Download f(R) Gravity and Cosmology - Workspace

I MPERIAL C OLLEGE L ONDON
Q UANTUM F IELDS AND F UNDAMENTAL F ORCES
MS C D ISSERTATION
f (R) Gravity and Cosmology
Author:
Christopher G ALLAGHER
Supervisor:
Dr. Ali M OZAFFARI
A thesis submitted in fulfillment of the requirements
for the degree of Master of Science
in the
Theoretical Physics Group
Physics Department
September 16, 2016
i
IMPERIAL COLLEGE LONDON
Abstract
Faculty of Natural Sciences
Physics Department
Master of Science
f (R) Gravity and Cosmology
by Christopher G ALLAGHER
This dissertation is intended to give an introduction to alternatives to
√
General Relativity based on Lagrangians of the form L = −g f (R). First,
an introduction to the Lagrangian formulation of General Relativity is given,
with a short exposition on the Gibbons-Hawking-York boundary term. This
provides an introduction to the techniques used in f (R) gravity theories.
Both metric f (R) theories and Palatini f (R) theories are investigated in
some detail, whereas metric-affine theories are discussed in less detail, since
there is proportionally less literature devoted to them. The equivalence of
these theories to specific versions of Brans-Dicke theory are discussed, and
the issue of the existence of well-posed initial value formulations for these
theories is also discussed. Finally, the application of these modified theories of gravity to the field of cosmology is discussed briefly, where they can
provide potential alternative explanations for the phenomena of late-time
accelerated cosmic expansion.
ii
Acknowledgements
I would like to thank Dr Mozaffari for guiding me through this fascinating project, my parents for making this year possible and finally, I would
like to acknowledge the help and support that my lovely girlfriend Felicity
Hall has provided through some of the tougher times this year.
iii
Contents
Abstract
i
Acknowledgements
ii
1
1
1
1
2
2
Introduction: General Relativity and ΛCDM Cosmology
1.1 Introduction: General Relativity . . . . . . . . . . . . . . . . .
1.1.1 Theoretical Underpinnings . . . . . . . . . . . . . . .
1.2 Introduction: Physical Cosmology . . . . . . . . . . . . . . .
1.2.1 Theoretical Underpinnings: The FRW spacetime, the
Friedmann equation . . . . . . . . . . . . . . . . . . .
1.2.2 The Mass-Energy Content of Our Universe . . . . . .
1.2.3 The ΛCDM Model . . . . . . . . . . . . . . . . . . . .
1.2.4 Problems with Big Bang Cosmology - The Need for
Inflation . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Alternative Solutions to Dark Matter and Dark Energy . . .
Gravity from an Action
2.1 The Einstein-Hilbert Action . . . . . . . . . . . . . . . . .
2.2 The Gibbons-Hawking-York Boundary Term . . . . . . .
2.2.1 The Non-Dynamical Term . . . . . . . . . . . . . .
2.3 Palatini Variation . . . . . . . . . . . . . . . . . . . . . . .
2.4 ADM: An Initial Value Formulation of General Relativity
.
.
.
.
.
.
.
.
.
.
4
5
6
7
8
9
9
12
16
17
18
3
Introduction to f (R) Gravity
24
3.1 Motivation: Modified Gravity and f (R) Theories . . . . . . .
24
3.2 Metric f (R) Gravity . . . . . . . . . . . . . . . . . . . . . . . .
25
3.2.1 Example: Starobinsky Inflation . . . . . . . . . . . . .
27
3.3 Legendre Transformations and the connection to Scalar-Tensor
Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.4 Conformal Transformations: f (R) Gravity in the Einstein Frame 30
3.5 Initial Value Formulation of Metric f (R) theories . . . . . . .
30
4
Palatini f (R) Gravity:
Different Approaches
4.1 Introduction to Palatini f (R) Theories . . . . . . . . . . . . .
4.2 The Role of Torsion in Palatini f (R) theories . . . . . . . . .
4.3 Field Equations for Palatini f (R) Theories . . . . . . . . . . .
4.4 The Relation between Palatini f (R) Theories and Brans-Dicke
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Initial Value Formulations of Palatini f (R) Theories . . . . .
4.6 Metric-Affine f (R) Gravity . . . . . . . . . . . . . . . . . . .
35
35
38
45
49
49
51
iv
5
6
Cosmological Solutions in f (R) Gravity
5.1 FRLW and Metric f (R) Gravity . . . . . . . . . . .
5.2 Accelerating Cosmologies in Metric f (R) Gravity:
sky Inflation . . . . . . . . . . . . . . . . . . . . . .
5.3 Late-time Acceleration from Metric f (R) Gravity .
5.4 Late-time Acceleration from Palatini f (R) Gravity
Discussion and Conclusions
Bibliography
. . . . . .
Starobin. . . . . .
. . . . . .
. . . . . .
53
53
57
59
61
63
64
1
Chapter 1
Introduction: General
Relativity and ΛCDM
Cosmology
1.1
Introduction: General Relativity
Einstein published his theory of General Relativity in 1915, adapting the
special relativity of local inertial reference frames to more general accelerated reference frames, and at the same time, recasting gravitational fields
as being nothing more than the fictitious forces felt by observers at rest in
these accelerated reference frames[1]. The theory posited that what had
previously been considered to be a "gravitational force field" that propagated at infinite speed (a concept sitting most uncomfortably with the notion that nothing physical can travel faster than the speed of light) was now
to be understood as resulting from the curvature of a 4-dimensional spacetime manifold, upon which unaccelerated observers moved on geodesics[2].
The theory perfectly explained the experimental curio of the precession of
the perihelion of Mercury that had baffled astronomers for decades[3], and
further predicted a number of striking new phenomena, including gravitational radiation, gravitational lensing and gravitational redshift[4][5][6].
The theory was initially met with a somewhat lukewarm reception within
the scientific community, with many feeling that the theory was unnecessarily complicated, and that its predictions were unmeasurable[7]. However Sir Arthur Eddington’s (now considered to be inconclusive) claim to
have measured gravitational deflection of light by the Sun during the Solar
eclipse of 1919, catapulted Einstein and his theory into the public imagination[6]. Since then, every theoretical prediction of General Relativity has
been experimentally verified[8] (most recently by the LIGO experiment that
detected gravitational waves[9]), and General Relativity has been considered to be one of the pinnacles of human scientific achievement[10].
1.1.1
Theoretical Underpinnings
General Relativity posits that the structure of the world that we live in can
be understood using the language of pseudo-Riemannian geometry[2][11], a
branch of the mathematical field of Differential Geometry. More specifically, Einstein’s theory asserts that spacetime is a 4-dimensional Lorentzian
manifold equipped with the Levi-Civita connection, with a symmetric metric tensor, whose components are determined dynamically by the Einstein
Chapter 1. Introduction: General Relativity and ΛCDM Cosmology
2
Field Equations[2][11][12]:
Gµν =
8πGN
Tµν ,
c4
(1.1)
where
1
Gµν = Rµν − gµν R
2
(1.2)
is the symmetric Einstein tensor, composed of the Ricci scalar, R, and the
symmetric Ricci tensor Rµν , which are constructed by taking partial derivatives of the metric[2]. Thus, Einstein’s equation is a symmetric rank-2 tensor
equation, comprising a system of ten coupled partial differential equations
for the components of the metric tensor. The equations are non-linear and
are extremely difficult to solve in practice; however some exact solutions
are known, and numerical techniques are now sufficiently advanced to be
of use in practical modelling[13].
Unaccelerated objects in this spacetime follow timelike geodesic trajectories that can be determined using the geodesic equation[2]:
ẍµ + Γµρσ ẋρ ẋσ = 0,
(1.3)
where xµ represents the coordinates of the object in question, and ẋµ etc.
are derivatives taken with respect to the time coordinate.
µ dxν
Light rays travel along null geodesics: i.e. geodesics that satisfy dx
dλ dλ gµν =
0, for some affine parameter λ. In this case, the derivatives in the geodesic
equation must be taken with respect to the affine parameter, rather than the
time coordinate, yielding a geodesic equation:
dxµ
dxρ dxσ
+ Γµρσ
= 0.
dλ
dλ dλ
(1.4)
Using these theoretical postulates and the mathematics of pseudo-Riemannian
geometry, one can extract the physical predictions of General Relativity[12].
1.2
Introduction: Physical Cosmology
Modern theoretical cosmology begins with General Relativity[14]. Initially,
Einstein believed in a static universe, and sought to show that a static spacetime geometry could solve his field equations, and thus provided a model
for the geometry and history of the entire universe[14]. He was unsuccessful - in its original formulation, General Relativity does not permit the
existence of a static universe filled with matter, since the matter gravitationally attracts over time[15]. Famously however, in what he later described as
his "greatest blunder" Einstein realised that the field equations of General
Relativity permitted the introduction of a so-called "cosmological constant",
that could counteract the attractive force of gravity on a cosmic scale, thus
allowing the existence of a static universe[16]. The Einstein equation with
the cosmological constant included is:
Gµν + Λgµν =
8πGN
Tµν ,
c4
(1.5)
Chapter 1. Introduction: General Relativity and ΛCDM Cosmology
3
where Λ is the cosmological constant.
Einstein first published a model based on this field equation in 1917; it
is now known as the "Einstein universe", and was the first relativistic cosmological model. The model itself however was unattractive theoretically,
since it proved to be unstable to small perturbations, and would eventually begin to either expand or contract when subjected to such perturbations[17].
Friedmann was the first to find the full set of cosmological solutions
to General Relativity; assuming the "Cosmological Principle", namely that
space was homogenous and isotropic, he found the Friedmann-RoberstonWalker (FRW) spacetime, and derived the Friedmann equation that governed the evolution of such spacetimes, which may contract or expand, and
whose geometry may be either closed (like a sphere, with positive curvature), open (like a hyperboloid, with negative curvature) or flat (with no
curvature)[18]. However it was still believed that the Universe was static
at this time, and so it was not until observational support for an expanding
universe materialised that the subject progressed.
Parallel in development to the theoretical advances in cosmology, were
the groundbreaking observations made at the beginning of the 20th century. In 1929, by determining the distance to far-away galaxies using measurements of the brightness Cepheid variable stars, Hubble was able to
show that the "spiral nebulae" measured by previous astronomers such as
Vesto Slipher, were in fact new galaxies, outside of the Milky Way[19]. By
comparing the redshift of these galaxies to their distance from Earth, Hubble was able to show the existence of a relationship between these quantities: the further away a galaxy was, the faster it seemed to be receding
from Earth[19]. These revolutionary measurements provided the observational basis needed for an expanding universe. In 1927, Lemaître had rederived the Friedmann equations, and constructed a theory in which the
universe began with a so-called "Big Bang" and had continued to expand
ever since[20].
Two competing cosmological models proceeded to develop, with both
models garnering support from different groups of cosmologists; the first of
these was the Steady-State theory of Fred Hoyle (who ironically coined the
originally derogatory "Big Bang" moniker for the opposing theory), which
proposed that new matter was generated as galaxies receded from each
other, and the Big Bang theory of Lemaître, which proposed that the universe had a beginning. The discovery of the Cosmic Microwave Background (CMB) by Penzias and Wilson in 1965 provided observational support for the Big Bang model however, and as the precision of CMB measurements has increased, a scientific consensus has been reached in favour
of the Big Bang theory[20].
Chapter 1. Introduction: General Relativity and ΛCDM Cosmology
1.2.1
4
Theoretical Underpinnings: The FRW spacetime, the Friedmann equation
The "Cosmological Principle" refers to the hypothesis that the universe is
approximately spatially homogenous and isotropic on the largest scales.
One can derive the most general metric that satisfies these criteria; this is
the FRW metric[21]:
dr2
2
2
2
2
2
2
2
2
(1.6)
ds = −dt + a (t)
+ r dθ + r sin θ dφ ,
1 − kr2
where we have written the metric in standard spherical polar coordinates,
a(t) is the time varying scale factor of the universe, and k is the Gaussian
curvature of space. This unit convention implies that ds is a measure of socalled "comoving distance". There are three possibilities for the value of
k:
• k > 0 : The universe has positive spatial curvature and is closed.
• k = 0 : The universe is flat and open.
• k < 0 : The universe has negative spatial curvature and is open.
Recent measurements lend support to a flat universe[22], so we will focus
on the case where k = 0 for the remainder of this report.
To determine the class of functions a(t) for which this metric is a solution of the Einstein equation, one calculates each component of the Einstein
equation. Friedmann found that the resulting constraint equations on a(t),
referred to as the Friedmann equations, in the case of a universe filled with
a perfect fluid of density ρ and pressure P . In such a case, the stress-energy
tensor takes the form


−ρ 0 0 0
 0 P 0 0

T=
(1.7)
 0 0 P 0.
0 0 0 P
By calculating the Einstein equations directly, Friedmann derived the equations bearing his name, which are:
ȧ2
8πGN ρ
k
Λ
=
− 2+
a2
3
a
3
ä
4πGN
Λ
=−
(ρ + 3P ) + .
a
3
3
(1.8)
(1.9)
These ordinary differential equations govern the evolution of the scale factor, a(t), with time, and hence provide a model for a dynamic universe
satisfying the Cosmological Principle. We can use these two equations to
derive a third equation, called the fluid equation, that is not independent of
the previous two, but is more convenient to use. The fluid equation is
ρ̇ +
3ȧ
(ρ + P ) = 0.
a
(1.10)
Chapter 1. Introduction: General Relativity and ΛCDM Cosmology
5
There is one further ingredient necessary to specify a cosmology: one
must provide an equation of state for the matter/energy that comprises ρ.
An equation of state is of the form:
P = P (ρ),
(1.11)
where P (ρ) is some function that specifies the pressure dependancy of matter. For example in cosmology, for a pressure-less perfect fluid, we simply
take the equation of state P = 0. One then solves the fluid equation for the
density as a function of the scale factor, and then solves the first Friedmann
equation, to obtain the scale factor as a function of time. This specifies a
cosmological model. More generally, for a perfect fluid one writes
P = wρ,
(1.12)
where w is a dimensionless number that characterises the equation of state
of the matter under consideration. From thermodynamic considerations,
one can show that w = 0 corresponds to the case of non-relativistic matter
and w = 1/3 corresponds to relativistic matter.
1.2.2
The Mass-Energy Content of Our Universe
It is an observational fact that our universe contains more matter than meets
the eye. A large raft of observational evidence for the existence of dark matter has accumulate over the past century, including but not limited to, the
mismatch between measured spiral galaxy rotation curves and the mass
profiling from luminosity densities, measurements of the velocity dispersions in elliptical galaxies, gravitational lensing of galaxy clusters, and cosmic microwave background measurements[23]. However, as of yet, particle
physics is yet to provide a candidate dark matter particle whose characteristics adequately fit the data[24].
We can categorise potential dark matter candidates by their "free streaming length", i.e how far particles moved in their thermal random walk in the
hot dense state of the early universe, before cosmic expansion reduced temperatures and slowed the particles down[25]. The free streaming length of
the matter in the universe is known to have important consequences for
the formation of structure via density perturbations at later times in the
universe - suggesting that we can use the statistics and characteristics of
density perturbations to probe the nature of dark matter[26]. In particular,
primordial density fluctuations smaller than the free streaming length are
smoothed as particles spread from overdense to underdense regions, while
density fluctuations larger than the free streaming length are unaffected.
Dark matter candidate particles are classified as:
• "cold dark matter" :
of a protogalaxy
The free streaming length is smaller than that
• "warm dark matter" : The free streaming length is approximately
the same as that of a protogalaxy
Chapter 1. Introduction: General Relativity and ΛCDM Cosmology
• "hot dark matter" :
a protogalaxy.
6
The free streaming length is larger than that of
Current measurements of structure formation strongly favour a universe that is predominantly composed of cold dark matter[15][26].
In 1999, Perlmutter, Reiss and Schmidt published data suggesting the
the universe is in fact undergoing a period of accelerated expansion[27]. This
was in conflict with expectations; most cosmologists had assumed that the
universe was likely to be undergoing a period of decelerated expansion,
based on cosmological models. The simplest explanation for accelerated
expansion was to include a positive cosmological constant in the Einstein
field equation, however the nature of the physics motivating such an inclusion is obscure, to say the least[28]. In fact, if one tries to model accelerated expansion in the traditional way using the Friedmann equations given
above, one is forced to arrive at the conclusion that the "dark energy" that
would be fuelling the accelerated expansion would be required to have negative pressure[26]! Indeed one can show that to have accelerated expansion,
one requires some type of exotic matter or energy for which w < −1/3. The
"cosmological constant explanation" leads to an equation of state with parameter w = −1 that satisfies this property, however in truth, understanding the physics behind "dark energy" remains as one of the great challenges
for physics in the 21st century.
1.2.3
The ΛCDM Model
The ΛCDM model is based on the FRW metric and Friedmann equations,
and equations of state given above[15]. In particular, we assume that dark
energy is explained by a cosmological constant, and that the dark matter
in the universe is cold. It is possible to rewrite the Friedmann equation in
terms of density parameters[15][26], Ωx , as
q
ȧ
H(a) ≡ = H0 (Ωc + Ωb )a−3 + Ωrad a−4 + Ωk a−2 + ΩDE a−3(1+w) ,
a
(1.13)
where Ωc is the fraction of matter comprised of cold dark matter, Ωb is
the fraction comprised of ordinary "baryonic" matter, Ωrad is the fraction
comprised of radiation (or relativistic matter), ΩDE is the fraction associated with dark energy, and Ωk is the energy density associated with the
curvature of the universe (one can consider the curvature term to provide
an effective energy density on the right hand side of the Friedmann equation)[21].
One can integrate this expression to obtain the cosmic expansion history of the universe in terms of the density parameters. The model that
is best-fit to the observational data constraining these densities is called the
"Minimal ΛCDM" model, which has six independent paramaters[15]: physical baryon density parameter; physical dark matter density parameter; the
age of the universe; scalar spectral index; curvature fluctuation amplitude;
Chapter 1. Introduction: General Relativity and ΛCDM Cosmology
7
and reionization optical depth. Within this framework, the currently accepted values for the percentages of baryonic matter, dark matter and dark
energy are[29]:
• Baryonic Matter : Ωb = 0.0486 ± 0.0010
• Cold Dark Matter : Ωc = 0.2589 ± 0.0057
• Dark Energy : ΩDE = 0.6911 ± 0.0062
1.2.4
Problems with Big Bang Cosmology - The Need for Inflation
Despite its incredible successes, several problems were highlighted in Big
Bang cosmology in the 1970s. The horizon problem arises from the fact that
it observationally appears to us that different areas of the night sky are in
thermal equilibrium with each other. Naively, this might be seen to be expected, considering the Cosmological Principle demands homogeneity and
isotropy; however we quickly run into problems when we consider that in
the Big Bang model, two patches of space that are far enough separated
such that a light signal could not be sent from one to the other after photon
decoupling could never have been in causal contact, and so could never
have interacted to come into thermal equilibrium. It is therefore extremely
surprising that different areas of the Universe are indeed in thermal equilibrium[30].
The flatness problem refers to the fine-tuning issue that arises when one
considers that the density of the Universe is extremely close to that of a
critical density universe. This implies that the Universe is very close to, or
indeed is, flat. The problem arises since universes that aren’t flat are unstable in the long term - density evolution with time is such that any densities
that are not exactly equal to the critical density diverge away from the critical density rapidly with time. Thus, if the universe is not flat, but very close
to being flat, it would have had to have been even closer to being flat in the
past than it is now. One can extrapolate this back to the early universe, and
calculate that the density would have to have differed from the critical density by around one part in 10−69 . This suggests arbitrary fine tuning, and is
unsatisfactorily explained[30].
Inflation was first proposed by Guth as a solution to the monopole problem - if Grand Unified Theories predict magnetic monopoles at high energies, why are no monopoles found in nature? His idea was that a period of
exponential expansion would "dilute" away any monopoles, so their relic
density from the Big Bang would be negligible. Guth found that the solutions in General Relativity which corresponded to exponentially expanding
spacetimes could be realised in the case of a "false vacuum" in a scalar potential. It was subsequently realised that a period of exponential inflation
in the very early Universe would also solve both the horizon problem and
the flatness problem. Furthermore, a period of cosmic inflation in the early
Universe provides a very compelling mechanism for generating a universe
that displays homogeneity and isotropy on the largest scales, as density
perturbations tend to become "smoothed out" during the inflationary period[30].
Chapter 1. Introduction: General Relativity and ΛCDM Cosmology
1.3
8
Alternative Solutions to Dark Matter and Dark Energy
Despite its successes, the ΛCDM model is far from providing a complete description of cosmology. Since neither "dark matter" nor "dark energy" have
ever been successfully directly detected, it is reasonable to consider alternatives to ΛCDM[31]. For example, it has been suggested that both dark
matter and dark energy could in fact be nothing more than a signal that
General Relativity is in fact invalid on the largest scales. Since the inception
of General Relativity, a whole host of different gravitational theories have
been posited, many of which are yet to be excluded experimentally[31].
The suggestion that General Relativity should be replaced by such a theory
should be taken seriously if such a theory could explain either the dark matter or the dark energy problem without having to resort to the introduction
of as-of-yet undiscovered types of matter. It has also been suggested that
such modified theories of gravity might provide the key to understand the
mechanism behind an initial period of cosmic inflation[32].
In this report we will explore the alternative explanations for dark energy and inflation that exist within a specific type of modified gravity theory, called f (R) theory, which arises when one replaces the Ricci scalar
in the Einstein-Hilbert action formulation of General Relativity with some
scalar function of the Ricci scalar, resulting in the inclusion of higher order
invariants in the theory, and leading to field equations more complicated
than the Einstein equation.
9
Chapter 2
Gravity from an Action
2.1
The Einstein-Hilbert Action
True to the spirit of modern theoretical physics, General Relativity can be
formulated as an action based classical field theory of the metric tensor, using the famous Einstein-Hilbert Action [2][11].
S = SEH + SM atter
Z
√
1
SEH =
d4 x −gR
2κ
(2.1)
For clarification of notation, note that we shall represent the determinant of
the metric tensor by g, i.e. g = det(gµν ). This factor arises as a natural part
of the integration measure in the Riemannian geometry of manifolds[33].
κ = 8πGN c−4 is Einstein’s constant.
Action based formalisms are easy to understand and extend - to obtain the
field equations, just apply the variational principle, and to modify the theory, just play around with different Lagrangians! Let us now see how the
Einstein Field Equations can be derived from the action given above:
Firstly we shall vary with respect to the inverse metric tensor g µν , considering a small variation δg µν . One must be careful to make the distinction
between raised and lowered indices when dealing with variations of the
metric, since gµν g µν = 2 implies that δgµν g µν + gµν δg µν = 0, so lowering the
indices on the variation costs us a minus sign[2].
Now let us vary:
δSEH
Z
√
√
1
=
d4 x δ( −g)R + −gδR
2κ
Z
√
1
−1
d4 x √ δ(det(gµν ))R + −gδR.
=
2κ
2 −g
(2.2)
Recall that the Ricci scalar is obtained by contraction of the Ricci tensor
with the inverse metric[2]: R = Rµν g µν . We can now calculate a variation
of the Ricci scalar in terms of the inverse metric tensor, the Ricci tensor, and
variations of both objects.
Chapter 2. Gravity from an Action
δR = δRµν g µν + Rµν δg µν .
10
(2.3)
Therefore:
δSEH
1
=
2κ
Z
√
−1
d4 x √ δ(det(gµν ))R + −g(δRµν g µν + Rµν δg µν ). (2.4)
2 −g
Derivatives of determinants of matrices can be calculated using Jacobi’s
formula[2]:
d
d
det(A) = Tr Adj A(t)
A(t) .
dt
dt
(2.5)
In this case we obtain δg = δ det(gµν ) = g g µν δgµν . Taking care to remember
the change in sign that arises from raising the indices, we obtain
√
Z
−g µν
1
4 √
µν
µν
d x −g(δRµν g + Rµν δg ) −
g δgµν R,
δSEH =
2κ
2
Z
√
√
1
1
=
d4 x −gδg µν (Rµν − gµν R) + −gg µν δRµν .
(2.6)
2κ
2
One can already see the left hand side of the Einstein equation beginning to emerge. However, the δRµν term must be dealt with appropriately.
Variations of the Ricci tensor can be calculated by considering variations of
the Riemann curvature tensor and then contracting appropriate indices[11].
Rρσµν = ∂ µ Γρ νσ − ∂ ν Γρ µσ + Γρ µα Γανσ − Γρ να Γαµσ
(2.7)
δRρσµν = ∂ µ δΓρ νσ − δΓρ να Γαµσ + δΓρ µα Γανσ
− ∂ ν δΓρ µσ + Γρ µα δΓανσ − Γρ να δΓαµσ
(2.8)
At this point, one can see that it is likely that the variation of the Riemann tensor could be rewritten in terms of covariant derivatives of variations of the connection. Fortunately, a variation in the connection is a difference between two connections!
Γρ µν + δΓρ µν = Γ̃ρ µν
It is, hence, a tensorial object (since two connections always differ by a
third rank tensor[2]). This allows the appropriate covariant derivatives to
be constructed, and a variation of the Riemann tensor to be expressed in
terms of those covariant derivatives.
∇λ (δΓρ µν ) = ∂ λ δΓρ µν + Γρ λα δΓαµν − Γαλµ δΓρ αν − Γαλν δΓρ αµ
(2.9)
Due to a nice cancellation of terms between covariant derivatives, we
obtain the Palatini Identity [34]:
Chapter 2. Gravity from an Action
11
δRρσµν = ∇µ (δΓρ σν ) − ∇ν (δΓρ µσ ).
(2.10)
Contracting indices, we therefore obtain the variation of the Ricci tensor:
δRµν = ∇ρ (δΓρ νµ ) − ∇ν (δΓρ µρ ).
The full variation of the Einstein-Hilbert action is therefore
Z
√
1
1
d4 x −gδg µν (Rµν − gµν R)
δSEH =
2κ
2
Z
√
1
+
d4 x −gg µν ∇ρ (δΓρ νµ ) − ∇ν (δΓρ µρ ) .
2κ
(2.11)
(2.12)
Shuffling indices appropriately and applying the well known identity (easily proven via the definition of the covariant derivative),
√
√
−g ∇µ Aµ = ∂µ ( −gAµ ),
we find that the second term can be written as the integral of a total derivative;
Z
√
1
d4 x −g∇σ g µν δΓσνµ − g µσ δΓν νµ
2κ
Z
√
1
(2.13)
=
d4 x ∂σ −g(g µν δΓσνµ − g µσ δΓν νµ ) .
2κ
As usual in field theory, total derivative terms can be discarded as long as
the endpoints of the variations are held fixed (δg µν (ti ) = δg µν (tf ) = 0), and
the metric is assumed to be asymptotically flat (i.e gµν (x) → 0 as x → ∞).
This leaves us with
δSEH =
1
2κ
Z
d4 x
√
1
−gδg µν (Rµν − gµν R).
2
(2.14)
Defining the matter action,
Z
δSM atter =
d4 x
√
−gLm
(2.15)
and taking the usual definition of the Hilbert stress-energy tensor[35],
√
2 δ( −gLM )
Tµν = − √
,
(2.16)
−g
δg µν
we demand that the variation of the total action is extremised, δS = 0, and
obtain the familiar Einstein Field Equations,
1
Rµν − gµν R = κTµν .
2
(2.17)
Chapter 2. Gravity from an Action
2.2
12
The Gibbons-Hawking-York Boundary Term
The astute reader, concerned with practically applying GR and creating
gravitational models, will have noticed that the previous derivation doesn’t
quite tell the full story. In modern models one often "glues" regions of
spacetime together; to do this one is obliged to consider manifolds with
boundaries[36]. Since ∂M 6= 0 in this case, the step taken above where the
total derivative was discarded is no longer valid. In the case of a closed
manifold (a compact manifold with no boundary), ∂M = 0, and one can
apply the generalised Stokes Theorem,1
Z
Z
ω,
(2.18)
dω =
M
∂M
in order to discard the unwanted term. In the more general case of a manifold with a boundary however, we must evaluate this non-zero term. In
order to ensure that the variational principle remains well defined[37], and
the Einstein equation is still recovered after applying the variational principle, it is necessary and beneficial to add a boundary term, SGHY to the
Einstein-Hilbert action[37],
SGravity = SEH + SGHY
I
√
1
SGHY =
d3 y hK.
κ ∂M
(2.19)
The following notation is in use in SGHY :
• y a (where a = 1, 2, 3) are the 3 coordinates on the boundary ∂M,
• hab is the induced metric on ∂M, h = det(hab ),
• = +1 if ∂M is a timelike hypersurface, = −1 if ∂M is a spacelike
hypersurface,
• K is the trace of the second fundamental form, II, which is defined on
the hypersurface as II(V, W ) = hn, ∇V W in. V and W are tangent vectors to the hypersurface, n is a normal vector of the hypersurface and
∇V is the covariant derivative on the ambient manifold (the manifold
within which the hypersurface is situated).
This is the famous Gibbons-Hawking-York boundary term (first conceived of by York[38], later modified by Gibbons and Hawking[39]) which
has recently found application in Loop Quantum Gravity[40], and has been
important in calculation of the entropy of black holes using semi-classical
approaches to quantum gravity (Euclidean Quantum Gravity)[41][42] (where
the term is necessary to ensure that the path integral has the right composition properties[37]).
We will
show that δSGHY cancels off the unwanted contribution
R now
4 √
1
from 2κ
d
x
−gg µν δRµν on ∂M, provided that variations in the metric
M
are subject to the condition δgµν |∂M = 0.
1
Which is in the author’s opinion, the most elegant formula in all of calculus.
Chapter 2. Gravity from an Action
13
R
4 √
1
µν
First we must evaluate 2κ
M d x −gg δRµν on ∂M. This is achieved
using the generalised Stokes theorem[12]. However we must first make
a small detour in order to define some notation and concepts that will be
useful. Let us first define a generic family of hypersurfaces, labelled by
values of C by the condition,
f (xµ ) = C,
(2.20)
for some function f . The vector field g µν ∂ν f is then normal to a member
of this family of hypersurfaces at any point. We can therefore define a unit
normal, nµ , to a particular hypersurface, Σ, such that;
−1 : if Σ is spacelike
µ
nµ n = =
+1 : if Σ is timelike
provided that Σ is not null (i.e. nµ nµ 6= 0) by,
∂µ f
nµ = p
.
|g ρσ (∂ρ f )(∂σ f )|
(2.21)
We can define three vectors, eµa (where a = 1, 2, 3) that are tangent to the
boundary ∂M,
eµa =
∂xµ ∂y a
∂M
.
(2.22)
Using these, we can pull back the metric tensor of the whole spacetime to
an induced metric on Σ:
hab = gµν eµa eνb .
(2.23)
Note that since eµ1 , eµ2 and eµ3 are tangential to ∂M, they must be orthogonal
to the normal vector field nµ , i.e nµ eµa = 0.
We also introduce the transverse metric;
hµν = gµν − nµ nν .
(2.24)
This allows us to project out any part of a 4-vector that is parallel to the
normal vector nµ . Hence, since all three eµa ’s are orthogonal to nµ , we can
write the induced metric in terms of the transverse metric,
hab = hµν eµa eνb .
(2.25)
Note further that the condition δgµν |∂M = 0 implies that the induced metric
on ∂M is held fixed during the variation, since hab = gµν eµa eνb .
We are now in a position to calculate the residual term in the variation
of the Einstein-Hilbert Action on ∂M:
Z
Z
√
1
1
4 √
µν
δSEH =
d x −gδg Gµν +
d4 x −gg µν δRµν .
(2.26)
2κ M
2κ M
Chapter 2. Gravity from an Action
14
Recall we found earlier that the second term could be written as a total
derivative;
Z
Z
√
1
1
4 √
µν
d x −gg δRµν =
d4 x −g∇σ g µν δΓσνµ − g µσ δΓν νµ
2κ M
2κ M
Z
√
1
=
d4 x ∂σ −g(g µν δΓσνµ − g µσ δΓν νµ ) .
2κ M
(2.27)
Now let us define g µν δΓσνµ − g µσ δΓν νµ = Y σ , so
Z
Z
√
1
1
4 √
µν
d x −gg δRµν =
d4 x ∂σ ( −gY σ ).
2κ M
2κ M
Now, evaluating this with Stokes’ Theorem, we obtain
Z
I
√
1
1
4
σ
d x ∂σ ( −gY ) =
dΣµ Y µ .
2κ M
2κ ∂M
p
Here dΣµ = nµ dΣ, dΣ = |h|d3 y and h = det(hab ), so
I
I
p
1
1
µ
dΣµ Y =
d3 y |h|nµ Y µ
2κ ∂M
2κ ∂M
(2.28)
(2.29)
(2.30)
We must now evaluate Y µ = g αβ δΓµαβ − g µα δΓααβ on the boundary ∂M,
recalling that δgµν |∂M = δg µν |∂M = 0, which allows us to discard terms
depending on the variation δgµν but not terms depending on derivatives of
this variation.
1
δΓµαβ |∂M = g µν (∂β δgνα + ∂α δgνβ − ∂ν δgαβ )
2
1
=⇒ Y µ = (∂ α δgνα g µν + g µν ∂β δgνβ − ∂µ δgαβ g αβ )
2
1 αµ ν
− (g ∂ δgνα + ∂ µ δgνβ g βν − ∂β δgαβ g µα )
2
=⇒ Y µ = g µν g αβ (∂α δgνβ − ∂ν δgαβ ).
(2.31)
We can now use the transverse metric defined earlier to evaluate Y µ nµ |∂M .
Y µ nµ |∂M = nµ g αβ (∂α δgµβ − ∂µ δgαβ )
= nµ (nα nβ + hαβ )(∂α δgµβ − ∂µ δgαβ ).
(2.32)
Now since
nµ (nα nβ ∂α δgµβ − nα nβ ∂µ δgαβ )
= nµ nα nβ ∂α δgµβ − nµ nα nβ ∂µ δgαβ
= 0,
we have
Y µ nµ |∂M = nµ hαβ (∂α δgµβ − ∂µ δgαβ ).
(2.33)
Since δgµν vanishes everywhere on ∂M, its tangential derivatives must also
vanish on ∂M, =⇒ ∂ρ δgµν eρa = 0.
Chapter 2. Gravity from an Action
15
This in turn implies that hµν ∂µ δgρν = hab eµa eνb ∂µ δgρν = 0.
We thus obtain
Y µ nν |∂M = −hνσ ∂µ δgνσ nµ ,
giving the final variation of the Einstein-Hilbert action as:
Z
I
p
1
1
4 √
µν
δSEH =
d x −g δg Gµν −
d3 y |h| hµν ∂σ (δgµν )nσ .
2κ M
2κ ∂M
(2.34)
The Gibbons-Hawking-York boundary term is chosen precisely so that its
variation cancels away the boundary contribution we have just calculated[37].
Let us now show that this is the case:
I
p
1
d3 y |h| δK.
δSGHY =
(2.35)
κ ∂M
Note that since the induced metric is kept fixed on the boundary ∂M, the
only object that is subjected to variation is K, the trace of the second fundamental form, or the extrinsic curvature.
K = ∇µ nµ = g µν ∇µ nν
= (nµ nν + hµν )∇µ nν
Combining the above expression with the identity ∇µ (nν nν ) = 0
nν ∇µ nν = 0 yields
K = hµν (∂µ nν − Γσµν nσ ).
(2.36)
=⇒
(2.37)
Using the fact that δnµ = 0 and that the metric is kept fixed on the boundary,
we obtain the variation of K:
δK = −hµν nρ gργ δΓγ µν nσ
1
= − hαβ nµ gµγ g γσ (∂β δgσα + ∂α δgσβ − ∂σ δgβα )
2
1 αβ µ
= − h n (∂β δgµα + ∂α δgµβ − ∂µ δgβα ).
2
(2.38)
(2.39)
Now since tangential derivatives of the variation of the metric must vanish
on ∂M, we have ∂β δgµα = 0 and ∂β δgµα = 0, leading to
1
δK = hαβ nµ ∂µ δgβα .
2
Therefore, the full variation of the Gibbons-Hawking-York term is
I
p
1
d3 y |h| hµν ∂σ (δgµν ),
δSGHY =
2κ ∂M
(2.40)
(2.41)
Chapter 2. Gravity from an Action
16
which, as claimed, perfectly cancels the boundary contribution from the
Einstein-Hilbert action, thus ensuring that
δS = δSGravity +δSM atter = δSEH + δSGHY + δSM atter = 0
1
=⇒ Rµν − gµν R = κTµν .
2
Thus for a closed manifold the appropriate action is of the form
Z
I
√
1
1
4 √
d x −gR +
d3 y hK.
S = SEH + SGHY =
2κ
κ ∂M
2.2.1
(2.42)
(2.43)
The Non-Dynamical Term
The need for the Gibbons-Hawking-York term (GHY term) arises because
the Ricci scalar contains second derivatives of the metric, which is atypical of
classical field theories[37]. One should note that there remains a degree of
ambiguity in this action - one can add an arbitrary functional of the induced
metric on the boundary to this action and still obtain the Einstein equations,
since the induced metric is kept fixed on the boundary.
It is sometimes necessary to subtract an additional term from the action in order to ensure that the boundary term does not diverge, keeping
the value of the action finite and ensuring that the variational problem is
well posed[37][39]. This becomes critical when considering the applications mentioned in the previous section (calculating black hole entropy in
Euclidean Quantum Gravity), where the entire contribution comes from the
GHY term[41]. It is also necessary to include the GHY term when calculating the ADM (Arnowitt-Deser-Misner) energy in the Hamiltonian formulation of General Relativity[43].
The non-dynamical term that we add to the action is
I
√
1
SGHY0 = −
d3 y hK0 .
κ ∂M
(2.44)
This term ensures that the numerical value of the action remains finite[44].
Here K0 is the extrinsic curvature of the boundary embedded in Minkowski
space,
K0 = ∇µ nµ = ∂µ nµ .
(2.45)
Here ∇µ refers to the covariant derivative with respect to the ambient manifold, however the ambient manifold in this case is simply M4 , which is flat,
so the covariant derivative just reduces to partial derivatives.
Since it is defined with reference to Minkowski space, this quantity will
depend only on the Minkowski metric, which is constant everywhere, and
hence variations δK0 /δg µν = 0.This ensures that the non-dymanical term
depends only on the determinant of the induced metric, and therefore, is
indeed non-dynamical, in the sense that it does not affect the derivation of
the field equations in any way.
Chapter 2. Gravity from an Action
2.3
17
Palatini Variation
In the previous sections, we have varied exclusively with respect to the 10
components of the metric. However one could consider an Einstein-Hilbert
action,
Z
√
1
SP alatini = SP =
d4 x −gRµν (Γ)g µν ,
(2.46)
2κ
where the Riemann and Ricci tensors are built out of an arbitrary connection[2]. We can then treat the 40 components of the connection and the 10
components of the metric as independent fields and vary with respect to
both of them[34],
Z
√
√
1
1
δSP alatini =
d4 x −gδg µν (Rµν − gµν R) + −gg µν δRµν (Γ). (2.47)
2κ
2
We consider Rµν = Rµν (∇ρ ), i.e the Ricci tensor as a function solely of a
derivative operator, ∇ρ . Recall that ∇ρ can be expressed in terms of an
arbitrary fixed derivative operator, ∇ρ , and a tensor field C λρσ [2][11]. We
will choose the arbitrary derivative operator to be the covariant derivative
with respect to the Levi-Civita connection, so that
∇ρ g µν = 0.
(2.48)
Now, since the Levi-Civita connection is fixed, variations in the connection
will simply be variations in this (1, 2) tensor field, δC λρσ . We can thus use
the Palatini identity as before to calculate a variation in the Ricci tensor:
δRµν = ∇ρ (δC ρνµ ) − ∇ν (δC ρρµ )
δRµν = −2∇[ν (δC ρρ]µ ).
(2.49)
We can now expand the original covariant derivative of the variation, taken
with respect to the arbitrary connection, in terms of the fixed covariant
derivative and the tensor field:
−2∇[ν (δC ρρ]µ ) = −2∇[ν (δC ρρ]µ ) + (C δνµ δC γγδ + C γγδ δC δνµ − 2C δγµ δC γνδ ).
(2.50)
We can therefore write the whole variation of the action in the form
Z
√
1
1
d4 x −g δg µν (Rµν − gµν R)
δSP alatini =
2κ
2
Z
√
1
−
d4 x −g ∇[ν (δC ρρ]µ )
κ
Z
√
1
+
d4 x −g (C δνµ δC γγδ + C γγδ δC δνµ − 2C δγµ δC γνδ ). (2.51)
κ
We can now discard the second term in this equation via Stokes’ Theorem,
since ∇µ is the covariant derivate with respect to the Levi-Civita connec√
√
tion, and we can the identity −g ∇µ Aµ = ∂µ ( −gAµ ) to write it as a total
Chapter 2. Gravity from an Action
derivative. This leaves the total variation as
Z
√
1
1
δSP alatini =
d4 x −g δg µν (Rµν − gµν R)
2κ
2
Z
√
1
d4 x −g (C βδδ δ αγ + C δδγ g αβ − 2C βγ α )δC γαβ ).
+
κ
18
(2.52)
Setting δSP + δSM = 0 we obtain the Einstein equation, Gµν = κTµν , along
with the constraint equation;
C βδδ δ αγ + C δδγ g αβ − 2C βγ α = 0,
(2.53)
which after some manipulation[2] forces
C γαβ = 0.
(2.54)
We have just derived ∇ρ = ∇ρ ; in other words, variation with respect
to the connection dynamically fixes the connection to be the Levi-Civita
connection. This is remarkable - as we shall see later, this no longer holds
true for actions more complicated than the Einstein-Hilbert action[34].
2.4
ADM: An Initial Value Formulation of General Relativity
For a classical physical theory to be useful, it is usually required to possess
an initial value formulation[2], that is to say, given a set of data (and potentially constraints) that completely determine the physical state of a system
at some point, the evolution of the system is completely and uniquely determined by the theory[34]. If small changes in the initial data inside a
compact region of spacetime induce only small changes to the solution inside that region, and if those small changes to the solution do not induce
changes to the region outside the causal future of the region, then the initial
value formulation is said to be well posed[2].
It is well known that General Relativity possesses an initial value formulation that is well posed[2]. Here, we will give the details of the so-called
"ADM formalism", a Hamiltonian-based formulation of General Relativity that has been established to provide a well posed initial value formulation[2][43]. We begin by decomposing the metric[45]:
ds2 = −(N 2 − Ni N i )dt2 + 2Ni dxi dt + γij dxi dxj ,
(2.55)
where we have defined the shift function,Ni , and the lapse function, N . Here,
Latin indices e.g. i, j, k refer only to the spatial dimensions, and γij is the
purely spatial part of the metric. Spatial indices can be raised and/or lowered as per normal using the spatial metric γij . What we are effectively
doing is decomposing the 4-dimensional spacetime into a foliation of spacelike 3-dimensional hypersurfaces, Σt . We consider the dynamic variables
to be the metric of three dimensional spatial slices γij (t, xk ) and their corresponding conjugate momenta π ij (t, xk ), which are constructed in the usual
fashion. This decomposition also introduces Lagrange multipliers[12], N
and Ni . The shift function and the lapse function describe how each of the
Chapter 2. Gravity from an Action
19
hypersurfaces, Σt , that foliate the total manifold, fit together. It is important to note that we can actually specify arbitrary equations of motion for
the shift and lapse function (precisely because they are Lagrange multipliers); this corresponds to the diffeomorphism invariance of General Relativity (i.e that it does not matter what coordinates we write the theory in), and
is equivalent to making a gauge choice[12].
The actual components of the metric can be given as follows:
g00 = −(N 2 − Ni N i ),
g0i = Ni ,
gij = γij .
(2.56)
It is easy to see by inspection that the inverse metric components are correspondingly
g 00 = −
1
,
N2
g 0i =
Ni
,
N2
g ij = γ ij −
N iN j
.
N2
(2.57)
We can also derive a useful relation between the determinants g = det(gµν )
and γ = det(γij ); applying the formula for a matrix inverse
A−1 =
1
Adj(A)
det(A)
and looking at the g 00 component, we arrive at the relation
√
√
−g = N γ.
We shall further introduce a unit normal vector
1 −N i
µ
n =
,
,
N N
(2.58)
(2.59)
(2.60)
and the corresponsding co-vector
gµν nν = nµ = (−N, 0).
(2.61)
The scalar product of these objects will not be generically conserved along
a path of parallel transport. We can calculate the extrinsic curvature of the
spatial slices as such:
Kij = −∇i nj = Γµij nµ = −N Γ0 ij .
(2.62)
In order to calculate Γ0 ij = g 0µ Γµij , we make the split
Γ0 ij = g 00 Γ0ij + g 0k Γkij ,
(2.63)
Using standard expressions for the Christoffel symbols, we have Γkij =
Γ̃kij , where Γ̃kij is the Christoffel symbol for the spatial submanifold, constructed in the usual way from derivatives of γij , and Γ0ij = 12 (∂j Ni +∂i Nj −
Chapter 2. Gravity from an Action
20
∂t γij . We can now construct the extrinsic curvature of the spatial slices:
Kij = −N Γ0 ij
= −N (g 00 Γ0ij + g 0k Γkij )
1
Nk
= −N −
(∂
N
+
∂
N
−
γ̇
+
Γ
i j
j i
ij
kij
2N 2
N2
1
=
(∂i Nj + ∂j Ni − 2Γkij − γ̇ij ),
2N
which can be more elegantly expressed using the covariant derivates taken
˜ i wj = ∂i wj − Γk wk :
with respect to the spatial submanifold, ∇
ij
Kij =
1 ˜
˜ j Ni − γ̇ij ).
(∇i Nj + ∇
2N
(2.64)
As usual in General Relativity, the next task is to compute the components of Riemann tensor. This is best done in terms of extrinsic curvatures,
Kij rather than time derivatives of the spatial metric, γ̇ij . We can use the
relation derived above to ensure that this is the case. We will need the following Christoffel symbols:
˜ j Ni = Γi0j
Γij0 = −N Kij + ∇
Γijk = Γ̃ijk
1
Γ0 00 = (Ṅ + N i ∂i N − N i N j Kij
N
1
Γ0 i0 = (∂i N − N j Kij ) = Γ0 0i
N
N i ∂j N
N iN k
ik
i
˜ j N i = Γi
−N γ −
Kkj + ∇
Γ 0j = −
j0
N
N2
1
Γ0 ij = − Kij
N
Ni
Γi jk = Γ̃i jk +
Kjk
N
We can now calculate the Riemann tensor, Rµνρσ , however let us first recall
the usual symmetries and use them to reduce the number of components
we have to calculate:
Rµνρσ = −Rνµρσ = −Rµνσρ = Rρσµν .
(2.65)
Chapter 2. Gravity from an Action
21
These symmetries imply that the only non-zero components of the Riemann
tensor are Rijkl , Ri0j0 and R0ijk . Let us begin with Rijkl :
Rijkl = giµ Rµjkl = giµ ∂k Γµjl − giµ ∂l Γµjk + Γikµ Γµjl − Γilµ Γµjk
(2.66)
m
1
1
N
m
˜ k Ni
= −Ni ∂k
Kjl + γim ∂k Γ̃ jl +
Kjl − Kjl − N Kik + ∇
N
N
N
1
Nm
Nm
m
m
+ Γ̃ikm Γ̃ lj +
Klj + Ni ∂l
Kjk − γim ∂l Γ̃ jk −
Kjk
N
N
N
m
1
˜ l Ni − Γ̃ilm Γ̃m + N Kkj .
+ Kjk − N Kil − ∇
kj
N
N
(2.67)
Now defining the Riemann tensor for each spacetime slice Σt as
R̃i jkl = ∂k Γi jl − ∂l Γi jk + Γi ka Γajl − Γi la Γakj ,
(2.68)
and cancelling off appropriate terms, we find that these components of the
Riemann tensor can be written very conveniently using the extrinsic curvature as:
Rijkl = R̃ijkl + Kik Kjl − Kil Kkj .
(2.69)
We could of course continue and calculate R0ijk and R0i0j individually at
this point, but we can use a clever trick to save ourselves some work. First
we calculate the contraction
nµ Rµiνj =
1
Nk
R0iνj −
Rkiνj .
N
N
(2.70)
This contains all the information that we are looking for:
nµ Rµijk = −N (∂j Γ0 ki + Γ0 jα Γαki ) + N (∂k Γ0 ji + Γ0 kα Γαji )
= ∂j Kki + Γ̃mki Kjm − ∂k Kji − Γ̃mji Kkm .
(2.71)
At these point we take note of the spatial covariant derivative of the extrinsic curvature,
˜ i Kjk = ∂i Kjk − Γ̃aij Kak − Γ̃a Kaj ,
∇
ik
(2.72)
and we observe that due to a fortuitous cancellation of terms, the previous
contraction can be written in terms of the difference of two spatial covariant
derivatives:
˜ j Kki − ∇
˜ k Kji .
nµ Rµijk = ∇
(2.73)
Isolating the nµ Rµi0j component, and using a little algebraic sleight of hand
we obtain
˜ i∇
˜ j N + N Kik Kkj − ∇
˜ j (Kik N k ) − Kkj ∇
˜ iN k ,
nµ Rµi0j = K̇ij + ∇
(2.74)
Chapter 2. Gravity from an Action
22
which we use along with our previous expressions to obtain
nµ nν Rµiνj =
1
˜ j N + N Kik Kkj − LN Kij ,
˜ i∇
K̇ij + ∇
N
(2.75)
where we note that the Lie derivative is given by LN Kij = N k ∂k Kij +
˜ j N k + Kjk ∇
˜ iN k .
Kik ∇
We can now derive an expression for the Ricci scalar, and hence write
the Einstein-Hilbert action in terms of the ADM formalism[45]. First we
should note the relation
∂t (Kij γ ij ) = K̇ij γ ij + Kij γ̇ ij
=⇒ γ ij K̇ij = ∂t (Kii ) + K ij γ̇ij ,
(2.76)
which uses the fact that ∂t (γij γ ij ) = 0 =⇒ γ ij γ̇ij = −γij γ̇ ij . Remembering
1 ˜
˜ j Ni − γ̇ij ), and the symmetry properties of the Riemann
Kij = 2N
(∇i Nj + ∇
tensor, we find:
R = g µν g αβ Rµανβ = γ ik γ jl Rijkl − 2nµ nν γ ij Rµiνj ,
j
2 ij
4
˜ j Ni + 2 N ∇
˜ j K i − 2 N
˜
γ K̇ij + K ij ∇
i
N
N
N
N
2
Nj ˜ i
2 ˜
= R̃ + Kii Kjj − K ij Kij − K̇ii + 2
∇j Ki − N,
(2.77)
N
N
N
= R̃ + Kii Kjj − 3K ij Kij −
˜ is the 3-dimensional version of the Laplace-Beltrami operator given
where ˜ i∇
˜ j.
˜
by = γ ij ∇
Using the determinant relation, we can write the Einstein-Hilbert action
as
Z
Z
√
4 √
SEHADM = d x −gR = d4 x γN R,
(2.78)
and using the relations
√
√
γ ij
γ γ̇ij
√2
γ ij ˜
˜
=
γ ∇i Nj + ∇j Ni − 2N Kij ,
2
∂t γ =
(2.79)
and,
√
√
√ ˜ j
−2 −g∇µ Tr(K)nµ = −2∂t γ Tr(K) + 2 γ ∇
Tr(K)N
j
(2.80)
we obtain:
Z
√
γN R̃ + Kii Kjj − K ij Kij
√
√ ˜
− 2 −g∇µ (Kii nµ ) − 2 γ N
.
SEHADM =
4
d x
(2.81)
As usual in field theory we can neglect total time derivatives and covariant divergence terms (provided they come multiplied by the correct tensor
Chapter 2. Gravity from an Action
23
density). Doing so, we obtain the Einstein-Hilbert action in the ADM formalism:
Z
√ SEHADM = d4 x γN R̃ + Kii Kjj − K ij Kij .
(2.82)
Upon variation of the action S = SEHADM + SM with respect to the lapse
function and shift vector, this action produces the set of constraint equations
R̃ + K 2 − Kij K ij = 2κρH
˜ i (K ij − γ ij K) = κsj ,
∇
(2.83)
(2.84)
where the matter source terms are defined as those seen by observers at
rest in the time slices, and can be obtained from the stress energy tensor by
taking the appropriate projections:
ρH = nµ nν Tµν
(2.85)
si = −γiµ nν Tµν
sij = γiµ γjν Tµν
(2.86)
(2.87)
(2.88)
The evolution equations are
∂t γij = LN γij − 2N Kij
(2.89)
˜ i∇
˜ jN
∂t Kij = LN Kij − ∇
1 k
1
k
k
+ N R̃ij − 2Kik Kj + Kk Kij − κ sij − sk γij + ρH γij . (2.90)
2
2
Note that this system of partial differential equations can be written in the
form
∂u
+ M i ∇i u = S(u)
∂t
(2.91)
where the matrix M i is called the characteristic matrix for the theory[46](This
proof is outlined in the book [2]). This shows that the theory is well formulated. Since the matrix M i is symmetric and the equations are hyperbolic,
the system is said to be well posed[2].
24
Chapter 3
Introduction to f (R) Gravity
3.1
Motivation: Modified Gravity and f (R) Theories
As evidenced by the recent discovery of gravitational waves by the LIGO
collaboration[9], the successes of General Relativity have rightfully earned
it a place of highest honour in the pantheon of physical theories1 . Its track
record of theoretically predicting phenomena that have subsequently been
experimentally discovered is unparalleled in the history of physics[10]. However, General Relativity (GR) is also famous for its incompatibility with
Quantum Field Theory (QFT), the standard paradigm for description of
phenomena at the energy scales accessible by modern particle accelerators. In particular, when GR is considered as a classical field theory of the
metric, it is clear that any quantisation of such a theory would be at least
non-renormalisable[47] (not to mention host to a variety of further perplexing conundrums relating to causality and the role of time in physical theories[48]), robbing it of predictive power in the high energy regime. If a
unified, field theoretical and quantum mechanical description of the four
fundamental physical forces is to be regarded as a desirable goal in theoretical physics, then the non-renormalisability of quantised versions of GR
alone is enough to motivate investigation into modified theories of gravitation.
Despite the successes of GR, there exist other theories of gravity which
are currently compatible with the observable data[31]. Although the standard cosmological model (ΛCDM, based on General Relativity) has been
successful so far, there are unanswered questions pertaining to the physical
origin of the late time energy content of the Universe, the validity of GR on
large scales, the composition and origin of dark matter, and the nature and
mechanism of inflation[49]. One natural attempt to extend ΛCDM is to replace General Relativity with a modified theory of gravity and examine the
consequences of this modification on cosmological models. By comparing
the predictions of these models to ΛCDM and observational data, one can
hope to shed light on the questions posed by ΛCDM, and at the very least,
constrain General Relativity[31].
The modified gravity theory of particular interest here is the so-called
"f (R) Gravity", which arises when one replaces the Ricci scalar, "R", in the
Einstein-Hilbert action with a scalar function of the Ricci Scalar, "f (R)".
There are three different approaches to variation of such actions[31]:
1
I appreciate this sentence feels comically bombastic, but that seems to be all the rage
when describing relativity these days, and who am I to buck the trend?
Chapter 3. Introduction to f (R) Gravity
25
• Variation with respect to the metric yields "Metric f (R) Gravity".
• Variation with respect to both the metric and the connection, assuming that the matter action is independent of the connection yields
"Palatini f (R) Gravity".
• Variation with respect to both the metric and the connection, relaxing
the assumption that the matter action is independent of the connection yields "Metric-Affine f (R) Gravity".
Unlike in General Relativity (where variation of the Einstein-Hilbert action with respect to either just the metric or both the metric and the connection yields the Einstein Field Equations[2]), the choice of method of variation has an impact on the field equations of the theory in question[46]. We
shall calculate the field equations for Metric f (R) Gravity in this chapter,
and go on to deal with Palatini formalism and Metric-Affine f (R) Gravity
in the next chapter.
3.2
Metric f (R) Gravity
Having established the basic machinery of variational calculus in the context of metric theories of gravity, using General Relativity as our example,
we shall move onto a more complicated2 theory. We will consider actions
of the form
S = SGravity + SM atter = SG + SM
Z
√
1
SGravity =
d4 x −gf (R),
2κ
(3.1)
where f (R) is some arbitrary function of the Ricci scalar and κ is some new
as-of-yet undefined gravitational constant. We will also assume that the
connection is the unique Lev-Civita connection
Actions of this form were first proposed by Buchdahl[50] in 1970. The
appeal arises from the fact that such non-linear actions may be able to explain the gravitational effects that are usually attributed to dark energy and
dark matter without the need for these as-of-yet unobserved forms of matter. Some of the forms that f (R) can take could be inspired by correction
terms from a potential quantum theory of gravity as well, keeping f (R)
relevant to early universe physics[51]. In particular, Starobinsky proposed
a model of cosmic inflation in 1980 that has generated an entire field of
research based on the Einstein equations with added quantum one-loop
contributions of conformally covariant matter fields[51]. These models are
shown to admit de-Sitter type solutions and are thus relevant to inflationary cosmology.
2
but less attractive!
Chapter 3. Introduction to f (R) Gravity
26
Let us calculate the field equations for such a theory; we will vary with
respect to the metric, as we did with the Einstein-Hilbert action.
Z
√
√
1
δSGravity =
d4 x δ( −g)f (R) + −gf 0 (R)δR
2κ
Z
√
1
1√
=
d4 x f (R)(−
−ggµν δg µν ) + f 0 (R) −g(Rµν δg µν + g µν δRµν ),
2κ
2
(3.2)
(R)
where f 0 (R) = dfdR
. Now, unlike in the case of the Einstein-Hilbert action,
the final term cannot be written as a total derivative and discarded. We
must therefore evaluate it using the Palatini identity.
Z
1
1√
δSGravity =
d4 x f (R)(−
−ggµν δg µν )
2κ
2
Z
√
1
4
µν
µν
ρ
ρ
0
∇ρ (δΓ νµ ) − ∇ν (δΓ µρ ) . (3.3)
+
d x f (R) −g Rµν δg + g
2κ
Since δΓλµν is tensorial, we can replace ordinary derivatives by covariant
derivatives to obtain an expression for the variation of the connection valid
in any reference frame,
1
δΓλµν = g λσ (∂µ δgσν + ∂ν δgσµ − ∂σ δgµν ).
2
(3.4)
We therefore obtain:
g µν ∇ρ (δΓρ νµ ) − ∇ν (δΓρ µρ ) =
1 µν ρα
1
g g ∇ρ ∇ν δgαµ + g µν g ρα ∇ρ ∇µ δgαν
2
2
1
1 µν ρα
− g g ∇ρ ∇α δgνµ − g µν g ρα ∇ν ∇ρ δgαµ
2
2
1 µν ρα
1 µν ρα
− g g ∇ν ∇µ δgαρ + g g ∇ν ∇α δgρµ .
2
2
(3.5)
Cancelling the first and fourth of these terms, adding the remaining terms
together, and remembering to switch signs when the indices on the variation of the metric are raised, we obtain the relation
g µν ∇ρ (δΓρ νµ ) − ∇ν (δΓρ µρ ) = gµν δg µν − ∇µ ∇ν δg µν ,
(3.6)
where = g αβ ∇α ∇β .
We are now in a position to obtain the field equations. This is done in
standard fashion, via integrating by parts and discarding total derivative
terms.
Z
√ 1
1
d4 x −g f 0 (R)Rµν − f (R)gµν δg µν
δSGravity =
2κ
2
Z
√
1
+
d4 x f 0 (R) −g gµν δg µν − ∇µ ∇ν δg µν .
(3.7)
2κ
Chapter 3. Introduction to f (R) Gravity
27
Now, we integrate by parts on the terms in the second line of the previous
expression; the first term can be rewritten as
Z
Z
√
√
d4 x f 0 (R) −ggµν δg µν = − d4 x −g gµν g ρσ ∇ρ f 0 (R) ∇σ δg µν ,
since we have the identities,
f 0 (R)∇ρ ∇σ δg µν = ∇ρ f 0 (R)∇σ δg µν − ∇ρ f 0 (R) ∇σ δg µν
(3.8)
√
√
and −g ∇µ Aµ = ∂µ ( −gAµ ), and we can discard the total derivative that
results in the action. Integrating by parts again we obtain
Z
Z
√
4 √
ρσ
0
µν
− d x −g gµν g ∇ρ f (R) ∇σ δg = d4 x −g gµν f 0 (R) δg µν .
Repeating the same procedure on the other term, we obtain the full variation of the gravitational part of the action:
Z
√
1
1
δSG =
d4 x −g g µν f 0 (R)Rµν − f (R)gµν + gµν f 0 (R) − ∇µ ∇ν f 0 (R) .
2κ
2
(3.9)
Using the same definition of the stress-energy tensor used in the last section,
this implies that the field equations for f (R) gravity with the action S =
SG + SM are
1
f 0 (R)Rµν − f (R)gµν + (gµν − ∇µ ∇ν )f 0 (R) = κTµν .
2
3.2.1
(3.10)
Example: Starobinsky Inflation
As an example of a particular case of an f (R), we shall present the field
equations for Starobinsky’s model[51]. Starobinsky chose
f (R) = R +
R2
6M 2
(3.11)
as his f (R), where M 2 has the dimension of mass[52][53]. Calculating the
derivative,
f 0 (R) = 1 +
R
,
3M 2
(3.12)
allows us to write the field equation:
(1 +
R R
1
R2
)R
−
(R
+
)g
+
(g
−
∇
∇
)
= κTµν ;
µν
µν
µν
µ ν
3M 2
2
6M 2
3M 2
(3.13)
or rearranging things into the form of (Einstein tensor) + (Corrections) =
(Stress-Energy tensor):
Gµν +
R
R2
1
R
−
gµν +
gµν − ∂µ ∂ν (R) = κTµν .
µν
2
2
2
3M
12M
3M
(3.14)
Chapter 3. Introduction to f (R) Gravity
28
We will return to this model in the context of inflationary cosmology in
subsequent chapters.
3.3
Legendre Transformations and the connection to
Scalar-Tensor Theories
The familiar procedure of Legendre transformation provides us with a new
perspective on the Metric f (R) theory of gravity; in particular showing it
to be equivalent to a different type of modified gravity theory, known as
Scalar-Tensor theory (albeit a quite strange version of it)[54]. Scalar-Tensor
theories operate on the premise that in addition to being a field theory of the
metric (a rank-2 tensor), gravitation is also dependent on a scalar field[55].
They are often considered to be prototypical modified theories of gravity
due to their ease of construction and use. Furthermore these theories are
often found in the context of effective actions obtained via dimensional reduction of Kaluza-Klein theories and string models[31], making them objects of independent interest.
A general action for a Scalar-Tensor gravity theory takes the form[31]:
Z
√ 1
SST =
d4 x −g f (φ)R + h(φ)∇µ φ∇µ φ − 2Λ(φ) + SM ψ, y(φ)gµν ,
16π
(3.15)
where f (φ), h(φ), y(φ) and Λ(φ) are arbitrary functions of the scalar field φ.
The symbol ψ is taken to be representative
all matter fields that are nongravitational in origin, and SM ψ, y(φ)gµν is the matter action (where we
have not excluded the possibility of coupling the non-gravitational matter
to the metric or the gravitational scalar field). We can actually absorb the
function y(φ) into the metric by making a conformal transformation:
y(φ)gµν → gµν .
(3.16)
This has the effect of picking out a particular conformal reference frame
where there is no interaction between the matter fields, ψ, and the gravitational scalar, φ. This special conformal frame is referred to as the Jordan
frame[31]. Changes to the action can just be absorbed into redefinitions of
the arbitrary functions f (φ), h(φ) and Λ(φ).
Via a further redefinition, f (φ) → φ, we can write this action in the form
Z
ω(φ)
1
4 √
µ
SST =
d x −g φR +
∇µ φ∇ φ − 2Λ(φ) + SM ψ, gµν ,
16π
φ
(3.17)
where ω(φ) is a dimensionless quantity referred to as the "coupling parameter", and Λ(φ) is a scalar-dependent generalisation of the cosmological constant.
Chapter 3. Introduction to f (R) Gravity
29
The special case of this action where ω(φ) = const and Λ = 0 gives
the well known Brans-Dicke theory[56], where the constant ω is constrained
as ω ≥ 40000 by observational data collected by the Cassini-Huygens experiment[57]. General Relativity is recovered as ω → ∞, ω 0 /ω 2 → 0 and
Λ → const [31].
How does all of this relate to the f (R) theory under consideration here
though? First, consider introducing an auxiliary field σ and writing the
action in a "massaged" form:
Z
√
Sf (R) = d4 x −g f (σ) + f 0 (σ)(R − σ) .
(3.18)
Now we take independent variations with respect to the inverse metric,
δg µν , and the auxiliary field, δσ,
Z
√
√ δR
δSf (R) = d4 x δ −g f (σ) + f 0 (σ)(R − σ) + −gf 0 (σ) µν δg µν
δg
√
√
√
0
00
0
+ −gf (σ)δσ + −gf (R − σ)δσ − −gf (σ)δσ.
(3.19)
This works out to be
Z
δR
1
4 √
µν
0
0
δSf (R) = d x −g δg
f (σ) µν − gµν f (σ) + f (σ)(R − σ)
δg
2
Z
√
+ d4 x −gf 00 (σ)(R − σ)δσ.
(3.20)
Setting δSf (R) = 0 yields the constraint equation,
f 00 (σ)(R − σ) = 0,
(3.21)
which is satisfied either when f 00 (σ) = 0 =⇒ f = Aσ + B, recovering
the Einstein-Hilbert action when substituted back into Sf (R) , or when R =
σ, when we recover the standard f (R) gravity action. We now make the
definition,
f 0 (σ) = φ,
(3.22)
and, assuming that φ(σ) is a smooth and invertible function, we can define
a potential
Λ(φ) =
1
σ(φ)φ − f σ(φ) .
2
This allows us to write Sf (R) in the form
Z
√
Sf (R) = d4 x −g φR − 2Λ(φ) ,
(3.23)
(3.24)
demonstrating that Sf (R) , the action for f (R) gravity, is equivalent to a
Brans-Dicke theory with ω = 0[54].
Chapter 3. Introduction to f (R) Gravity
3.4
30
Conformal Transformations: f (R) Gravity in the
Einstein Frame
Under conformal transformations of the metric, i.e transformations of the
form,
ygµν → g µν ,
(3.25)
one finds that it is also possible to write the field equations of Metric f (R)
gravity in the form of the Einstein equations of General Relativity with an
additional scalar field. The conformal frame under consideration is often
referred to as the Einstein frame. This is sometimes referred to as "Bicknell’s
theorem" in the context of f (R) theories[31]. We will consider conformal
transformations where the function y takes the form y = f 0 (R), define a
scalar;
r
3
df
φ=
,
(3.26)
log
κ
dR
and a potential;
V =
df
R dR
−f
.
df 2
κ dR
(3.27)
ρ
The next step is to calculate the conformally transformed connection, Γ µν
in terms of the original connection, Γρ µν (This simple calculation can be
found in [2]). This allows one to calculate the conformally transformed Rieρ
mann tensor, R µσν , conformally transformed Ricci tensor, Rµν , and conformally transformed Ricci scalar, R, in terms of the original quantities, Rρµσν ,
Rµν and R. One can then write the field equations in terms of the conformally transformed quantities[31], finding
1
1
Rµν − g µν R = κ ∂µ φ∂ν φ − g µν φ − g µν V + κT µν ,
(3.28)
2
2
where = g µν ∇µ ∇ν is the conformal Laplace-Beltrami operator. It is important to remember that the conformal stress-energy tensor, κT µν , is not
conserved[58], and that we have abandoned to the notion of a metric coupling for the scalar. This shows that any theory derived via metric variation
from an f (R) action will be conformally equivalent to General Relativity
and a massless scalar field, with a non-metric coupling[58].
3.5
Initial Value Formulation of Metric f (R) theories
It has been established that metric f (R) theories do indeed possess a well
formulated initial value formulation which is analogous to the ADM formalism for General Relativity[59][60]. Scalar tensor actions of the form,
Z
ψ(φ)R 1 α
4√
S = d −g
− ∂ φ∂α φ − W (φ) + SM ,
(3.29)
2κ
2
Chapter 3. Introduction to f (R) Gravity
31
have been shown by Salgado to possess well posed initial value formulations[61]. These results can be extended to more general actions, as was
shown by Lanahan-Tremblay and Faraoni in 2006[59] for the action
Z
ψ(φ)R ω(φ) α
4√
S = d −g
−
∂ φ∂α φ − W (φ) + SM ,
(3.30)
2κ
2
with the exception of ω = −3/2 (which coincidentally is precisely the case
that corresponds to Palatini f (R) theories). Since metric f (R) is dynamically equivalent to Brans-Dicke theory with ω = 0, and this action reduces
to the Brans Dicke action when ω(φ) = ω = const, metric f (R) therefore
possesses a well-posed initial value formulation.
Let us now show how the theory described by the scalar-tensor action
given above can be put into an ADM-style formalism. The field equations
for such a theory (setting κ = 1 for convenience) are[55]
1 00
f (∇µ φ∇ν φ − gµν ∇ρ φ∇ρ φ) + f 0 (∇µ ∇ν φ − gµν φ)
Gµν =
f
1
1
ρ
M
+
ω ∇µ φ∇ν φ − gµν ∇ φ∇ρ φ − V (φ)gµν + Tµν
(3.31)
f
2
ω 0 (φ) ρ
f0
∇ φ∇ρ φ = 0.
(3.32)
ω(φ)φ + R − V 0 (φ) +
2
2
By making the convenient definitions
(ef f )
Tµν
=
1 f
φ
M
T + Tµν
+ Tµν
),
f µν
(3.33)
where
f
Tµν
= f 00 (∇µ φ∇ν φ − gµν ∇ρ φ∇ρ φ) + f 0 (∇µ ∇ν φ − gµν φ)
1
φ
Tµν
= ω ∇µ φ∇ν φ − gµν ∇ρ φ∇ρ φ − V (φ)gµν ,
2
(3.34)
(3.35)
we can write this field equation in the form of the Einstein equation, but
with a modified stress-energy tensor[59]:
(ef f )
Gµν = Tµν
.
Taking the trace of the first field equation we obtain
3 00 ρ
1
0
R=
f ∇ φ∇ρ φ + f φ + ω∇ρ φ∇ρ φ + 4V (φ) .
f
f
(3.36)
(3.37)
Rearranging the second field equation we obtain:
R=
2V 0 ω 0 ∇ρ φ∇ρ φ 2ωφ
−
−
.
f0
f0
f0
(3.38)
Chapter 3. Introduction to f (R) Gravity
32
Thus, by eliminating the Ricci scalar, we can solve for φ as:
φ =
f 0T M
2
ω0 f
f0
2 − 2 (ω
3f 02 2f
− 2f 0 V (φ) + f V 0 (φ) + −
f ω+
+ 3f 00 ) ∇ρ φ∇ρ φ
. (3.39)
We can now proceed and set the ADM formalism for this scalar tensor theory[59]. This is a lengthy procedure detailed in references[59] and [61], but
we end up with the following set of ADM-style setup. The constraint equations are
R̃ + K 2 − Kij K ij = 2E,
(3.40)
Dl K l i − Di K = Ji ,
(3.41)
and,
the evolution equation is
∂t Kji + N l ∂l Kji + Kli ∂j N l − Kjl ∂l N i + Di Dj N − R̃ji N − N KKji
Nf0
N 00 2
f Q − Π2 + 2V (φ) + f 0 φ δji +
Di Qj + ΠK i j
+
2f
f
h
i
N
N
ω + f 00 Qi Qj =
S (m) − E (m) δji − 2S (m) i j ,
(3.42)
+
f
2f
where R̃ is the 3-dimensional Ricci scalar, Kij is the extrinsic curvature
of a hypersurface, Dc , is a 3-dimensional covariant derivative defined in
the usual way, Π is the momentum of the scalar field, Π = N1 (∂t φ + N c Qc ),
Qc is the gradient of the scalar field, Qc = Dc φ, and N and N a are the
lapse function and shift vectors respectively. The ADM decomposition of
ef f
is as follows:
the effective stress-energy tensor Tab
(ef f )
Tab
=
1
(Sab + Ja nb + Jb Na + Ena nb )
f
(ef f )
Sab = hca hdb Tcd
Ja =
E=
(f )
(φ)
(M )
= (Sab + Sab + Sab )
(ef f )
−hca Tcd nd = (Ja(f ) + Ja(φ) + Ja(M ) )
(ef f )
na nb Tab
= (E (f ) + E (φ) + E (M ) ),
(3.43)
(3.44)
(3.45)
(3.46)
where all the contributions to the effective stress-energy tensor have been
separated out as detailed above, hca is the projector associated with the spatial metric of a hypersurface Σt , hab and na is the unit vector normal to that
hypersurface. Because of the standard properties of normal vectors, the
trace of the effective stress energy tensor is simply T ef f = S − E.
We can proceed to calculate the specific quantities relating to f and φ
that appear in the decomposition of the effective stress tensor. These calculations have been carried out in detail in [61]. First we deal with the (f )
Chapter 3. Introduction to f (R) Gravity
33
quantities. This yields
E (f ) = f 0 (Dc Qc + KΠ) + f 00 Q2 ,
Ja(f )
(f )
Sab
= −f
0
(Kac Qc
(3.47)
00
+ Da Π) − f ΠQa ,
0
= f (Da Qb + ΠKab − hab φ) − f
(3.48)
00
2
2
hab Q − Π
− Qa Qb , (3.49)
where Q2 = Qc Qc . Taking traces yields further useful quantities.
S (f ) = f 0 (Dc Qc + KΠ − 3φ) + f 00 3Π2 − 2Q2 ,
S (f ) − E (f ) = −3f 0 φ − 3f 00 Q2 − Π2 .
(3.50)
(3.51)
Moving onto the (φ) quantities we obtain:
ω
Π2 + Q2 + V (φ) ,
2
= −ωΠQa ,
i
hω
= ωQa Qb − hab
Q2 − Π2 + V (φ) .
2
ω
2
2
3Π − Q − 3V (φ)
=
2
= ω Π2 − Q2 − 4V (φ) .
E (φ) =
(3.52)
Ja(φ)
(3.53)
(φ)
Sab
S (φ)
S (φ) − E (φ)
(3.54)
(3.55)
(3.56)
Plugging in all these expressions we obtain the full constraint equations
and evolution equation. The Hamiltionian constraint is therefore
ω 2 Q2
2 0
c
00
2
ij
f (Dc Q + KΠ) + Π +
ω + 2f
R̃ + K − Kij K −
f
2
2
2 (m)
=
E
+ V (φ) ,
(3.57)
f
the momentum constraint is
Dl K l i − Di K +
J (m)
1 0
f (Ki c Qc + Di Π) + ω + f 00 ΠQi = i ,
f
f
(3.58)
and the dynamical equation takes on the form
∂t Kji + N l ∂l Kji + Kli ∂j N l − Kjl ∂l N i + Di Dj N − R̃i j N − N KK i j
N 00 2
Nf0
+
f Q − Π2 + 2V (φ) + f 0 φ δji +
Di Qj + ΠK i j
2f
f
h
i
N
N
+
ω + f 00 Qi Qj =
S (m) − E (m) δji − 2S (m) i j ,
(3.59)
f
2f
whilst the trace equation becomes
Nf0
(Dc Qc + ΠK)
f
N 00 2
N +
f Q − 2ω + 3f 00 Π2 =
−2V (φ) − 3f 0 φ + S (m) + E (m) .
2f
2f
(3.60)
˜ − N Kij K ij −
∂t K + N l ∂l K + ∆N
Chapter 3. Introduction to f (R) Gravity
34
Finally, we can express the equation for the second-derivative "box operator" acting on the scalar field as
Ln Π − ΠK − Qc Dc (logN ) − Dc Qc = −φ.
(3.61)
The "Cauchy Problem" in this case is phrased as "Given an initial data set
comprising (Σt , φ, Π, Kij , Qi , hij ), which are subject to the constraint equations, determine the subsequent evolution of the system"[61]. One can
clearly see that this is a well-formulated problem. To demonstrate that it
is well-posed, the author of [61] goes on to choose specific coordinates (make
explicit choices for the lapse function and shift vector - which is equivalent
to making a gauge choice) and reduce this system to a system of first order
hyperbolic partial differential equations.
In [59], these results are translated into the language of Brans-Dicke theory, by setting ω(φ) = ω0 /φ, f (φ) = φ and V → 2V . The constraint equations then read:
2
ω0
R̃ + K 2 − Kij K ij −
Π2 + Q2
Dc Qc + KΠ +
φ
2φ
h
i
2
=
E (m) + V (φ) ,
(3.62)
φ
(m)
J
ω0
1
l
l
Ki Ql + Di Π + ΠQi = i
,
(3.63)
Dl K i − Di K +
φ
φ
φ
and the dynamical equations reduce to
∂t K i j + N l ∂l K i j + K i l ∂j N l − Kj l ∂l N i + Di Dj N
N i
N
− R̃i j N − N KK i j +
δj (2V (φ) + φ) +
Di Qj + ΠK i j
2φ
φ
N (m)
N ω0 i
S
− E (m) δji − 2S (m) i j ,
(3.64)
+ 2 Q Qj =
φ
2φ
˜ − N Kij K ij − N (Dc Qc + ΠK) − ω0 N Π2
∂t K + N l ∂l K + ∆N
φ
φ2
h
i
N
=
−2V (φ) − 3φ + S (m) + E (m) ,
(3.65)
2φ
with the Brans-Dicke field equation that governs the scalar dynamics being,
3
T (m)
ω0
φ =
− 2V (φ) + φV 0 (φ) +
Π2 − Q2 .
ω0 +
2
2
φ
(3.66)
Metric f (R) gravity is dynamically equivalent to Brans-Dicke theory with
ω0 = 0. In [59], the authors set ω0 = 0, and then eliminate any terms involving φ using the Brans-Dicke field equation. This enables them to repeat
the analysis of [61] and thus reduce the system to first order, demonstrating
that metric f (R) gravity has an initial-value formulation that is well-posed.
35
Chapter 4
Palatini f (R) Gravity:
Different Approaches
4.1
Introduction to Palatini f (R) Theories
In Chapter 2 we showed that it made no difference whether one varied with
respect to the 10 components of the metric or the 10 + 40 components of the
metric and the connection - in both cases the Einstein equation was recovered[12], and in the case of Palatini variation, the Levi-Civita connection
was enforced as a dynamic constraint (effectively another field equation)[2].
However once we generalise actions to the form of f (R) gravity, this ceases
to be the case[34]. We will now show that an entirely different field equation is recovered when one varies the f (R) action with respect to the metric
and the connection independently (assuming that the matter Lagrangian is
independent of the connection).
The action under consideration here takes the form
Z
√
1
SP =
d4 x −g f Rµν (Γ)g µν + SM atter (ψ, gµν ).
2κ
(4.1)
We cannon a priori assume Γαµν is the Levi-Civita connection since it
is determined dynamically[34]. Therefore, Γαµν 6= Γανµ , and furthermore,
we cannot make any assumptions about symmetries of the lower indices of
Rαβµν . We will however assume that gµν = gνµ since using a non-symmetric
metric tensor takes us away from the realms of pseudo-Riemannian geometry[33]. Non-symmetric metrics have been considered, most notably by
Moffat in [62], however such theories are not considered to be mainstream
and are outside the scope of this review.
Now we vary the action as before:
Z
√
1
df
f (R)
δS = δSP = δSM =
d4 x −g
R(µν) −
gµν δg µν
2κ
dR
2
Z
√
1
df µν
+
d4 x −g
g δR(µν) + δSM ,
(4.2)
2κ
dR
noting that the Ricci tensor is no longer required to be symmetric, since
the connection is undetermined. However since gµν and hence δgµν are
symmetric, we note that the contractions with Rµν and δRµν simply select
out the symmetric part of each of these tensors, leaving the antisymmetric
part unconstrained. Returning to material from Chapter 2, we consider the
Chapter 4. Palatini f (R) Gravity:
Different Approaches
36
derivation of the Palatini identity, but this time with an arbitrary connection
that is not necessarily symmetric in the lower indices:
δRρσµν = ∂ µ δΓρ νσ − δΓρ να Γαµσ + δΓρ µα Γανσ
− ∂ ν δΓρ µσ + Γρ µα δΓανσ − Γρ να δΓαµσ
(4.3)
As before, we consider the covariant derivative with respect to an arbitrary connection,
∇λ (δΓρ µν ) = ∂ λ δΓρ µν + Γρ λα δΓαµν − Γαλµ δΓρ αν − Γαλν δΓρ αµ ,
(4.4)
but when we subtract the two covariant derivatives this time, there is no
cancellation between terms because Γαµν 6= Γανµ . This leaves us with an
extra term,
δRρσµν = ∇µ (δΓρ σν ) − ∇ν (δΓρ µσ ) + 2T λµν δΓρλσ ,
(4.5)
where T λµν = 12 (Γλµν − Γλνµ ) is the torsion tensor (the antisymmetric part of
the connection).
Contracting indices, we obtain the total variation of a Ricci tensor built
from an arbitrary connection,
δRµν = ∇ρ (δΓρ νµ ) − ∇ν (δΓρ µρ ) + 2T λρν δΓρ λµ .
(4.6)
Now consider the piece of the varied action with the contribution from
δRµν :
Z
√
1
df µν
I=
d4 x −g
g δRµν
2κ
dR
Z
√
df µν
1
d4 x −g
g ∇ρ (δΓρ νµ ) − ∇ν (δΓρ µρ ) + 2T λρν δΓρ λµ . (4.7)
=
2κ
dR
We define the vector
J λ = f 0 (R)(g µν δΓλµν − g µλ δΓσσµ ),
(4.8)
√
and compute ∇λ ( −gJ λ ) using the Leibniz rule:
√
√
√
∇λ ( −gJ λ ) = ∇λ −gf 0 (R)g µν δΓλµν − ∇λ −gf 0 (R)g µλ δΓσσµ
√
√
= δΓλµν ∇λ −gf 0 (R)g µν + −gf 0 (R)g µν ∇λ δΓλµν
√
√
− δΓσσµ ∇λ −gf 0 (R)g µλ − −gf 0 (R)g µν ∇λ δΓσσµ . (4.9)
This allows us to rewrite the relevant part of the action in terms of the variation of the connection δΓλνµ :
√
√
d x ∇λ ( −gJ λ ) + δΓλνµ − ∇λ −gf 0 (R)g µν
√
√
ν
0
µρ
0
µσ ν
+ δ λ ∇ρ −gf (R)g
+ 2 −gf (R)g T λσ .
1
I=
2κ
Z
4
(4.10)
Chapter 4. Palatini f (R) Gravity:
Different Approaches
37
√
Finally, we can evaluate the covariant derivative ∇λ ( −gJ λ ), remember√
√
ing that the covariant derivative of the tensor density −g is ∇µ ( −g) =
√
√
∂µ ( −g) − Γσµσ ( −g), leaving a total derivative that can be discarded, and
two independent variations with respect to the metric and the connection:
√
√
√
∇λ ( −gJ λ ) = ∂λ ( −gJ λ ) + −g(Γλλα − Γλαλ )J α
√
√
= ∂λ ( −gJ λ ) + 2 −gT λλα J α .
(4.11)
Now plugging in the definition of J λ we obtain
√
√
√
∇λ ( −gJ λ ) = ∂λ ( −gJ λ ) + 2 −gf 0 (R) g µν T σσλ − δ νλ g µρ T σσρ δΓλνµ .
(4.12)
Feeding this back into the relevant part of the varied action, and discarding
Z
√
1
I=
d4 x ∂λ ( −gJ λ ) → 0
2κ
we obtain,
√
−gf 0 (R)g µν + δλν ∇ρ −gf 0 (R)g µρ
√
0
µσ ν
µν σ
σ
ν µρ
+ 2 −gf (R) g T λσ + g T σλ − T σρ δ λ g
.
(4.13)
1
I=
2κ
Z
4
d x
δΓλνµ
− ∇λ
√
Since independent variations must vanish in order to extremise the action,
taking the usual definition of the Hilbert Stress-Energy tensor, and recalling
that we assumed the matter action SM was indepenent of the connection so
δSM
= 0, we obtain the field equations:
δΓλ
νµ
1
df
Rµν − f (R)gµν = κTµν ,
dR 2
√
√
− ∇λ −gf 0 (R)g µν + δλν ∇ρ −gf 0 (R)g µρ
√
+ 2 −gf 0 (R) g µσ T νλσ + g µν T σσλ − T σσρ δ νλ g µρ = 0.
(4.14)
(4.15)
We can proceed further if we impose the condition that the connection is
torsion-free (which is done in almost all of the literature, but is not strictly
necessary[34])[31], setting T λµν = 0. Then the second field equation reduces
to
√
√
−∇λ −gf 0 (R)g µν + δλν ∇ρ −gf 0 (R)g µρ = 0
√
=⇒ (δ νλ δ ρσ − δ ρλ δ νσ )∇ρ −gf 0 (R)g µσ = 0
√
=⇒ ∇ρ −gf 0 (R)g µσ = 0,
Chapter 4. Palatini f (R) Gravity:
Different Approaches
38
since this must hold for every value of the indices possible. We therefore
obtain the field equations for a torsionless theory:
df
1
Rµν − f (R)gµν = κTµν ,
dR
√ 2 µν 0
∇ρ −g g f (R) = 0.
(4.16)
(4.17)
In particular, in the second of these two equations, the covariant derivative
should be taken with respect to Γρ µν which is now a general connection.
One can note that this reduces to the Levi-Civita connection if f 0 (R) = 1 (as
in the case of General Relativity).
4.2
The Role of Torsion in Palatini f (R) theories
In the previous section we discarded the terms in the field equation relating
to the torsion in order to simplify the analysis. It is not necessary to do
this however[34], and it is possible to glean some insight into the role that
torsion plays in these Palatini f (R) theories by examining the possibilities
a little more deeply. We begin by taking the trace over ν and λ in the second
field equation:
√
√
− ∇λ −gf 0 (R)g µλ + 4∇ρ −gf 0 (R)g µρ
√
+ 2 −gf 0 (R) g µσ T λλσ + g µλ T σσλ − 4T σσρ g µρ = 0.
√
√
(4.18)
=⇒ 3∇ρ −gf 0 (R)g µρ = 4 −gf 0 (R)g µν T σσρ .
We can now re-insert this condition back into the field equation:
4√
√
− ∇λ −gf 0 (R)g µν + δλν
−gf 0 (R)g µρ T σσρ
3
√
µν σ
0
µσ ν
+ 2 −gf (R) g T λσ + g T σλ − T σσρ δ νλ g µρ = 0
√
√
1
∇λ −gf 0 (R)g µν = 2 −gf 0 (R) g µσ T νλσ + g µν T σσλ − T σσρ δ νλ g µρ .
3
(4.19)
The next step is to split the connection Γλµν into its symmetric part, C λµν ,
λ:
and antisymmetric (torsion) part Tµν
λ
Γλµν = C λµν + Tµν
.
(4.20)
We now define a new covariant derivative with respect to the symmetric
α
part of the connection, ∇C
µ Aν = ∂µ Aν − C µν Aα , which allows us to separate out terms in the full covariant derivative which depend only on the
√
torsion. Remembering that −g is a tensor density and applying the covariant derivative accordingly, we find
√
√
0
µν
∇λ −gf 0 (R)g µν =∇C
−gf
(R)g
λ
√
+ −gf 0 (R) g µσ T νλσ + g νσ T µλσ + g µν T σσλ . (4.21)
Chapter 4. Palatini f (R) Gravity:
Different Approaches
39
Having
two expressions involving the full covariant derivative,
√ obtained 0
µν
∇λ −gf (R)g
, we can now eliminate this and obtain an expression
for the covariant derivative with respect to the symmetric part of the connection in terms of the torsion:
√
√
0
µν
0
−gf
(R)g
=
−gf
(R)
g µσ T νλσ − g νσ T µλσ
∇C
λ
2
+ g µν T σσλ − δ νλ g µρ T σσρ .
(4.22)
3
Now we can exploit the symmetry properties of each side of this equation
a little further; since g µν = g νµ , by permuting the µ and ν indices in this
equation and then adding the result to the original equation we obtain
√
√
2∇C
−gf 0 (R)g µν = −gf 0 (R) g µσ T νλσ − g νσ T µλσ + g µν T σσλ
λ
2
2
− δ νλ g µρ T σσρ + g νσ T µλσ − g µσ T νλσ + g νµ T σσλ − δ µλ g νρ T σσρ . (4.23)
3
3
This simplifies to the expression
∇C
λ
√
√
1
−gf 0 g µν = −gf 0 g µν T σσλ − (δ νλ g µρ + δλµ g νρ )T σσρ ,
3
(4.24)
where f 0 (R) has been shortened to f 0 for brevity. This equation shows us
that the symmetric part of the connection couples to the torsion only via
the contraction T σσρ . We can derive another useful equation via a similar
procedure, permuting the µ and ν indices but then subtracting the resulting
equation from the original one. This yields:
0=
√
2
−gf 0 (R) g µσ T νλσ − g νσ T µλσ + g µν T σσλ − δ νλ g µρ T σσρ
3
2
µ
−g νσ T λσ + g µσ T νλσ − g νµ T σσλ + δ µλ g νρ T σσρ
3
1 ν µρ
µσ ν
νσ µ
=⇒ g T λσ − g T λσ = (δλ g − δλµ g νρ )T σσρ .
3
(4.25)
Equations (3.52) and (3.53) together suggest that an intelligently chosen
change of variables might be able to clear things up. We consider new variables:
2
Γ̃λµν = Γλµν + δ λν T σσµ .
3
(4.26)
The new variables are chosen specifically so that the contraction T̃ σσν = 0,
where T̃ λµν is the torsion of the shifted connection, Γ̃λµν ; let us now prove
this is the case:
1
1 λ
T̃ λµν =
Γ µν − Γλνµ +
δνλ T σσµ − δµλ T σσν
2
3
1
(4.27)
= T λµν +
δνλ T σσµ − δµλ T σσν .
3
Chapter 4. Palatini f (R) Gravity:
Different Approaches
40
Now taking the trace over λ and µ we obtain
T̃ λλν = T λλν +
1 σ
T σν − 4T σσν
3
(4.28)
= 0.
We can also construct the symmetric part of the shifted connection:
C̃ λµν = C λµν +
1 λ σ
δν T σµ + δµλ T σσν .
3
(4.29)
This enables us to rewrite the equations we derived above for the symmetric and antisymmetric parts of the connection in terms of the new variables,
yielding
ν
+
g µσ T̃ νλσ − g νσ T̃ µλσ = g µσ Tλσ
µ
−g νσ Tλσ
+
1 ν ρ
(δ σ T ρλ − δ νλ T ρρσ )
3
1 µ ρ
(δ σ T ρλ − δ µλ T ρρσ )
3
1
= g µσ T νλσ − g νσ T µλσ − (δ νλ g µσ − δ µλ g νσ )T ρρσ = 0
3
=⇒ g µσ T̃ νλσ − g νσ T̃ µλσ = 0.
(4.30)
We can use the definition of the covariant derivative with respect to the
shifted symmetric connection,
√
√
√
µ
0
µν
0
µν
0
αν
∇C̃
−gf
(R)g
=
∂
−gf
(R)g
+
C̃
−gf
(R)g
λ
λ
αλ
√
√
+ C̃ ναλ −gf 0 (R)g αµ − C̃ ααλ −gf 0 (R)g µν
(
√
√
√
= ∂λ −gf 0 (R)g µν + C µαλ −gf 0 (R)g αν + C ναλ −gf 0 (R)g αµ
)
√
1 µ σ
δλ T σα + δαµ T σσλ −gf 0 (R)g αν
3
√
√
1 ν σ
1 α σ
+
δλ T σα + δαν T σσλ −gf 0 (R)g αµ −
δα T σλ + δλσ T ρρσ
−gf 0 (R)g µν ,
3
3
− C ααλ
√
−gf 0 (R)g µν
+
to write this covariant derivative in terms of a covariant derivative with
respect to the original symmetric connection and some terms that depend
on torsion:
√
1
√
µ σ
0 µν
C √
0 µν
µ σ
∇C̃
−gf
g
=
∇
−gf
g
+
δ
T
+
δ
T
−gf 0 g αν
α σλ
λ
λ
3 λ σα
√
√
1 ν σ
1 α σ
+
δλ T σα + δαν T σσλ −gf 0 g αµ −
δα T σλ + δλσ T ρρσ
−gf 0 g µν .
3
3
(4.31)
Now, substituting in for the equation derived earlier from symmetry considerations,
√
√
1
∇C
−gf 0 g µν = −gf 0 g µν T σσλ − (δ νλ g µρ + δλµ g νρ )T σσρ ,
(4.32)
λ
3
Chapter 4. Palatini f (R) Gravity:
Different Approaches
41
we obtain:
√
√
1 ν µρ
µ νρ
0 µν
0
µν σ
σ
−gf
g
=
−gf
g
T
−
∇C̃
(δ
g
+
δ
g
)T
σρ
σλ
λ
λ
3 λ
√
√
1 µ σ
1 ν σ
+
δλ T σα + δαµ T σσλ −gf 0 g αν +
δλ T σα + δαν T σσλ −gf 0 g αµ
3
3
√
1 α σ
σ ρ
0 µν
δ T
+ δλ T ρσ
−gf g .
−
3 α σλ
1
√
1
5
= −gf 0 (δ µλ T σσα g αν + T σσλ g µν ) + (δ νλ T σσα g αµ + T σσλ g µν ) − g µν T σσλ
3
3
3
√
1
+ −gf 0 g µν T σσλ − (δλν g µρ + δλµ g νρ )T σσρ
3
= 0.
(4.33)
Hence, we see that the coupling equations (3.52) and (3.53) written in terms
of the new variables are
√
0 µν
−gf
g
= 0,
(4.34)
∇C̃
λ
g µσ T̃ νλσ − g νσ T̃ µλσ = 0.
(4.35)
The second of these equations implies a further index symmetry on the
torsion:
gµδ gν g µσ T̃ νλσ − gµδ gν g νσ T̃ µλσ = 0
=⇒ T̃λδ = T̃δλ .
(4.36)
This symmetry, combined with the antisymmetry of the last two indices of
the torsion tensor, forces T̃λδ = 0:
T̃λδ = −T̃δλ = −T̃λδ = T̃λδ
= T̃δλ = −T̃δλ = −T̃λδ .
We can therefore write the torsion in the original coordinates in terms of a
vector, Aµ = T σσµ , since equation (3.55) simply reduces to
1
T λµν = (δµλ Aσσν − δ λν Aσσµ ).
3
(4.37)
We have therefore managed to constrain the original connection to the following form:
2
Γαµν = C̃ αµν − Aµ δ αν .
3
(4.38)
Chapter 4. Palatini f (R) Gravity:
Different Approaches
42
The next task is to calculate the effect that our torsion generating vector has
on the Riemann tensor for the manifold and the corresponding Ricci tensor.
Rαβµν (Γ) = ∂ µ Γανβ − ∂ ν Γαµβ + Γαµλ Γλνβ − Γανλ Γλµβ
= (∂ µ C̃ ανβ − ∂ ν C̃ αµβ + C̃ αµλ C̃ λνβ − C̃ ανλ C̃ λµβ )
2
2
2
+ δ αβ ∂ν Aµ − δ αβ ∂µ Aν + Aν δ λβ C̃ αµλ
3
3
3
2
2
2
− Aν δ λβ C̃ αµλ + Aµ δ αλ C̃ λνβ − Aµ δ αλ C̃ λνβ
3
3
3
4
4
α
α
+ Aµ Aν δ β − Aν Aµ δ β
3
3
4
=⇒ Rαβµν (Γ) = Rαβµν (C̃) − ∂[µ Aν] δ αβ
3
4
=⇒ Rµν (Γ) = Rµν (C̃) − ∂[µ Aν] .
3
(4.39)
(4.40)
We have just proved that the symmetric part of the Ricci tensor (that appears in the field equations) is, unsurprisingly, unaffected by the presence
of torsion since R(µν) (Γ) = R(µν) (C̃). We can also go one step further and
calculate the Ricci scalar:
R(Γ) = R(C̃).
(4.41)
Contraction with the symmetric metric screens out the antisymmetric part
of the Ricci tensor, so the scalar curvature is insensitive to the presence of a
torsion term. This result is known as the project invariance of scalar curvature[34].
√
0 (R)g µν
We are now in a position to solve ∇C̃
−gf
= 0. At first
λ
glance this looks intimidating since the Ricci scalar involves second derivatives of the metric and we are taking further derivatives still, however we
will see that a judicious transformation simplifies this equation into a familiar form. We start by taking the trace of the field equation to obtain the
relation
Rf 0 (R) − 2f (R) = κT,
(4.42)
where T = Tµν g µν . This equation can be considered to give T as a function
of R. Assuming that this function is smooth and invertible, we can invert
this to obtain an algebraic equation for R in terms of T . We define a new
metric, which is conformally related to the original metric:
df (T )
gµν
dR
= f 0 (T )−1 g µν
hµν =
=⇒ hµν
=⇒ f 0 (R)2 hµν = f 0 (R)g µν
(4.43)
(4.44)
Chapter 4. Palatini f (R) Gravity:
Different Approaches
43
Since we are operating in four dimensions, the determinants are related by
4
det(hµν ) = det f 0 (R)gµν = f 0 (R) det(gµν )
√
√
1
−h.
=⇒ −g = 0
f (R)
(4.45)
(4.46)
Together, these relations imply
∇C̃
λ
√
√
−gf 0 (R)g µν = ∇C̃
−hhµν = 0,
λ
(4.47)
which is conveniently solved by the Levi-Civita connection for the conformal metric,
1
C̃ αµν = hαλ ∂µ hνλ + ∂ν hµλ − ∂λ hµν .
2
(4.48)
We can thus write the full connection (which has now been determined
dynamically) in terms of the conformal metric as
2
Γαµν = Lαµν − δ λν Aµ ,
3
(4.49)
where Lαµν is the standard Levi-Civita connection. We can now transform
back to the original frame to get an expression for the connection in terms
of f (R) and gµν :
1 λα g
∂µ (gαν )f 0 + gαν ∂µ f 0 + ∂ν (gαµ )f 0 + gαµ ∂ν f 0
2f 0
2
− (∂α gµν )f 0 − gµν ∂α − δ λν Aµ
3
1
1
1 λ
0
λ
= L µν + 0 δ ν ∂µ f + 0 δ λµ ∂ν f 0 − 0 gµν ∂ λ f 0
2f
2f
2f
1
2 3
= Lλµν + 0 (δ λµ δν f 0 − gµν ∂ λ f 0 ) − δ αν Aµ − 0 ∂µ f 0
2f
3
4f
Γαµν =
(4.50)
Since the field equation is insensitive to the presence of the torsion term, we
can choose Aµ = 4f3 0 ∂µ f 0 , in which case the connection reduces to
Γλµν = Lλµν +
1 λ
(δ δν f 0 − gµν ∂ λ f 0 ).
2f 0 µ
(4.51)
Having made this choice for the torsion vector, we can calculate the torsion
tensor that it generates by using the expression
1
T αµν = (δ αµ Aν − δ αν Aµ )
3
∂λ f 0 α λ
=⇒ T αµν =
δ µ δ ν − δ αν δ λµ .
0
4f
(4.52)
We find that we can rewrite the connection calculated earlier using a tensorial quantity constructed from the torsion; the contorsion tensor:
K αµν = T αµν + Tµ α ν + Tν
α
µ.
(4.53)
Chapter 4. Palatini f (R) Gravity:
Different Approaches
44
Tµ α ν = gµρ g ασ T ρσν
∂λ f 0 α λ
(δ µ δ ν − gµν g αλ )
4f 0
∂λ f 0 α λ
(δ ν δ µ − gµν g αλ ).
=
4f 0
=
Tµ α ν
(4.54)
(4.55)
Summing the terms, we find the full expression for the contorsion:
∂λ f 0 α λ
δ µ δ ν − δ αν δ λµ
0
4f
∂λ f 0 α λ
+
(δ µ δ ν − gµν g αλ )
4f 0
∂λ f 0 α λ
+
(δ ν δ µ − gµν g αλ )
4f 0
1
= 0 (δ λµ δν f 0 − gµν ∂ λ f 0 ).
2f
K αµν =
=⇒ K αµν
(4.56)
The dynamically determined connection is therefore simply
Γαµν = Lαµν + K αµν .
(4.57)
With a bit of effort, it is in fact possible to show that this is a metric connec√
tion, i.e that ∇µ −gg µν = 0. Using the field equation
√
√
1
∇λ −gf 0 (R)g µν = 2 −gf 0 (R) g µσ T νλσ + g µν T σσλ − T σσρ δ νλ g µρ
3
(4.58)
and the Leibniz rule we can isolate the desired term:
√
√
∇λ −gg µν f 0 (R) + ∂λ f 0 g µν −g
√
1 σ ν µρ
µν σ
0
µσ ν
= 2 −gf (R) g T λσ + g T σλ − T σρ δ λ g
.
3
(4.59)
At this point we trace over λ and µ:
√
√
∇µ −gg µν f 0 (R) + ∂µ f 0 g µν −g
√
1 σ ν µρ
0
µσ ν
µν σ
= 2 −gf (R) g T µσ + g T σµ − T σρ δ µ g
3
√
2
= 2 −gf 0 (R) T σσ ν + T νσσ .
3
√
√
√
2 σν
−g ∂ ν f 0
µν
νσ
=⇒ ∇µ −gg
= 2 −g T σ + T σ −
.
(4.60)
3
f0
At this point we can introduce our explicit expression for the torsion to
calculate T νσσ and T σσ ν :
Chapter 4. Palatini f (R) Gravity:
Different Approaches
45
∂λ f 0 α λ
(δ µ δ ν − δ αν δ λµ )
4f 0
∂λ f 0 ν λ
= T νµσ g µσ = g µσ
(δ µ δ σ − δ νσ δ λµ )
4f 0
∂λ f 0 νλ
=
(g − g λν ) = 0,
4f 0
= T σµρ g ρν δ µσ
T αµν =
T νσσ
T σσ ν
∂λ f 0 σ λ
(δ µ δ ρ − δ σρ δ λµ )g ρν δ µσ
4f 0
∂λ f 0
3∂ ν f 0
λν
νσ λ
=
(4g
−
g
δ
)
=
.
σ
4f 0
4f 0
=
Inserting these expressions back into our expression for ∇µ
we obtain the desired result:
√
√
√
2 3 ∂ν f 0
∂ν f 0
µν
= 2 −g
·
−
−g
= 0.
∇µ −gg
3 4 f0
f0
√
−gg µν ,
(4.61)
This result shows that torsionless Palatini f (R) is dynamically equivalent to
an f (R) theory with a predetermined connection that is metric-compatible
and includes torsion, and furthermore, as was shown earlier, both of these
theories are equivalent to a general Palatini f (R) theory with arbitrary torsion generated by a vector field[34].
4.3
Field Equations for Palatini f (R) Theories
Since we have now dynamically solved for the connection, we can return to
the field equations and rewrite them in a more illuminating form. We will
choose our torsion vector to be Aµ = 4f3 0 ∂µ f 0 so that any dependence in the
connection is removed. The connection is therefore
Γαµν = Lαµν +
1 α
(δ δν f 0 − gµν ∂ α f 0 ),
2f 0 µ
(4.62)
and we must now calculate the field equation
df
1
Rµν − f (R)gµν = κTµν ,
dR
2
(4.63)
Chapter 4. Palatini f (R) Gravity:
Different Approaches
46
in terms of this connection. In order to do this we must calculate the Riemann tensor, the Ricci tensor and the Ricci scalar.
Rαβµν = ∂µ Γανβ − ∂ν Γαµβ + Γαµλ Γλνβ − Γανλ Γλµβ
= ∂µ Lανβ − ∂ν Lαµβ + Lαµλ Lλνβ − Lανλ Lλµβ
1 α
1 α
0
α 0
0
α 0
+ ∂µ
− ∂ν
δ ν δβ f − gνβ ∂ f
δ µ δβ f − gµβ ∂ f
2f 0
2f 0
1 α
1 λ
α
λ
0
α 0
0
α 0
+ L µλ + 0 δ µ δλ f − gµλ ∂ f
L νβ + 0 δ ν δβ f − gνβ ∂ f
2f
2f
1 λ
1
λ
α
α
0
α 0
0
α 0
L µβ + 0 δ µ δβ f − gµβ ∂ f
.
− L νλ + 0 δ ν δλ f − gνλ ∂ f
2f
2f
After some tedious algebra, one obtains
1
α
α
α
0
α
0
α 0
α 0
R βµν = R βµν (g) + 0 δ ν ∇µ ∇β f − δ µ ∇ν ∇β f + gµβ ∇ν ∇ f − gνβ ∇µ ∇ f
2f
1
+ 02 δ αµ 3∂ν f 0 ∂β f 0 − gνβ (∂f 0 )2 − δ αν 3∂µ f 0 ∂β f 0 − gµβ (∂f 0 )2
4f
0 α 0
0 α 0
+ 3gµν ∂µ f ∂ f − 3gµβ ∂ν f ∂ f ,
(4.64)
where
Rαβµν (g) = ∂µ Lανβ − ∂ν Lαµβ + Lαµλ Lλνβ − Lανλ Lλµβ ,
is the Riemann tensor with respect to the Levi-Civita connection, and we
consider all covariant derivatives to be taken with respect to the Levi-Civita
connection (e.g ∇µ ων = ∂µ ων − Lλµν ωλ ). This notational set up will allow
us to cast the field equation in terms of the Einstein equation of General
Relativity with added corrections, where the added corrections look like
additional matter terms. To continue, we contract over α and µ to obtain
the Ricci tensor:
1
1
3
0
0
Rβν = Rβν (g) − 0 ∇ν ββ f + gνβ f + 02 ∂ν f 0 ∂β f 0 .
(4.65)
f
2
2f
Finally, we can take the trace to obtain the Ricci scalar1 :
R = R(g) −
3
3
0
0 2
f
+
∂f
.
f0
2f 02
(4.66)
We can now express the left hand side of the field equation in terms the
regular objects in General Relativity and additional terms that represent
deviations away from General Relativity:
f 0 R(µν) −
1
f
1
3
f
gµν = f 0 Rµν (g) − ∇µ ∇ν f 0 − gµν f 0 + 0 ∂µ f 0 ∂ν f 0 − gµν
2
2
2f
2
= κTµν
(4.67)
Please note that this formula is quoted incorrectly in reference [34]
Chapter 4. Palatini f (R) Gravity:
Different Approaches
47
Gathering all the non-familiar terms onto the right hand side, we seek to
isolate the Einstein tensor:
κTµν
f
1
1
3
+ 0 gµν + 0 ∇µ ∇ν f 0 + 0 gµν f 0 − 02 ∂µ f 0 ∂ν f 0
0
f
2f
f
2f
2f
κTµν
f
1
1
1
+ 0 gµν + 0 ∇µ ∇ν f 0 + 0 gµν f 0
Rµν (g) − gµν R(g) =
2
f0
2f
f
2f
3
3
1
3
0
0
0
0 2
.
− 02 ∂µ f ∂ν f − gµν R + 0 f − 02 ∂f
2f
2
f
2f
(4.68)
Rµν (g) =
We can now write the field equation:
κTµν
1
Rf 0 − f
1
Rµν (g) − gµν R(g) =
−
gµν + 0 ∇µ ∇ν f 0 − gµν f 0
0
0
2
f
2f
f
2
1
3
.
(4.69)
− 02 ∂µ f 0 ∂ν f 0 − gµν ∂f 0
2f
2
It is important to remember that in this equation, the quantities R and f 0 are
to be interpreted as an algebraic functions of the trace of the stress-energy
tensor, T . This is because we used the trace relation
Rf 0 − 2f = κT
(4.70)
in order to solve for the connection.
Let us now consider the vacuum field equations; setting Tµν = 0, =⇒
T = 0, we note that the trace relation (which is sometimes referred to as the
structural equation[34]) reduces to the algebraic relation
df R(0)
R(0)
− 2f R(0) = 0,
(4.71)
dR
which can be solved to find R(0) = Rvacuum = const[34]. We note that this
equation may have more than one solution for Rvacuum and comment that
these different solutions would correspond to different possible realisations
of spacetimes in this formalism[34][63]. Having shown that the Ricci scalar
is constant over spacetime in the case of a vacuum, we note that all the
terms containing derivatives on the right hand side of the field equations
vanish, and the vacuum field equation becomes
1
Rf 0 − f
Rµν (g) − gµν R(g) = −
gµν
2
2f 0
(4.72)
which can be written in the form
Gµν = Λef f ective gµν
(4.73)
where
Λef f ective
R(0)f 0 (0) − f (0)
=−
.
2f 0 (0)
(4.74)
This shows that Palatini f (R) gravity only differs from General Relativity
Chapter 4. Palatini f (R) Gravity:
Different Approaches
48
with an effective cosmological constant in the presence of matter, and when
the Ricci scalar is allowed to vary throughout spacetime, hence ∂µ f 0 6= 0.
We can therefore deduce that in regions devoid of matter, the solutions of
the field equation should be the same as those from General Relativity with
a cosmological constant. Furthermore, Birkhoff’s theorem will hold[64],
gravitational waves (in the usual sense) will be able to propagate[65] and
there will be no instabilities (for example like those found in the regular
metric form of these theories[66][67])[34].
We must be careful not to extend this correspondence too far; although
the solutions outside of matter will be the same as those from General Relativity, nothing can be said about the boundary conditions at the border
between matter filled regions and the vacuum, because the solutions in the
regions with matter are emphatically not the same as those of General Relativity[34].
We will further note that conservation of energy momentum is ensured
by the diffeomorphism invariance of the matter action. This can be shown
as follows: First, make an infinitesimal change of coordinates xµ → xµ +
λξ µ . Then consider the effect of this on the variation of the matter action:
√
Z
−gLM
1
4 δ
δSM =
d x
δgµν
2
δgµν
Z
√
1
d4 x −gT µν δgµν ,
(4.75)
=
4
M
using the canonical definition of the stress-energy tensor, Tµν = √2−g δL
δg µν .
The change in the metric induced by the diffeomorphism is δgµν =
Lξ gµν = −2∇(µ ξν) (N.B. ξ µ is NOT a Killing vector here - it is an arbitrary
small element of the tangent space). Thus, relabelling indices and using the
fact that the stress-energy tensor is symmetric we obtain
Z
√
1
δSM = −
d4 x −gT µν ∇µ ξν ,
(4.76)
2
which, after an integration by parts leads to
Z
√
1
δSM = −
d4 x ∇µ −gT µν ξν .
2
(4.77)
Since this quantity must vanish for arbitrary displacements ξν , we must
conclude that
√
∇µ −gT µν = 0,
(4.78)
which simply reduces to
∇µ T µν = 0
(4.79)
when one recalls that the connection in consideration is metric compatible.
One therefore must conclude that energy and momentum are conserved
Palatini f (R) theories[34].
Chapter 4. Palatini f (R) Gravity:
Different Approaches
4.4
49
The Relation between Palatini f (R) Theories and
Brans-Dicke Theory
As might be expected considering the corresponding result in Metric f (R)
theories, Palatini f (R) can also be related to scalar-tensor gravity[31]. Consider the well known Brans-Dicke field equations:
dV
− 2V (φ)
dφ
1
ω
1
−
V (φ)gµν + 2 (∂µ φ∂ν φ − gµν ∂ρ φ∂ ρ φ)
2φ
φ
2
(2ω + 3)φ = κT + φ
κ
Tµν
φ
1
+ (∇µ ∇ν φ − gµν φ).
φ
Gµν =
(4.80)
(4.81)
It is easy to see that if one sets the parameter ω = −3/2, then the field
equations just reduce to
dV
= κT
dφ
1
3
1
−
V (φ)gµν − 2 (∂µ φ∂ν φ − gµν ∂ρ φ∂ ρ φ)
2φ
2φ
2
2V (φ) − φ
κ
Tµν
φ
1
+ (∇µ ∇ν φ − gµν φ).
φ
Gµν =
(4.82)
(4.83)
Now making the notational changes φ = f 0 and V (φ) = Rf 0 − f we obtain
κTµν
Rf 0 − f
1
−
gµν + 0 ∇µ ∇ν f 0 − gµν f 0
0
0
f
2f
f
3
1
0
0
0 2
− 02 ∂µ f ∂ν f − gµν ∂f
2f
2
Gµν =
(4.84)
along with the trace relation
Rf 0 − 2f = κT.
(4.85)
This establishes that Palatini f (R) is equivalent to Brans-Dicke theory
with ω = −3/2.
4.5
Initial Value Formulations of Palatini f (R) Theories
Although the Palatini incarnations of the f (R) theories are clearly interesting, they have traditionally been regarded with a degree of suspicion.
In particular, concerns regarding the formulation of the Cauchy problem
have been raised, calling into question the predictive capability of such theories[59][31]. Let us survey these arguments now. Recall from Chapter 3
Chapter 4. Palatini f (R) Gravity:
Different Approaches
50
that the ADM decomposition for Brans-Dicke theory was given by the following set of constraint equations
2
ω0
2
ij
c
2
2
R̃ + K − Kij K −
D Qc + KΠ +
Π +Q
φ
2φ
i
2 h (m)
=
E
+ V (φ) ,
(4.86)
φ
(m)
J
1
ω0
l
l
Dl K i − Di K +
Ki Ql + Di Π + ΠQi = i
,
(4.87)
φ
φ
φ
the dynamical equation
∂t K i j + N l ∂l K i j + K i l ∂j N l − Kj l ∂l N i + Di Dj N
N
N i
δj (2V (φ) + φ) +
Di Qj + ΠK i j
− R̃i j N − N KK i j +
2φ
φ
N ω0 i
N
(m)
(m)
i
(m) i
+ 2 Q Qj =
S
−E
δj − 2S
,
(4.88)
j
φ
2φ
˜ − N Kij K ij − N (Dc Qc + ΠK) − ω0 N Π2
∂t K + N l ∂l K + ∆N
φ
φ2
i
h
N
=
(4.89)
−2V (φ) − 3φ + S (m) + E (m) ,
2φ
and the Brans-Dicke field equation that governs the scalar dynamics,
3
T (m)
ω0
ω0 +
φ =
− 2V (φ) + φV 0 (φ) +
Π2 − Q2 .
2
2
φ
(4.90)
Palatini f (R) gravity is demonstrably equivalent to Brans-Dicke theory with
ω0 = −3/2. However when we insert this choice into the equations above,
things are no longer as simple as they were for metric f (R) gravity; naively
speaking, this is in fact precisely the value eliminates the φ term from the
field equation, making it impossible to eliminate the second derivatives and
reduce the system to first order unless φ = 0, which corresponds to either
φ = const in which case the theory just reduced to General Relativity, or φ
solving the wave equation. It was asserted in [59] that apart from in these
special cases, this constitutes a "no-go theorem" for Palatini f (R) gravity,
rendering it predictively useless due to the lack of a well-formulated and
well-posed Cauchy problem. This was based on the assertion that the φ
field was non-dynamical and that it could be freely assigned on any region
within the spacetime so long as the degenerate field equation (which reduces to a constraint with the absence of the φ term) is satisfied.
Recently, however, it has been argued by Olmo in [34] that Palatini f (R)
does in fact admit a well-formulated initial value problem, and furthermore
that the formulation is likely to be well-posed. It was argued that the perspective outlined in [59] and [60] was in fact misleading, and that the field
φ which corresponds to f 0 (φ) is in fact algebraically determined from the
trace of the field equation,
Rφ − 2f (R) = κT.
(4.91)
Chapter 4. Palatini f (R) Gravity:
Different Approaches
51
Using a different approach to the ADM decomposition to the one based
on the work of Salgado in [61], which was originally intended to apply to
scalar-tensor theories and not f (R) theories, the author is able to arrive at a
first order system of equations, in his opinion demonstrating that the Palatini f (R) is at the very least well formulated. Although the equations derived in [34] are apparently not suitable for demonstrating that the Cauchy
problem is well-posed (i.e that the equations are hyperbolic), the author argues that one could potentially exploit the resemblances between the equations for General Relativity and generic Brans-Dicke theories to argue that
the Cauchy problem is likely to be well-posed, and at the very least, that
there is no reason to suspect that it is not. The problem of establishing
whether Palatini f (R) gravity theories are in fact well-posed appears to be
open.
4.6
Metric-Affine f (R) Gravity
In the previous section we investigated the consequences of treating the
connection as a set of independent fields to be determined by the variational
procedure. Although this method produces the same results as standard
metric variation for General Relativity, as we saw in the last section, it leads
to startlingly different theories once one includes higher order terms. One
could further relax the condition that the matter action is independent of the
connection and investigate the consequences[31]. This approach is known
as Metric-Affine f (R) gravity, and it has not been investigated in as much
detail, compared to the other two approaches[68]. The action now becomes
Z
√
1
SM A =
d4 x −g f Rµν (Γ)g µν + SM atter (ψ, gµν , Γ).
(4.92)
2κ
At this point, one should note the invariance of the Ricci scalar under
transformations of the type
Γρ µν → Γρ µν + λµ δ ρν .
(4.93)
This is a projective transformation[31], and this type of invariance is not
usually displayed by matter fields, which renders the action in its current
form inconsistent. To "cure" the inconsistency, one can add a Lagrange multiplier type term, leading to the following action:
Z
√
1
SM A =
d4 x −g f Rµν (Γ)g µν + B µ Λν[νσ] + SM (ψ, gµν , Γ). (4.94)
2κ
Variation of this action now leads to the following field equations[68], (the
first of which bears resemblance to the field equations derived from Palatini
f (R) gravity),
Chapter 4. Palatini f (R) Gravity:
Different Approaches
52
1
f 0 (R)Rµν − f (R)gµν = κTµν ,
(4.95)
2
Γµ[µν] = 0
(4.96)
√
√
1
√
∇ρ ( −gf 0 (R)g µρ )δ νσ −∇σ ( −gf 0 (R)g µν ) + 2f 0 (R)g µσ Γν [σρ]
−g
2 σ[ν µ]
µν
= κ ∆ρ − ∆σ δ ρ .
(4.97)
3
√
[µν]
Here ∆ρ µν = (2/ −g)δSM /δΓρ µν . One then finds that ∆ρ
= 0 corre(µν)
sponds to vanishing torsion and ∆ρ
6= 0 corresponds to non-zero nonmetricity of the theory.
Metric-Affine theories have not been as well investigated as either metric f (R) gravity or Palatini f (R) gravity. There is much scope for further
investigation of the predictions and properties of these theories[68].
53
Chapter 5
Cosmological Solutions in f (R)
Gravity
5.1
FRLW and Metric f (R) Gravity
To investigate the cosmological implications of f (R) modifications to gravity, the natural first step is to calculate the equivalent of the Friedmann
equation[31][69]. We will restrict ourselves to the case of a flat universe
for simplicity. We accomplish this by taking the field equations for Metric
f (R) gravity calculated in Chapter 3,
1
f 0 (R)Rµν − f (R)gµν + (gµν − ∇µ ∇ν )f 0 (R) = κTµν ,
2
(5.1)
and insert the flat Friedmann-Roberston-Walker metric:
ds2 = −dt2 + a2 (t)g̃ij dxi dxj ,
where g = det(gµν ) = −a6 and therefore
√

1 0 0
g̃ = 0 1 0 ,
0 0 1
(5.2)
−g = a3 , and

g̃ij = δij
(5.3)
since we are using Cartesian coordinates.
Proceeding as usual, we calculate the non-zero components of the connection (using Cartesian coordinates this is simple, since Γ̃i jk vanish):
Γ0ij = aȧδij ,
Γij =
ȧ 1
δij .
a
(5.4)
Continuing, one finds the relevant components of the Ricci tensor:
ä
= −3(Ḣ + H 2 )
a
Rij = (2ȧ2 + aä)δij .
R00 = −3
Then, contracting with the metric we obtain the Ricci scalar:
ä ȧ2
00
ij
R = g R00 + δ Rij = 6
+
.
a a2
(5.5)
(5.6)
(5.7)
Chapter 5. Cosmological Solutions in f (R) Gravity
54
It is important to note at this point that the Ricci scalar is independent of
space, meaning any terms like ∂µ R can be discarded. We can use this expression to write the R00 component in a more convenient way as well:
1
R00 = − R + 3H 2 .
2
(5.8)
At least to begin with, we will consider a perfect fluid in thermodynamic
equilibrium, in which case the stress-energy tensor takes on a particularly
simple form[15][2]:
T µν = (ρM + PM )uµ uν + pg µν
(5.9)
where ρM is the energy density of the fluid, PM is the hydrostatic pressure and uα is the fluid’s 4-velocity. Since we are using a unit system where
c = G = 1, and 4-velocity is a timelike 4-vector, the 4-velocity satisfies
uµ uν gµν = −1.
(5.10)
If we make the obvious Lorentz transformation to an inertial frame of reference that is comoving with the 4-velocity the 4-velocity is
(5.11)
u = (1, 0, 0, 0),
the inverse metric is simply

g −1
−1
0
= 
0
0
0
1
0
0
0
0
1
0

0
0

0
1
(5.12)
and the stress-energy tensor is conveniently diagonal:

ρM
 0
T =
 0
0
0
PM
0
0
0
0
PM
0

0
0 
.
0 
PM
(5.13)
Now, to derive the fluid equation, we simply note that ∇µ T µν = 0 implies
∂µ T µ0 +Γµµλ T λ0 − Γλµ0 T µλ = 0
ȧ
=⇒ ρM
˙ + 3 (ρM + PM ) = 0,
a
(5.14)
(5.15)
which is the familiar fluid equation from standard FRW cosmology using
General Relativity[2][12]. In essence we are noting that the effect of f (R)
modifications is to alter the gravitational part of the theory rather than the
matter content[31].
We are now in a position to calculate the analogue of the Friedmann
equation for Metric f (R) Gravity (which we will refer to simply as the
Chapter 5. Cosmological Solutions in f (R) Gravity
55
Friedmann equation for brevity from now on). The field equation is
1
f 0 (R)Rµν − f (R)gµν + (gµν − ∇µ ∇ν )f 0 (R) = κTµν .
2
(5.16)
First we attack the Laplace-Beltrami term using an identity from Riemannian geometry:
√
1
f 0 = √ ∂µ −gg µν ∂ν f 0
−g
√
√
1
00
0
ij
0
∂0 −gg ∂0 f + ∂i −gg ∂j f
.
=√
−g
(5.17)
(5.18)
Then, we take note of the fact that since the Ricci scalar is independent of
the spatial coordinates, we have
∂ df
= 0,
∂xi dR
and so the previous expression reduces to
√
1
00
0
0
∂0 −gg ∂0 f
f = √
−g
1 ∂
∂f 0 (R)
1 ∂
3
= 3
a (−1)
= − 3 (a3 f˙ 0 ).
a ∂t
∂t
a ∂t
(5.19)
(5.20)
(5.21)
Continuing, we find
ȧ ˙ 0 ¨0
f = − 3 f + f = −(3H f˙ 0 + f¨0 ).
a
0
(5.22)
Now, as in the derivation of the Friedmann equation, we consider the (00)
component of the field equation:
1
f 0 R00 − f g00 + 3H f˙ 0 = ρM
2
1
1
f 0 − R + 3H 2 + f + 3H f˙ 0 = ρM
2
2
1
f 0R
3f 0 H 2 = ρM − f − 3H f˙ 0 +
2
2
0
f R−f
ρM
f˙ 0
H2 =
+
−
H
6f 0
3f 0
f0
2
ä
ȧ
f
ρM
f˙ 0
H2 = +
− 0 + 0 − H 0.
a
a
6f
3f
f
(5.23)
(5.24)
(5.25)
(5.26)
(5.27)
Now we consider the spatial equation:
1
f 0 Rij − f gij − ∇i ∇j f 0 + gij f 0 = a2 δij PM
2
(5.28)
Chapter 5. Cosmological Solutions in f (R) Gravity
56
Since
√
1
gij f 0 = a2 δij √ ∂µ −gg µν ∂ν f 0
−g
!
1 ∂
∂f 0 (R)
1 ∂
3
=
a (−1)
=−
δij
a ∂t
∂t
a ∂t
= (−2a2 H f˙ 0 − a2 f¨0 )δij
Inserting the relevant expressions we obtain
a2
f 0 Rij + δij − f − 2a2 H f˙ 0 − a2 f¨0 = a2 δij PM
2
a2
=⇒ f 0 (2ȧ2 + aä) − f − 2a62H f˙ 0 − a2 f¨0 = a2 PM
2
aä − ȧ2
ȧ2 0 1
+ 3 2 f − f − 2H f˙ 0 − f¨0 = PM
a2
a
2
(5.29)
(5.30)
(5.31)
(5.32)
(5.33)
(5.34)
Now using the other field equation in the form
−
f
f0
= 3H 2 f 0 −
+ 3H f˙ 0 − ρM ,
2
R
(5.35)
we can substitute in for − f2 to obtain:
Ḣf 0 + 6H 2 f 0 − 3f 0 (2H 2 + Ḣ) + H f˙ 0 − f¨0 = ρM + PM
=⇒ −2Ḣf 0 = f¨0 − H f˙ 0 + ρM + PM .
(5.36)
These Friedmann equations are usually written together in the following
form:
1
1
f 0 R + 3H 2 + f + 3H f˙ 0 = ρM
(5.37)
2
2
−2Ḣf 0 = f¨0 − H f˙ 0 + ρM + PM .
(5.38)
For comparison, we give the regular Friedmann equations from ΛCDM cosN
= 1):
mology (in units where 8πG
c
ρM
3
ä
1
= − (ρM + 3PM ).
a
6
H2 =
(5.39)
(5.40)
Both of these theories satisfy the same continuity equation[70]
∇µ T µν = 0.
(5.41)
Chapter 5. Cosmological Solutions in f (R) Gravity
5.2
57
Accelerating Cosmologies in Metric f (R) Gravity:
Starobinsky Inflation
In 1980, Alexander Starobinsky made the proposal that particular quantum
corrections to the Einstein-Hilbert action in the form of R2 terms could result in cosmological acceleration[51]. In Starobinsky’s model, the R2 corrections arose as a consequence of of making "one-loop" quantum corrections
to the stress-energy tensor in the Einstein equations, essentially taking a
semi-classical approach to quantum gravity[71]. For this acceleration to
be relevant to inflation, the assumption is that the total energy density of
the universe is low enough at the start of the inflationary period so that a
full theory of quantum gravity is not necessary and semi-classical quantum
gravity suffices. In this section however, we will consider the alternative
viewpoint - that the R2 correction terms arise from the f (R) modification
that is made to the Einstein-Hilbert action, and that this is interpreted as the
true gravitational action for purely classical gravity[31][72].
A period of inflation can be characterised by ä(t) > 0 during some interval of time [ti , tf ], where ti is the time at which inflation began, and tf
is the time at which inflation ended[30]. Note that this definition requires
that ȧ(ti ) = ȧ(tf ). The "amount" of inflation that occurs is traditionally
quantified by "e-folds"[30], which are defined by the relation
a(tf )
N = log
,
(5.42)
a(ti )
where N is the number of "e-folds". Some quick calculations show that the
required number of e-folds to solve the horizon problem and the flatness
problem described in the introduction is in the region
(5.43)
N > 70
where the uncertainty arises as a result of the lack of knowledge of the exact
energy at the end of the inflationary period[30]. We will consider the well
known case of "slow-roll" inflation[73], which can be parameterised by the
slow-roll parameters, , η and ζ, which are defined as
=
Ḣ
,
H2
η =−
˙
,
2H
ζ=
˙ − η̇
.
H
(5.44)
For our choice of model, the function f (R) is given by
f (R) = R +
R2
.
6M
(5.45)
For this choice of f (R), the field equations of metric f (R) gravity will reduce to the Friedmann-like equations
Ḧ −
Ḣ
1
+ M 2 H = −3H Ḣ
2H
2
R̈ + 3H Ṙ + M 2 R = 0.
(5.46)
(5.47)
Chapter 5. Cosmological Solutions in f (R) Gravity
58
Periods of cosmic inflation are characterised by slow evolution of the Hubble parameter[73] - accordingly we can neglect the first two terms in the
equation, yielding the simply solvable equation
1 2
M H = −3H Ḣ
2
M2
Ḣ = −
.
6
(5.48)
We can integrate up this equation:
Z
H
Z
t
dH = −
Hi
ti
=⇒ H = Hi −
M2
dt
6
(5.49)
M2
(t − ti ),
6
(5.50)
where Hi is the Hubble parameter at the beginning of the inflationary epoch,
and ai is the scale factor at the same time. Now integrating again, we obtain
the time dependence of the scale factor:
a
t−ti
M2
Hi dt −
(t − ti )d(t − ti )
6
ai
0
ti
M2
2
(t − ti )
=⇒ a = ai exp Hi (t − ti ) −
12
Z
da
=
a
Z
t
Z
(5.51)
(5.52)
Inflation continues for as long as the "slow-roll" condition is satisfied[73]:
=−
Ḣ
M2
≈
<1
H2
6H 2
=⇒ H 2 > M 2 .
(5.53)
When the slow roll parameter is ≈ 1, we are at time tf . From the expression
for the slow roll parameter given above
H˙f
≈1
H2
M
Hf ≈ √ .
6
(5.54)
Using the results just derived, we can obtain an approximate expression for
the length of the inflationary period:
M
M2
√ = Hi −
(tf − ti )
6
6
M2
M2
M
tf = Hi +
ti − √
6
6
6
6Hi
tf ≈
+ ti .
M2
√
(5.55)
Here we have discarded the term M6 , since physical measurements have
constrained this term to be extremely small. We are now in a position to
Chapter 5. Cosmological Solutions in f (R) Gravity
59
calculate the number of e-folds this model will give us:
6Hi M 2
N ≈ Hi 2 −
M
12
6Hi
M2
2
≈
3Hi2
1
≈
,
2
M
2(tf )
(5.56)
where we have used that = M 2 /6H 2 , which comes from the definition.
Since we require N > 70 to solve the flatness problem and the horizon
problem, this constrains the slow-roll parameter to be (tf ) < 7 × 10−3 .
5.3
Late-time Acceleration from Metric f (R) Gravity
Since 1999 it has been observationally known that the rate at which the
Universe is expanding is accelerating[27][74]. This phenomena is difficult
to explain within the standard cosmological ΛCDM model without resorting to artificially introducing exotic matter (i.e matter that would violate
the strong energy condition[21] in the form of so called dark energy[75]. The
standard approach so far has been to retain General Relativity as the basis for cosmology, and to look to particle physics and quantum field theory for a solution to the dark energy problem, or to investigate so-called
"quintessence" scenarios[76]. However one could also consider the possibility that General Relativity is only an approximation to a more complicated
theory of gravity, such that the predictions were the same at small scales
but differed on the largest scales. If General Relativity could be replaced by
a such a theory that was amenable to late time acceleration without having to introduce an artificial cosmological constant, but still agreed with the
observational data, this could provide an alternative explanation for dark
energy[77]. Much of the recent interest in Metric f (R) gravity has been
fuelled by its suitability for this purpose[70][31].
Let’s see how it might be possible to obtain late-time acceleration from
Metric f (R) gravity. The Friedmann equations we derived earlier can be
expressed in the form
κ
Rf 0 − f
2
00
H = 0 ρ+
− 3H Ṙf
(5.57)
3f
2
κ
1
2Ḣ + 3H 2 = − 0 P + Ṙ2 f 000 + 2H Ṙf 00 + R̈f 00 + (f − Rf 0 ).
(5.58)
f
2
At this point we will make the assumption that f 0 > 0, which is necessary to
ensure that the effective gravitational coupling is positive (i.e that gravity
is attractive!), and we will also assume that f 00 > 0 in order to avoid the
introduction of a particular type of instability, first documented by Dolgov
and Kawasaki[66]. With the Friedmann equations in this form, we see that
with a convenient redefinition of the curvature terms on the right hand side
in terms of an effective energy density and effective pressure induced by the
geometry,
3H Ṙf 00
Rf 0 − f
−
2f 0
f0
Ṙ2 f 000 + 2H Ṙf 00 + R̈f 00 + 12 (f − Rf 0 )
=
,
f0
ρef f =
(5.59)
Pef f
(5.60)
Chapter 5. Cosmological Solutions in f (R) Gravity
60
we can rewrite these Friedmann equations as the standard Friedmann equations from ΛCDM cosmology:
κ ρ
2
H =
+ ρef f
(5.61)
3 f0
"
#
ä
κ
ρ
P
=−
+ ρef f + 3 0 + Pef f .
(5.62)
a
6
f0
f
Consider the vacuum limit, ρ → 0. This immediately forces ρef f ≥ 0 for
a flat FRW spactime, due to the first of the two Friedmann equations. The
equations then reduce to
κ
ρef f
3
i
κh
ä
= − ρef f + 3Pef f .
a
6
H2 =
(5.63)
(5.64)
From this perspective, the corrections to the geometry can be viewed as
equivalent to some effective fluid. Note that we should not push this analogy too far however - were we to include matter in this description, the
coupling is proportional to 1/f 0 , which is obviously inequivalent to standard cosmology[46]. We can then compute the effective equation of state
parameter, wef f , in standard fashion, by computing the ratio of the effective pressure and effective density:
wef f =
Ṙ2 f 000 + 2H Ṙf 00 + R̈f 00 + 12 (f − Rf 0 )
Pef f
.
=
Rf 0 −f
ρef f
− 3H Ṙf 00
(5.65)
2
We know from the condition on ρef f that the denominator of this expression must be strictly positive. In order to recreate behaviour similar to the
de-Sitter solutions, we need an effective state parameter of w = −1[26].
From this, we can derive a condition on the numerator of the effective state
parameter:
f 000
ṘH − R̈
.
=
00
f
Ṙ2
(5.66)
This condition must be satisfied in order for a metric f (R) model to produce dark energy-like behaviour.
Let us now investigate the classes of model in which this behaviour can
be realised. We first consider functions of the form f (R) = αRn . If we
assume the scale factor is some general power law, a(t) = a0 (t/t0 )β (since a
general a(t) leads to a time-varying wef f , as shown in [78]), it is then simple
to calculate the effective state parameter:
wef f = −
6n2 − 7n − 1
,
6n2 − 9n + 3
n 6= 1.
(5.67)
and
β=
−2n2 + 3n − 1
.
n−2
(5.68)
Chapter 5. Cosmological Solutions in f (R) Gravity
61
We can check this against our results for Starobinsky inflation; setting n = 2
we obtain wef f = −1 and β = ∞ as expected[51].
The second type of model we will consider here is of the form f (R) =
R−µ2(n+1) /Rn where the parameter µ is chosen to have appropriate dimensions[79]. If we again make the assumption that the scale function displays
generic power law behaviour, we can calculate that the effective parameter
of state should be
wef f = −1 +
2(n + 2)
.
3(2n + 1)(n + 1)
(5.69)
The most commonly considered model of this form is n = 1, where wef f =
−2/3. Models of this type are considered more amenable to late-time acceleration modelling due to the presence of terms that are inversely proportional to R. This is because it is desirable to have the modifications be
negligible at early times, but to have them dominate at late times[79]. Since
R is large at early times, and R → 0 at late times, these models have the
desired characteristics.
In light of the previously discussed equivalence of metric f (R) theories
to Brans-Dicke theories with ω = 0, we can recast the equations for the
effective state parameter, the effective density and the effective pressure in
terms of the scalaron: f 0 (R) = φ(R),
(φ̈ − H φ̇)
κ(φ̈ − H φ̇)
= −1 +
,
3φH 2
Rφ − f − 6H φ̇
!
φ̈ − H φ̇
φ̇ d
φ̇
+ Pef f =
.
=
log
φ
φ dt
a
wef f = −1 + 2
ρef f
(5.70)
(5.71)
We can see from this that the behaviour reduces to a regular de-Sitter type
solution when the derivative φ̇ = 0, as expected[54].
5.4
Late-time Acceleration from Palatini f (R) Gravity
In order to describe the Universe by the FRW metric, space must be homogenous and isotropic. Measurements of the isotropy of the cosmic microwave background have been carried out to an extremely high degree
of precision. Furthermore, there exists a formal result in General Relativity, known as the Ehlers-Geren-Sachs theorem[80], that guarantees that if
all freely falling observers measure the cosmic microwave background radiation to be isotropic, then that universe must be described by the FRW
spacetime. Even more astoundingly, there exists a so called "almost-EhlersGeren-Sachs" theorem, first detailed by Stoeger, Maartens and Ellis[81], that
guarantees that if all freely falling observers measure the CMB to be "almost" isotropic, then the universe will be described by an "almost-FRW"
spacetime. It was shown in 1996 by Rippl et al that this theorem can be extended to metric f (R) gravity[82].
Chapter 5. Cosmological Solutions in f (R) Gravity
62
In the paper [46] it was claimed that there was some concern that the
Palatini variant of f (R) gravity would not satisfy an EGS type theorem,
and that using the FRW line element should be regarded with a degree of
suspicion. However one of the authors of that paper subsequently used
the equivalence of Palatini f (R) to Brans-Dicke theory with ω = −3/2 to
prove such a theorem in the article [80]. We will therefore proceed in standard fashion and insert the FRW line element (for a flat geometry, where
k = 0) into the Palatini field equations. We obtain the modified Friedmann
equation
H+
1 f˙ 0 2 1 κ(ρ + 3P ) 1 f
=
+
.
2 f0
6
f0
6 f0
(5.72)
Assuming that the matter in the universe is composed of both pressure-less
dust and radiation, we can now use the trace relation,
f 0 (R)R − 2f (R) = κT,
(5.73)
along with the convenient fact that for radiation T = 0 to derive an algebraic relation between R and ρM where ρM is the energy density of the
dust, and ρ = ρM + ρradiation . This relation can be combined with energy
conservation to yield a relation for the time derivative of the Ricci scalar:
Ṙ = −
3H(Rf 0 − 2f )
.
Rf 00 − f 0
(5.74)
This relation can now be used to rewrite the modified Friedmann equation
as the standard Friedmann equation with a modified source term:
H2 =
1
2κρ + Rf 0 − f
.
6f 0 1 − 3 f 000 (Rf000 −2f0 ) 2
2 f (Rf −f )
(5.75)
By examining different choices of f (R) we can change the cosmological
α2
implications of the theory. A model in which f (R) = R − 3R
has been
shown to approach de-Sitter type solutions in the limit as the density goes
to zero in the paper [83].
63
Chapter 6
Discussion and Conclusions
In this report we have surveyed modified gravity theories based on the
f (R)-type modifications to the Einstein-Hilbert action. We have covered
both the metric f (R) formalism and the Palatini f (R) formalism in some
detail, with particular focus on the contentious issue of the Cauchy problem. The ADM formalism for metric f (R) theories was presented via the
equivalence to Brans-Dicke theories with ω0 = 0, and it was concluded that
metric f (R) theories must have a well-posed Cauchy problem by analogy
to Brans-Dicke. We conclude that the pessimism expressed about the wellposedness of the Cauchy problem in Palatini f (R) theories in references
such as [59] and [60] might be unfounded, and that future investigation
into this area is needed.
The cosmological implications of these versions of modified gravity were
also briefly surveyed, and it is concluded that certain choices of the function f R) in either formalism may lead to behaviour that mimics the effect
of dark energy and/or inflation. However we should further note here,
that in order for these models to be viable alternatives to General Relativity,
there are significantly more obstacles that must be overcome[46]. In particular, having the correct Newtonian and Post Newtonian weak field limits
is necessary for these theories to be under consideration at all. We have not
discussed this issue in this report. Furthermore, the existence of instabilities in these theories (notably of the type discussed in [66]) has not been
covered in any detail.
f (R) gravity in all its forms remains an active area of research, and
much more theoretical and experimental work needs to be done before it
can be excluded.
64
Bibliography
[1]
T. Sauer, “Albert einstein’s 1916 review article on general relativity”,
2004. arXiv: physics/0405066 [physics].
[2]
R. Wald, General Relativity. University of Chicago Press, 1984, ISBN:
9780226870328. [Online]. Available: https://books.google.co.
uk/books?id=ibSdQgAACAAJ.
[3]
C. Magnan, “Complete calculations of the perihelion precession of
mercury and the deflection of light by the sun in general relativity”,
2007. arXiv: 0712.3709 [gr-qc].
[4]
M. Anyon and J. Dunning-Davies, “Some comments on the tests of
general relativity”, ArXiv e-prints, Jun. 2008. arXiv: 0806.0528 [physics.gen-ph].
[5]
A. Le Tiec and J. Novak, “Theory of gravitational waves”, 2016. arXiv:
1607.04202 [gr-qc].
[6]
C. M. Will, “The 1919 measurement of the deflection of light”, Class.
Quant. Grav., vol. 32, no. 12, p. 124 001, 2015. DOI: 10.1088/02649381/32/12/124001. arXiv: 1409.7812 [physics.hist-ph].
[7]
J. van Dongen, “Reactionaries and einstein’s fame: ’german scientists
for the preservation of pure science,’ relativity, and the bad nauheim
meeting”, Phys. Perspect., vol. 9, pp. 212–230, 2007. DOI: 10.1007/
s00016-006-0318-y. arXiv: 1111.2194 [physics.hist-ph].
[8]
C. M. Will, “The confrontation between general relativity and experiment”, Living Rev. Rel., vol. 17, p. 4, 2014. DOI: 10 . 12942 / lrr 2014-4. arXiv: 1403.7377 [gr-qc].
[9]
B. P. Abbott et al., “Observation of gravitational waves from a binary
black hole merger”, Phys. Rev. Lett., vol. 116, no. 6, p. 061 102, 2016.
DOI : 10.1103/PhysRevLett.116.061102. arXiv: 1602.03837
[gr-qc].
[10]
T. Padmanabhan, “One hundred years of general relativity: summary,
status and prospects”, Curr. Sci., vol. 109, pp. 1215–1219, 2015. arXiv:
1512.06672 [gr-qc].
[11]
S. M. Carroll, “Lecture notes on general relativity”, 1997. arXiv: grqc/9712019 [gr-qc].
[12]
C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, Misner,
C. W., Thorne, K. S., & Wheeler, J. A., Ed. 1973.
[13]
D. Garfinkle, “Numerical relativity beyond astrophysics”, 2016. arXiv:
1606.02999 [gr-qc].
[14]
H. Kragh, “Historical aspects of post-1850 cosmology”, American Institute of Physics Conference Series, vol. 1632, pp. 3–26, Nov. 2014.
DOI : 10.1063/1.4902842. arXiv: 1410.2753.
[15]
L. A. R., An introduction to modern cosmology. Wiley, 2003.
BIBLIOGRAPHY
65
[16]
G. Weinstein, “George gamow and albert einstein: did einstein say
the cosmological constant was the ”biggest blunder” he ever made in
his life?”, ArXiv e-prints, Oct. 2013. arXiv: 1310.1033.
[17]
D. Soares, “Einstein’s static universe”, ArXiv e-prints, Mar. 2012. arXiv:
1203.4513 [physics.gen-ph].
[18]
M. Jones, R. Lambourne, and D. Adams, An Introduction to Galaxies and Cosmology. Open University, 2004, ISBN: 9780521546232. [Online]. Available: https://books.google.co.uk/books?id=
36K1PfetZegC.
[19]
R. P. Kirshner, “Hubble’s diagram and cosmic expansion”, Proceedings of the National Academy of Science, vol. 101, pp. 8–13, Dec. 2003.
DOI : 10.1073/pnas.2536799100.
[20]
J.-P. Luminet, “The rise of big bang models, from myth to theory and
observations”, 2007. arXiv: 0704.3579 [astro-ph].
[21]
G. Ellis, R. Maartens, and M. MacCallum, Relativistic cosmology. United
Kingdom: Cambridge University Press, 2012, ISBN: 9780521381154.
DOI : 10.1017/CBO9781139014403.
[22]
S. S. Bayin, “Is the universe flat?”, 2013. arXiv: 1309.5815 [gr-qc].
[23]
K. Freese, “Review of observational evidence for dark matter in the
universe and in upcoming searches for dark stars”, EAS Publ. Ser.,
vol. 36, pp. 113–126, 2009. DOI: 10 . 1051 / eas / 0936016. arXiv:
0812.4005 [astro-ph].
[24]
J. L. Feng, “Dark matter candidates from particle physics and methods of detection”, Ann. Rev. Astron. Astrophys., vol. 48, pp. 495–545,
2010. DOI: 10 . 1146 / annurev - astro - 082708 - 101659. arXiv:
1003.0904 [astro-ph.CO].
[25]
V. Lukovic, P. Cabella, and N. Vittorio, “Dark matter in cosmology”,
Int. J. Mod. Phys., vol. A29, p. 1 443 001, 2014. DOI: 10.1142/S0217751X14430015.
arXiv: 1411.3556 [astro-ph.CO].
[26]
P. Coles and F. Lucchin, Cosmology: The Origin and Evolution of Cosmic Structure. Wiley, 2002, ISBN: 9780471489092. [Online]. Available:
https://books.google.co.uk/books?id=uUFVb-DHtCwC.
[27]
S. Perlmutter and B. P. Schmidt, “Measuring cosmology with supernovae”, Lect. Notes Phys., vol. 598, pp. 195–217, 2003. DOI: 10.1007/
3-540-45863-8_11. arXiv: astro-ph/0303428 [astro-ph].
[28]
P. J. E. Peebles and B. Ratra, “The cosmological constant and dark
energy”, Rev. Mod. Phys., vol. 75, pp. 559–606, 2003. DOI: 10.1103/
RevModPhys.75.559. arXiv: astro-ph/0207347 [astro-ph].
[29]
E. Di Valentino, A. Melchiorri, and J. Silk, “Beyond six parameters: extending Λcdm”, Phys. Rev., vol. D92, no. 12, p. 121 302, 2015. DOI: 10.
1103/PhysRevD.92.121302. arXiv: 1507.06646 [astro-ph.CO].
[30]
A. D. Linde, “Particle physics and inflationary cosmology”, Contemp.
Concepts Phys., vol. 5, pp. 1–362, 1990. arXiv: hep - th / 0503203
[hep-th].
[31]
T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, “Modified gravity and cosmology”, Phys. Rept., vol. 513, pp. 1–189, 2012. DOI: 10.
1016/j.physrep.2012.01.001. arXiv: 1106.2476 [astro-ph.CO].
BIBLIOGRAPHY
66
[32]
K. Bamba and S. D. Odintsov, “Inflationary cosmology in modified
gravity theories”, Symmetry, vol. 7, no. 1, pp. 220–240, 2015. DOI: 10.
3390/sym7010220. arXiv: 1503.00442 [hep-th].
[33]
M. Nakahara, Geometry, topology, and physics, ser. Graduate student
series in physics. Bristol, Philadelphia: Institute of Physics Publishing, 2003, ISBN: 0-7503-0606-8. [Online]. Available: http://opac.
inria.fr/record=b1120046.
[34]
G. J. Olmo, “Palatini approach to modified gravity: f(r) theories and
beyond”, Int. J. Mod. Phys., vol. D20, pp. 413–462, 2011. DOI: 10 .
1142/S0218271811018925. arXiv: 1101.3864 [gr-qc].
[35]
M. Leclerc, “Canonical and gravitational stress-energy tensors”, Int. J.
Mod. Phys., vol. D15, pp. 959–990, 2006. DOI: 10.1142/S0218271806008693.
arXiv: gr-qc/0510044 [gr-qc].
[36]
D. Marolf and S. Yaida, “Energy conditions and junction conditions”,
Phys. Rev., vol. D72, p. 044 016, 2005. DOI: 10.1103/PhysRevD.72.
044016. arXiv: gr-qc/0505048 [gr-qc].
[37]
E. Dyer and K. Hinterbichler, “Boundary terms, variational principles and higher derivative modified gravity”, Phys. Rev., vol. D79,
p. 024 028, 2009. DOI: 10 . 1103 / PhysRevD . 79 . 024028. arXiv:
0809.4033 [gr-qc].
[38]
J. W. York, “Role of conformal three-geometry in the dynamics of
gravitation”, Phys. Rev. Lett., vol. 28, pp. 1082–1085, 16 Apr. 1972. DOI:
10 . 1103 / PhysRevLett . 28 . 1082. [Online]. Available: http :
//link.aps.org/doi/10.1103/PhysRevLett.28.1082.
[39]
G. W. Gibbons and S. W. Hawking, “Action integrals and partition
functions in quantum gravity”, Phys. Rev. D, vol. 15, pp. 2752–2756, 10
May 1977. DOI: 10.1103/PhysRevD.15.2752. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevD.15.2752.
[40]
N. Bodendorfer and Y. Neiman, “Imaginary action, spinfoam asymptotics and the transplanckian regime of loop quantum gravity”, Class.
Quant. Grav., vol. 30, p. 195 018, 2013. DOI: 10.1088/0264-9381/
30/19/195018. arXiv: 1303.4752 [gr-qc].
[41]
B. R. Majhi, “Entropy function from the gravitational surface action
for an extremal near horizon black hole”, Eur. Phys. J., vol. C75, no. 11,
p. 521, 2015. DOI: 10.1140/epjc/s10052- 015- 3744- 7. arXiv:
1503.08973 [gr-qc].
[42]
S. W. Hawking, “Euclidean quantum gravity”, in Recent Developments
in Gravitation: Cargèse 1978, M. Lévy and S. Deser, Eds. Boston, MA:
Springer US, 1979, pp. 145–173, ISBN: 978-1-4613-2955-8. DOI: 10 .
1007 / 978 - 1 - 4613 - 2955 - 8 _ 4. [Online]. Available: http : / /
dx.doi.org/10.1007/978-1-4613-2955-8_4.
[43]
R. Arnowitt, S. Deser, and C. W. Misner, “Dynamical structure and
definition of energy in general relativity”, Phys. Rev., vol. 116, pp. 1322–
1330, 5 Dec. 1959. DOI: 10 . 1103 / PhysRev . 116 . 1322. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRev.
116.1322.
BIBLIOGRAPHY
67
[44]
S. W. Hawking and G. T. Horowitz, “The gravitational hamiltonian,
action, entropy and surface terms”, Class. Quant. Grav., vol. 13, pp. 1487–
1498, 1996. DOI: 10 . 1088 / 0264 - 9381 / 13 / 6 / 017. arXiv: gr qc/9501014 [gr-qc].
[45]
A. Golovnev, “Adm analysis and massive gravity”, pp. 171–179, 2013.
arXiv: 1302.0687 [gr-qc]. [Online]. Available: https://inspirehep.
net/record/1217542/files/arXiv:1302.0687.pdf.
[46]
T. P. Sotiriou and V. Faraoni, “F(r) theories of gravity”, Rev. Mod. Phys.,
vol. 82, pp. 451–497, 2010. DOI: 10.1103/RevModPhys.82.451.
arXiv: 0805.1726 [gr-qc].
[47]
A. Shomer, “A pedagogical explanation for the non-renormalizability
of gravity”, 2007. arXiv: 0709.3555 [hep-th].
[48]
C. J. Isham, “Canonical quantum gravity and the problem of time”,
in 19th International Colloquium on Group Theoretical Methods in Physics
(GROUP 19) Salamanca, Spain, June 29-July 5, 1992, 1992. arXiv: grqc/9210011 [gr-qc].
[49]
P. Bull et al., “Beyond Λcdm: problems, solutions, and the road ahead”,
Phys. Dark Univ., vol. 12, pp. 56–99, 2016. DOI: 10.1016/j.dark.
2016.02.001. arXiv: 1512.05356 [astro-ph.CO].
[50]
H. A. Buchdahl, “Non-linear lagrangians and cosmological theory”,
Monthly Notices of the Royal Astronomical Society, vol. 150, no. 1, pp. 1–
8, 1970. DOI: 10.1093/mnras/150.1.1. eprint: http://mnras.
oxfordjournals.org/content/150/1/1.full.pdf+html.
[Online]. Available: http://mnras.oxfordjournals.org/content/
150/1/1.abstract.
[51]
A. Starobinsky, “A new type of isotropic cosmological models without singularity”, Physics Letters B, vol. 91, no. 1, pp. 99–102, 1980, ISSN:
0370-2693. DOI: http://dx.doi.org/10.1016/0370-2693(80)
90670 - X. [Online]. Available: http : / / www . sciencedirect .
com/science/article/pii/037026938090670X.
[52]
A. Kehagias, A. M. Dizgah, and A. Riotto, “Remarks on the starobinsky model of inflation and its descendants”, Phys. Rev., vol. D89, no.
4, p. 043 527, 2014. DOI: 10.1103/PhysRevD.89.043527. arXiv:
1312.1155 [hep-th].
[53]
M. Artymowski, Z. Lalak, and M. Lewicki, “Inflationary scenarios in
starobinsky model with higher order corrections”, JCAP, vol. 1506,
p. 032, 2015. DOI: 10 . 1088 / 1475 - 7516 / 2015 / 06 / 032. arXiv:
1502.01371 [hep-th].
[54]
J. P. M. Graa and V. B. Bezerra, “Remarks about the correspondence
between f(r) and brans?dicke gravity theories”, Mod. Phys. Lett., vol.
A30, no. 30, p. 1 550 156, 2015. DOI: 10.1142/S0217732315501564.
arXiv: 1312.1659 [gr-qc].
[55]
J. S. Kim, C. J. Kim, S. C. Hwang, and Y. H. Ko, “Scalar - tensor gravity
with scalar -matter direct coupling and its cosmological probe”, 2015.
arXiv: 1506.09121 [astro-ph.CO].
BIBLIOGRAPHY
68
[56]
C. Brans and R. H. Dicke, “Mach’s principle and a relativistic theory
of gravitation”, Phys. Rev., vol. 124, pp. 925–935, 3 Nov. 1961. DOI:
10.1103/PhysRev.124.925. [Online]. Available: http://link.
aps.org/doi/10.1103/PhysRev.124.925.
[57]
J. G. Williams, S. G. Turyshev, and D. H. Boggs, “Progress in lunar
laser ranging tests of relativistic gravity”, Phys. Rev. Lett., vol. 93,
p. 261 101, 2004. DOI: 10.1103/PhysRevLett.93.261101. arXiv:
gr-qc/0411113 [gr-qc].
[58]
I. A. Brown and A. Hammami, “Gauge issues in extended gravity and
f(r) cosmology”, JCAP, vol. 1204, p. 002, 2012. DOI: 10.1088/14757516/2012/04/002. arXiv: 1112.0575 [gr-qc].
[59]
N. Lanahan-Tremblay and V. Faraoni, “The cauchy problem of f(r)
gravity”, Class. Quant. Grav., vol. 24, pp. 5667–5680, 2007. DOI: 10.
1088/0264-9381/24/22/024. arXiv: 0709.4414 [gr-qc].
[60]
S. Capozziello and S. Vignolo, “The cauchy problem for f(r)-gravity:
an overview”, Int. J. Geom. Meth. Mod. Phys., vol. 9, p. 1 250 006, 2012.
DOI : 10.1142/S0219887812500065. arXiv: 1103.2302 [gr-qc].
[61]
M. Salgado, “The cauchy problem of scalar tensor theories of gravity”, Class. Quant. Grav., vol. 23, pp. 4719–4742, 2006. DOI: 10.1088/
0264-9381/23/14/010. arXiv: gr-qc/0509001 [gr-qc].
[62]
J. W. Moffat, “Nonsymmetric gravitational theory”, Phys. Lett., vol.
B355, pp. 447–452, 1995. DOI: 10.1016/0370-2693(95)00670-G.
arXiv: gr-qc/9411006 [gr-qc].
[63]
G. Allemandi, M. Capone, S. Capozziello, and M. Francaviglia, “Conformal aspects of palatini approach in extended theories of gravity”,
Gen. Rel. Grav., vol. 38, pp. 33–60, 2006. DOI: 10 . 1007 / s10714 005-0208-7. arXiv: hep-th/0409198 [hep-th].
[64]
V. Faraoni, “The jebsen-birkhoff theorem in alternative gravity”, Phys.
Rev., vol. D81, p. 044 002, 2010. DOI: 10 . 1103 / PhysRevD . 81 .
044002. arXiv: 1001.2287 [gr-qc].
[65]
M. E. S. Alves, O. D. Miranda, and J. C. N. de Araujo, “Probing the
f(r) formalism through gravitational wave polarizations”, Phys. Lett.,
vol. B679, pp. 401–406, 2009. DOI: 10.1016/j.physletb.2009.
08.005. arXiv: 0908.0861.
[66]
A. D. Dolgov and M. Kawasaki, “Can modified gravity explain accelerated cosmic expansion?”, Phys. Lett., vol. B573, pp. 1–4, 2003.
DOI : 10.1016/j.physletb.2003.08.039. arXiv: astro- ph/
0307285 [astro-ph].
[67]
T. P. Sotiriou, “Curvature scalar instability in f(r) gravity”, Phys. Lett.,
vol. B645, pp. 389–392, 2007. DOI: 10.1016/j.physletb.2007.
01.003. arXiv: gr-qc/0611107 [gr-qc].
[68]
T. P. Sotiriou and S. Liberati, “Metric-affine f(r) theories of gravity”,
Annals Phys., vol. 322, pp. 935–966, 2007. DOI: 10 . 1016 / j . aop .
2006.06.002. arXiv: gr-qc/0604006 [gr-qc].
BIBLIOGRAPHY
69
[69]
T. Clifton and P. K. Dunsby, “On the emergence of accelerating cosmic expansion in f(r) theories of gravity”, Phys. Rev., vol. D91, no.
10, p. 103 528, 2015. DOI: 10.1103/PhysRevD.91.103528. arXiv:
1501.04004 [gr-qc].
[70]
A. Mukherjee and N. Banerjee, “Acceleration of the universe in f (r)
gravity models”, Astrophys. Space Sci., vol. 352, pp. 893–898, 2014. DOI:
10.1007/s10509-014-1949-0. arXiv: 1405.6788 [gr-qc].
[71]
D. Muller and S. D. P. Vitenti, “About starobinsky inflation”, Phys.
Rev., vol. D74, p. 083 516, 2006. DOI: 10 . 1103 / PhysRevD . 74 .
083516. arXiv: gr-qc/0606018 [gr-qc].
[72]
L. Sebastiani and R. Myrzakulov, “F(r) gravity and inflation”, Int. J.
Geom. Meth. Mod. Phys., vol. 12, no. 09, p. 1 530 003, 2015. DOI: 10 .
1142/S0219887815300032. arXiv: 1506.05330 [gr-qc].
[73]
A. R. Liddle, P. Parsons, and J. D. Barrow, “Formalizing the slow roll
approximation in inflation”, Phys. Rev., vol. D50, pp. 7222–7232, 1994.
DOI : 10.1103/PhysRevD.50.7222. arXiv: astro-ph/9408015
[astro-ph].
[74]
A. G. Riess et al., “Observational evidence from supernovae for an
accelerating universe and a cosmological constant”, Astron. J., vol.
116, pp. 1009–1038, 1998. DOI: 10 . 1086 / 300499. arXiv: astro ph/9805201 [astro-ph].
[75]
P. J. E. Peebles and B. Ratra, “The cosmological constant and dark
energy”, Rev. Mod. Phys., vol. 75, pp. 559–606, 2003. DOI: 10.1103/
RevModPhys.75.559. arXiv: astro-ph/0207347 [astro-ph].
[76]
G. D’Amico, T. Hamill, and N. Kaloper, “Quantum field theory of interacting dark matter/dark energy: dark monodromies”, 2016. arXiv:
1605.00996 [hep-th].
[77]
A. Joyce, L. Lombriser, and F. Schmidt, “Dark energy vs. modified
gravity”, 2016. arXiv: 1601.06133 [astro-ph.CO].
[78]
S. Capozziello, S. Carloni, and A. Troisi, “Quintessence without scalar
fields”, Recent Res. Dev. Astron. Astrophys., vol. 1, p. 625, 2003. arXiv:
astro-ph/0303041 [astro-ph].
[79]
S. M. Carroll, A. De Felice, V. Duvvuri, D. A. Easson, M. Trodden, and
M. S. Turner, “The cosmology of generalized modified gravity models”, Phys. Rev., vol. D71, p. 063 513, 2005. DOI: 10.1103/PhysRevD.
71.063513. arXiv: astro-ph/0410031 [astro-ph].
[80]
V. Faraoni, “Extension of the egs theorem to metric and palatini f(r)
gravity”, 2008. arXiv: 0811.1870 [gr-qc].
[81]
W. R. Stoeger, R. Maartens, and G. F. R. Ellis, “Proving almost-homogeneity
of the universe: an almost ehlers-geren-sachs theorem”,, vol. 443, pp. 1–
5, Apr. 1995. DOI: 10.1086/175496.
[82]
S. Rippl, H. van Elst, R. Tavakol, and D. Taylor, “Kinematics and dynamics off(r) theories of gravity”, General Relativity and Gravitation,
vol. 28, no. 2, pp. 193–205, 1996, ISSN: 1572-9532. DOI: 10 . 1007 /
BF02105423. [Online]. Available: http : / / dx . doi . org / 10 .
1007/BF02105423.
BIBLIOGRAPHY
[83]
70
D. N. Vollick, “1/r curvature corrections as the source of the cosmological acceleration”, Phys. Rev., vol. D68, p. 063 510, 2003. DOI:
10 . 1103 / PhysRevD . 68 . 063510. arXiv: astro - ph / 0306630
[astro-ph].