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String Theory: An
Introduction and
Application to Cosmology
Project Report
By
Paramita Barai
In course Phys 8610:
High Energy Nuclear and Particle Physics
Instructor: Dr. Xiaochun He
Department of Physics and Astronomy
Georgia State University
10th May, 2004
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Abstract
It has been human curiosity from ancient times to understand the underlying mechanisms
of the physical processes going on in the world around us. Through our search we have learnt
several exciting facts about some of the very large scales (galaxies) in the universe and also very
small atomic and nuclear scales. Powerful physical theories have been formulated like Relativity,
Quantum Field Theory etc. which have been extremely successful to describe the nature in their
limiting cases. We have learnt that the macroscopic bodies we see around us are made up of tiny
particles in a hierarchical structure (e.g. protons, electrons, quarks etc). Space-time and matter
have been found to be integrally coupled. However, there are still several questions about the
Universe unanswered, several mysteries unsolved. One such aspect is the Grand Unification
Theory, the Holy Grail of Physics. This theory is supposed to explain all particles, their
interactions and the forces in a single integrated framework. But no such concrete theory has
been found till date.
STRING THEORY is a theory to explain the nature, which is presently at a preliminary
theoretical developmental stage. Originally proposed to describe strong forces, String Theory
today is the strongest candidate of a Unified Theory, as it has brought Relativity and Quantum
Mechanics into the same footing. The basic concept behind String Theory is that all particles of
the universe are composed of tiny (~ Planck length) fundamental strings, whose different modes
of vibrations produce the particle zoo and the interactions between them. To do so, String Theory
demands the strings to vibrate in 10 spacetime dimensions. String Theory accounts for
Gravitation by predicting a massless spin-2 particle (graviton), which no other theory does.
Hence in this aspect String Theory holds the promise of being a Grand Unified Theory sometime
in future. The problem with String Theory is that, it is not mature enough to predict something
that can be tested by present experiments and observations, to decide if the theory is right or
wrong. Whatever it predicts is at so high energy that it is much beyond the reach of our
technology today.
Still, the wonderful theoretical implications of String Theory (if it is true) makes it so
attractive. We hope that eventually it can be decided if String Theory is right or not; and if right
more and more successful physical predictions can be done with it.
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Table of Contents
Abstract …………………………………………………………………………………………...2
List of figures ……………………………………………………………………………………..5
Chapter 1
String Theory: Basic Concepts
1.1 Introduction: What is String Theory? ………………………………………………………...7
1.2 Brief history and Timeline ……………………………………………………………………8
1.3 Basic characteristics of Strings ……………………………………………………………….9
1.4 From String to Particle ………………………………………………………………………12
Chapter 2
Revolutionary features and Successful Predictions
2.1 More than just strings: Branes ………………………………………………………………16
2.2 How many dimensions ………………………………………………………………………16
2.3 How many string theories? M Theory ………………………………………………………17
2.4 T-duality ……………………………………………………………………………………..18
2.5 Predictions and advantages of String Theory ……………………………………………….20
Chapter 3
Cosmology: A Review ..
3.1 Historical developments ……………………………………………………………………..21
3.2 The Big Bang Model ………………………………………………………………………...22
3.2.1 Qualitative Overview of the model ………………………………………………………..22
3.2.2 The Starting point …………………………………………………………………………24
3.2.3 Phase transitions …………………………………………………………………………...25
3.2.4 Topological defects ………………………………………………………………………..26
Chapter 4
Problems in Big Bang Theory: Inflationary Solution
4.1 Initial Condition Problems …………………………………………………………………..26
4.2 Relic Problems ………………………………………………………………………………30
4.3 Inflation ……………………………………………………………………………………...30
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4.4 Why Inflation Occurs ………………………………………………………………………..31
4.5 Some Consequences of Inflation: Solution to Flatness and Relic problems ………………..33
4.6 Reheating ……………………………………………………………………………………36
4.7 Inflation and Particle Physics ………………………………………………………………..37
4.8 Perturbations in Inflation ……………………………………………………………………38
4.9 Initial conditions and eternal inflation ………………………………………………………39
Chapter 5
Stringy cosmology: String Theory applied to Cosmology
5.1 The beginning of time ……………………………………………………………………….40
5.2 Extra dimensions and Compactification in Cosmology ……………………………………..42
5.3 The late universe …………………………………………………………………………….46
Chapter 6
Conclusions ……………………………………………………………………………………...48
References ……………………………………………………………………………………….49
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List of figures
1.1 – Closed String Interaction: The world-sheet of 2 strings joining to form the world-sheet of
single string ………………………………………………………………………………….10
1.2 – The Feynman diagram for particle interaction …………………………………………….11
1.3 – The String Interaction diagram when consider higher order terms in perturbation theory..11
1.4 – A mode of string oscillation (assumed to do the calculations simpler) …………………...13
2.1. – The length scale showing the grand unifying feature ……………………………………21
4.1. – Infra-red map of the sky (as observed by COBE and WMAP) showing the smoothness
and fluctuations in the CMBR …………………………………………………………………..27
4.2. – 2-Dimensinal analogue of the Possible Curvature of the Universe ………………………28
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Acknowledgement
I would like to thank Dr. He for his cooperation and advice during the course of my
project. I learnt to organize my thoughts in a natural rational flow from the several discussions
that I had with him, which also helped me to gain more insight about such a complicated topic.
His help and guidance about organizing the references and constructing a coherent picture of the
topic have been very helpful. I have also gained from his course lectures, to get motivated for
this kind of project.
Paramita Barai
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Chapter 1
Introduction
1.1 What is String Theory?
The physical idea behind String Theory is utterly simple. The macroscopic objects we
see around us have been found to be composed of several small particles. In String Theory we go
a step further and say that the fundamental particles are composed of tiny vibrating strings.
Instead of many types of elementary point–like particles, we postulate that there is only a single
variety of string–like object in nature. The string is not "made up of anything", rather, it is
fundamental and other things are made up of it. Similar to musical strings, this basic string can
vibrate, and each vibrational mode can be viewed as a point-like elementary particle, just as the
modes of a musical string are perceived as distinct notes! In this way all the matter in the
universe and their interactions by various forces are unified under string theory as everything is
described by vibration of the same string.
String theory solves the deep problem of the incompatibility of the two fundamental
theories: General Relativity and Quantum Field Theory, by modifying the properties of General
Relativity when it is applied to scales on the order of the Planck length. Hence it is the most
powerful potential candidate for a Grand Unified Theory (human’s ultimate dream!).
But as of now, string theory is only at a very preliminary theoretical stage. There are no
experimental or observational proofs that the theory is true. This is because we have not yet
developed the technology to approach the high energy range where stringy behavior of particles
can come to play. Modern accelerators can only probe down to distance scales around 10–16 cm
and hence these loops of strings appear to be point objects. However, the string theoretic
hypothesis that all the particles are actually composed tiny loops, changes drastically the way in
which the particles interact on the shortest of distance scales. This modification is what allows
gravity and quantum mechanics to form a harmonious union.
In this chapter I report the historical developments leading to String Theory and then
some conceptual foundations of it. Chapter 2 gives some of the revolutionary mind boggling
ideas of string theory, which makes it so unique. There I also touch upon the successful
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predictions of string theory and how it is a potential candidate of the Grand Unified Theory.
Then to give a concrete idea on how strings can be applied to solve some of the cosmological
problems, I give a brief review of present cosmology, the Big Bang Model in Chapter 3. Then
Chapter 4 describes some of the problems of conventional Big Bang Theory, and giving the
solution through Inflation, and how does physics of strings come to play there. Chapter 5 gives a
very brief introduction to Stringy Cosmology, the kinds of calculations done when apply string
theory to cosmology. Finally, Chapter 6 gives the conclusions.
1.2 Brief history and Timeline
This section gives a very brief outline of the development of string theory [1].
In 1921 Kaluza-Klein theory was proposed which stated that electromagnetism can be
derived from gravity in a unified theory if there are 4 space dimensions instead of 3, and the 4 th
dimension is curled into a tiny circle. Kaluza and Klein made this discovery independently. But
the idea did not become so popular till 1970 which year marks the official birth of string theory.
Three particle theorists; Yoichiro Nambu, Leonard Susskind and Holger Nielsen realized that the
dual theories of 1968 to describe particle spectrum could also describe quantum mechanics of
oscillating strings. Originally String Theory was formulated to explain the strong force. But it
predicted a massless spin–2 particle (the graviton), which did not fit into the standard picture as
of that time. Later it was found that Quantum Chromo Dynamics (QCD) could describe strong
interactions successfully. String Theory made its revival after some more years. In 1971,
Supersymmetry (SUSY) was invented in two contexts at once: in ordinary particle field theory
and as a consequence of introducing fermions into string theory. SUSY holds the promise of
resolving many problems in particle theory, but when applied to string theory requires equal
numbers of fermions and bosons, so it cannot be an exact symmetry of Nature. In 1974 gravitons
were predicted to be the carrier of gravitational force. String theory using closed strings fails to
describe hadronic physics because the spin–2 excitation has zero mass. However, that makes it
an ideal candidate for the missing theory of quantum gravity!! This marks the advent of string
theory as a proposed unified theory of all four observed forces in Nature. In 1976 Supergravity
was proposed, namely Supersymmetry is added to gravity, making Supergravity. 1980 marks the
beginning of the 1st Superstring Revolution. Michael Green and John Schwarz combined string
theory with supersymmetry to yield an excitation spectrum that has equal numbers of fermions
and bosons, showing that string theory can be made totally supersymmetric. The resulting
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objects are called Superstrings. Finally in the late 1980’s and 1990’s string theory was accepted
by the mainstream physics community as an actual candidate theory uniting quantum mechanics,
particle physics and gravity. In 1991–95 several interesting works on stringy black holes in
higher dimensions lead to a revolution in understanding how different versions of string theory
are related through duality transformations. This brought in a surge of progress towards a deeper
nonperturbative picture of string theory. This period also marks the 2nd Superstring Revolution,
when Edward Witten and Townsend discovered that the 5 different prevalent string theories were
actually manifestations of a single Superstring theory called the M Theory. In 1996 Black Hole
entropy was described using Einstein’s relativity and Hawking radiation. There were hints in the
past that black holes have thermodynamic properties that need to be understood microscopically.
A microscopic origin for black hole thermodynamics was finally achieved in string theory, when
Andy Strominger and Cumrun Vafa accounted for black hole entropy using string theory.
And the search is going on. Now, in the beginning of this century, physicists all over the
world are engaged in studying string theory and predict something meaningful from it.
1.3 Basic characteristics of Strings
We are used to thinking of fundamental particles (like electrons) as point–like 0–
dimensional objects. A generalization of this is a fundamental string which is an 1–dimensional
object [1]. Strings have no thickness but do have a length of the order of the Planck Length,
LString  LPlanck 
GN
~ 10 33 cm .
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c
The strings are stretched under tension (as expressed) in order to become excited.
TString 
1
2 
The parameter  is called the string parameter and the square root of this number represents
the approximate distance scale at which string effects should become observable. In other words,
2
 is equal to the square of the string length scale, i.e.   ~ LString . The string tension is
fantastically high – equivalent to a loading of about 1039 tons.
These strings have certain vibrational modes which can be characterized by various
quantum numbers such as mass, spin, etc. The basic idea is that each mode carries a set of
quantum numbers that correspond to a distinct type of fundamental particle. This is the ultimate
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unification: all the fundamental particles we know can be described by one object, a string! A
string oscillates in space and time, and as it oscillates, being a 1–dimensional entity, it sweeps
out a 2–dimensional surface in spacetime that we call a world sheet (compared to the 1–
dimensional world line path of a particle). The dynamics of a free relativistic particle is
described by requiring the path swept out by it to be extremal in length. By analogy, the
dynamics for the string is based on the requirement that the area of the surface swept out is
extremal. In addition, there are quantum fields which are defined on the world-sheet, and which
enter into the dynamics too [2].
There is essentially only one adjustable constant in the model, the tension of the string,
which determines the characteristic mass scale. The characteristic length scale; related to the
tension of the string, which is the one free parameter in the theory; is the Planck length. This is
so tiny that even on the scale of particle physics the tube is so narrow as to resemble a line – just
like the world-line of an elementary particle.
Strings interact by splitting and joining. For example the annihilation of two closed
strings into a single closed string occurs with an interaction that looks like the following [4].
Fig 1.1. – Closed String Interaction:
The world-sheet of 2 strings joining to form the world-sheet of a single string
Here the worldsheet of the interaction is a smooth surface. This essentially accounts for another
nice property of string theory. It is not plagued by infinities in the way that point particle
quantum field theories are. The analogous Feynman diagram (for the above interaction of
strings) in a point particle field theory is the following.
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Fig 1.2. – The Feynman diagram for particle interaction
In this case the interaction point occurs at a topological singularity in the diagram (where the 3
world-lines intersect). This leads to a break down of the point particle theory at high energies.
The leading contribution to this process of constructing Feynman diagram analogues of
string interactions is called a tree level interaction. To compute quantum mechanical
amplitudes using perturbation theory we add contributions from higher order quantum
processes. Perturbation theory provides good answers as long as the contributions get smaller
and smaller as we go to higher and higher orders. Then we only need to compute the first few
diagrams to get accurate results. In string theory, higher order diagrams correspond to the
number of holes (or handles) in the world sheet, as depicted nicely in the following diagram,
which shows several higher order terms.
Fig 1.3. – The String Interaction diagram when consider higher order terms in
perturbation theory
The nice thing about this is that at each order in perturbation theory there is only one
diagram. [For a comparison, in point particle field theories the number of diagrams grows
exponentially at higher orders.] The problem lies in the fact that to extract answers from
diagrams with more than about two handles is very difficult due to the complexity of the
mathematics involved in dealing with these surfaces. Perturbation theory is a very useful tool for
studying the physics at weak coupling, and most of our current understanding of particle physics
and string theory is based on it. However it is far from complete. The answers to many of the
deepest questions will only be found once we have a complete non-perturbative description of
string theory.
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Hence it is clear that in the string-theory analogues to Feynman diagrams there are
none of the singularities that in field theory lead to infinities. So the Quantum Gravity that is
contained in string theory is free from infinities; i.e. string theory contains a consistent quantum
theory of gravity, something that had eluded physicists for at least half a century. If one goes to
the limit in which the world-sheet tube resembles a line to see how this comes about, it is
apparent that Einstein's theory has to be modified, but in a very strictly determined way. The
corrections that need to be introduced are quite negligible when one is concerned with the
gravitational phenomena so far observed, because quantum effects are far too small to be
relevant.
String theories are classified according to whether or not the strings are required to be
closed loops, and whether or not the particle spectrum includes fermions. In order to include
fermions in string theory, there must be a special kind of symmetry called Supersymmetry
(SUSY), which means that for every boson (a particle, of integral spin that transmits a force)
there is a corresponding fermion (a particle, of half-integral spin that makes up matter). So
SUSY relates the particles that transmit forces to the particles that make up matter.
Supersymmetric partners to currently known particles have not been observed as of date in
particle experiments, but theorists believe this is because supersymmetric particles are too
massive to be detected using present-day technology. Particle accelerators could be on the verge
of finding evidence for high-energy supersymmetry in the next decade. Evidence for
supersymmetry at high energy would be a very compelling evidence that string theory is a good
mathematical model for nature at the smallest distance scales.
In string theory, all of the properties of elementary particles – charge, mass, spin, etc –
come from vibration of the same string. The easiest to see is mass. The more frantic the
vibration, the more energy. And since mass and energy are the same thing, higher mass comes
from higher vibration.
1.4 From String to Particle
String theory proclaims that all the observed particle properties (mass, charge, spin etc)
are reflections of the various ways in which a string can vibrate. Just as the strings on a piano or
violin have some resonant (natural) frequencies at which they prefer to vibrate – the same holds
true for the loops of string theory. Each of the preferred patterns of vibration of a string
appears as a particle whose mass, spin and charge are determined by the string's
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oscillatory pattern. The electron is a string vibrating one way; the up-quark is a string vibrating
another way. Force mediator particles like photons, weak gauge bosons, and gluons are yet other
resonant patterns of string vibration. There is even a mode describing the graviton which is the
particle carrying the force of gravity. Particle properties in string theory are the manifestations of
one and the same physical feature: the resonant patterns of vibration of fundamental loops of
string. The same idea applies to the forces of nature as well. Hence everything, all matter and all
forces, is unified under the microscopic string oscillations – the notes that strings can play [1, 2,
3, 4].
The strings are so tiny (~1016 smaller than the present distances probed by most powerful
accelerators) that when we see them over large scale their vibration appears to us as point
particles. We are not yet technologically powerful to resolve the stringy nature of particles. But
this formalism can be followed mathematically too. In this section, a crude review of the
equations of motion of the strings are given starting with the well-known notion of classical
strings, aiming to describe particle behavior from there [1].
Consider the wave equation for a classical macroscopic string with a tension T and a
mass per unit length . If the string is described in coordinates as in fig. 1.4, where x is the
distance along the string and y is the height of the string, and the string oscillates in time t.
Fig 1.4. – A mode of string oscillation (assumed to do the calculations simpler)
Then the equation of motion is the one-dimensional wave equation
2
 2 y  x, t 
T  2 y  x, t 
2  y  x, t 


v
,
w
 x 2
t 2
x 2
where vw is the wave velocity along the string. For the first case, assume this is a nonrelativistic
string, i.e. one where the wave velocity much smaller than the speed of light.
When solving the equations of motion, we need to know the boundary conditions of
the string. If we suppose that the string is fixed at each end and has an unstretched length L. The
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general solution to this equation can be written as a sum of normal modes, here labeled by the
integer n, such that

nv w t
nv w t 
nx

.
yx, t     a n cos
 bn sin
 sin
L
L 
L
n 1 
The condition for a normal mode is that the wavelength be some integral fraction of twice the
string length, or  n 
nv
2L
. The frequency of the normal mode is then f n  w . The string wave
n
2L
velocity vw increases as the tension of the string is increased, and so the normal frequency of the
string increases as well.
Now consider relativistic strings, i.e. where the wave velocity is comparable to the speed
of light. According to Einstein's theory, a relativistic equation has to use coordinates that have
the proper Lorentz transformation properties. In the nonrelativistic string, there exists a clear
difference between the space coordinate along the string, and the time coordinate. But in a
relativistic string theory, we wind up having to consider the world sheet of the string as a twodimensional spacetime of its own, where the division between space and time depends upon the
observer. The classical equation can be written as,
2

 2 X   , 
2  X  , 

c
,
 2
 2
where  and  are coordinates on the string world sheet representing space and time along the
string, and the parameter c2 is the ratio of the string tension to the string mass per unit length.
These equations of motion can be derived from Euler-Lagrange equations from an
action based on the string world sheet,
S 
1
dd
4  
h h nm  m X   n X  .
In this expression, the spacetime coordinates X of the string are also fields X in a twodimension field theory defined on the surface that a string sweeps out as it travels in space. The
partial derivatives are with respect to the coordinates  and  on the world sheet and hmn is the 2dimensional metric defined on the string world sheet.
The general solution to the relativistic string equations of motion looks very similar to the
classical nonrelativistic case above. The transverse space coordinates can be expanded in normal
modes as,
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
1 
nc
nc
X i  ,   x i  x i  i 2    ni  cos
 i sin
L
L

n0 n
n

.
 cos
L

The string solution above is for an open string with floppy ends i.e. it isn't tied down at either
end and so travels freely through spacetime as it oscillates. For a closed string, the boundary
conditions are periodic, and the resulting oscillating solution looks like two open string
oscillations moving in the opposite direction around the string. These two types of closed string
modes are called right-movers and left-movers, and this difference is important in the
supersymmetric heterotic string theory. The analysis till now was for classical string.
When we add quantum mechanics by making the string momentum and position obey
quantum commutation relations, the oscillator mode coefficients have the commutation relations,


m

,  n  m m  n  .
The quantized string oscillator modes wind up giving representations of the Poincaré group (the
inhomogeneous Lorentz Group), through which quantum states of mass and spin are classified
in a relativistic quantum field theory. So this is how the elementary particle arises in string
theory.
In the generic quantum string theory, there are quantum states with negative norm, also
known as ghosts. This happens because of the minus sign in the spacetime metric, which implies


that,  m0 ,  n0  m m  n .
So there ends up being extra unphysical states in the string spectrum. In 10 spacetime
dimensions, these extra unphysical states wind up disappearing from the spectrum. Therefore
string quantum mechanics is only consistent if the dimension of spacetime is 10.
By looking at the quantum mechanics of the relativistic string normal modes, one can
deduce that the quantum modes of the string look just like the particles we see in spacetime,
with mass (MJ) that depends on the spin (J) according to the formula,
J   M J ,
2
which is the well known relation obtained in particle physics from Regge plots.
Boundary conditions are important for string behavior. Strings can be open, with ends
that travel at the speed of light, or closed, with their ends joined in a ring. One of the particle
states of a closed string has zero mass and 2 units of spin, the same mass and spin as a
graviton, the particle that is supposed to be the carrier of the gravitational force.
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For string theory to be a successful theory of quantum gravity, the average size of a string
should be somewhere near the length scale of quantum gravity, called the Planck length, which
is about 10–33 cm.
Chapter 2
Revolutionary features and Successful Predictions
2.1 More than just strings: Branes
Superstring theory is not just a theory of 1-dimensional objects called strings. There are
higher dimensional objects in string theory with dimensions from zero to nine, called p-branes
(objects with p dimensions). In terms of branes, what we usually call a membrane would be a 2brane, a string would be a 1-brane and a point would be called a 0-brane [2, 3].
Now the question arises how do we visualize a p-brane. In very simple words to explain
physically, we can think of a brane as being a slice through the higher dimensional world
that string theory says exists.
Mathematically speaking, a p-brane is a spacetime object that is a solution to the
Einstein’s equations in the low energy limit of superstring theory, with the energy density of the
non-gravitational fields confined to some p-dimensional subspace of the 9 space dimensions in
the theory. (Point to note, superstring theory lives in 10 spacetime dimensions, which means 1
time dimension plus 9 space dimensions.) For example, in a solution with electric charge, if the
energy density in the electromagnetic field was distributed along a line in spacetime, this 1dimensional line would be considered a p-brane with p=1.
A special class of p-brane in string theory is called D-brane. Roughly speaking, a Dbrane is a p-brane where the ends of open strings are localized on the brane. A D-brane is like a
collective excitation of strings. D-branes are important in understanding black holes in string
theory, especially in counting the quantum microstates that lead to black hole entropy, which
was a very big accomplishment for string theory.
2.2 How many dimensions?
Before string theory won the full attention of the theoretical physics community, the most
popular unified theory was an 11–dimensional theory of supergravity, which is supersymmetry
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combined with gravity. The 11–dimensional spacetime was to be compactified on a small 7–
dimensional sphere, leaving 4 spacetime dimensions visible to observers at large distances.
This theory didn't work as a unified theory of particle physics, because it doesn't have a
sensible quantum limit as a point particle theory. But this 11 dimensional theory did not die. It
eventually came back to life in the strong coupling limit of superstring theory in 10 dimensions
[5].
Now the question is, how could a superstring theory with 10 spacetime dimensions
turn into a supergravity theory with 11 spacetime dimensions? We will see in subsequent
sections that, duality relations between superstring theories relate very different theories, equate
large distance with small distance, and exchange strong coupling with weak coupling. So there
must be some duality relation that can explain how a superstring theory that requires 10
spacetime dimensions for quantum consistency can really be a theory in 11 spacetime
dimensions after all.
But in our known world around we see only 3–space (x, y, z) and 1–time dimension (t).
Then where are these extra 6 (or 7) dimensions which must exist if string theory is to be true.
There are two main ways to explain these missing dimensions. One is to propose that these extra
dimensions have compactified into a tiny space of size of string scale, which our present
technology cannot resolve. The other idea is that we are trapped on the surface of a higher
dimensional brane, for which we can’t leave the brane and hence can’t perceive the extra
dimensions.
2.3 How many string theories? M Theory
At one time, string theorists believed that there were 5 distinct superstring theories: type
I, types IIA and IIB, heterotic SO(32) and E8XE8 string theories. Each of these was a valid
String Theory differing in the ways of formulation, and each gave valid predictions. The thinking
was that out of these five candidate theories, only one was the actual correct Theory of
Everything, and that theory was the theory whose low energy limit, with 10 dimensions
spacetime compactified down to 4, matched the physics observed in our world today. The other
theories would be nothing more than rejected string theories, mathematical constructs not blessed
by Nature with existence.
But now it is known that the five superstring theories are connected to one another as
if they are each a special limiting case of some more fundamental theory. In the mid-nineties it
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was learned that superstring theories are related by duality transformations known as T-duality
and S-duality. These dualities link the quantities of large and small distance, and strong and
weak coupling, limits that have always been identified as distinct limits of a physical system in
both classical and quantum physics. These duality relationships between string theories have
sparked a radical shift in our understanding of string theory, and have led to the reasonable
expectation that all five superstring theories -- type I, types IIA and IIB, heterotic SO(32) and
E8XE8 -- are special limits of a more fundamental theory. This more fundamental theory,
called the M-Theory exists in 11 spacetime dimensions [1, 3].
2.4 T-duality
The duality symmetry that obscures our ability to distinguish between large and small
distance scales is called T-duality. This comes about from the compactification of extra space
dimensions in a 10–dimensional superstring theory. Let's take one of the 9 directions, Xi in the
flat 10–dimensional spacetime, and compactify it into a circle of radius R, so that,
X i  X i  2R .
A particle traveling around this circle will have its momentum quantized in integer multiples of
1/R, and a particle in the nth quantized momentum state will contribute to the total mass squared
of the particle as,
mn 
2
n2
.
R2
A string can travel around the circle, too, and the contribution to the string mass squared is the
same as above.
A closed string can also wrap around the circle, something a particle cannot do. The
number of times the string winds around the circle is called the winding number, denoted as w,
and w is also quantized in integer units. Tension is energy per unit length, and the wrapped string
has energy from being stretched around the circular dimension. The winding contribution Ew to
the string energy is equal to the string tension Tstring times the total length of the wrapped string,
which is the circumference of the circle multiplied by the number of times w that the string is
wrapped around the circle.
E w  2wRTstring 
wR
,

19
where as already introduced before,    Ls , tells us the length scale Ls of string theory.
2
The total mass squared for each mode of the closed string is

n 2 w2 R 2 2
~
m  2 

N  N 2
2

R

~
with, N  N  nw
2

The integers N and Ñ are the number of oscillation modes excited on a closed string in the rightmoving and left-moving directions around the string. The above formula is invariant under the
exchange:
R

R
,n  w.
In other words, we can exchange compactification radius R with radius '/R if we exchange the
winding modes with the quantized momentum modes. This mode exchange is the basis of the
duality known as T-duality. If the compactification radius R is much smaller than the string
scale Ls, then the compactification radius after the winding and momentum modes are exchanged
is much larger than the string scale Ls. So T-duality obscures the difference between
compactified dimensions that are much bigger than the string scale, and those that are much
smaller than the string scale.
T-duality relates type IIA superstring theory to type IIB superstring theory, and it
relates heterotic SO(32) superstring theory to heterotic E8XE8 superstring theory. A duality
relationship between IIA and IIB theory is very unexpected, because type IIA theory has
massless fermions of both chiralities, making it a non-chiral theory, whereas type IIB theory is a
chiral theory and has massless fermions with only a single chirality.
T-duality is something unique to string physics. It's something point particles cannot do,
because they don't have winding modes. If string theory is a correct theory of Nature, then this
implies that on some deep level, the separation between large vs. small distance scales in physics
is not a fixed separation but a fluid one, dependent upon the type of probe we use to measure
distance, and how we count the states of the probe.
This idea may sound that it goes against all traditional physics, but this is indeed a
reasonable outcome for a quantum theory of gravity, because gravity comes from the metric
tensor field that tells us the distances between events in spacetime. There is a similar duality
concept called S-duality which relates strong and weak coupling constants with each other [1].
20
Hence String Theory seems to be very interesting and rich theory where distance scales,
coupling strengths and even the number of dimensions in spacetime are not fixed concepts but
fluid entities that shift with our point of view.
2.5 Predictions and Advantages of String Theory
String Theory has predicted several solutions mechanisms in various fields of physics and
hence has turned to be the most powerful potential candidate for the Grand Unified Theory
[5, 6, 7, 8]. In low energy limit, string theory predicts the right types of particles and the right
types of interactions among them, which corresponds to the precise particle behavior we see in
the Standard Model. The string solutions also generally include Yang–Mills gauge theories.
String theory explains gravitation, by predicting the massless spin–2 particle, graviton,
the quantum of gravitation, approximated at low energies by General Relativity. No other
quantum theory has this feature of unifying gravity, which makes string theory ahead of all.
String theory predicts Supersymmetry (SUSY) at low energies (the electroweak scale),
which says for every fermions existing in the universe there are corresponding bosons.
String theory also has several successful predictions in cosmology, to be discussed in the
next chapter.
String theory holds the potential to combine General Relativity with Quantum theory
and give a Grand Unified Theory. This Grand Unification is a kind of Holy Grail of Physics.
This aims to formulate a theory that will describe virtually everything of the universe. It will
predict the fundamental particles we see around us, and the various forces and interactions
between them which combine to give the world around us. Another important feature of the
unification is, there should be a very less number of input parameters to the unified theory.
Some of the grand unifying feature of String theory are as following. It unifies in one
single 4–dimensional quantum theory all known forces and elementary particles. In the process,
it reduces the number of parameters that need to be determined by experiments from between
20 and 30 (assuming massive neutrinos) to 1 (the string length or tension), not counting
Planck's constant and the velocity of light [9, 10].
Hence, Superstring theory holds out the promise of a unification of all the successes of
the standard model with a quantum theory of gravity, with no awkward infinities and with no
arbitrary constants to be adjusted once the string tension has been chosen.
21
Fig 2.1 – The length scale showing the grand unifying feature
Chapter 3
Cosmology: A Review
In most simple words, Cosmology is the study of structure and evolution of the universe;
how does the universe look like, how did it begin and evolve to came to the present universe as
we see it today [11, 12].
3.1 Historical developments
When Einstein formulated the General Theory of Relativity, he found that it was
incompatible with a static universe; as the equations predicted that the universe must either be
expanding or shrinking. That time, the notion that the universe is static, was so strong that
Einstein altered the equations of relativity, by forcing the Cosmological Constant in General
Relativity equation to become zero, in order to allow for a static solution. But then Edwin Hubble
found that the universe was indeed expanding by observing far off galaxies receding away from
us. So Einstein retracted this alteration, calling it the biggest blunder of his life. From that point
onwards the prevailing scientific viewpoint has been that of an expanding universe that at
earlier times was much hotter and denser than it is today. Extrapolating this expansion
backwards, we find that at a specific time in the past the universe would have been infinitely
22
dense. This time, the beginning of the universe's expansion, has come to be known as the event
of BIG BANG.
The Big Bang model has been extremely successful at explaining known aspects of the
universe and correctly predicting new observations. Nonetheless, there are certain problems with
the model. There are several features of our current universe that seem to emerge as strange
coincidences in Big Bang theory. There are some predictions of the theory that are in
contradiction with observation. These problems have motivated people to look for ways to
extend or modify the theory without losing all of the successful predictions it has made. In 1980
a theory was developed by Alan Guth, that solved many of the problems plaguing the big bang
model while leaving intact its basic structure. More specifically, this new theory modified our
picture of what happened in the 1st fraction of a second of the universe's expansion. This change
in our view of that 1st fraction of a second has proven to have profound influences on our view of
the universe and the Big Bang itself. This new theory is called INFLATION.
3.2 The Big Bang Model
At first I give a broad qualitative discussion of the Big Bang model, followed by some
mathematical intricacies.
3.2.1 Qualitative Overview of the Model
The most popular and accepted theory about the large-scale structure and history of the
universe is the Big Bang model. According to this model the universe is expanding and at very
early times was a nearly uniform expanding collection of high energy, high temperature
particles. The small random differences in density that existed from one point to another created
tiny inhomogeneities called density fluctuations. As the universe expanded and cooled these
small inhomogeneities were then amplified by gravity. The matter in regions with slightly higher
than average density gravitationally collapsed to form the structures we see today such as
clusters and galaxies. Extrapolating the expansion backwards, (i.e. effectively contracting the
size of the universe) we get at earlier times a nearly homogeneous fireball having higher
temperatures and densities, ultimately reaching infinite density at a moment about 13.7 billion
years ago. That moment is called the Big Bang, the ‘moment of creation of our present
universe’ in a very crude way.
There are no direct evidences of the Big Bang event (as it involves the moment of
creation of the Universe, about which we have no idea of). However, we can still extrapolate
23
further backwards and assume the Big Bang model to be an accurate description of the universe
all the way back to the Big Bang Singularity (when everything was crunched into a point at
infinite temperature and density). But, such a complete extrapolation of the theory is not
possible, because of several limitations of our theories of high energy physics. When we think
about extrapolating the Big Bang model backwards, we are referring to running the equations of
General Relativity backwards to earlier times and higher densities. We know, however, that
General Relativity ceases to be valid when we try to describe a region of spacetime whose
density exceeds a certain value known as the Planck density, roughly 1093 g/cm3. If we try to
consistently apply Quantum Mechanics and General Relativity at such a density we find that
quantum fluctuations of spacetime become important, and we have no theory that describes such
a situation, as Quantum Mechanics and General Relativity, by themselves, are incompatible with
each other. The Planck density is enormous. It corresponds to the mass of 100 billion galaxies
being squeezed into a space the size of an atomic nucleus. If we could extrapolate General
Relativity all the way back to the Big Bang the universe would have gone from infinite density to
the Planck density in roughly 10–43 seconds (the Planck time). So considering something had
happened, let 3 minutes after the Big Bang is equivalent to considering that it happened 3
minutes after the time the universe was at Planck density, i.e. the Planck Time.
At any time before the Planck time all existing theories break down. This means that all
accepted theories like, General Relativity, Quantum Field Theory etc. can’t be defined any more
as they give infinite quantities as answers.
Since we can't describe physics at densities higher than the Planck density, the best we
can do in using the Big Bang model to describe the very early universe is the following. We
assume, at some point in the past the density of the universe was above the Planck density. We
don't know what physics governs such a case so we can make no predictions based on it.
Somehow this super-Planckian state (sometimes called Spacetime Foam) gave rise to at least
one region of sub-Planckian density with the right initial conditions to produce the universe we
currently see. Here the term "initial conditions" for our universe, means that the state in which
the observable part of the universe was in after the density first became sub–Planckian and the
universe (or at least this region of it) could thus be described by known theories of physics. So
we apply these known theories (Relativity, Quantum Mechanics etc.) from after the Planck Time
of the Big Bang Singularity.
24
This Big Bang model after the Planck Time is in perfect accord with the theory of
General Relativity, which predicts that a homogeneous universe would expand and cool in
exactly that way. Moreover, there have been many observational confirmations of the Big Bang
model. Some of these are as following. Distant objects have been found to be moving away from
us radially, obtained from measuring redshifts of distant galaxies observing their spectral line
shift. The microwave radiation left over from the early universe from the epoch of recombination
is seen today in the sky as isotropic blackbody radiation at ~ 2.7 ° K, called the CMBR (Cosmic
Microwave Background Radiation). The abundances of light elements formed in the first few
minutes after the Big Bang have been measured, and hence an estimate of primordial
nucleosysthesis (discussed in next paragraph) have been done.
For roughly 3 minutes after the Big Bang the temperature of the universe was so high that
protons and neutrons couldn't bind together into nuclei; all the particles had so much energy that
the forces that hold nuclei together were too weak to make them stick to each other. Thus for
those first 3 minutes the only element in the universe was hydrogen, i.e. single protons not bound
to anything else. As the universe expanded and cooled it eventually reached a temperature where
the protons and neutrons could bind together, and different elements were formed. The formation
of these nuclei from their constituent particles (i.e. protons and neutrons) is known as
Nucleosynthesis.
String Theory is applicable before the Planck Time.
3.2.2 The Starting point
Here I give a review on some of the mathematical principles applied from the time after
Planck Time of the Big Bang [15]. This very early universe, had it’s energy density as radiationdominated. The universe on large scales can be accurately described by a perturbed RobertsonWalker metric. On thermodynamic grounds (backed up by evidence from CMB anisotropy) it
seems likely that these perturbations are growing rather than shrinking with time, at least in the
matter-dominated era; it would require extreme fine-tuning of initial conditions to arrange for
diminishing matter perturbations. Thus, the early universe was smoother as well.
Let us trace the history of the universe as we reconstruct it given these conditions (in the
previous paragraph) plus our current best guesses at the relevant laws of physics. We start at a
25
time infinitesimally after the reduced Planck scale (which corresponds to an energy of,
EPlanck  1/ 8G ~ 1018 GeV ).
[The “ordinary” Planck scale is simply 1 / G ~ 1019 GeV . It is only an accident of history (Newton's law of
gravity predating general relativity, or for that matter Poisson's equation) that it is defined this way".]
We imagine an expanding universe with matter and radiation in a thermal state, perfectly
homogeneous and isotropic (we will put in perturbations later), and all conserved quantum
numbers set to zero (no chemical potentials). Note that asymptotic freedom makes our task much
easier; at the high temperatures we are concerned with, QCD (and possible grand unified gauge
interactions) are weakly coupled, allowing us to work within the framework of perturbation
theory.
3.2.3 Phase transitions
The high temperatures and densities characteristic of the early universe typically put
matter fields into different phases than they are in at zero temperature and density, and often
these phases are ones in which symmetries are restored. Consider a simple theory of a real scalar
field  with a Z2 symmetry   . The potential at zero temperature can be written as
V  , T  0  
2
2

 2   4 . Interactions with a thermal background typically give positive
4
contributions to the potential at finite temperature: V  , T   V  ,0  T 2 2   , where  = /8
in the theory.
1 

At high T, the coefficient of 2 in the effective potential,  T 2   2  , will be a
2 

positive number, so the minimum-energy state will be one with vanishing expectation value,
  0 . The Z2 symmetry is unbroken in such a state. As the temperature declines, eventually
the coefficient will be negative and there will be two lowest-energy states, with equal and
opposite values of  . A zero – temperature vacuum will be built upon one of these values,
which are not invariant under the Z2; we therefore say the symmetry is spontaneously broken.
The dynamics of the transition from unbroken to broken symmetry is described by a phase
transition, which might be either first–order or second–order. A first–order transition is one in
which first derivatives of the order parameter (in this case ) are discontinuous; they are
26
generally dramatic, with phases coexisting simultaneously, and proceed by nucleation of bubbles
of the new phase. In a second-order transition only second derivatives are discontinuous; they are
generally more gradual, without mixing of phases, and proceed by “spinodal decomposition”.
3.2.4 Topological defects
Note that, post–transition, the field falls into the vacuum manifold (the set of field values
with minimum energy – in our current example it’s simply two points) essentially randomly. It
will fall in different directions at different spatial locations x1 and x2 separated by more than one
correlation length of the field. In an ordinary FRW universe, the field cannot be correlated on
scales larger than approximately H–1, as this is the distance to the particle horizon (as will be
discussed later in the section on inflation). If  x1   v and  x2   v , then somewhere in
between x1 and x2,  must climb over the energy barrier to pass through zero. Where this happens
there will be energy density; this is known as a topological defect (in this case a defect of
codimension one, a domain wall). The argument that the existence of horizons implies the
production of defects is known as the Kibble mechanism.
Chapter 4
Problems in Big Bang Theory: Inflationary Solution
Despite the great success of the conventional cosmology, there remain certain problems
with extending the Big Bang model back to the Planck Time. These can be categorized as initial
condition problems and relic problems. These problems and their solutions are discussed in the
following sections. The initial condition problem embodies two interesting conceptual puzzles:
flatness and isotropy. The leading solution to these problems is the inflationary universe
scenario, which has become a central organizing principle of modern cosmology.
4.1 Initial Condition Problems
We want to understand, what state was the universe in when it first dropped below the
Planck density; whether it was just a random collection of energy. In order to give rise to a
universe at all like the one we see the early universe had to have a number of very precisely
tuned features. Since there is no present accepted theory that can predict about what gave rise to
the initial conditions for our universe; these features are simply assumed, or introduced into the
27
theory by hand. [This is something standard Big Bang theory can’t predict. But the Grand
Unified Theory (GUT) (when formulated) is supposed to give the initial conditions. This is
a great triumph of the GUT theory, to predict the initial conditions of the early universe
which led to its universe in its present state as we see today].
Two of these features unexplained by Big Bang are homogeneity and flatness. The early
universe was nearly homogeneous, i.e. the same or uniform in all places. Our most direct
measure of this uniformity comes from observing the Cosmic Microwave Background Radiation
(CMBR) that was emitted when the universe was roughly 300,000 years old. The intensity of
this radiation is a direct measure of how dense the universe was at that time. Looking at this
background radiation coming to us from different directions, and examining its smoothness;
shows that the largest density differences from one point to another were about only about one
part in 100,000. If the universe really had been such less homogeneous it would not have given
rise to the smooth distribution of galaxies we see filling the sky. However if it had been exactly
homogeneous, then clumps of matter like galaxies and us would never have emerged at all. The
big bang model offers no explanation for why the universe emerged in this nearly—but not
perfectly—homogeneous state; so that matter were formed and also at the same time we see the
CMBR significantly smooth.
Fig. 4.1. – Infra-red map of the sky (as observed by COBE and WMAP) showing the
smoothness and fluctuations in the CMBR.
28
The second initial condition has to do with something called curvature. General
Relativity says that the universe can be closed, i.e. curved inward like the surface of a ball; or
open, meaning curved outward like the surface of a saddle, or flat, meaning it has no curvature.
Fig 4.2. – 2-Dimensinal analogue of the Possible Curvature of the Universe
These different kinds of curvature cause the universe to evolve in different ways. A
closed universe will eventually stop expanding and recollapse, while an open universe will tend
to fly apart more and more quickly. A flat universe (i.e. with no curvature) must have a particular
density value; mathematically,  = 1 (where  is the Density Parameter). For the Big Bang
model to work the universe at the time of Planck density must have been almost precisely flat;
the curvature couldn't have exceeded one part in 1059. If it were slightly more curved than
this (closed) it would have recollapsed long ago and if it were slightly less so (open) it would
have flown apart so quickly galaxies would never have formed. This apparent coincidence—the
universe initially having exactly the curvature required to survive to later times and form
galaxies—is known as the flatness problem.
29
Mathematically, the flatness problem comes from considering the Friedmann equation in
a universe with matter and radiation but no vacuum energy: H 2 
1
 M   R   k2 .
3
3M P
a
The curvature term –k / a2 is proportional to a-2, while the energy density terms fall off faster
with increasing scale factor, M  a–3 and R  a–4. This raises the question of why the ratio
(ka–2)/(/3MP2) isn't much larger than unity, given that a has increased by a factor of perhaps
1028 since the grand unification epoch. Said another way, the point  = 1 is a repulsive fixed
point – any deviation from this value will grow with time, so why do we observe  ~ 1 today?
The isotropy problem is also called the horizon problem since it stems from the
existence of particle horizons in Friedmann Robertson Walker cosmologies. Horizons exist
because only a finite amount of time has passes since the Planck Time, and thus there is only a
finite distance that photons can travel within the age of the universe. Consider a photon moving
along a radial trajectory in a flat universe.A radial null path obeys;
0  ds 2  dt 2  a 2 dr 2 .
So the comoving distance traveled by such a photon between times t1 and t2 is,
t2
dt
.


a
t
t1
r  
To get the physical distance as it would be measured by an observer at time t1, we need to
multiply r by a(t1). For a universe dominated by an energy density   a–n, this becomes,
r 


1  2 
n / 21
,

 a
n

2
a H 

n/2

where the  subscripts refer to some fiducial epoch (the quantity an / 2 H  is a constant). The
horizon problem is simply the fact that the CMB is isotropic to a high degree of precision, even
though widely separated points on the last scattering surface are completely outside each other’s
horizons. Choosing a0 = 1, the comoving horizon size today is approximately H0 –1, which is also
the approximate comoving distance between us and the surface of last scattering (since, of the
comoving distance traversed by a photon between a redshift of infinity and a redshift of zero, the
amount between z = ∞ and z = 1100 is much less than the amount between z = 1100 and z = 0).
Meanwhile, the comoving horizon size at the time of last scattering was approximately aCMB H0–1
~ 10–3 H0–1, so distinct patches of the CMBR sky were causally disconnected at recombination.
30
Nevertheless, they are observed to be at the same temperature to high precision. The question
then is, how did they know ahead of time to coordinate their evolution in the right way, even
though they were never in causal contact? We must somehow modify the causal structure of the
conventional Friedmann Robertson Walkar cosmology.
4.2 Relic Problems
Another set of problems with the big bang model has to do with the production of exotic
particles at high energies. According to our current physical theories we believe that in the hot,
dense environment prevalent in the early universe a number of exotic particles would have been
produced. The current universe is far too cold to produce the reactions required to make these
particles, but if they had been produced in the early universe we would expect some of them to
be still detectable today. But we do not see any such exotic particles around us. Although these
particles could only have been produced in the first very small fraction of a second after Planck
density we would nonetheless expect so many of them to have been produced that they would be
quite abundant today. Any particle left over from the early, hot stages of the universe is called a
relic particle. The big bang model predicts that we should see such relics, but we don't.
4.3 Inflation
The basic idea of INFLATION has to do with the rate at which the universe is
expanding. In an expanding universe the distances between galaxies are increasing, and the rate
of expansion essentially refers to how long it takes for all of those distances to double. In the
standard Big Bang model the universe experiences power law expansion, meaning the doubling
time gets longer as the universe expands. For example, in our current power law expansion
distances in the universe were roughly half their current value about 10 billion years ago, but
they won't be twice their current value until about 30 billion years from now. By contrast, if the
doubling time stays constant then the expansion is referred to as exponential. Inflationary
theory says that before our current power law expansion there was a brief period of
exponential expansion.
Exponential growth can be much faster than power-law growth. In the simplest models of
inflation the universe would have expanded by a factor of over ten to the ten million in a fraction
of a second. There are two obvious questions raised by this idea: What mechanism would cause
such an expansion to occur and what would be the consequences if it did?
31
4.4 Why Inflation Occurs
In general relativity the rate at which the universe expands depends on the average energy
density in the universe. If the density is high the expansion is rapid and the doubling time is
small. The relation is of the form:
Doubling _ time ~
1
.
Energy _ Density
(This relation is for a flat universe. For an open or closed one it is slightly different, but the basic
implications do not change). Here the energy density also includes the density of matter because
relativity says mass is a form of energy.
In general the expansion rate slows down as the universe expands because the average
density decreases. If there are 1000 galaxies in some region of space and all distances double
then the volume of space occupied by those galaxies will increase eight times. Since the galaxies
have the same total mass as before their density will decrease by eight times. If the mass of
galaxies were the only form of energy in the universe then every time distances doubled the
doubling time would increase by a factor of the square root of eight. In short a universe whose
energy consists entirely of mass will experience power law expansion.
It turns out, however, that other forms of energy behave differently as the universe
expands. For example, the energy density contained in light (which is a form of electromagnetic
radiation) decreases faster than the energy density of mass. Every time the universe expands by a
factor of two the energy density of light decreases not by eight times, but actually by sixteen. So
if there is a lot of light energy in the universe the doubling time increases faster than it would for
a universe with only mass energy.
Now a problem might occur if we think conventionally with the above facts. This is that
the total energy of the universe is apparently not conserved. If a region of space doubled in
radius and the energy density in that region did anything other than decrease by a factor of eight
then the total energy would change. The resolution of this problem is a somewhat subtle issue in
General Relativity and involves a kind of gravitational energy, which we cannot directly observe,
associated with the expansion of the universe. I’ll not go to get into this issue in any detail here.
It suffices to say that; the total energy including gravitational potential energy is still
conserved, however, the amount of energy that we can observe in the universe can change
as the universe expands. This gravitational energy is not included in the energy density that
32
determines the expansion rate, and from here on when I refer to the energy density of the
universe I will be referring to observable energy, whose density can change in a variety of ways
as the universe expands.
We have never observed a kind of energy that acts like this, but according to our current
theories of physics there is one. This kind of energy is in the form of a field. It is well known that
different kinds of fields react differently to the expansion of the universe. For example, the
energy density of electromagnetic radiation decreases faster than that of ordinary matter. It turns
out that there is one particular kind of field with the property that when its energy density is very
large that density decreases very slowly as the universe expands. When its energy density
decreases past a certain point it stops behaving this way and starts decreasing at the same rate as
ordinary matter. Such a field is called a scalar field.
So in order for inflation to have occurred it requires that some scalar field exists and at
some point in the history of the universe it had a very large energy density. It's true that we have
never to date observed a scalar field, but physicists believe for a variety of theoretical reasons
that many of them probably do exist and that we will start to see them in our next generation of
particle accelerators (which will be able to probe higher energies).
However there is a second requirement of inflation. Having a scalar field with a large
energy density is in many ways like having a very strong magnetic field. The idea that the early
universe was filled with strong magnetic fields (actually scalar fields) must be justified by
proper reasons in a successful theory. The initial conditions for our universe were set by physics
that we don't know occurring above the Planck scale. So it might be that somewhere in the
universe, after the Planck time, a region emerged where the largest contribution to the energy
came from a high-energy scalar field. If that happened then that region (big or small in size)
would inflate, almost instantly growing much larger. Very soon this inflationary region would
occupy nearly 100% of the total volume.
In the standard Big Bang model the entire universe started expanding uniformly at the
same rate, at the same moment. In the inflationary scenario the expansion began with an
exponential growth in only one small part of the universe while the rest of it either grew in a
power law or started shrinking. However this one inflationary region became so big that
everything we can see, or will probably ever be able to see, lies within it. Thus the universe
appears to us to be uniform, even though on much larger scales it is not.
33
4.5 Some Consequences of Inflation: Solution to Flatness and
Relic Problems
The main consequences of inflation mostly stem from one simple fact, namely that the
expansion experienced during that time was enormous. In fact, during inflation the universe
expanded, i.e. all distances increased, by a factor of over 1010 million. This number comes from
assuming that the energy density of the scalar field driving inflation started near the Planck
density and inflation continued until it had decreased enough to no longer cause quasiexponential expansion. This number is so large that is out of our imagination and concept. By
comparison, the total number of elementary particles in the observable universe is much less than
10100.
In our universe the differences in energy density from one place to another smoothens
due to stretching and expansion of the universe. Suppose that before inflation there was a region
one foot wide in which the energy density varied smoothly such that it was twice as great on one
side as the other. During inflation the width of this region would expand by perhaps ten to the ten
million times, becoming so vast that we could never hope to see from one end of it to the other.
Suppose the region of the universe that we observe, which is roughly thirty billion light years
across, were somewhere in the middle of this inflated region. The difference in energy density
we would see from one side of the observable universe to the other would be far too small for us
to measure.
Inflation solves the flatness problem. It turns out that in exponential expansion the
curvature of the universe decreases. On an intuitive conceptual level, we think of this in terms of
rapidly stretching out a piece of rubber. For a closed universe the rubber is like the surface of a
sphere and for an open one it is like a saddle shape, but in either case once you stretch it out
enough it looks locally flat. This geometric picture is a somewhat simplified account of how
inflation solves the flatness problem. The detailed solution requires solving the equations of
General Relativity, but when you do you find that after inflation the curvature of the universe
will be far too small for us to measure.
Inflation also solves the problem of relic particles. Our current theories of particle
physics predict that in the hot, dense conditions of the early universe that existed before inflation
and even during its early stages various kinds of exotic particles would be produced that we don't
34
observe. There are numerous examples of such particles, like magnetic monopoles and
gravitinos, and many of them presumably were created before inflation began.
However, any particles that may have existed in this inflationary region before it began
inflating are reduced to a density of essentially zero. After inflation the density and temperature
were too small to produce these particles. Any monopoles and gravitinos produced before
inflation or during its early stages were spread out too thin for us to find, and by the time
inflation ends, the density and temperature had dropped too much for them to be produced.
In the example of the previous paragraph, if a one foot wide region contained 1030
monopoles before inflation, the odds of a single one ending up within 30 billion light years of us
is virtually zero. The same logic applies to particles produced during the early stages of inflation.
At some point during inflation the energy density would have dropped enough that such particles
could no longer be produced, which explains why we don't see any of them today.
All of these results of inflation are theoretically attractive as explanations of the features
we observe in the universe at large scales. Probably the most important success of inflation,
however, is its explanation of the origin of inhomogeneities in the universe. In the rubber
sheet analogy before it was pointed out that, any fluctuations existing before inflation get
stretched out so much that we can no longer detect them. However quantum mechanics predicts
that some fluctuations will always be produced at small scales. During inflation (specially in
the later phases) quantum mechanics causes microscopic fluctuations to be generated and
inflation stretches these fluctuations out to large distances. The quantum fluctuations
produced early in inflation get stretched out so much that we can't see them. As before it's as if
we were standing on a hill so wide it looks flat to us. Fluctuations generated close to the end of
inflation, however, produce "hills" whose width is still small enough today for us to see from one
end of them to the other. The height of these hills, i.e. the difference in energy density from one
place to another, is small because the quantum fluctuations that produced them represented small
perturbations in the energy density.
When inflationary theory was developed in the 1980s people used this theory of stretched
out quantum fluctuations to predict the differences that should be seen in the microwave
background radiation from one part of the universe to another. These differences are so small
that they weren't detected at all until the 1990s. Fluctuations in CMBR have been detected by the
spacecrafts COBE (COsmic microwave Background Explorer) in the 1990’s and by WMAP
35
(Wilkinson Microwave Anisotropy Probe) in the beginning of this century. More than just
predicting the existence of these fluctuations, inflation predicts in some detail the shape that they
will have. Only in the last few years of the 1990s were measurements taken with enough
accuracy to test the detailed predictions of inflation regarding these fluctuations, and to date the
data match these predictions perfectly.
It is primarily the successful prediction of the form of the microwave background
fluctuations that has caused inflation to be generally accepted today by most early universe
physicists. Since these small inhomogeneities are what later gave rise to galaxies, stars, and us,
all of the structure we see in the universe arose because of quantum fluctuations.
Some insight into the mathematical formulation of inflation is given now. Let's consider
a period in the early universe when it was dominated by vacuum energy rather than by matter or
radiation. (Robertson-Walker metric is assumed, which assumes isotropy from the start). Then
the flatness and horizon problems can be simultaneously solved. First, during the vacuumdominated era,  / 3MP2  a0 grows rapidly with respect to – k / a2 so the universe becomes
flatter with time ( is driven to unity). If this process proceeds for a sufficiently long period,
after which the vacuum energy is converted into matter and radiation, the density parameter will
be sufficiently close to unity, so that it will not have had a chance to noticeably change into the
present era. The horizon problem, meanwhile, can be traced to the fact that the physical
distance between any two comoving objects grows as the scale factor, while the physical horizon
size in a matter or radiation dominated universe grows more slowly, as rhor ~ a n / 21 H 01 . This can
again be solved by an early period of exponential expansion, in which the true horizon size
grows to a fantastic amount, so that our horizon today is actually much larger than the naive
estimate that it is equal to the Hubble radius H 01 .
In fact, a truly exponential expansion is not necessary; both problems can be solved by a
universe which is accelerated, a  0 . Typically we require that this accelerated period be
sustained for 60 or more e-folds, which is what is needed to solve the horizon problem. It is easy
to overshoot, and this much inflation generally makes the present-day universe spatially at to
incredible precision.
36
4.6 Reheating
Inflation occurs when in some region of space the largest contribution to the energy
density comes from a high energy scalar field. In the tradition of naming particles and fields with
the suffix on (proton, neutron, photon, etc) the scalar field that caused inflation is called the
inflaton. During inflation the universe expands exponentially and the energy density of
everything else drops to essentially zero, while the energy density of the inflaton field decreases
only very slowly. Inflation ends when this field reaches a low enough energy density that it starts
behaving like matter, i.e. when the universe starts experiencing power law expansion. So at the
end of inflation essentially all of the energy of the universe is contained in this one, nearly
homogeneous field.
So when inflation ends, the energy in the inflaton potential is converted into a
thermalized gas of matter and radiation, a process known as reheating. The universe after
inflation is very close to being homogeneous, flat, and empty of all particles. One of the
successes of inflationary theory is that it explains why we don't see relic particles like monopoles
and gravitinos today (as discussed in previous section), but somehow we still have to explain
how all the particles we do see got here. Since inflation reduces the particle density to virtually
zero, we know the particles in the universe today must all have been produced after inflation.
Once its energy density becomes small enough to allow power law inflation, the inflaton
field becomes highly unstable. (This is another fact about scalar fields that can be derived from
field theory). After inflation the energy in the inflaton field would have quickly decayed into
other particles and fields until eventually the universe consisted mainly of long-lived forms of
energy such as protons, neutrons, electrons, and electromagnetic radiation.
We don't know yet how long did it take after inflation for the inflaton to decay into the
particles we see today. But what we can say is that, the inflaton field must have decayed by the
time of nucleosynthesis, i.e. about three minutes after the end of inflation. Assuming that to
be true, the universe at the time of nucleosynthesis would have looked exactly like it does in the
Big Bang model (minus relic particles like monopoles). The key difference is that inflation
explains why the universe had many of the features it did at that time.
This decay of the inflaton field used to be modeled as a perturbative decay of –bosons
into other particles; this is a relatively inefficient process, and the temperature of the resulting
thermal state cannot be very high. More recently it has been realized that nonlinear effects
37
(parametric resonance) can efficiently transfer energy from coherent oscillations of  into other
particles, a process referred to as “preheating”. The resulting temperature can be quite a bit
higher than had been previously believed. (On the other hand, the inflaton tends to be weakly
coupled, which suppresses the reheat temperature.)
A proper understanding of the reheating process is of utmost importance, as it controls
the production of various relics that we may or may not want in our universe. For example, one
of the most beneficial aspects of inflation in the context of grand unification is that it can solve
the monopole problem. Essentially, any monopoles will be inflated away, leaving a relic
abundance well under the observational limits. It is therefore important that reheating does not
reproduce too many monopoles. On the other hand, we do want to reheat to a sufficiently high
temperature to allow for some sort of Baryogenesis (formation of baryons from elementary
particles) scenario.
4.7 Inflation and Particle Physics
It is nevertheless important to try to implement inflation within a believable particle
physics model. Some relevant issues are listed below.

A great deal of effort has gone into exploring the relationship between inflation and
supersymmetry, although simultaneously satisfying the strict requirements of inflation
and SUSY turns out to be a difficult task.

Hybrid inflation is a kind of model which invokes two scalar fields with a waterfall
potential. One field rolls slowly and is weakly coupled, the other is strongly coupled and
leads to efficient reheating once the first rolls far enough.

Another interesting class of models involve scalar-tensor theories and make intimate
use of the conformal transformations relating these theories to conventional Einstein
gravity.

The need for a flat potential for the inflaton, coupled with the fact that string theory
moduli can naturally have flat potentials, makes the idea of modular inflation an
attractive one. Specific implementations have been studied, but we probably don't
understand enough about moduli at this point to be confident of finding a compelling
model.
38
4.8 Perturbations in Inflation
A crucial element of inflationary scenarios is the production of density perturbations,
which may be the origin of the CMBR temperature anisotropies and the large-scale structure in
galaxies that we observe today (qualitatively discussed in the previous sections on Inflation).
The idea behind density perturbations generated by inflation is fairly straightforward.
Inflation will attenuate any ambient particle density rapidly to zero, leaving behind only the
vacuum. But the vacuum state in an accelerating universe has a nonzero temperature, the
Gibbons Hawking temperature, analogous to the Hawking temperature of a black hole. For a
universe dominated by a potential energy V, the Gibbons Hawking temperature is given by
TGH  H / 2 ~ V 1 / 2 / M P .
Corresponding to this temperature are fluctuations in the inflaton field  at each wavenumber k,
with magnitude,
 k  TGH .
Since the potential is by hypothesis nearly flat, the fluctuations in  lead to small fluctuations in
the energy density,
  V   .
Inflation therefore produces density perturbations on every scale. The amplitude of the
perturbations is nearly equal at each wavenumber, but there will be slight deviations due to the
gradual change in V as the inflaton rolls. We characterize the fluctuations in terms of their
spectrum which describes scalar fluctuations in the metric. These are tied to the energy
momentum distribution, and the density fluctuations produced by inflation are adiabatic (or,
better, isentropic) – fluctuations in the density of all species are correlated. The fluctuations are
also Gaussian, in the sense that the phases of the Fourier modes describing the fluctuations at
different scales are uncorrelated. These aspects of inflationary perturbations – a nearly scale-free
spectrum of adiabatic density fluctuations with a Gaussian distribution – are all consistent with
current observations of the CMBR and large-scale structure, and new data scheduled to be
collected over the next decades should greatly improve the precision of these tests.
Not only the nearly–massless inflaton gets excited during inflation, but any nearly–
massless particle does so. The other important example is the graviton, which corresponds to
tensor perturbations in the metric (propagating excitations of the gravitational field). The
39
existence of tensor perturbations is a crucial prediction of inflation, which may in principle
be verifiable through observations of the polarization of the CMBR. In practice, however, the
induced polarization is very small, and we may never detect the tensor fluctuations even if they
are there.
Our current knowledge of the amplitude of the perturbations can give us important
information about the energy scale of inflation. The tensor perturbations depend on V alone (not
its derivatives), so observations of tensor modes yields direct knowledge of the energy scale. If
the CMBR anisotropies detected by further space explorers, are due to tensor fluctuations
(possible, although unlikely), we can instantly derive,


4
Vinf lation ~ 1016 GeV .
(Here, the value of V being constrained is that which was responsible for creating the observed
fluctuations; namely, 60 e-folds before the end of inflation.) This is remarkably reminiscent of
the grand unification scale, which is very encouraging. Even in the more likely case that the
perturbations observed in the CMB are scalar in nature, we can still write,
1/ 4
1/ 4
Vinf
1016 GeV ,
lation ~ 
where  is the slow-roll parameter. Although we expect  to be small, the 1/4 in the exponent
means that the dependence on  is quite weak; unless this parameter is extraordinarily tiny, it is
1/ 4
15
 1016 GeV . The fact that we can have such information about such
very likely that Vinf
lation ~ 10
tremendous energy scales is a cause for great wonder.
4.9 Initial conditions and eternal inflation
The topic of initial conditions for inflation is an especially important as, although
inflation is supposed to solve the horizon problem, it is necessary to start the universe
simultaneously inflating in a region larger than one horizon volume in order to achieve
successful inflation. Presumably we must appeal to some sort of quantum fluctuation to get the
universe (or some patch thereof) into such a state.
Fortunately, inflation has the wonderful property that it is eternal. That is, once
inflation begins, even if some regions cease to inflate there will always be an inflating region
with increasing physical volume. This property holds in most models of inflation that we can
construct. It relies on the fact that the scalar inflaton field doesn't merely follow its classical
equations of motion, but undergoes quantum fluctuations, which can make it temporarily
40
roll up the potential instead of down. The regions in which this happens will have a larger
potential energy, and therefore a larger expansion rate, and therefore will grow in volume in
comparison to the other regions. One can argue that this process guarantees that inflation never
stops once it begins.
We can therefore imagine that the universe approaches a steady state (at least
statistically), in which it is described by a certain fractal dimension. This means that the universe
on ultra–large scales, much larger than the current Hubble radius, may be dramatically
inhomogeneous and isotropic, and even raises the possibility that different post-inflationary
regions may have fallen into different vacuum states and experience very different physics than
we see around us. Certainly, this picture represents a dramatic alteration of the conventional view
of a single Robertson Walker cosmology describing the entire universe.
Of course, it should be kept in mind that the arguments in favor of eternal inflation rely
on features of the interaction between quantum fluctuations and the gravitational field, which are
slightly outside the realm of things we claim to fully understand. It would certainly be interesting
to study eternal inflation within the context of string theory.
Chapter 5
Stringy cosmology:
String Theory applied to Cosmology
There are many things we do not understand about both cosmology and string theory to
make statements about the very early universe in string theory with any confidence. However, it
is still worthwhile to speculate about different possibilities, and work towards incorporating
these speculations into a more complete picture [15].
5.1 The beginning of time
As the correct place to start is not known, a simple guess can done by considering the
action as the low–energy effective action in D dimensions (bosonic, NS-NS part), which is
expressed in terms of the Ricci scalar, the dilaton, field strength tensor for the two-form gauge
field (which is typically set to zero in papers on cosmology). The existence of the dilaton implies
that the theory of gravity described by this action is a scalar-tensor model (reminiscent of BransDicke theory), not pure general relativity. Of course there are good experimental limits on scalar
41
components to the gravitational interaction, but they are only sensitive to low-mass scalars (i.e.,
long-range forces).
With such an action (as described in last paragraph), the cosmological solutions
(homogeneous but not necessarily isotropic) have a scale–factor duality symmetry, i.e. for any
solution ai t ,  t , there is also a solution with, a i 
1
,      2 ln ai .
ai
i
Thus expanding solutions are dual to contracting solutions. (In fact this is just T-duality, and is a
subgroup of a larger O(D – 1, D – 1) symmetry.) The solutions with decreasing curvature are
mapped to those with increasing curvature.
This feature of the low-energy string action has led to the development of the Pre-BigBang Scenario, in which the universe starts out as flat empty space, begins to contract (with
increasing curvature), until reaching a stringy state of maximum curvature, and then expands
(as curvature decreases) and commences standard cosmological evolution.
There are various questions about the Pre-BB scenario. One is a claim that significant
fine tuning is required in the initial phase, in the sense that any small amount of curvature will
grow fantastically during the evolutionary process and must be extremely suppressed. Another is
the role of the potential for the dilaton. We cannot set this potential to zero on the grounds that
the relevant temperatures are much higher than the SUSY–breaking scale TSUSY ~ 103 GeV;
supersymmetry is an example of a symmetry which is not restored at high temperatures. Indeed,
almost any state breaks supersymmetry. In a thermal background, this breaking is manifested
most clearly by the differing occupation numbers for bosons and fermions. More generally, the
SUSY algebra
Q, Q H  Z ,
with Z a central charge, implies that Q  0 whenever H  0, except in BPS states, which feature a
precise cancellation between H and Z. In the real world these are a negligible fraction of all
possible states. It is not clear how SUSY breaking affects the Pre-BB idea. Perhaps more
profoundly, it seems perfectly likely that the appropriate description of the high curvature stringy
phase will be nothing like a smooth classical spacetime. Evidence for this comes from matrix
theory, not to mention attempts to canonically quantize general relativity.
There are other, non-stringy, approaches to the very beginning of the universe, and it
would be interesting to know what light can be shed on them by string theory. One is quantum
42
cosmology, which by some definitions is just the study of the wave function of the universe,
although in practice it has the connotation of mini-superspace techniques (drastically truncating
the gravitational degrees of freedom and quantizing what is left). There is also the related idea of
creation of baby universes from our own. This is in principle a conceivable scheme, as closed
universes have zero total energy in general relativity. There is also the hope that string theory
will offer some unique resolution to the question of cosmological (and other) singularities;
studies to date have had some interesting results, but we don't know enough to understand the
Big Bang singularity of the real world.
5.2 Extra Dimensions and Compactification in Cosmology
Of all the features of string theory, the one with the most obvious relevance to cosmology
is the existence of 6 (or 7?) extra spatial (temporal?) dimensions. The success of our traditional
description of the world as a (3+1)-dimensional spacetime implies that the extra dimensions must
be somehow inaccessible, and the simplest method for hiding them is Compactification: the
idea that the extra dimensions describe a compact space of sufficiently small size that they can
only be probed by very high energies.
Of course in General Relativity (and even in string theory) spacetime is dynamical, and it
would be natural to expect the compact dimensions to evolve. However, the parameters
describing the size and shape of the compact dimensions show up in our low-energy world
as moduli fields whose values affect the Standard Model parameters. As discussed earlier,
we have good limits on any variation of these parameters in spacetime, and typically appeal to
SUSY breaking to fix their expectation values. This raises all sorts of questions. Why are three
dimensions allowed to be large and expanding while the others are small and essentially frozen?
What is the precise origin of the moduli potentials? What was the behavior of the extra
dimensions in the early universe?
For the most part these are baffling questions, although there have been some provocative
suggestions. One is by Brandenberger and Vafa, who attempted to understand the existence of 3
macroscopic spatial dimensions in terms of string dynamics. Consider an n–torus populated by
both momentum modes and winding modes of strings. The momentum and winding modes are
dual to each other under T-duality (R 1/R), and have opposite effects on the dynamics of the
torus: the momentum modes tend to make it expand, and the winding modes tend to make it
contract. (It's counterintuitive, but true.) We can therefore have a static universe at the self-dual
43
radius where the two effects are balanced. However, when wound strings intersect they tend to
intercommute and therefore unwind. Through this process, the balance holding the torus at the
self–dual radius can be upset, and the universe will begin to expand, hopefully evolving into a
conventional Friedmann cosmology.
But notice that in a sufficiently large number of spatial dimensions, 1-dimensional strings
will generically never intersect. (Just as 0-dimensional points will generically intersect in one
dimension but not in two or more dimensions.) The largest number in which they tend to
intersect is three. So we can imagine a universe that begins as a tiny torus in thermal equilibrium
at the self-dual point, until some winding modes happen to annihilate in some three-dimensional
subspace which then begins to expand, forming our universe. Of course a scenario such as this
loses some of its charm (because of increasing complexity) in a theory which has not only strings
but also higher-dimensional branes.
An alternate route is to imagine that we are living on a brane. That is to say, that the
reason why the extra dimensions are invisible to us is not simply because they are so very small
that low-energy excitations cannot probe them, but because we are confined to a threedimensional brane embedded in a higher-dimensional space. We know that we can easily
construct field theories confined to branes, for example a U(N) gauge theory by stacking N
coincident branes; it is not an incredible stretch to imagine that the entire Standard Model can
be constructed in such a way in principle (it hasn't been done yet, because of emergence of
surmounting complexities as we go on). Unfortunately, it seems impossible to entirely do away
with the necessity of compactification, since there is one force, which we don't know how to
confine to a brane, namely gravity (although see below).
We therefore imagine a world in which the Standard Model particles are confined to a 3–
brane, with gravity propagating in a higher-dimensional bulk which includes compactified extra
dimensions. In D spacetime dimensions, Newton's law of gravity can be written
~ mm
F D  r   G D  1D 22 ,
r
~
where G D  is the D–dimensional Newton's constant with appropriate factors of 4 absorbed. If
we compactify (D – 4) of the spatial dimensions on a compact manifold of volume V(D–4), the
effective 4 dimensional Newton's constant is,
44
~
G D 
~
.
G 4  ~
V D  4 
~
We can rewrite this in terms of what we will define as the Planck scale, M P  G41/ 2 , and the
~ 1 /( D2)
fundamental scale, M   G D 
,as
M P2 ~ M D2V D4  .
In conventional compactification, M  ~ M P and VD4  ~ M P D4  , so this relation is satisfied in
a straightforward way. But we can also satisfy it by lowering the fundamental scale and
increasing the compactification volume. Imagine that the compactification manifold has n large
dimensions of radius R and (D – 4 – n) dimensions of radius M 1 . Then,
M
R ~  P
 M



2 n
M 1 .
A scenario of this type was proposed by Horava and Witten, who suggested that the gravitational
coupling could unify with the gauge couplings of GUT’s by introducing a single large extra

dimension with R ~ 1015 GeV

1
.
But we can go further. The lowest value we can safely imagine the fundamental scale
having is M  ~ 10 3 GeV ; otherwise we would have detected quantum gravity at Fermilab or
CERN. This value is essentially the desired low-energy supersymmetry breaking scale (i.e. just
above the electroweak scale), so it is tempting to explain the apparent hierarchy
M  / M EW ~ 1015 by trying to move M  all the way down to 103 GeV. (Supersymmetry itself can
stabilize the hierarchy, but doesn’t actually explain it.). Then we have,
R ~ 10 30
n 3
GeV 1 ~ 10 30
n 17
cm.
For n = 1, we have a single extra dimension of radius R ~ 1013 cm, about the distance from the
Sun to the Earth. This is clearly ruled out, as such a scenario predicts that gravitational forces
would fall off as r–3 for distances smaller than 1013 cm. But for n = 2 we have R ~ 10–2 cm, which
is just below the limits on deviations from the inverse square law from laboratory experiments.
Larger n gives smaller values of R; these are not as exciting from the point of view of having
macroscopically big extra dimensions, but may actually be the most sensible from a physics
standpoint.
45
So we have a picture of the world as a 3-brane with Standard Model particles restricted to
it, and gravity able to propagate into a bulk with extra dimensions which are compactified but
perhaps of macroscopic size, with a fundamental scale M  ~ 103 GeV and the observed Planck
scale simply an artifact of the large extra dimensions. (There is still something of a hierarchy
problem, since R must be larger than M  to get the Planck scale right.) Such scenarios are
subject to all sorts of limits from astrophysics and accelerator experiments, from processes such
as gravitons escaping into the bulk. (In these models gravity becomes strongly coupled near 10 3
GeV.)
There are also going to be cosmological implications, although it is not precisely clear as
yet what these are. Our entire popular notion of the thermal history of the universe for T > 10 3
GeV would obviously need to be discarded. Baryogenesis will presumably be modified. Inflation
is a very interesting question, including the issues of inflation in the bulk vs. inflation in the
boundary. Of course we don’t know what stabilizes the large extra dimensions, but then again we
don't know much about moduli stabilization in conventional scenarios. There is also the
interesting possibility of a 3–brane parallel to the one we live on, which only interacts with us
gravitationally, and on which the dark matter resides. There are some cosmological problems,
though – most clearly, the issue of why the bulk is not highly populated by light particles that
one might have expected to be left over from an early high-temperature state; presumably
reheating after inflation cannot be to a very high temperature in these models (although we must
at the very least have Treheat > 1 MeV to preserve standard nucleosynthesis). Clearly there is a
good deal of work left to do in exploring these scenarios.
Randall and Sundrum found a loophole in the conventional wisdom that gravity cannot
be confined to a brane. They showed that a single extra dimension could be infinitely large, but
still yield an effective 4-dimensional gravity theory on the brane, if the bulk geometry were antide Sitter rather than flat. The curvature in the extra dimension can then effectively confine
gravity to the vicinity of the brane. In the subsequent times a great deal of effort has gone into
understanding cosmological and other ramifications of Randall-Sundrum scenarios.
5.3 The late universe
The behavior of gravity and particle physics on extremely short length scales and high
energies is a largely uncharted territory, and it is clear that string theory, if correct, will play an
46
important role in understanding this regime. But it is also interesting to contemplate the
possibility of new physics at ultra-large length scales and low energies. You might guess that
experiments in the zero-energy limit are straightforward to perform, but in fact it requires great
effort to isolate yourself from unwanted noise sources in this regime. Cosmology offers a way to
probe physics on the largest observable length scales in the universe, and it is natural to take
advantage.
We have evidence from observations that the expansion of the universe is accelerating.
Explaining the observations with a positive vacuum energy  V  M V4 requires M V ~ 10 3 eV ,
which is remarkably small in comparison to M SUSY ~ 10 3 GeV  1012 eV , not to mention
M Planck ~ 1018 GeV  10 27 eV . It does, of course, induce the irresistible temptation to write
MV ~
2
M SUSY
.
MP
This is a numerological curiosity without a theory that actually predicts it, although it has the
look and feel of similar relations familiar from models in which SUSY breaking is
communicated from one sector to another by gravitational interactions. Another provocative
relation is
M V ~ e 1 / 2 M P ,
where  is the fine-structure constant. Again, it falls somewhat short of the standards of a
scientific theory, but it does suggest the possibility that the vacuum energy would be precisely
zero if it were not for some small nonperturbative effect.
More generally, we can classify vacuum energy as coming from one of three categories:
true vacua, which are global minima of the energy density; false vacua, which are local but not
global minima; and non-vacua, which is a way of expressing the idea that we have not yet
reached a local minimum value of the potential energy. For example, we could posit the
existence of a scalar field  with a very shallow potential. From our analysis in previous sections,
the field will be overdamped when V < H, and its potential energy will dominate over its kinetic
energy (exactly as in inflation). Such a possibility has been termed quintessence. For a
quintessence field to explain the accelerating universe, it must have an effective mass,
m  V   H 0 ~ 10 33 eV ,
and a typical range of variation over cosmological timescales,
47
 ~ M P ~ 1018 GeV .
From a particle-physics point of view, these parameters seem somewhat contrived, to say the
least. Quintessence models have the benefit of involving dynamical fields rather than a single
constant, and it may be possible to take advantage of these dynamics to ameliorate the
coincidence problem that  ~ M today (despite the radically different time dependences of
these two quantities). In addition, there may be more complicated ways to get a time-dependent
vacuum energy that are also worth exploring. The moduli fields of string theory could provide
potential candidates for quintessence, and the acceleration of the universe more generally
provides a rare opportunity for string theory to provide an explanation of an empirical fact.
We could also imagine that string theory may have more profound late-time
cosmological consequences than simply providing a small vacuum energy or ultralight scalar
fields. An interesting move in this direction is to explore the implications of the holographic
principle for cosmology. This principle was inspired by our semiclassical expectation that the
entropy of a black hole, which in traditional statistical mechanics is a measure of the number of
degrees of freedom in the system, scales as the area of the event horizon rather than as the
enclosed volume (as we would expect the degrees of freedom to do in a local quantum field
theory). In its vaguest (and therefore most likely to be correct) form, the holographic principle
proposes that a theory with gravity in n dimensions (or a state in such a theory) is equivalent in
some sense to a theory without gravity in n–1 dimensions (or a state in such a theory). Making
this statement more precise is an area of active investigation and controversy; see Susskind's
lectures for a more complete account. The only context in which the holographic equivalence has
been made at all explicit is in the AdS/CFT correspondence, where the non gravitational theory
can be thought of as living on the spacelike boundary at conformal infinity of the AdS space on
which the gravitational theory lives.
Regrettably, we don't live in anti-de Sitter space, which corresponds to a Robertson –
Walker metric with a negative cosmological constant and no matter, since our universe seems to
feature both matter and a positive cosmological constant. How might the holographic principle
apply to more general spacetimes, without the properties of conformal infinity unique to AdS, or
for that matter without any special symmetries? A possible answer has been suggested by
Bousso, building upon ideas of Fischler and Susskind. The basic idea is to the area A of the
boundary of a spatial volume to the amount of entropy S passing through a certain null sheet
48
bounded by that surface. (For details of how to construct an appropriate sheet, see the original
references.) Specifically, the conjecture is that S  A / 4G .
This is more properly an entropy bound, not a claim about holography; however, it seems to be a
short step from limiting entropy (and thus the number of degrees of freedom) to claiming the
existence of an underlying theory dealing directly with those degrees of freedom. Does this
proposal have any consequences for cosmology? It is straightforward to check that the bound is
satisfied by standard cosmological solutions7, and a classical version can even be proven to hold
under certain assumptions. One optimistic hope is that holography could be responsible for the
small observed value of the cosmological constant. Roughly speaking, this hope is based on the
idea that there are far fewer degrees of freedom per unit volume in a holographic theory than
local quantum field theory would lead us to expect, and perhaps the unwarranted inclusion of
these degrees of freedom has been leading to an overestimate of the vacuum energy. It remains
to be seen whether a workable implementation of this idea can capture the successes of
conventional cosmology.
Chapter 6
Conclusions
The last several years have been a very exciting time in string theory, as we have learned
a great deal about non-perturbative aspects of the theory, most impressively the dualities
connecting what were thought to be different theories. They have been equally exciting in
cosmology, as a wealth of new data have greatly increased our knowledge about the constituents
and evolution of the universe. The two subjects still have a long way to go, however, before their
respective domains of established understanding are definitively overlapping. One road towards
that goal is to work diligently at those aspects of string theory and cosmology which are best
understood, hoping to enlarge these regions until they someday meet. Another strategy is to leap
fearlessly into the murky regions in between, hoping that our current fumbling attempts will
mature into more solid ideas. Both approaches are, of course, useful and indeed necessary.
Cosmology is a wonderful interplay between general relativity and particle physics, the
physics of the very large and of the very small. The theory of superstrings has not yet answered
some of the open questions (of detail, rather than of principle) in Big Bang cosmology, but I
believe it will. In any case it is really awesome that there are such intimate links between the
49
physics at the smallest scales, set by the Planck time and the Planck length, and the physics of
the cosmos as a whole with a time scale of the order of fifteen billion years and a length scale of
as many light-years. There are after all some sixty orders of magnitude separating the Planck
from the cosmic scales. These links have only emerged in this century, and will surely enrich
physics in the next.
In the meanwhile we can look forward to a lot of good tests of early universe physics in
the next couple of decades. High sensitivity probes of the microwave background, searches for
waves of gravity surviving from the early universe, and many other experiments are going to
give us excellent tests, not only of inflation, but of our understanding of the universe in general.
References
[1] http://superstringtheory.com/ [The Official String Theory Website]
[2] http://www.pbs.org/wgbh/nova/elegant/ [The Elegant Universe]
[3] http://www.theory.caltech.edu/people/jhs/strings/ [The Second Superstring Revolution]
[4] http://www.sukidog.com/jpierre/strings/ [Superstrings Homapage]
[5] http://turing.wins.uva.nl/~rhd/string_theory.html
[6] http://www.lassp.cornell.edu/GraduateAdmissions/greene/greene.html
[7] http://theory.tifr.res.in/~mukhi/Physics/string.html
[8] http://www.damtp.cam.ac.uk/user/gr/public/qg_ss.html
[9] http://tena4.vub.ac.be/beyondstringtheory/
[10] http://fy.chalmers.se/tp/Egroup/stringtheory.html
[11] http://www.nuclecu.unam.mx/~alberto/physics/string.html
[12] http://www.crystalinks.com/superstrings.html
[13] http://monopole.ph.qmul.ac.uk/~jmc/EUni/Strings.html
[14] http://www.hyper-mind.com/hypermind//universe/content/gsst.htm
[15] arXiv:hep-th/0011110 – TASI Lectures: Cosmology for String Theorists