Bachelor research
Inflationary models
based on conformal symmetry
Anno Touwen
Supervisor/ First examiner prof. dr. Diederik Roest
Second examiner
prof. dr. Rien van de Weijgaert
4th July 2016
Abstract
This provides a general overview of inflation, leading to a conformal inflation model, consistent with the common predictions for the observables in the Cos- mic Microwave Background related to inflation. A short introduction to Big Bang expansion is provided. The horizon and flatness problem are described and it is shown how a decreasing Hubble radius, predicted by inflation, provides a solution to these problems in the Big Bang theory. A derivation of the slow- roll parameters ε and η based on the Friedmann equations is provided. The implications of freeze in of quantum fluctuations in the inflaton field are shown and the resulting observables in the CMB, the spectral indices for both scalar and tensor power-spectrum ns, nt and the tensor-to-scalar ratio r, are derived from this. The recent observational constraints by the Planck satellite on these parameters are described, including implications for the credibility of some mod- els. An introduction in Conformal Field Theory is provided and a model for inflation based on conformal symmetry is suggested involving a SO(1,1) field combination of conformal field χ and inflaton field φ. The SO(1,1) symmetry is broken and a general potential is introduced. In the large field expansion for ϕ the prediction for the observables of this class of models are derived to be in accordance with recent observations.
Acknowledgements
Writing a Bachelor thesis is more than a first encounter with doing research. I have encountered many new aspects of my scientific carrier during the last three months. As for everything it was great not to have to do this alone, therefore I would like to thank everyone who contributed to this work. At first, I would like to thank my direct colleagues Alexander, Floris and Guus. It was great to discuss both my research and that there is more to life, when I got stuck again. Secondly, many thanks to those who supported me from further away on the campus: Bas, Daan, Jesper, Jos and Syb. You never failed to rescue me from my office at the end of the day. Thank you for helping me empty the cookie jar at my office, enabling me to keep a varied diet. Furthermore, I thank my sister, Emmie, for daring to dive into this puzzling Physics thesis and helping to improve my English. Most of all, I would like to thank my supervisor Diederik, not only for his great advice on doing the research, but mainly for making the last couple of months so much fun. I enjoyed our meetings a lot, I believe to have always filled the complete hour of our meeting, discussing much more than only my project. Also, many thanks for the invitation to multiple presentations, seminars and even a promotion. It was wonderful really feeling part of your group.
Contents
1 Introduction 4
2 Inflation as a solution to Big Bang problems 5
2.1 An expanding universe . . . 5
2.2 Friedmann-Robertson-Walker metric . . . 6
2.3 Friedmann equations . . . 6
2.4 Cosmic Microwave Background . . . 7
2.5 Problems in an expanding universe . . . 8
2.5.1 Horizon problem . . . 8
2.5.2 Flatness problem . . . 9
2.6 Finetuning of initial conditions . . . 9
2.7 Inflation . . . 10
2.8 Monopole problem . . . 11
3 Slow-roll inflation 12 3.1 Number of e-folds . . . 12
3.2 Inflaton field . . . 12
3.3 First slow-roll parameter . . . 14
3.4 Second slow-roll parameter . . . 15
3.5 Duration of slow-roll inflation . . . 16
4 Quantum fluctuations and related observables 17 4.1 Perturbations due to quantum fluctuations . . . 18
4.1.1 Gauge invariance . . . 18
4.2 Freeze in of perturbation modes . . . 19
4.3 Evolution of perturbation . . . 20
4.3.1 Scalar perturbations . . . 20
4.3.2 Tensor perturbations . . . 20
4.3.3 Quantization . . . 21
4.3.4 de Sitter solution . . . 21
4.4 Power spectra . . . 22
4.4.1 Scalar power spectrum . . . 22
4.4.2 Tensor power spectrum . . . 23
4.5 Observables . . . 23
4.5.1 Tensor-to-scalar ratio . . . 23
4.5.2 Spectral index . . . 23
4.6 Observational constraints . . . 24
5 Toy models for inflation 27
5.1 Cosmological constant potential . . . 27
5.2 Quadratic potential . . . 28
5.3 Starobinsky potential . . . 29
6 Conformal inflation models 31 6.1 Conformal symmetry . . . 31
6.2 Conformon field . . . 32
6.2.1 Proof of conformal symmetry . . . 33
6.2.2 Gauge fixing . . . 34
6.3 Inflaton field . . . 34
6.4 Breaking the SO(1,1) symmetry . . . 34
6.4.1 Monomial V . . . 35
6.4.2 General large φ expansion . . . 35
6.5 Non-canonical coordinate frame . . . 37
7 Conclusion 39 7.1 Recommendations for further research . . . 39
Bibliography 41
Chapter 1
Introduction
One of the most frequently answered questions, perhaps after ”What will be for dinner tonight?”, concerns the origin of the universe. One of the most com- monly accepted answers in the current universe is provided by cosmology in the form of the Big Bang theory. Many observations have provided reason to be- lieve in this theory, but have also provided some major problems challenging the accuracy of this vague concept of a Big Bang. Fortunately, inflation provides convincing answers to these issues, such as the horizon and flatness problem.
This theory of inflation predicting a period of accelerating expansion in the very early universe does not only solve the problems mentioned before, but also pro- vides an explanation for the composition of cosmological objects via quantum fluctuations. These testable predictions with observational confirmation have lead to for the wide acceptance of this theory and to years of research spent on this period lasting approximately 10−34 seconds.
In chapter 2 an overview is provided of the discoveries leading to the theory of inflation. The concept of slow-roll inflation is explored in chapter 3, including some quantitative parameters related to this. These parameters are related to observable quantities in the Cosmic Microwave background via quantum fluc- tuations in during inflation. A derivation of this relation and an overview of the observational restrictions to the parameters is provided in chapter 4. Many models for inflation have been suggested over the last few years. To show the extend of possibilities, some are discussed in chapter 5. The predictions for the observables from the previous chapter are derived, whereby some models are found to be more accurate than others. One class of models providing results in consistence with observations is based on Conformal Field Theory. In chapter 6 this class of models and its limitations are studied.
Chapter 2
Inflation as a solution to Big Bang problems
In this chapter a semi historical overview is presented of the developments lead- ing to inflation. The aim of this chapter is to show how inflation is a logical consequence of historical discoveries and current observations. This overview is mainly based on [1, 2, 3, 4, 5, 6].
2.1 An expanding universe
Over most of human history the universe was considered to be static, hardly changing as a whole at all. Many ideas on how the universe was created were considered, but all had one idea in common: the present universe is complete.
Figure 2.1: Observation of the expansion of the universe as presented in the original publication by Hubble [7].
Direct evidence that this was not the case was the observa- tion by Hubble. He used the redshift of starlight from stars at different distances to mea- sure the velocity of these ob- jects away from us, which re- sulted in the conclusion that the universe is not static, but expanding as time proceeds.
v = H0d. (2.1) The proportionality factor H0 is the current Hubble param- eter, valued 2.2 × 1018 s−1. This relation was predicted some years before the obser-
vation by Lemaˆıtre, who concluded that, as the universe is expanding now, the universe must have been smaller in the past. This brings us to a singularity as the origin of the universe, later disapprovingly called the Big Bang by Hoyle.
2.2 Friedmann-Robertson-Walker metric
An attempt was made at describing the spacetime structure of this observed to be expanding universe using the theory of relativity by Einstein. For simplicity the cosmological principle was assumed to hold: the universe is isotropic, equal in all directions, and homogeneous, the same everywhere. In the end this prin- ciple seemed to hold so well, according to experiments, that it causes problems by itself. Since the solution of the Einstein equation for this isotropic, homo- geneous, expanding universe was derived by multiple physicists independently around the same time the resulting metric is called the Friedmann-Robertson- Walker metric (FRW-metric for short).
ds2= gµνdxµdxν,
= −c2dt2+ a2(t)
dr2
1 − kr2 + r2(dθ2+ sin2θdφ2)
. (2.2)
Figure 2.2: Three dimensional analogy to illustrate what is meant by open, closed and flat spacetime. Picture from NASA.
In this equation the coordinates t, r, θ and φ are comoving with the expansion of the universe. The curvature parameter k describes the curvature of the spacetime of the universe, where k = −1 im- plies an open, k = 0 a flat and k = +1 a closed universe. The general curvature of the space- time of the universe is determined by the density parameter Ω, re- lated to the total energy content of the universe. The expansion of the universe in this equation is described by the (positive) scale factor a(t). For an expanding universe the scale factor increases over time,
da
dt > 0. (2.3)
2.3 Friedmann equations
The rate at which the scale factor of the universe increases is determined by the content of the universe. A good way to describe this content is by considering its energy density ρ and pressure p. How these influence the expansion rate can be found by filling in the FRW-metric into the Einstein equation, where the cosmological constant term, introduced to create a static universe solution, is ignored. The results for a perfect fluid are the Friedmann equations.
H2= ˙a a
2
= 8πG
3 ρ −kc2
a2 , (2.4)
H + H˙ 2= ¨a
a= −4πG 3
ρ +3p
c2
. (2.5)
The second is mostly called the acceleration equation. In these equations the Hubble parameter appears, describing the expansion, H ≡ aa˙. The continuity equation can be derived from a specific combination of (2.4) and (2.5), describing the time evolution of the energy density.
dρ
dt = −3H ρ + p
c2
. (2.6)
This continuity equation can be solved to be:
ρ ∝ a−3(1+ω), (2.7)
with equation of state parameter ω = ρcp2. Using the Friedmann equation (2.4) the Comoving Hubble radius, that will become relevant later, can be found,
1 Ha= 1
˙a =
H02a−(1+3ω)− kc2−12
. (2.8)
Here H0 is the Hubble constant, as was already encountered in Hubble‘s law.
For a flat universe, k = 0, the comoving Hubble radius is thus:
1 aH = 1
H0
a12(1+3ω). (2.9)
2.4 Cosmic Microwave Background
As the universe expands, the energy density of the universe decreases. Thus the universe cools down over time, which implies a hot Big Bang. In the very early universe the temperature was so high that no atoms could form. High energy photons were so numerous that any electron bound to a proton quickly uncoupled again by ionization. As the universe expanded, it cooled down, until at some point the energy of most photons was too low to ionize, after which stable atoms formed. This is called the epoch of recombination.
Since at this point the protons, neutrons and electrons formed neutral atoms, the photons did not constantly interact with these particles anymore and thus the universe became transparent. Due to the lack of interaction of these photons with any other particles, these are still present in the universe, as a surface of last scattering (last scattering just before the formation of atoms). Due to the further expansion of the universe these photons have been redshifted, now being in the order microwaves. The omnipresent surface of last scattering is also called the Cosmic Microwave Background. This CMB has been observed to have an average temperature of:
TCMB= 2.725 K. (2.10)
Figure 2.3: The Cosmic Microwave Back- ground is homogeneous up to very high ac- curacy. The temperature fluctuations illus- trated by the color spectrum are of the or- der of 10−5compared to the all-sky average.
Further study into the implication of these fluctuations are presented in chapter 4. Pic- ture from ESA.
The surface of last scattering is of great importance in modern cosmology. Multiple satellites have been built to study the Cos- mic Microwave Background at in- creasing accuracy. The photons in the CMB can still be traced back to their creation during re- combination, not long after the creation of the universe. Thereby it provides a perfect base for do- ing measurements on the very early universe. Therefore, it is commonly described by physicists as a baby-picture of the universe.
Many of the experimental evi- dence that is provided in this the- sis is based on measurements on the CMB.
2.5 Problems in an expanding universe
Up to this point normal Big Bang expansion seems to be in perfect agreement with the observed universe. The universe turned out not to be static, as was assumed before, but is expanding over time. After some computations this did not seem to be a problem, it even provided some prediction on periods in the early universe. In the description of the expanding universe it was assumed that the universe could be approximated to be homogeneous and isotropic. This even seemed to be in agreement with direct observations. It turned out that this approximation is even more accurate than would be theoretically expected. The universe turned out to be too perfect to be in agreement with the predictions of normal Big Bang expansion. In the following sections two problems are explained to illustrate this.
2.5.1 Horizon problem
Two objects can only influence each other by being in causal contact, in other words, there is no action at a distance. Since the introduction of the theory of relativity, it has been well known that information can only be transferred at a speed smaller than or equal to the speed of light. This gives rise to the concept of a horizon, a maximum distance at which events can be influenced by each other. The size of the horizon of an event happening at t0 = 0, at a later time t0= t is:
R(t) = c Z t
0
dt0 a(t0) = c
Z a 0
da Ha2 ∝ 1
aH. (2.11)
Thus, the comoving horizon is proportional to the comoving Hubble radius, which was introduced in (2.9). There it was shown that for ω > −13 the Hubble radius, and therefore thus the horizon, increases as the size of the universe increases.
Let us now consider the CMB again. As mentioned before, the CMB is almost homogeneous, with only small temperature fluctuations. This is a problem with the idea of causality. In normal Big Bang expansion at the time of decoupling the universe consisted of 104 causally independent patches. As the horizon size increases with the expansion of the universe, this would imply that regions on the sky now separated by 2.3o could never have been in causal contact [5].
However, observations show that these regions do have the same temperature up to very high accuracy. How can this enormous finetuning exist if no information has been transferred to make any comparison possible?
2.5.2 Flatness problem
The curvature of the general spacetime of the universe is not determined by the theory of an expanding universe, the FRW-metric (2.2) is a solution for both an open, closed and flat universe, described by the different values for k. The flatness can be studied in terms of the energy content of the universe if the Friedmann equation (2.4) is rewritten to the form:
Ω − 1 = kc2
(Ha)2. (2.12)
Here Ω = ρρ
c is the density parameter, with ρc= 3H8πG2 the critical density. From this it follows that a flat universe k = 0 needs a density parameter Ω = 1.
Consider the time evolution of the density parameter, d
dt(Ω − 1) =2kc2 Ha
d dt
1 aH
. (2.13)
Again the comoving Hubble radius appears, which was shown to be increasing for normal Big Bang expansion. Therefore we see that the flat universe with Ω = 1 is an unstable solution.
Nowadays, in observations the universe is found to be flat up to percentage level
|Ω − 1| . 0.01 [8]. This implies even more accurate flatness of the universe at earlier times, up to some dozens of orders of magnitude higher flatness in the early universe. Where does this initial flatness come from?
2.6 Finetuning of initial conditions
As seen in the horizon and flatness problem, there are some problems in the normal Big Bang theory that involve very accurate finetuning of conditions for the early universe. Of course, the universe could have started under these very specific conditions, but they have to be finetuned so accurately that the formation of this universe would be very improbable. There is no physical reason know why the universe would need the initial parameters to be exactly as they appear to be from normal Big Bang expansion. The temperature anisotropies in the CMB could have been much larger within the theory and the FRW-metric does hold for non-flat universes as well. Is our universe just a very improbable coincidence or is there some reason for these specific initial conditions?
2.7 Inflation
Although it is possible that our universe has been formed from very specific initial conditions, another, more beautiful solution to the finetuning problem is provided by introducing a period of inflation at a very early stage in the evolution of the universe. The period of inflation is defined by an accelerated expansion of the scale of the universe, so that, opposite to during normal Big Bang expansion, the Hubble radius decreases over time,
d dt
1 aH
= d dt
1
˙a
= − ¨a
˙a2 < 0. (2.14) This formula shows why this period of inflation is defined by the accelerated ex- pansion (¨a > 0). The acceleration equation (2.5) therefore implies the following conditions on the content of the universe during the phase of inflation:
ρ +3p
c2 < 0 ⇔ ω < −1
3. (2.15)
Due to this decreasing Hubble radius a period of inflation is a solution to both the horizon and the flatness problem, which both occurred because the Hubble radius was thought to be increasing during the whole evolution of the universe.
No specific initial conditions are needed, only a period of inflation lasting long enough to provide an explanation for the observed finetuning.
Figure 2.4: Expansion of the universe with and without an epoch of inflationary expansion. All scales are approximate and only meant to give a feeling for the size of the events. Based on similar pictures by NASA and [3].
2.8 Monopole problem
Next to the horizon and flatness problem, other problems can be solved by in- flation as well. To show the strength of the theory of inflation, one somewhat unrelated problem will be presented here: the so called monopole problem.
While trying to find a Grand Unified Theory, in which the electromagnetic, strong and weak force combine to one at very high energy levels, it was pre- dicted that an enormous amount of magnetic-monopoles were formed in the high energy universe, that should still be present today. However, up to now no magnetic-monopole has been discovered during any experiment. Where did all these monopoles go to?
Again, inflation presents a solution. If the universe vastly expands during a period after the energy of the universe had dropped below the point were these monopoles could be formed, the density of these particles would rapidly de- crease. If this expansion is large enough, the density could decrease up to a point were it is probable that there are no monopoles present in the observable universe to be observed today.
By now hopefully you are convinced of the power of he theory of inflation.
In the next sections a more rigorous description of inflationary models will be presented.
Chapter 3
Slow-roll inflation
For the problems of normal Big Bang expansion to be solved by inflation this period has to last long enough. Long enough in this context means either slowly accelerating expansion over a long period of time, or a shorter period of rapid accelerating expansion. Longer lasting inflation is referred to as slow- roll inflation, since it can be understood as being driven by a slow-roll down a potential energy function. Historically slow-roll inflation was also known as New inflation, in relation to old inflation, which was a model introduced earlier, wherein the universe tunneled from being in a false vacuum state to a true vacuum state. A more quantitative description of what slow-roll inflation means will be given in this section, based on [3, 5, 9, 10]. But first, let us introduce a useful measure of time during inflation.
3.1 Number of e-folds
Since inflation is a period of expansion a natural measure of time is the size of the universe. The size of the universe is described by the scale factor a.
It turns out that during inflation the evolution of the scale factor is almost exponential a ∝ eHt, where H is approximately constant. Therefore evolution during inflation can accurately be described by the number of e-folds N , which is a measure for the number of times the universe has expanded by a factor e. It is common to count e-folds still needed until the end of inflation, thus eHdt= edN, which implies,
N (t) = Z tf
t
Hdt. (3.1)
Here tf is the end of the inflationary epoch, which corresponds to N (tf) = 0.
3.2 Inflaton field
For deriving a simple description of slow-roll inflation a model is considered where inflation is driven by a scalar field called the inflaton field φ. To simplify many expressions from now on the constant Planck-mass is set to one, mp = 1/√
8πG = 1. For now consider the evolution of the inflaton field to be described
by the following general Lagrangian consisting of a kinetic and potential energy term:
L = 1
2gµν∂µφ∂νφ − V (φ). (3.2) In this description no influences of coupling to the gravitational background are taken into consideration. Since it was shown that in inflation a flat universe is an attractor, we will consider k = 0 everywhere during the following derivation.
This implies that the energy-momentum tensor for flat space can be considered, Tµν= ∂L
∂(∂µφ)∂νφ − δνµL. (3.3) If now the field φ is assumed to behave as a perfect, isotropic fluid, the energy- momentum tensor needs to fulfill,
T00= ρ = 1 2
φ˙2+ V (φ), (3.4)
Tii= −p c2 = −1
2
φ˙2+ V (φ). (3.5)
Where i = 1, 2, 3 is a spatial index. The perfect, isotropic fluid implies that
∇φ = 0, which therefore does not appear in the equations above.
The above assumptions may seem dangerous. The universe might not be flat at all at the beginning of inflation. And why would the scalar field, which nature is not even well known, behave as a perfect fluid? It is well known by now that a period of inflation would lead to a perfectly isotropic, flat universe near the end of inflation, but this does not imply that these conditions do hold during the period of inflation itself. Using computer simulations however it is shown that deviations in initial conditions before inflation are of minor influence on observable quantities that will be derived. This was for example considered in [11]. Thus assumptions as mentioned above lead to accurate approximations, and are useful to study to develop some insight in the meaning of slow-roll inflation.
Slow-roll inflation appears when looking at the equation of state parameter,
ω = p ρc2 =
1
2φ˙2− V (φ)
1
2φ˙2+ V (φ). (3.6)
For inflation ω ≈ −1 is needed, thus we must find ˙φ2 V , which is the slow-roll approximation.
From the Friedmann equations introduced in section 2.3, the following relations can be derived, using the relations above for the energy-momentum components:
H2= 1 3
1 2
φ˙2+ V (φ)
, (3.7)
H + H˙ 2= −1
3 ˙φ2− V (φ)
, (3.8)
φ + V¨ φ= −3H ˙φ. (3.9)
Which were derived from the Friedmann equation (2.4), the acceleration equa- tion (2.5) and the continuity equation (2.6) respectively. Here Vφ denotes the the derivative of the potential with respect to the field φ: Vφ= dVdφ.
3.3 First slow-roll parameter
For slow-roll inflation to hold some testable parameters can be introduced. The first parameter is general for inflation to hold, it is related to the defining acceler- ating expansion condition. The first slow-roll parameter that is often considered is defined as:
εH≡ − H˙
H2 = −a
˙a
2 d dt
˙a a
= −aa¨
˙a2 + 1. (3.10) From the accelerating expansion condition (2.14), where a, ˙a, ¨a > 0, it is con- cluded that a strict condition for inflation is,
εH < 1. (3.11)
The subscript H denotes the parameter to be described by the Hubble param- eter. This description is useful since it is fully exact. It does however have as a major disadvantage, it can only be used when the equations of motion and thereby the time evolution of the scale factor is known explicitly. The parame- ter can also be approximated to a form depending on the potential V , and its derivatives to the scalar field φ. The scalar field thereby takes over the function of time evolution. This requires that the time dependence of the scalar field does not change sign, so let‘s consider the arbitrary choice where ˙φ > 0. If the approximation | ¨φ| |Vφ|, which is the first time derivative of the slow-roll approximation ˙φ2 V , also holds, then equation (3.9) can be approximated to be,
Vφ≈ −3H ˙φ. (3.12)
Next to the velocity of the inflation field the acceleration is thus also small. The change of the Hubble parameter with time, which is related to the accleration of the expansion of the universe, can be derived from taking the difference between (3.7) and (3.8).
H = −˙ φ˙2
2 ≈ − Vφ2
18H2. (3.13)
Now the first slow-roll parameter can be derived following the definition in 3.10 using (3.7) and (3.13) in terms of the inflaton field φ,
εH= φ˙2 2H2 = 3
1 2φ˙2
1
2φ˙2+ V. (3.14)
This exact slow-roll parameter can be approximated in terms of the the potential V and its derivative. The reason for this is the trouble of finding the exact slow- roll parameter, for which the equation of motion for the inflaton field are needed.
The approximate slow-roll parameter is:
εV = 1 2
Vφ
V
2
. (3.15)
Thus the first slow-roll parameter is related to the steepness of the potential V . In the slow-roll approximation εH ≈ εV, and εH, εV 1, where the last is clear from (3.14).
Now let us have a closer look at what this flat potential means. If the inflaton
very slowly rolls over a plateau in the potential, this potential takes the role of a cosmological constant. In this regime where ω ≈ −1 we do thus by filling in equation 2.9 find exponential expansion of the universe with approximately constant Hubble parameter ( ˙H 1), as was assumed in the definition of the number of e-folds before. In number of e-folds N this first slow-roll parameter can be expressed as:
εH= − 1 H2
dH dt = −1
H dH
dN = −d ln H
dN . (3.16)
Thus the number of e-folds is,
N (t) = Z tf
t
Hdt = Z φf
φ
H
φ˙ dφ ≈ − Z φf
φ
V
Vφdφ = − Z φf
φ
√dφ 2εV
. (3.17)
The end of inflation tf is defined by violation of condition (3.11), εH= 1 ≈ εV.
3.4 Second slow-roll parameter
In the previous subsection a parameter was derived in which a strict condition was found for inflation to occur. Also an approximation was derived in which inflation would last over a longer period. To let the statement ε 1 last long enough it is also required that it does not rapidly increase over time (measured in N ). To ensure this a second slow roll parameter η is introduced,
d ln εH dN = ε˙H
HεH
= H¨
H ˙H − 2 H˙
H2 = −2ηH+ 2εH, (3.18) which implies,
ηH = −1 2
H¨ H ˙H = −
φ¨
H ˙φ. (3.19)
In the the slow-roll approximation | ¨φ| |Vφ| ≈ |3H ˙φ| it is immediately clear that this second slow-roll parameter should also be small: ηH 1. Again some similar parameter can be introduced as a function of the potential V .
ηV = Vφφ
V . (3.20)
Using the slow-roll approximation once again the following relations can be derived from (3.7) and (3.9) respectively,
V ≈ 3H2, Vφφ≈ −3 ˙H − 3H ¨φ
φ˙ . (3.21)
From which we can conclude that in the slow-roll approximation ηV ≈ ηH+ εH. Since ηH, εH 1 we also find that for slow-roll ηV 1.
Even higher order slow-roll parameters could be introduced, but these are mostly not considered since they do not contribute much to observable quantities.
3.5 Duration of slow-roll inflation
It turns out that many problems can be solved by a period of inflation that lasts long enough? But how long is long enough? Although the procedure of getting the answer is completely different, it turns out that the results for the different problems is similar. In this section the number of e-folds needed to solve the horizon problem is derived since the method is rather intuitive, but the same result is obtained from measurements on the flatness of the universe [2].
It is clear from observations that the complete CMB sky is homogeneous up to tiny fluctuations. Therefore the largest scale observed today must have been n causal contact at some point. This scale is the size of the observable universe, given by the current Hubble parameter H0,
λ(t0) = c H0
. (3.22)
This scale must therefore have been inside the Hubble radius at some point during inflation. Since the Hubble radius decreases during inflation this must have been a long enough period before the end of inflation, at some time ti,
λ(ti) = λ(t0)ai a0
= c H0
af a0
ai af
< c HI
. (3.23)
Here HI is the approximately constant Hubble parameter during inflation. The fraction aai
f relates the the scale factor at the moment that there is still causal contact between points that will be at the ends of the current observable universe to the scale factor at the end of inflation. This relation defines the number of e- folds, thus the smallest period of inflation needed to solve the horizon problem.
The relation between these scales is thus af = eNai. The fraction aaf
0 relates the scale factor at the end of inflation to the scale of the universe now. Using that the universe has been in a radiation dominated state during this period the Stefan-Boltzmann’s law can be used, relating the scale of the universe and temperature as T ∝ a−1.
These relations can be combined to find the lower bound for the number of e-folds,
c H0
T0
Tfe−N < c
HI ⇒ N > ln T0
H0
− ln Tf
HI
≈ 60, (3.24) from filling in the observed quantities. This provides a lower bound to the num- ber of e-folds, almost independent on the model for inflation. Of course the number of e-folds could be much higher, this can however not be confirmed by observations on the current universe, all problems can be solved with approxi- mately 60 e-folds of inflation.
Chapter 4
Quantum fluctuations and related observables
Up to this point only a perfect, classical universe was considered during inflation.
Many problems were solved by looking at the evolution of such a universe dur- ing a period of slow-roll inflation, resulting in a rather homogeneous universe.
If however quantum fluctuations are taken into account a description is ob- tained were next to a general homogeneity small anisotropies in the background are predicted. The important observation underlying this description including quantum fluctuations are is the small but clearly present space dependent de- viations in the temperature of the CMB. The power spectrum obtained from these anisotropies can be very well explained by consideration of quantum fluc- tuations of the inflaton field during inflation.
Figure 4.1: Slow-roll inflation driven by a inflaton field rolling down a potential
Inflation stops at a clearly defined point, when expansion is no longer acceler- ating, as was the definition of inflation. In chapter 3 it was shown that this also implies breaking of some slow-roll parameter conditions, εV ≈ εH = 1. For a completely homogeneous universe this would imply a fixed point in time were inflation stops everywhere. Quantum fluctuations do however allow the inflaton to move up and down the potential, which results in a space dependent ending
of inflation. In terms of physical observables this can be interpreted as fluctu- ations in energy density, leading to locally stronger gravitational attraction. In these regions the inflaton will take longer to slow-roll, resulting in a locally later ending of inflation. What these quantum fluctuations do imply for observable quantities will be studied following [10].
4.1 Perturbations due to quantum fluctuations
The quantum fluctuations impose small deviations, depending on both space and time, from a homogeneous background, which is only dependent on time, in both the energy density and the pressure of the inflaton field.
φ(x, t) = ¯φ(t) + δφ(x, t) ⇒
(ρ(x, t) = ¯ρ(t) + δρ(x, t)
p(x, t) = ¯p(t) + δp(x, t). (4.1) The homogeneous background is denoted by an overbar and the fluctuations by a δ. Following the Einstein equations these fluctuations imply perturbations in the here considered flat FRW-metric (2.2). In general this first-order perturbed flat FRW-metric is of the form [10],
ds2= gµνdxµdxν,
= −(1 + 2Φ)c2dt2+ 2aBidxicdt + a2[(1 − 2Ψ)δij+ Eij]dxidxj. (4.2) Here perturbations of three different natures can be distinguished, scalar, vector and tensor perturbations. To make this distinction clear Bi and Eij can be decomposed,
Bi≡ ∂iB − Si, Eij ≡ ∂i∂jE + 2∂[iFj]+ hij, (4.3) where ∂iSi, ∂iFi = 0 and hii, ∂ihij = 0, so that Φ, Ψ, B and E correspond to scalar, Si and Fi to vector and hij to tensor perturbations. In inflation the scalar perturbations cause temperature fluctuations. Vector perturbations do not occur during inflation. Tensor perturbations will cause density fluctuations and gravitational waves [5]. It can be shown [10] that these different modes can be studied independently, the different modes are not coupled by the Einstein equations for a flat FRW-metric.
4.1.1 Gauge invariance
Important in consideration of perturbations is that the distinction between back- ground and perturbation is dependent on the coordinate or gauge choice. A frame could be chosen where the energy and pressure are completely homoge- neous, so that there do not appear to be any fluctuations at all. On the other hand a frame can be chosen whereby perturbations appear without any physical meaning. The solution to this problem is taking into account all perturbations in a consistent way, since no perturbations are really lost by a gauge choice, only transferred between matter and metric perturbations. This is as would be expected, since there is no physical meaning in imposing a particular coordi- nate system. Tensor perturbations are gauge invariant, vector perturbations do not play a role in inflation, but the transformations of the scalar perturbations therefore have to be handled with some care.
4.2 Freeze in of perturbation modes
Since the perturbations on a flat FRW-metric are translation invariant they can be split in non-interacting perturbation modes by a Fourier decomposition.
Since the different modes are non-interacting they can be treated independently, δφk(t) =
Z
δφ(x, t)eikxd3x. (4.4) Here k is the wavenumber, inversely proportional to the scale of the perturbation mode. As was shown before the Hubble radius shrinks during inflation. Thereby perturbation modes that were inside the causal horizon during formation can over time loose causal contact and thereby their amplitude can freeze into the background.
Figure 4.2: Freezing in of a perturbation mode by exciting sub-horizon scales.
Based on [10].
As discussed in section 4.1.1 one can not simply consider perturbations in the inflaton field only, since by a coordinate transformation these perturbations can be changed or even put to zero. A quantity related to scalar perturbations that can be shown to be invariant [10] is the comoving curvature perturbation,
Rk= Ψk−H φ˙¯
δφk. (4.5)
From the Einstein equations the time evolution of R can be found to be ap- proximately constant, ˙R = 0 at super-horizon scales k < aH [10]. Thus the co- moving curvature perturbation mode does not change after exciting the Hubble radius, leaving the amplitude of the perturbation mode unchanged, until hori- zon re-entry. But since the perturbation mode does only re-enter the horizon after inflation has ended the later time evolution is very predictable. Thereby a direct observable related to inflation is created in the later created Cosmic Microwave Background.
4.3 Evolution of perturbation
To find the evolution of the perturbations the general action for the scalar inflaton field weakly coupled to gravity is considered, where for simplicity the Planck mass is again set to be one,
S = Z
d4x√
−g 1 2R −1
2∂µφ∂µφ − V (φ)
. (4.6)
The comoving curvature and metric perturbations are independent on gauge transformations, therefore without loss of generality the gauge can be consid- ered, where,
δφ = 0 ⇒ φ(x, t) = ¯φ(t), gij = a2[(1 − 2R)δij+ hij]. (4.7) Here R is the comoving curvature scalar perturbation and hij corresponds to the metric perturbations, thus hii, ∂ihij = 0. These two will now be considered separately, but using comparable methods.
4.3.1 Scalar perturbations
To study the scalar fluctuations the action, for which no exact solution for the equations of motion can be obtained, can be expanded. The second order expansion in scalar perturbation R of the action for this gauge is:
S(2)= Z
d4x a
√−g 2
φ˙2 H2
h
a2R˙2− (∇R)2i
. (4.8)
To make the structure of this equation more clear it is useful to introduce the Mukhanov variable v and switch to conformal time τ ,
v ≡ zR, z ≡ a φ˙
H, dτ = dt
a. (4.9)
Then the expansion of the action is:
S(2)= Z
dτ d3x
√−g 2
(v0)2+ (∇v)2+z00 z v2
. (4.10)
The prime represents a derivative with respect to conformal time. This is clearly the action corresponding to a harmonic oscillator with time dependent fre- quency. After Fourier expansion of the Mukhanov variable the equations of motion corresponding to the action are found using Euler-Lagrange’s method to be:
vk00+
k2−z00
z
vk= 0. (4.11)
4.3.2 Tensor perturbations
The action (4.6) can also be expanded to second order with respect to the tensor perturbation hij, resulting in:
S(2)= 1 8 Z
dτ d3xa2[(h0ij)2− (∂lhij)2]. (4.12)
After the Fourier expansion of the tensor perturbation, hij=
Z d3k (2π)3
X
s=+,×
sij(k)hk(τ )eikx, (4.13) and redefinition,
vsk≡a
2hsk, (4.14)
an equation of motion very similar to (4.11) can be obtained:
(vsk)00+
k2−z00 z
vsk= 0. (4.15)
4.3.3 Quantization
For quantization of the scalar perturbation, which by now is described by (the Fourier decomposition of) the Mukhanov variable vk. Quantization is done by promoting the scalar field perturbation mode vk to a quantum operator ˆvk,
ˆ
vk= vkˆak+ v−k∗ ˆa†−k. (4.16) Here ˆak and ˆa†−k are annihilation and creation operators, obeying the usual commutation relation,
[ˆak, ˆa†k0] = (2π)3δ(3)(k − k0). (4.17)
4.3.4 de Sitter solution
As discussed before the background can be approximated by a perfect de Sitter space, where the Hubble parameter H is constant and potential is flat, thus φ = 0. This solves the equation of motion with time dependent frequency,¨
z00 z = a00
a + φ˙00
φ˙ + H 1 H
00
=a00
a = 2H2a2= 2
τ2. (4.18) Therefore the equation of motion (4.11) in the de Sitter limit becomes a har- monic oscillator with time dependent frequency,
v00k+
k2− 2
τ2
vk= 0. (4.19)
It can be checked that the general solution to this is:
vk= αe−ikτ
√2k
1 − i
kτ
+ βeikτ
√2k
1 + i
kτ
. (4.20)
Locally spacetime has to be flat, a Minkowski solution should be found at the sub-horizon level, which corresponds to the lowest energy state solution. Thus in the sub-horizon limit k aH ⇒ kτ −1 the mode functions only contains the positive frequency solution,
lim
kτ −1vk= e−ikτ
√
2k . (4.21)
This bounds the mode equation to the solution where α = 1, β = 0, thus, vk= e−ikτ
√ 2k
1 − i
kτ
. (4.22)
4.4 Power spectra
To provide a quantitative description of the perturbation modes the power spec- trum is considered. To compute this power spectrum related to the perturbation v following the solution for a de Sitter background as derived in the previous sec- tion this solution (4.22) has to be filled into the operator decomposition (4.16).
Using the definitions of the vacuum ˆa|0i and h0|ˆa†and the commutation relation between the ladder operators (4.17) the correlation function between the mode k and k0 is found to be:
hvkvk0i ≡ h0|ˆvkvˆk0|0i = h0|(vkˆak+ vk∗ˆa†−k)(vk0ˆak0+ v∗k0aˆ†−k0)|0i,
= vkv−k∗ 0h0|ˆakaˆ†−k0|0i = vkv−k∗ 0h0|[ˆak, ˆa†−k0]|0i,
= (2π)3|vk|2δ(3)(k + k0).
(4.23)
Here |vk|2 is found from taking the product of the solution for vk (4.22) with its complex conjugate,
|vk|2= vkv∗k= 1 2k3
k2+ 1
τ2
= |vk|2= 1
2k3 k2+ a2H2 . (4.24) Since modes freeze in at super-horizon scales, the super-horizon limit has to be considered, where k aH. In this limit the correlation function becomes:
hvkvk0i = (2π)3δ(3)(k + k0)a2H2
2k3 . (4.25)
4.4.1 Scalar power spectrum
The power spectrum related to scalar fluctuations can be calculated using the correlation function between two scalar fluctuation modes, which can be related to the correlation function for v as derived before, using relation (4.9),
hRkRk0i = H2
a2φ˙2hvkvk0i = (2π)3 δ(3)(k + k0)H2 2k3
H2
φ˙2. (4.26) From this equation it is clear that the modes do indeed freeze in at super-horizon scales, the correlation function is constant in time. The power spectrum for scalar fluctuations is therefore:
PR(k) = H2 2k3
H2 φ˙2 ≡ 2π2
k3 ∆2R(k), (4.27)
where the dimensionless scalar power spectrum is thus,
∆2s(k) = ∆2R(k) = H2 (2π)2
H2
φ˙2 = H2
8π2εH. (4.28) Here the first slow-roll parameter εH appears, which was defined in section 3.3. Since we are interested in the power spectrum that was frozen into the background by exiting the causal horizon the power spectrum of interest is at horizon crossing, k = aH.
4.4.2 Tensor power spectrum
Again the treatment for tensor fluctuations is very similar to the treatment of scalar fluctuations. Important again is that there are two polarizations of tensor fluctuations (s = +, ×), which have to be considered separately. The following relation is obtained for the correlation function between two tensor fluctuation modes, related to v via equation (4.14),
hhskhsk0i = 4
a2hvksvks0i = (2π)3 δ(3)(k + k0)2H2
k3 . (4.29)
Resulting in the following (dimensionless) power spectrum:
Ph(k) =2H2
k3 , ∆2h(k) =H2
π2. (4.30)
Again, we are interested in the power spectrum at horizon exit. Since there are two polarizations the total dimensionless tensor power spectrum is:
∆2t(k) = 2∆2h(k) =2H2
π2 . (4.31)
4.5 Observables
4.5.1 Tensor-to-scalar ratio
An interesting model dependent measure on inflation is the ratio between tensor and scalar perturbations. The related observable is the tensor-to-scalar ratio r, which is the ratio between the (dimensionless) tensor and scalar power spectrum,
r = ∆2t(k)
∆2s(k) = 16εH≈ 16εV. (4.32) One can thus compute the predicted tensor-to-scalar ratio for a model via cal- culating the corresponding slow-roll parameter at horizon exit.
4.5.2 Spectral index
An other observable related to the power spectrum is the spectral index n. This is a measure of the general form of the power spectrum related to the the scale of perturbation modes. The spectral index is defined as the power of k in the dimensionless power spectrum. Since the scalar spectral index is close to one, mostly ns− 1 is considered.
∆2s(k) ∝ kns−1, ∆2t(k) ∝ knt, (4.33) which can be solved for the spectral index to be
ns− 1 = d ln ∆2s
d ln k , nt= d ln ∆2t
d ln k . (4.34)
These can again be related to the slow-roll parameters as introduced in chapter 3. A possible way to see this is by changing to derivatives with respect to the
number of e-folds N . Standard rules for logarithms can be used to rewrite the spectral indices.
ns− 1 = d ln ∆2s dN
dN d ln k =
2d ln H
dN −d ln εH
dN
dN
d ln k, (4.35) nt=d ln ∆2t
dN dN
d ln k = 2d ln H dN
dN
d ln k. (4.36)
The definitions of the slow-roll parameters in terms of the number of e-folds can be used to both first terms. The second terms can be solved using the fact that we are interested in in the spectral indices at horizon exit. We can therefore use that k = aH, which implies,
ln k = ln a + ln H = N + ln H. (4.37) This can be used to evaluate the last terms in (4.35) and (4.36),
dN
d ln k = d ln k dN
−1
= (1 − εH)−1= 1 + εH+ O(ε2H) ≈ 1. (4.38) The last approximation holds since εH 1 during slow-roll inflation. Therefore the following relations can be obtained for slow-roll inflation:
ns− 1 = −4εH+ 2ηH≈ −6εV + 2ηV, nt= −2εH≈ −2εV. (4.39)
4.6 Observational constraints
As for any physical theory the only measure of ’truth’ for the theory of inflation is accordance to experimental results. It has been mentioned before that the most direct measurements on inflation can be performed by looking at the Cosmic Microwave Background. Multiple telescopes have been set up to measure the tiny fluctuations in the CMB with increasing accuracy, as can be seen in figure 4.3. In this section the most resent observations by the Planck satellite are discussed, following the publications of results in 2013 [8] and 2015 [12], also based on the von K´arm´an lecture by one of the collaborators, presenting a very clear overview of the recent development [13]. The most important results will be presented in this section, meant to convince the reader of the evidence for the credibility of the theory of inflation, including restrictions on the range of inflationary models.
Figure 4.3: The increasing accuracy in determination of the temperature fluctu- ations in the Cosmic Microwave Background measured by the COBE, WMAP and Planck satellites, including the year of the measurement. Image credit:
Chris North, Cardiff University.
The Planck satellite scans the entire sky in bands, rotating around its axis once per minute. The scans are made in a wide frequency range from 30 to 857 GHz to be able to measure, next the CMB, also many other cosmological object influencing the low temperature picture. The measurements identifying these objects can then be used to subtract their influences on the observed CMB to get a cleaner picture of the surface of last scattering. Mapping the three dimen- sional scan to a two dimensional sky map after subtraction of secondary sources results in plots as shown in figure 4.3. In these plots the color is chosen to illus- trate the small fluctuations in temperature of the CMB at fractional deviation 10−5 from the all-sky temperature average of the CMB. It is important to keep in mind that the wavelengths corresponding to these temperatures are far to small to be observable with the human eye. And even if the wavelengths would be shifted to a visible region of the spectrum the fluctuations would be far to small to be distinguishable.
From this sky map multiple correlation functions can be obtained related to the power spectrum of the CMB as it is measured today. One example of such a correlation function is the temperature power spectrum, as shown in figure 4.4.
A further description of the exact meaning of this temperature correlation func- tion and how it is obtained is beyond the scope of this text, for further reading one is referred to [8, 12]. A best fit to this power spectrum can be explained by a minimal six parameter model, both parameters related to the initial creation of the power spectrum at the epoch of recombination and parameters related to the evolution of the universe thereafter. One of these parameters is the spectral index ns, that was encountered before. This spectral index or spectral tilt is related to the deviation from scale invariance ns= 1 of the initial power spec- trum of the perturbation modes creating these temperature fluctuation. After careful computation this procedure results in a experimental constraint on the spectral index ns[12],
ns= 0.968 ± 0.006. (4.40)
Figure 4.4: Angular TT power spectrum, a correlation function related to the current CMB. From [12].
Up to this point only the T-polarization, which is only induced by scalar per- turbations was considered. There exists an other polarizations, that can be split in E-modes and B-modes. This polarization is partially induced by tensor perturbations, but can also be caused by scalar perturbations. Angular power spectra can be studied, similar to the TT spectrum in figure 4.4, but then re- lated to these E-modes and B-modes. Since these power spectra can be induced by tensor perturbations, but do not necessarily only consist of these, an upper limit to the tensor-to-scalar ratio r can be obtained from these measurements.
In 2015 the Planck satellite results constrained the tensor-to-scalar ratio to be [12],
r < 0.11. (4.41)
These two observables, the spectral index nsand tensor-to-scalar ratio r, are of great importance in the distinction between more and less accurate models for inflation. In figure 4.5 the constraint to both of these parameters are plotted, including some common models for inflation. This leads to a strong tendency to a particular class of models, which will be studied in more detail in the following chapters.
Figure 4.5: Constraints by the Planck satellite on the tensor-to-scalar ratio r and spectral tilt ns. The theoretical predictions of different models for inflation are included in this graph to illustrate the correspondence to the experimental results. From [12].
Chapter 5
Toy models for inflation
In the long derivation presented in the previous section two observable quantities were found. In this section two simple inflation models will be presented for which the observable quantities will be derived. This is useful to get a better feeling for those quantities, before taking a look at a more complicated class of inflationary models.
5.1 Cosmological constant potential
One of the most trivial models for inflation is the inflaton scalar field with constant potential. Nevertheless it is an interesting case to study since it is the limit approached by many slow-roll inflation models.
Figure 5.1: Constant potential Since the potential is constant all its derivatives
are zero and therefor the slow-roll parameters are zero as well. For the observable quantities related to inflation this implies that the spectral index is constant and one for scalar perturba- tions (ns= 1) and for for tensor perturbations zero (nt = 0). It is thus also obvious that the tensor-to-scalar ratio is zero as well (r = 0), in this model no tensor perturbations are being created.
This model is clearly not a good candidate for describing inflation. On a constant potential the scalar field does not evolve, inflation never ends. Because he potential is constant there is no influence by quantum fluctuations and no tensor modes appear. Nevertheless this is a use- ful model to consider, since it is the limit case of many inflationary models. From the obser- vational restrictions presented in section 4.6 it
is clear that there is a small but finite deviation from scale invariance in infla- tion, the scalar spectral index nsis close to but not equal to zero. This predicts an almost flat, but not constant potential for inflation, further referred to as a plateau.
5.2 Quadratic potential
In this section a completely different model of inflation is considered, where the potential is much steeper. For this simple quadratic potential V = m2φ2the ap- proximate slow-roll parameters can be calculated,
Figure 5.2: Quadratic potential εV =1
2
Vφ
V
2
= 2
φ2, (5.1) ηV =Vφφ
V = 2
φ2. (5.2)
Now take a look at how this is related to the number of e-folds (3.17),
N = − Z φf
φ
√dφ
2εV = −1 2
Z φf φ
φdφ,
= −1
4 φ2f− φ2 .
(5.3)
Inflation ends approximately when εV = 1, φf = 1
√2. (5.4)
Thus the scalar field value at a particular number of e-folds is, φ(N ) =√
4N + 2. (5.5)
Therefore, the slow-roll parameters in this model are:
εV = ηV = 1
2N + 1≈ 1
2N, (5.6)
which is indeed small for more e-folds, thus confirms the slow-roll approximation (N ≈ 60, which was derived before to correspond to the minimal length of inflation, leads to εV = ηV = 0.00826). For the spectral indices and tensor-to- scalar ration this implies,
ns− 1 = 2ηV − 6εV = −2
N ≈ −0.033, (5.7)
nt= −2εV = −1
N ≈ −0.017, (5.8)
r = 16εV = 8
N ≈ 0.13. (5.9)
The constant approximation is the predicted value for the observable after 60 e- folds of inflation. From this it is clear that that the model based on a quadratic potential is not in accordance to the observational constraints from section 4.6.
The potential is too steep, which results in more tensor fluctuations than ob- served, the upper bound to the tensor-to-scalar ratio is r < 0.11. Again the conclusion must be that a potential should be considered approaching a plateau.
5.3 Starobinsky potential
An other interesting model for inflation is the Starobinsky model, which can be written as [5]:
L =√
−g 1 2R −1
2∂µφ∂µφ −3 4
1 − e−
√
2/3φ2
. (5.10)
The last term corresponds to the inflaton potential, depicted in figure 5.3.
Figure 5.3: Starobinsky potential [5]
This potential is interest- ing since for large values of the inflaton field φ the potential approaches a con- stant value of −3/4, as stud- ied in 5.1. But closer to φ = 0 it can be approx- imated by a quadratic po- tential, as studied in section 5.2. The plateau has there- fore a slight negative devia- tion from the constant po- tential for all φ, resulting in non-zero spectral indices and tensor-to-scalar ratio.
The Starobinsky potential can be used to calculate the slow-roll parameters directly,
εV = 1 2
Vφ
V
2
= 4 3
1 e
√
2/3φ− 1
!2
, (5.11)
ηV = Vφφ
V = 4 3
−e
√
2/3φ+ 2
e
√
2/3φ− 12. (5.12)
Since slow-roll can take place at the edge of the plateau the large field approxi- mation φ 1 can be used. In this approximation the slow-roll parameters can be expanded to be:
εV = 4 3e−2
√
2/3φ, ηV = −4 3e−
√
2/3φ. (5.13)
Inflation ends approximately when εV = 1, φf= − 1
2p2/3ln 3 4
, (5.14)
which can be used to compute the number of e-folds dependence of φ,
N ≈ − Z φf
φ
√dφ 2εV
= −e12 +3 4e
√
2/3φ(N ). (5.15)
Therefore the scalar field can for large numbers of e-foldings be approximated:
φ(N ) ≈ 1
p2/3ln 4N 3
, (5.16)
resulting in the slow-roll parameters, εV = 3
4N2, ηV = −1
N. (5.17)
This corresponds to the following spectral indices and tensor-to-scalar ratio:
ns− 1 = −2
N ≈ −0.033, nt= − 3
2N2 ≈ −4.2 10−4, r = 12
N2 ≈ −3.3 10−3. (5.18) These values for the observables are accurate according to the observational constraints from section 4.6. In figure 4.5 the observables are the same as for R2inflation, denoted in orange.
It turns out that this Starobinsky model for inflation is not unique in predicting these accurate values for the observables, in the next chapter a more general class of models is described resulting in the same predictions.
