
152
6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT
and this result implies
4(β − 1) = 0 .
(6.99)
This means the complete compatibility of the f (R) solutions between the PPN-parameters
β and γ.
Now we want to see if the parameter values, obtained for these models, are compatible
with bounds coming from the stochastic background of GWs achieved by interferometric
experiments.
6.5
Stochastic backgrounds of gravitational waves to constrain f (R)-gravity
As we said before, also the stochastic background of GWs can be taken into account in order to constrain models. This approach could reveal very interesting because production
of primordial GWs could be a robust prediction for any model attempting to describe
the cosmological evolution at primordial epochs. However, bursts of gravitational radiation emitted from a large number of unresolved and uncorrelated astrophysical sources
generate a stochastic background at more recent epochs, immediately following the onset
of galaxy formation. Thus, astrophysical backgrounds might overwhelm the primordial
one and their investigation provides important constraints on the signal detectability
coming from the very early Universe, up to the bounds of the Planck epoch and the
initial singularity [384, 393, 394, 396].
It is worth stressing the unavoidable and fundamental character of such a mechanism.
It directly derives from the inflationary scenario [397, 398], which well fits the WMAP
data with particular good agreement with almost exponential inflation and spectral
index ≈ 1, [399, 400].
The main characteristics of the gravitational backgrounds produced by cosmological
sources depend both on the emission properties of each single source and on the source
rate evolution with redshift. It is therefore interesting to compare and contrast the
probing power of these classes of f (R)-models at hight, intermediate and zero redshift
[401].
To this purpose, let us take into account the primordial physical process which
gave rise to a characteristic spectrum Ωsgw for the early stochastic background of relic
scalar GWs by which we can recast the further degrees of freedom coming from fourthorder gravity. This approach can greatly contribute to constrain viable cosmological
models. The physical process related to the production has been analyzed, for example,
in [393, 394, 395] but only for the first two tensorial components due to standard General
Relativity. Actually the process can be improved considering also the third scalar-tensor
component strictly related to the further f (R) degrees of freedom [386].
At this point, using the above LIGO, VIRGO and LISA upper bounds, calculated
for the characteristic amplitude of GW scalar component, let us test the f (R)-gravity
6.5. STOCHASTIC BACKGROUNDS OF GRAVITATIONAL WAVES TO CONSTRAIN F (R)-GRAVITY
153
models, considered in the previous sections, to see whether they are compatible both
with the Solar System and GW stochastic background.
Before starting with the analysis, taking into account the discussion in the Chapter
5, section 5.4. As above, for the considered models, we have to determine the values
of the characteristic parameters which are compatible with both Solar System and GW
stochastic background.
Let us start, for example, with the model (6.12). Starting from the definitions (??),
it is straightforward to derive the scalar component amplitude
ΦIII = h
sRc
2s+1
λ
s(2s + 1) RRc
i
h
Rc 2s
λ − R log 2 − 2s
R
i.
(6.100)
Rc 2s+1
λ
R
Such an equation satisfies the constraints in Table.4.26 for the values s = 0.5, RRc ∼ 1,
√
λ = 1.53 and s = 1, RRc ∼ 1, λ = 0.95 (LIGO); s = 2, RRc = 3, λ = 2 (VIRGO); s = 1,
λ = 2 and RRc = 3.38 (LISA).
It is important to stress the nice agreement with the figures achieved from the PPN
constraints. In this case, we have assumed Rc ∼ ρc ∼ 10−29 g/cm3 , where ρc is the
present day cosmological density.
Considering the model (6.9), we obtain
2n
2n−1
R
n (2n + 1) Rc
− 2n + 1 RRc
λ
.

ΦI = −
)
“ ”2n−1
( 2
2n
2n
2n
λ
2n RR
R
R RRc
+1
+ 1 − n RRc
Rc λ log 1 − „“ ”c 2n «2 
Rc
R
Rc
+1
(6.101)
The expected constraints for GW scalar amplitude are fulfilled for n = 1 and λ = 2 and
for n = 1 and λ = 1.5 when 0.3 < RRc < 1.
Furthermore, considering the model (6.11), one gets
2 −p
2p 1 + R
Rc (1 + 2p) R2 − Rc2 λ
R2c
ΦI = −
(6.102)
„
«−1−p 
«−1−p  .
„


2
(R2
−
Rc2 )2 2
−
2p 1+ R2
Rc
Rc
λ
2
 ln 2 −
The LIGO upper bound is fulfilled for p = 1,
R
Rc
R
Rc
>
√
2pR 1+ R2
Rc
λ
Rc
3, λ >
R
Rc
8
√
; the
3 3√

VIRGO one for
p = 1,
= 3.38, λ = 2; finally, for LISA, we have p = 2,
= 3 and 0.944 < λ <
0.966. Besides, considering LISA in the regime R >> Rc , we have λ = 2 and p = 1.5.
Finally, let us consider Models VI and VII. We have
b2 α tanh 12 b(R − R0 )
h
i,
ΦV I =
(6.103)
bα
[bα + cosh(b(R − R0 )) + 1] ln cosh(b(R−R
0 ))+1
154
6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT
and
ΦV II = log 0.5 bαsech2 (0.5b(R − R0 )) − bI αI sech2 (0.5bI (R − RI )) + 2
× bαsech2 (0.5b(R − R0 )) − bI αI sech2 (0.5bI (R − RI )) + 4
× b2 αsech2 (0.5b(R − R0 )) tanh(0.5b(R − R0 )) − b2I αI sech2 (0.5bI (R − RI )) tanh(0.5bI (R − RI )) .
(6.104)
These equations satisfy the constraints for VIRGO, LIGO and LISA for b = 2, bI =
0.5, α = 1.5 and αI = 2 with RI valued at Solar System scale and R0 at cosmological
scale.
6.5. STOCHASTIC BACKGROUNDS OF GRAVITATIONAL WAVES TO CONSTRAIN F (R)-GRAVITY
155
I model
II model
III model
IV model
f(R)=R
1
0.8
f(R)/Rc
0.6
0.4
0.2
0
−0.2
0
0.5
1
1.5
R/Rc
Figure 6.1: Plots of four different F (R) models as function of RRc . Model I in Eq. (6.9)
with n = 1 and λ = 2 (dashed line). Model II in Eq.(6.11) with p = 2, λ = 0.95 (dashdot
line). Model III in Eq.(6.12) with s = 0.5 and λ = 1.5 (dotted). Model IV in Eq.(6.13)
with q = 0.5 and λ = 0.5 (solid line). We also plot F (R) = R (solid thick line) to see
whether or not the stability condition F,R > 0 is violated.
156
6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT
1.5
V model
f’(R) of V model
f(R)
1
0.5
R=0.64
R=1
0
−0.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
R
Figure 6.2: Plots of Model V (6.38) (solid line) and its first derivative (dashed line).
Here n = 2 and α, β, γ are assumed as in (6.42) with the value of R0 taken in the Solar
System. f ′ (R) is negative for 0 < R < 0.64. f (R) is given in the range 0 < R < 1 where
we have adopted suitable units.
6.5. STOCHASTIC BACKGROUNDS OF GRAVITATIONAL WAVES TO CONSTRAIN F (R)-GRAVITY
157
1.6
VI model
VII model
1.4
1.2
f(R)
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
R
Figure 6.3: Plots of Model VI (6.45) (solid line) and Model VII (6.51) (dashed line).
Here b = 2 and bI = 0.5 with α = 1.5 and αI = 2. The value of RI is taken in the Solar
System while R0 corresponds to the present cosmological value.
158
6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT
1.2
f’(R) of I model
f’(R) of II model
f’(R) of III model
f’(R) of IV model
1
0.8
f’(R)/R c
0.6
0.4
0.2
x=0.06
x=0.293
x=1
0
−0.2
−0.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
R/Rc=x
Figure 6.4: Plots of the first derivatives of four different models as function of x = RRc .
Model I (dashed) is drawn for n = 1 and λ = 2. Model II (dashdot), for p = 2, λ = 0.95.
Model III (dotted), for s = 0.5 and λ = 1.5. Model IV (solid) is Ω143.280TdΩ[(n)-4.76908]TJΩ/R4610.9091
160
6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT
f’(R) of VI model
f’(R) of VII model
2
f’(R
1.5
1
0.5
0
0
1
2
3
R
4
5
Figure 6.6: Plots represent the first derivatives of functions (6.50) (solid line) and (6.51)
(dashed line). Here, b = 2, bI = 0.5, α = 1.5 and αI = 2 with RI with the Solar System
value and R0 the today cosmological value. It is F, R > 0 for R > 0.
6
6.5. STOCHASTIC BACKGROUNDS OF GRAVITATIONAL WAVES TO CONSTRAIN F (R)-GRAVITY
161
1.5
f’’(R) of VI model
f’’(R) of VII model
1
0.5
f’’(R)
R=2.35
0
R=4
−0.5
−1
0
1
2
3
4
5
R
Figure 6.7: Second derivatives of Model VI (solid line) and VII (dashed line). Here F,RR
is negative in the range 0 < R < 4 for Model VI and in the range 0 < R < 2.35 for
Model VII. As above, we have used b = 2, bI = 0.5, α = 1.5 and αI = 2 with the value
of RI taken in the Solar System and R0 for the today cosmological value.
6
162
6. FURTHER PROBE: PARAMETRIZED POST NEWTONIAN LIMIT
Chapter 7
Future perspectives and
conclusions
7.1
Brief summary
Before concluding this thesis or discussing future perspectives of the work presented
here, let us attempt to summarize in this section some of the results presented so far.
The motivation of this thesis has been thoroughly discussed in Chapter 1 and a general
discussion about modifications of gravity was laid out in Chapter 2. In Chapter ??,
a compact, self-contained approach to gravitation, based on the local Poincare gauge
invariance, is proposed and we have shown as generate the gravitation by a simple
simmetry break. We discussed the Invariance Principle as our starting point in order
to derive gravity as a local Poincaré gauge theory. Global Poincaré invariance and the
local one are discussed, showing how a local transformation is related to the gauge
fields. Spinors, vectors and tetrads which transform under Lorentz transformations are
discussed. In particular we discuss the Fock - Ivanenko connection in the framework
of the local Poincaré transformations. Starting from the Fock - Ivanenko covariant
derivative, curvature, torsion and metric tensors are derived after field equations for
gravity are discussed in the framework of the present approach. In Chapter 4 in we
defined the space-time perturbations in the framework of the metric formalism giving
the notion of first and second deformation matrices. we do the main properties of
deformations. In particular, we discuss how deformation matrices can be split in their
trace, traceless and skew parts. We derive the contributions of deformation to the
geodesic equation and, starting from the curvature Riemann tensor, the general equation
of deformations. In We discuss the notion of linear perturbations under the standard
of deformations. In particular, we recast the equation of gravitational waves and the
transverse traceless gauge under the standard of deformations and discussed the action
of deformations on the Killing vectors. The result consists in achieving a notion of
approximate symmetry. Finally in the Chapters 5 and 6 focused on the cosmological
163
164
7. FUTURE PERSPECTIVES AND CONCLUSIONS
and astrophysical aspects of these theories and on their viability. Infact we wanted to
face the problem of how the GW stochastic background and f (R) gravity can be related
showing, vice-versa, that a revealed stochastic GW signal could be a powerful probe for
a given effective theory of gravity. Our goal was to show that the conformal treatment
of GWs can be used to parameterize in a natural way f (R) theories.we review the field
equations of f (R) gravity in the metric approach and their scalar-tensor representation,
useful to compare the theory with observations. We review and discuss some viable
f (R) models capable of satisfying both local gravity prescriptions as well as the observed
cosmological behavior. In particular, we discuss their stability conditions and the field
values which have to achieved to fulfill physical bounds. We derived the values of model
parameters in agreement with the PPN experimental constraints while, we deal with the
constraints coming from the stochastic background of GWs. These latter ones have to
be confronted with those coming from PPN parameterization. As a general remark, we
find out that bounds coming from the interferometric ground-based (VIRGO, LIGO) and
space (LISA) experiments could constitute a further probe for f (R) gravity if matched
with bounds at other scales.
As mentioned in the Introduction, these theories were introduced as tools that could
help us to examine how much and in which ways one can deviate from General Relativity.
Our intention was neither to tailor a model within the framework of extended theories
that would fit the data adequately nor to pick out a specific well-motivated low-energy
effective action from some fundamental theory and to confront it with observations.
The task which we undertook was to consider theories that were easy to handle, each of
them deviating from the framework of General Relativity , and to exploit them in order
to get a deeper understanding of the difficulties and limitations of modified gravity. In
the light of this, it is probably preferable to provide here a qualitative summary of our
results which summarizes the lessons learned from this procedure, instead of repeating
in detail the results already presented in the previous chapters.
Starting from the theoretical side we have shown that all the necessary ingredients for
a theory of gravitation can be obtained from a gauge theory of local Poincaré symmetry.
Gauge fields have been obtained by requiring the invariance of the Lagrangian density
under local Poincaré transformations.
The resulting Einstein-Cartan theory describes a space endowed with non-vanishing
curvature and torsion. The lowest order gravitational action is one that is linear in
the curvature scalar while being quadratic in torsion. However, the scheme can be
immediately extended to more general gravitational theories as in [246].
The Dirac spinors can be introduced as matter sources and it has been found that
they couple to gravity via the torsion stress form Tµν component of the total energymomentum Σµν . The field equations obtained from the action by means of a standard variational principle describe a nonlinear equation of the Heisenberg-Pauli type
in the matter sector, gravitational field equations similar to the Einstein equations
and a constraint equation relating torsion to spin energy potential. The generalized
energy-momentum tensor is comprised of the usual canonical energy-momentum tensor
7.1. BRIEF SUMMARY
165
of matter in addition to a torsion stress form. The stress form contains a torsion divergence term as well as a term similar to an external non-spinor source to gravity. In
view of the structure of the generalized energy-momentum tensor, we remark that the
gravitational field equations here obtained are similar to the equations of motion found
in Einstein-Yang-Mills theory, the torsion tensor playing the role of the Yang-Mills field
strength.
The Bianchi identities of Einstein-Cartan gravity differ from those of General Relativity since the Riemann curvature tensor characterizing the non-Riemannian geometry
does not exhibit the usual symmetry properties. In the limit of vanishing torsion, the
Bianchi identities reduce to their usual form. The conservation laws for the angular momentum and the energy-momentum has been obtained. From the former, it has
been found that the generalized energy-momentum tensor contains a non-vanishing antisymmetric component proportional to the divergence of the spin-energy potential. For
the latter, we found that the generalized energy-momentum tensor is only divergenceless
in the limit of vanishing torsion.
We have proposed a novel definition of space-time metric deformations parameterizing them in terms of scalar field matrices. The main result is that deformations can
be described as extended conformal transformations. This fact gives a straightforward
physical interpretation of conformal transformations: conformally related metrics can
be seen as the ”background” and the ”perturbed” metrics. In other words, the relations
between the Jordan frame and the Einstein frame can be directly interpreted through
the action of the deformation matrices contributing to solve the issue of what the true
physical frame is [347, ?].
Besides, space-time metric deformations can be immediately recast in terms of perturbation theory allowing a completely covariant approach to the problem of gravitational waves.
The discussion about the cosmological and astrophysical aspects of the theories examined here and the confrontation of the theories with cosmological, astrophysical and
Solar System observations hopefully clarified that it is very difficult to construct a simple
viable model in an alternative theory of gravity.
In summary, we have shown that amplitudes of tensor GWs are conformally invariant
and their evolution depends on the cosmological background. Such a background is
tuned by conformal scalar field which is not present in the standard general relativity.
Assuming that primordial vacuum fluctuations produce stochastic GWS, beside scalar
perturbations, kinematical distortions and so on, the initial amplitude of these ones is a
function of the f (R)-theory of gravity and then the stochastic background can be, in a
certain sense “tuned” by the theory. Vice versa, data coming from the Sachs-Wolfe effect
could contribute to select a suitable f (R) theory which can be consistently matched with
other observations. However, further and accurate studies are needed in order to test
the relation between Sachs-Wolfe effect and f (R) gravity. This goal could be achieved
very soon through the forthcoming space (LISA) and ground-based (VIRGO, LIGO)
interferometers.
166
7. FUTURE PERSPECTIVES AND CONCLUSIONS
We have investigated the possibility that some viable f (R) models could be constrained considering both Solar System experiments and upper bounds on the stochastic
background of gravitational radiation. Such bounds come from interferometric groundbased (VIRGO and LIGO) and space (LISA) experiments. The underlying philosophy is
to show that the f (R) approach, in order to describe consistently the observed universe,
should be tested at very different scales, that is at very different redshifts. In other
words, such a proposal could partially contribute to remove the unpleasant degeneracy
affecting the wide class of dark energy models, today on the ground.
Beside the request to evade the Solar System tests, new methods have been recently
proposed to investigate the evolution and the power spectrum of cosmological perturbations in f (R) models [363]. The investigation of stochastic background, in particular
of the scalar component of GWs coming from the f (R) additional degrees of freedom,
could acquire, if revealed by the running and forthcoming experiments, a fundamental
importance to discriminate among the various gravity theories [401]. These data (today only upper bounds coming from simulations) if combined with Solar System tests,
CMBR anisotropies, LSS, etc. could greatly help to achieve a self-consistent cosmology
bypassing the shortcomings of ΛCDM model.
Specifically, we have taken into account some broken power law f (R) models fulfilling
the main cosmological requirements which are to match the today observed accelerated
expansion and the correct behavior in early epochs. In principle, the adopted parameterization allows to fit data at extragalactic and cosmological scales [355]. Furthermore,
such models are constructed to evade the Solar System experimental tests. Beside these
broken power laws, we have considered also two models capable of reproducing the effective cosmological constant, the early inflation and the late acceleration epochs [359].
These f (R)-functions are combinations of hyperbolic tangents.
We have discussed the behavior of all the considered models. In particular, the
problem of stability has been addressed determining suitable and physically consistent
ranges of parameters. Then we have taken into account the results of the main Solar
System current experiments. Such results give upper limits on the PPN parameters
which any self-consistent theory of gravity should satisfy at local scales. Starting from
these, we have selected the f (R) parameters fulfilling the tests. As a general remark,
all the functional forms chosen for f (R) present sets of parameters capable of matching
the two main PPN quantities, that is γexp and βexp . This means that, in principle,
extensions of GR are not a priori excluded as reasonable candidates for gravity theories.
The interesting feature, and the main result of this thesis, is that such sets of parameters are not in conflict with bounds coming from the cosmological stochastic background
of GWs. In particular, some sets of parameters reproduce quite well both the PPN upper
limits and the constraints on the scalar component amplitude of GWs.
Far to be definitive, these preliminary results indicate that self-consistent models
could be achieved comparing experimental data at very different scales without extrapolating results obtained only at a given scale.
7.2. CONCLUDING REMARKS
7.2
167
Concluding remarks
To conclude, even though some significant progress has been made with developing
alternative gravitation theories, one cannot help but notice that it is still unclear how to
relate principles and experiments in practice, in order to form simple theoretical viability
criteria which are expressed mathematically. Our inability to express these criteria and
also several of our very basic definitions in a representation-invariant way seems to have
played a crucial role in the lack of development of a theory of gravitation theories. This
is a critical obstacle to overcome if we want to go beyond a trial-and-error approach in
developing alternative gravitation theories.
It is the author’s opinion that such an approach should be one of the main future
goals in the field of modified gravity. This is not to say, of course, that efforts to propose
or use individual theories, such as f (R) gravity, in order to deepen our understanding
about the gravitational interaction should be abandoned or have less attention paid to
them. Such theories have proved to be excellent tools for this cause so far, and there
are still a lot of unexplored corners of the theories mentioned in this thesis, as well as
in other alternative theories of gravity.
The motivation for modified gravity coming from High Energy Physics, Cosmology
and Astrophysics is definitely strong. Even though modifying gravity might not be
the only way to address the problems mentioned in Chapter 1, it is our hope that the
reader is by now convinced that it should at least be considered very seriously as one
of the possible solutions and, therefore, given appropriate attention. The path to the
final answer is probably long. However, this has never been a good enough reason for
scientists to be discouraged.
168
7. FUTURE PERSPECTIVES AND CONCLUSIONS
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