Universiteit van Amsterdam. Cosmological Inflation

Transcription

1 Universiteit van Amsterdam Institute for Theoretical Physics Master s Thesis Cosmological Inflation Author: Jason Dekdebrun Supervisor: Prof. Dr. Kostas Skenderis August 2nd, 2010

2 To Dr. Richard Eykholt

3 Preprint typeset in JHEP style - HYPER VERSION Revised August 10, 2010 Cosmological Inflation Jason Dekdebrun Abstract: An introductory analysis of the most prominent theory for the early stages of the universe: cosmological inflation. The theory explores the dynamics of a universe that undergoes an early stage of rapid, exponential expansion, known as inflation. This answers the big problems plaguing big bang cosmology, while also providing an origin for the large-scale structure of the cosmos. The expansion itself relies heavily on the presence of scalar fields, whose energy density dominates that of the universe; thus, an extensive analysis of the dynamics of scalar fields is given. To study the important implications of inflation, one must consider the effects of primordial perturbations to the spacetime, and the subsequent evolution of the universe during inflation. This analysis requires a detailed study of gauge transformations, as only gauge invariant quantities will be physically observable. Specifically, this thesis aims at deriving the full expressions for the second order, gauge invariant Einstein equations, from which the time evolution of the primordial density perturbations may be calculated. This may then be checked by experimental observations.

4 Contents 1. Introduction 4 2. Dynamics Of Inflation Basics Of General Relativity The Friedmann-Robertson-Walker Metric Conservation Of Energy And Momentum The Einstein Equations Introduction To Inflation The Friedmann Equations Co-moving Coordinates, Particle Horizons, & The Hubble Radius Problems With Hot Big Bang Cosmology Pure And Quasi de Sitter Stages Scalar Field Dynamics The Energy-Momentum Tensor Of A Scalar Field The Slow-roll Approximation Conformal Time Scalar Field Fluctuations Fluctuations In A Pure de Sitter Space-time Exact Solution For Wave Modes Fluctuations In A Quasi de Sitter Space-time The Power Spectrum Perturbations General Theory Metric Perturbations The Covariant Metric The Contravariant Metric Scalar Field Perturbations Perturbations To A Perfect Fluid 58 1

5 5. Gauge Transformations Taylor Expansion For Tensor Fields Gauge Transformations Of The FRW Metric First Order Results Second Order Results Gauge Transformations Of A Scalar Field Gauge Invariant Variables First Order Results Second Order Results Higher Order Results Curvature Perturbation On Different Hypersurfaces Multiple Fluids Gauge Choices Perturbed Einstein Equations Preliminary Analysis Background Equations First Order Vector Modes First Order Einstein Equations Gauge Invariant Formalism Second Order Einstein Equations Poisson Gauge Gauge Invariant Formalism Simplification Of Quadratic Terms Summary Of Results Perturbed Klein-Gordon Equation Gauge Invariant Formalism First Order Results Second Order Results Exact Solution Of The First Order Klein-Gordon Equation Connection With Observation Non-Gaussian Statistics Relationship To Cosmological Perturbations Current And Future Constraints On f NL Conclusion 148 2

6 A. Second Order Perturbed Quantities 149 A.1 Cosmic Time 149 A.2 Conformal Time 150 A.2.1 The Christoffel Symbols 153 A.2.2 The Ricci Tensor 155 A.2.3 The Four-Tensor Ξ αβµ ν 159 A.2.4 The Einstein Tensor 161 B. Notation 165 3

7 1. Introduction Knowledge and understanding of the origin of the universe has long been sought by the human race. Ever since man first looked up at the stars he wondered at their meaning, and pondered the ineffable mystery of the heavens. With the advent of scientific research, we have come closer and closer to elucidating the complexity of the cosmos, while admittedly revealing even greater enigmas to cover our discoveries paths. From the time Georges Lemaitre first proposed the idea of the big bang, this quaerere veritatem has made a most radical change in our fundamental perception of reality: a beginning to time as we know it; a beginning to the universe. Ever since, the annals of cosmological evolution itself has been the prevailing conquest of cosmologists. Today, a new chapter in this epic chronicle, recently unearthed, has been the focus of modern research. A time when the universe, in but comparatively the tiniest moment of its lifetime, underwent a most sweeping metamorphosis: an accelerated expansion of its size and shape by an unimaginable factor. This period of inflation has been scientist s missing link in the understanding of the evolution of the cosmos. Forsooth, just such an epoch in time has been able to explain many of the dilemmas that have arisen over the past few decades. The underlying dynamics of a period of accelerated expansion in the universe is the study of what is known as inflationary theory, and is currently the hottest area of cosmological research. In it, the accelerated expansion is caused by a scalar field, the inflaton, whose potential energy dominates that of the universe. In the inflationary paradigm, primordial density perturbations on tiny size scales are subsequently amplified by the acceleration of the universe, and finally frozen into the night sky. These are believed to be the source of large-scale structure in the distribution of galaxies and clumps of dark matter, originally proposed by [1], and brought into a modern context again later in [2]. In addition to this, a simple period of inflation in the universe solves three of the major cosmological problems, known as the flatness, horizon, and magnetic monopole problems. Not just a compelling answer to some of the deep mysteries of the cosmos, inflationary theory can and has been verified with experiment. Primordial density perturbations were first detected by the Cosmic Background Explorer as anisotropies in the cosmic microwave background CMB [3], and subsequently by the Wilkinson Microwave Anisotropy Probe [4]. More recently, the Planck satellite was launched in 2009 with aims at unprecedented accuracy in measurements of the CMB. While the reliance on a scalar field is unequivocally ingrained in inflationary theory, there are several models in which the precise details vary. In the so-called standard scenario, the primordial density perturbations are a result of the quantum fluctuations 4

8 of the inflaton field itself. The inflationary expansion is generated while the inflaton field slowly rolls down its potential, subsequently oscillating about its global minimum, and finally decaying into radiation. Though the standard scenario may be the paragon of inflationary theory, other models may be applicable as well. The most prominent of these are the curvaton mechanism - whose energy is negligible during inflation, but nonetheless produces the density perturbations - and the inhomogeneous reheating, ghost inflationary, and the D-cceleration scenarios to name a few. The primordial density perturbations are, of course, very tiny, and thus the focus of inflationary theory started at linear order in the perturbations. It has been recently realized that the best way to distinguish amongst the various models, is to study the density perturbations beyond linear order; namely, in the realm of second order perturbation theory. What is more, observable imprints in the night sky correspond only to gauge invariant quantities. It is thus an important undertaking to develop the gauge invariant equations of motion for such perturbations at second order, as will be the main focus of this thesis. We will start by setting up the framework for the analysis of second order perturbations, including a brief review of the standard tools and language arising from general relativity; this comprises the bulk of Chapter 2. As the presence of a scalar field is an essential part of inflation, in Chapter 3 we will study in detail the dynamics of an arbitrary scalar field. In Chapter 4, we provide a formulation for the metric and scalar field perturbations, up to second order. To tackle the issue of gauge invariance, Chapter 5 looks at the characteristics of gauge transformations in full, leading to a definition of the relevant gauge invariant quantities. Finally, reaching the pinnacle of this thesis, in Chapters 6 and 7 we will develop the second order equations of motion in a completely gauge invariant way. As a last anchor to physical interpretation, Chapter 8 draws a short connection between theory and observation; in particular the connection between the second order perturbations and experimental measurements of the CMB. Chapter 9 will end with a brief summary of what was done in the thesis, and what prospects await the future. Additionally, Appendix A derives the second order, perturbed general relativity quantities, while Appendix B tabulates the notation used throughout. 5

9 2. Dynamics Of Inflation A competent understanding of cosmology, or the more narrowed focus of inflationary theory, all require a solid background in general relativity. Since the basic linguistics of general relativity are used continually throughout this paper, we will start in this chapter with a brief summary of what one is expected to know. We define the spacetime metric we will be using, as well as defining and constructing the standard tensor quantities that make up most of the backbone of general relativity. Next we will introduce the idea of inflation, and how it may arise from the dominance of a single scalar field in the universe. It is then practical to discuss the dynamics of such a scalar field, by deriving its standard equation of motion. We also introduce and discuss conditions on the scalar field that guarantee and quantify periods of inflation. Finally, we will end by introducing the handy conformal time variable, which will be the new fundamental time-like coordinate throughout this paper. 2.1 Basics Of General Relativity The Friedmann-Robertson-Walker Metric The universe looks to a great degree of accuracy isotropic. Based on the assumption of isotropy and homogeneity in the universe, one can derive the Friedmann-Robertson- Walker FRW spacetime metric: [ ds 2 g µν xdx µ dx ν = dt 2 + at 2 dr 2 ] 1 Kr + 2 r2 dω Since the derivation of the FRW metric is based on the assumption of homogeneity and isotropy alone, we use this metric as a first approximation to our own universe for the entirety of this paper. Perturbations to the metric correspond to the slight deviations from the nearly complete isotropy we know to exist in the universe in the form of galaxies and other clumps of matter. In the FRW metric, the Robertson-Walker scale factor, at, arises from assuming spaces for our universe consisting of a sphere of radius a, which could possibly vary with time [5]. The Hubble rate, H ȧ, quantifies the rate of expansion of the universe, a as well as its dynamics. If H < 0, we must have that at is decreasing with time; similarly if H > 0 then at increases with time. If H = 0 then the universe remains the shape and size it is. It can easily be shown [5] that the distance between two points in space is proportional to at; so, if at is increasing, which it is, then the distance between those two points grows with time. 6

10 The constant K arises from incorporating the possibility of these spaces to be either spherical, hyper-spherical, or spatially flat: +1, spherical K = 1, hyper-spherical 0, spatially flat Proven by several theorems, if the universe is spherically symmetric and isotropic to freely falling observers which it is, then these are the only possibilities for the metric. Throughout the entirety of this paper, we will find that the curvature constant K can often safely be neglected see Section This will be detailed further on, but for now we will simply focus on a spatially flat K = 0 universe, in which the FRW metric 2.1 becomes: ds 2 g µν xdx µ dx ν = dt 2 + at 2 [ dr 2 + r 2 dω 2 ] 2.2 This is the metric we will often work from, and is commonly referred to as the background spacetime, as it has not yet been adjusted in a perturbative expansion. This metric has the matrix form: If we now define γ ij such that, we then have: at g µν x = 0 0 at 2 r at 2 r 2 sin 2 θ γ ij 0 r r 2 sin 2 θ [ ] 1 0 g µν x = 0 at 2 γ ij The components of the metric are then simply given by: 2.5 g 00 = 1, g 0i = g i0 = 0, g ij = at 2 γ ij Note that Latin indices are raised or lowered using γ ij, while γ i j acts as a Kronecker delta γ i j δ i j; as such γ ij γ ij = δ i i = 3. 7

11 The inverse, or contravariant metric, is found by requiring that g νλ g λµ = δ µ ν, which simply gives 1 : This has components: g µν x = [ ] a 2 tγ ij g 00 = 1, g 0i = g i0 = 0, g ij = a 2 tγ ij Note that from here on we will often denote at simply by a Conservation Of Energy And Momentum 2.6 The assumption of homogeneity and isotropy in the last section also leads to the perfect fluid form of the energy-momentum tensor. Often the perfect fluid approximation will be used, in which case the energy-momentum tensor defined formally later takes the diagonal form 2, ρ T ν µ 0 P 0 0 = 0 0 P P 2.7 where ρ is the energy density of the fluid, and P is its pressure. In general relativity, conservation of energy and momentum is encapsulated within the following covariant expression, D µ T µ ν = where here the covariant derivative D µ is used. We say in this case that the energymomentum tensor is covariantly conserved. D µ is defined through the standard Christoffel symbols or connection coefficients, given in terms of the metric by: Γ α βγ 1 2 gαρ β g ργ + γ g βρ ρ g βγ 2.9 From this definition we see that they are clearly symmetric in their lower two indices. The covariant derivative D λ then acts on an arbitrary tensor as: 1 Again the inverse γ ij of γ ij is found by requiring γ νλ γ λµ = δ µ ν. 2 The perfect fluid form follows from the more general expression in Additionally, note that the overall sign on the energy-momentum tensor is a matter of convention, and is due to the fact that our metric convention is to have ds 2 = dt ; i.e. the negative sign in front of dt 2. 8

12 D λ T µ 1µ 2 ν 1 ν 2 λ T µ 1µ 2 ν 1 ν 2 + Γ µ 1 λσ T σµ 2 ν 1 ν 2 + Γ µ 2 λσ T µ 1σ ν 1 ν Γ σ λν 1 T µ 1µ 2 σν 2 Γ σ λν 2 T µ 1µ 2 ν 1 σ Thus in our case for the energy-momentum tensor we have: D λ T µ ν = λ T µ ν + Γ µ λρ T ρ ν Γ ρ λν T µ ρ The Christoffel symbols can easily be computed using the metric, 2.5,2.6, and the non-zero components are given by: Γ 0 ij = a 2 Hγ ij, Γ i 0j = Hδ i j 2.12 Setting ν = 0 in 2.8 leads to the energy continuity equation: ρ = 3Hρ + P 2.13 From the continuity equation, if the energy and density satisfy an equation of state, then we can solve for ρ exactly: P = ωρ 2.14 dρ ρ = 31 + ωȧ a dt ρ a 31+ω There are three extreme cases for the equation of state: Non-relativistic, cold matter e.g. dust: P = 0 hence ω = 0 ρ a 3 Relativistic, hot matter e.g. radiation: ω = 1 3 ρ a 4 9

13 Vacuum Energy: P = ρ hence ω = 1 ρ = constant In this case ρ is known as the vacuum energy. These states are all still true for different materials separately, as long as they do not exchange energy The Einstein Equations From the Christoffel symbols, we can construct the Riemann tensor, defined by: R α βµν µ Γ α βν ν Γ α βµ + Γ α λµγ λ βν Γ α λνγ λ βµ 2.15 The Riemann tensor is antisymmetric in its first and last two indices, while symmetric under an exchange of the first pair of indices with the second when all indices are lowered: R αβµν = R βαµν = R αβνµ = R µναβ 2.16 In particular this leads to the Bianchi identity, which states: 0 = D λ R αβµν + D α R βλµν + D β R λαµν 2.17 The Ricci tensor is then a contraction of the Riemann tensor: R µν R α µαν = α Γ α µν ν Γ α µα + Γ α λαγ λ µν Γ α λνγ λ µα 2.18 Note that it is a contraction of the first and third index which is the only independent choice using the Christoffel connection. Contracting the Ricci tensor itself leads to the Ricci scalar: If one twice contracts indices in equation 2.17, using the symmetries in 2.16 gives, R R µ µ = g µλ g αν D λ R αβµν + D α R βλµν + D β R λαµν = D µ g αν R αβµν + D ν g µλ R βλµν + D β R 10

14 0 = D µ R βµ D ν R βν + D β R which results in a Bianchi identity for the Ricci tensor: D µ R βµ = 1 2 D βr 2.20 The final quantity we are interested in is the Einstein tensor, defined through a relation to the Ricci scalar and tensor: G µν R µν 1 2 g µνr 2.21 Acting with the covariant derivative D µ shows that the Einstein tensor is also conserved in a covariant sense 3 : D µ G µν = D µ R µν 1 2 g µνd µ R = 1 2 D νr 1 2 D νr = In this paper the Einstein tensor and the Christoffel symbols are the only ones of importance, for the Christoffel symbols are used in deriving the energy continuity equation, and the Einstein tensor appears in the fundamental set of Einstein s field equations, G µ ν = κ 2 T µ ν 2.23 where κ 2 8πG N, and G N is Newton s gravitational constant. Using the values for the Christoffels in 2.12, one can compute the components of the Einstein tensor: G 0 0 = 3H 2, G i 0 = G 0 i = 0, G i j = H 2 + 2ä a δ i j 2.24 Also, using the values of the perfect fluid energy-momentum tensor given by 2.7, we can find each of the three separate Einstein equations 4 : 0, 0 G 0 0 = κ 2 T 0 0 3H 2 = κ 2 ρ 3 In fact, the conservation of the energy-momentum tensor, 2.8, is a consequence of the conservation of the Einstein tensor, as it is directly proportional to the energy-momentum tensor in Einstein s field equations. 4 Due to the symmetry of the Einstein and energy-momentum tensors in their two indices, the i, 0 and 0, i equations are identical. 11

15 This gives the very important relation, κ 2 = 3H2 ρ , i Since both G 0 i and Ti 0 are zero, the 0, i equation gives us no information. When we later consider perturbations to the scalar field and background metric, however, this will no longer be the case. i, j H 2 + 2ä a G i j = κ 2 Tj i δj i = κ 2 P δj i ä a = 1 2 H 2 + κ 2 P Introduction To Inflation The Friedmann Equations The Einstein equations in a FRW background spacetime as presented in the last section, are referred to as the Friedmann equations, though they are often cast in a slightly different form. Using 2.25 in 2.26 we have: ä a = 4πG N ρ + 3P This is just one of the two Friedmann equations. To find the second Friedmann equation, we must first step back a bit. In the above derivations we have been neglecting the curvature constant K. As explained below, this is a very good approximation for the purposes of our future analysis; however, the second Friedmann equation usually contains the curvature constant K, as it is merely the 0, 0 component of Einstein s equations, using the full metric given by 2.1, retaining K throughout the entire calculation. We need here only give the result: H 2 = 8πG N ρ K a 2 This is the other Friedmann equation, and though presently we will revert back to the K = 0 case, it will be used again in Section

16 Friedmann equation 2.28 can be exactly solved for the three universe constituent cases mentioned earlier, in the case that K = 0. The equation we will solve reads: Non-relativistic Matter: ρ a 3 ȧ 2 ρa = ȧ 2 1 a = a 1 2 ȧ 1 = d dt a = a 3 2 t = a t This provides a very simple relation between H and the age of the universe, a t 2 ȧ 3 = a 2 3 t 1 = t H so that we may easily determine the present age of the universe by an accurate measurement of the current Hubble rate. Relativistic Matter: ρ a 4 = ȧ 2 1 a 2 = a2 ȧ 2 1 = aȧ 1 = a 2 t = a t This again provides a simple relation between time and the Hubble rate: a t 1 ȧ 2 = a t 1 = t H Vacuum Energy: ρ = constant Solving for the K = 0 case gives: 2.33 H 0 = ȧ a = a eh 0t 2.34 Based upon these three scenarios for the constituents of the energy density, the history of the universe is often divided into periods based on what source of energy is dominant in the universe at that time. Thus there are three distinct types of ages of the universe, denoted by matter dominated, radiation dominated, and vacuum energy dominated periods 5. 5 A period dominated by non-relativistic, or cold, matter simply consists of baryons and electrons along with their their more complicated atomic combinations, while a relativistic, or hot, matter dominated era is dominated by electromagnetic radiation, or light. 13

17 If the scale factor has the exponential form of the vacuum energy case above, the universe is said to go through a period of inflation; i.e. inflation is an accelerated expansion of the universe. Thus vacuum energy dominated periods of the universe are synonymous with periods of inflationary expansion. Inflationary theory is the study of the evolution of the universe during just such a period. In the standard scenario of inflation, there exists a scalar field usually the inflaton, whose potential energy dominates the total energy density in the universe, at the very earliest moments of its history. The universe thus starts out very early in a vacuum energy dominated state, and consequently undergoes inflationary expansion. During this period quantum fluctuations in the scalar field are amplified by the expansion of the universe. After this period, the scalar field oscillates about the global minimum of its potential, decaying into light particles the radiation fluid, in a process known as reheating [6]. The time at which the radiation finally spreads out, creating what we see as the cosmic microwave background, is known as the time of last scattering. Once the reheating stage is complete, the universe cools off, and is predominantly composed of matter and dark energy. Thus we see how the universe, from the moment of the big bang until our present epoch, traverses the three cases of constituent energy density outlined above. In what follows we mostly consider the case of a vacuum energy dominated universe. As such at is often supposed to increase exponentially; however, this is not the case in other periods of the universe, and when discussing these different ages the distinction should always be noted. None the less, in all cases the size of the universe is increasing, be it even at various rates Co-moving Coordinates, Particle Horizons, & The Hubble Radius The spatial coordinates, x, found in our FRW metric, 2.2, are commonly called comoving coordinates, for they move along with the expansion of the universe. In general, the distance between any two points may be written as s = atx, where x is referred to as the comoving distance, and s is simply the physical distance between the two points. The comoving distance is then always equal to the physical distance at the present moment in time. In these coordinates one may then calculate the particle horizon, which is simply the maximum distance light could have travelled since time t = 0. This is also the maximum distance at which past events can be observed. If one looks at the FRW metric in 2.2, for a null path in which ds 2 = dω 2 = 0, we have: 0 = dt 2 + at 2 dr 2 14

18 This may be simply solved to give the particle horizon: d tf t i dη aη 2.35 Fundamentally, cosmological calculations are based on the scale size of the universe. This is more concretely described by the Hubble radius, defined to be H 1. The Hubble radius commonly referred to as the Hubble horizon represents a characteristic length scale beyond which causal processes cannot operate. In other words it is a measure of how far light can travel while the universe expands [7]. The comoving Hubble radius is then ah 1 6. This quantity is of crucial importance for cosmology, as it is one of only two relevant size scales possible for a universe that is homogeneous and isotropic the other being a measure of curvature. Causal processes such as the establishment of thermal equilibrium, or the propagation of quantum waves, may only operate on scales smaller than the Hubble radius. For inflationary theory, this plays a crucial role, as both of these processes are of key interest. Inflationary theory thus relies heavily on comparing given quantities to the scale of the Hubble radius. If we consider a given length scale λ, then so-called super-horizon scales are cases where λ ah 1, and sub-horizon scales when λ ah 1. In particular, in the evolution of the universe, due to inflation it is possible for given length scales to become larger or smaller than the Hubble radius, commonly termed leaving or entering the horizon Problems With Hot Big Bang Cosmology There are several problems with hot big bang cosmology which inflationary theory poses a great possibility of solving. In fact, in 1981 it was Alan Guth who originally recognized in [8] that a period of inflationary expansion in the universe could solve three of the major dilemmas at the time 7. They are referred to as the flatness, horizon, and magnetic monopole problems. Of these three, the first two are the most relevant to the framework of this paper, and a discussion of both is given below. The Flatness Problem Suppose that there is a known value of the Hubble rate today, H 0. From Friedmann equation 2.28 we may define a critical density, corresponding to K = 0. 6 The physical Hubble radius is simply a times the comoving Hubble radius, so that aah 1 = H 1. 7 The actual idea of inflation had actually already been motivated for alternate reasons in [9]. 15

19 ρ crit 3H2 0 8πG N 2.36 Say we measure the density of the universe today, and find it to be greater than ρ crit ; then, from 2.28, in order to keep the same value of H 0, K must equal +1 to stabilize the equality; the antonymic relation being that if the measured value of the density is less than ρ crit, we must have K equal to 1. If the measured value is in fact the critical density, then K must be equal to 0. Suppose now that the universe is dominated by matter, both relativistic and non-relativistic. Rearranging 2.28 we have: a 2 H 2 = 8πG N 3 ρa 2 K 2.37 Now, in the universe we are considering, recall that the hot and cold matter contributions to the energy density go as a 4 and a 3 respectively. In the early moments of the universe, at is believed to be very tiny, so that early on the quantity ρa 2 grows at least as fast as a 1 in the sense that a 0, while at the same time the quantity K remains constant. Thus, we may safely neglect K for times early on in the universe. Often in this case we say that K is red-shifted away. There is more to the story, however, as this approximately puts the density at that time equal to the critical density. A problem arises when we measure the current density of the universe, a great deal of time after the universe was created, where we find that it in fact is still a large fraction of the critical density. The problem? How could the density of the universe have remained so nearly the same after so much time? Recall that in matter or radiation dominated periods of the universe, the energy density falls off at least as fast as a 3. So as the universe expands during these periods, the energy density must be decreasing rapidly. If the density is very close to the critical density today, this indicates that towards the beginning of the universe the energy density must have been incredibly close to the critical density; almost as if it had been finely tuned. This is referred to as the flatness problem. To see how inflationary theory solves this problem it is easiest to first define Ω ρ/ρ c as the ratio of the actual density of the universe to the critical density. If the universe is flat, recall ρ = ρ c and hence Ω = 1. Now, we may divide 2.37 by a 2 H 2 to get: 16

20 1 = Ω K a 2 H During inflation recall that H is constant, so that the term K/a 2 H 2 decays exponentially to zero. This must mean that as this term vanishes, Ω gets closer and closer towards value unity. In other words, during a period of inflation, the density of the universe is driven towards the critical density. What is more is that Ω may be driven arbitrarily close to 1, depending on how long inflation sets in. This is important, because as [10] pointed out, at time t = 1 second, for example, when the process of nucleosynthesis was just beginning, Ω must have equalled unity to an accuracy of one part in Thus, if inflation lasts for a sufficient amount of time, Ω will be so close to unity that even after inflation stops and the density starts decreasing, when we measure it today it may still be very close to the critical density. To quantify this, cosmologists define the number of e-foldings as: N = tf t i Hdt 2.39 The number of e-foldings is in a sense a good measure of the amount of inflation that has taken place from time t i to t f. Also useful is the number of e-foldings that have taken place since a field of a particular wavelength λ leaves the horizon, until the end of inflation: N λ = tf tλ Hdt 2.40 This can be more explicitly written out in terms of the expansion parameter at as: N λ = tf t λ ȧ tf a dt = t λ da a = ln atf at λ If during this period H is nearly constant, we simply have, 2.41 N λ = H t 2.42 where t t f t λ. 17

21 Inflationary theory then solves the flatness problem without assuming a flat spacetime, while at the same time showing it is a reasonable approximation to take K = 0. The Horizon Problem When one measures the temperature of the cosmic microwave background radiation, one finds it to be roughly the same temperature no matter what part of the sky is observed. This is a blatant indication that as this radiation was spreading out from its source, it had time to causally interact with itself, exchanging energy and settling into the standard Planck distribution. There is a problem, however, when one considers how long ago this radiation was last scattered, and what the value of the particle horizon was at that time. To make this more concrete, let us suppose there never was a period of inflation, and the universe was always dominated by either cold or hot matter, in any given abundances. We found then in equations 2.30 and 2.32 that the scale factor goes as a t p p = 2/3 for matter and p = 1/2 for radiation. The best way to quantify the pending dilemma is to consider the distance that light may have travelled in a given interval of time [5]: d = tf t i dη tf aη = dηη p t 1 p f t 1 p i 2.43 t i There are three relevant times in the history of the universe: an early time very close to that of the big bang, t i ; the time of last scattering, t ls ; and finally, our present time at which we measure the radiation, t f. The relevant periods of interest are then: d ls t 1 p f t 1 p ls, d i t 1 p ls t 1 p i 2.44 Now, it is well known that the difference in time between the time of last scattering and the big bang is much smaller than the difference in time between our present age and that of last scattering. In fact, the time of last scattering is estimated to have occurred roughly 400,000 years after the big bang, while the present age of the universe is some 14 billion years old. This gives a time scale difference of order Thus d ls d i by the same order of magnitude. The impact of this is best seen in Figure 1, in which the light cones of two events are plotted. The figure shows how two points observed by us, maximum distance apart, will have light cones that do not overlap, even if extended all the way to the big bang. 18

22 Figure 1: Diagram showing the impossibility of distant points in space having interaction at the time of the big bang, if one assumes a completely matter or radiation dominated history. Thus, two observers located at these points could not have received information from the same source. In fact, [5] estimates that that the angular distance with which the radiation had causal contact was a mere 1.6 ; not nearly enough to produce isotropy throughout the whole sky. How is it then that all this radiation we see in the sky could have coincidentally ended up at exactly the same temperature, with virtually no anisotropy in its distribution, if it never had the ability to causally interact? This is known as the horizon problem, and inflationary theory provides yet another simple solution to the puzzle. They key to solving it is that we assumed the universe expanded like a t p for all of history. How does this change then during inflation, in which the universe expands like a e Ht? The particle horizon in this case becomes: d in = tf t i dη tf aη = dηe Hη = 1 e Ht i e Ht f t i H 2.45 Now, let us consider once again the interval d i, in which time starts near the big bang, and ends at the moment of last scattering. In this case notice that since t i 1, and t f 1, we have almost exactly that: d i 1 H 2.46 Thus, during a period of inflation the particle horizon remains nearly constant over time for recall during inflation H is also constant in time. This is the solution to the horizon problem; if t i is made small enough or if H is made large 19

23 Figure 2: Diagram showing how inflation stretches the particle horizon to remain constant over time, allowing light cones of two events to overlap at the time of the big bang. enough, then one may result in d ls d i. This is again shown in Figure 2, in which the period of inflation from t i to t ls shows how the particle horizon is held constant in time, so that the light cones are then stretched to overlap, allowing causal contact if extended back to the time of the big bang. 2.3 Pure And Quasi de Sitter Stages The accelerated expansion of the universe, ät > 0, is taken as the definition of inflation. Since a period of early inflation can possibly solve several cosmological problems, recognizing when inflation occurs is of the utmost importance. The first obvious characteristic of inflation is that the universe is indeed undergoing accelerated expansion; i.e. ä > 0. Looking back at Friedmann equation 2.27, we see that this occurs when aρ + 3P > 0; or, since a > 0, when: P < 1 3 ρ 2.47 This is a necessary condition for inflation. In particular, since we take the energy density to be positive, this implies the pressure P is always negative. The precise relationship between P and ρ may be crucial, and can be broken down into different scenarios. Mainly we will consider the cases of a pure and quasi de Sitter stage. A pure de Sitter stage is one in which the pressure and the density of the universe have an equal but opposite relationship; i.e. P = ρ 8. As already noted in the last section, a period when the universe is dominated by an energy density that is constant over time ρ = 0, gives rise to inflation. From the energy continuity equation, 2.13, we see that this is accomplished in a period when P = ρ; precisely that of a 8 Note that a de Sitter stage without a prefixing adjective, refers to a pure de Sitter stage. 20

24 pure de Sitter stage. Thus, a pure de Sitter stage will always be a period of inflation. Additionally, if in a pure de Sitter stage ρ is constant, from 2.28 we see that H is also constant in time. In a quasi de Sitter stage, the Hubble rate does vary slightly with time, for we have only approximately that P = ρ P ρ. This case will be elaborated on in Section Scalar Field Dynamics We have hinted at the importance of a scalar field whose energy density dominates the universe in the realm of inflationary theory. It is thus necessary to analyze the properties of a scalar field itself, and how it behaves in a FRW spacetime. The action for a scalar field ϕ, with potential V ϕ, is generally given by: S = d 4 x gl d 4 x [ g 1 ] 2 gµν µ ϕ ν ϕ V ϕ 2.48 Requiring that the action be minimized along the classical path of the scalar field implies small variations of the action with respect to the field itself must vanish. Variation of the action with respect to ϕ gives: δ ϕ S = d 4 x [ g g µν ν ϕδ µ ϕ + V ] ϕ δϕ = [ gg d 4 x µν µ ν ϕ g V ] δϕ ϕ Setting δ ϕ S = 0 gives: gg µν µ ν ϕ = g V ϕ Or, using a suitable definition for the covariant D Alembert operator, ϕ 1 gg µν µ ν ϕ g 2.49 we derive the classical equation of motion for the scalar field, which is in the form of the Klein-Gordon equation: ϕ = V ϕ 2.50 As we have just derived, equation 2.50 is the equation of motion for the scalar field ϕ, however, plugging in the form of the metric can make this equation more 21

25 explicit. Using the K = 0 form of the FRW metric, equations 2.5 and 2.6, we have g detg µν = a 6 detγ ij. Looking at 2.4 we then see that γ detγ ij = r 4 sin 2 θ. For now, however, we will simply note that g = a 3 γ. With this we have for the definition of the D Alembert operator, ϕ = 1 a 3 γ µ a 3 γg µν ν = ϕ 3H ϕ + 1 a 2 γ i γ i ϕ = ϕ 3H ϕ + 1 a 2 2 ϕ 2.51 γ where we defined the Laplacian, 2 ϕ i i ϕ / γ. From 2.50 it then follows that the equation of motion for the scalar field is given by: ϕ + 3H ϕ 1 a 2 2 ϕ + V ϕ = There is a term of particular interest here, namely the frictional, or drag term 3H ϕ. Since this term is proportional to the velocity, the scalar field will incur a frictional force as it rolls down its potential. Since this term is also proportional to the Hubble rate, the motion of the scalar field is thus affected by the expansion of the universe. 2.5 The Energy-Momentum Tensor Of A Scalar Field By definition, the energy-momentum tensor is given by, T µν 2 g δs δg µν 2.53 where S is the action, in our case of a scalar field, given by In this case δs is given by 9 : δs = = d 4 x 1 2 L 1 δg + g L g g µν δgµν d 4 x 1 2 Lg L µν g + g δg µν g µν 9 Note that the variation is with respect to the metric g µν. 22

26 The expression used for δg is proven as follows: By definition, for an n dimensional matrix: Thus: g detg µν = 1 n! ɛα 1 α n ɛ β 1 β n g α1 β 1 g αnβ n 2.54 δg = 1 n! ɛα 1 α n ɛ β 1 β n δg α1 β 1 g αnβ n + g α1 β 1 δg α2 β 2 g αnβ n +... g α1 β 1 δg αnβ n Since these are dummy indices, this simplifies to: δg = 1 n 1! ɛα 1 α n ɛ β 1 β n g α1 β 1 δg αnβ n 2.55 Next note that g αβ g αβ = δ α α = n, and thus, by differentiating both sides, g αβ δg αβ + g αβ δg αβ = 0, or g αβ δg αβ = g αβ δg αβ. With this, we then see that 2.55 can be obtained by multiplying 2.54 by g α 1β 1 δg α1 β 1. Thus, Going back we now have, δg = gg αβ δg αβ, or δg = gg αβ δg αβ 2.56 which gives: δs δg = 1 µν 2 Lg L µν g + g g µν T µν 2 δs g δg = Lg µν µν + 2 L 2.57 g µν In our particular case, L = 1 2 gαβ α ϕ β ϕ V ϕ, and so, since V ϕ is a function of ϕ only, we have: Raising an index with g αµ we have: 1 T µν = µ ϕ ν ϕ g µν 2 gαβ α ϕ β ϕ + V ϕ T ν µ = g αµ α ϕ ν ϕ δ ν µ 2 gαβ α ϕ β ϕ + V ϕ 2.59 We put the energy-momentum tensor into this particular form because the Einstein equations become simpler with a raised and lowered index. 23

27 Often we will consider our scalar field to behave as a perfect fluid; i.e. its pressure and energy density are given by comparing 2.7 with the expression for the energymomentum tensor given by We have the following definitions from 2.7, while the components of 2.59 are given by: T 0 0 ρ, T i 0 = T 0 i 0, T i j δ i jp 2.60 T 0 0 = 1 2 ϕ2 1 2 a 2 ϕ 2 V ϕ T0 i = ϕ i ϕ, Ti 0 = ϕ i ϕ 1 Tj i = δj i 2 ϕ2 1 2 a 2 ϕ 2 V ϕ 2.61 Comparing the i, 0 components we see that: ϕ i ϕ = This tells us that the scalar field must not depend on space, and is a function of time only, ϕt. Note that since G 0 i is also zero, the Einstein equations themselves tell us this must be the case. Since our universe is one in which isotropy and homogeneity are the rule, the background scalar field must necessarily not depend on space. Using this result, we have the following equalities: ρ = 1 2 ϕ2 + V ϕ 2.63 P = 1 2 ϕ2 V ϕ 2.64 In particular note that the addition of 2.63 and 2.64 leads to the very useful identity: ρ + P = ρ1 + ω = ϕ These identities between the pressure and energy, and the scalar field and its potential, are extremely useful because they allow us to always convert equations pertaining to energy and pressure into equations concerning only the scalar field. As an example, let us rewrite the two Friedmann equations in terms of the scalar field itself. Looking at 2.28 and 2.27, and making use of 2.63 and 2.64, we have: H 2 = 8πG N ϕ2 + V ϕ K 2.66 a 2 24

28 ä a = 8πG N ϕ 2 V ϕ Also, we may rewrite the energy continuity equation, given by 2.13, 0 = ϕ ϕ + V ϕ ϕ t + 3H ϕ2 = ϕ + V + 3H ϕ ϕ which gives precisely the equation of motion we obtained from the Klein-Gordon equation in the last section The Slow-roll Approximation As we have discovered, a sufficient condition for inflation is a universe in a de Sitter stage, in which P = ρ. The scalar field we have been studying is assumed to be the dominant the source of energy in the universe. Furthermore, suppose that ϕ 2 V ϕ i.e. it s potential energy is far greater than its kinetic. In this case the scalar field will slowly roll down its potential. This is called a slow-roll period, and the inequality is referred to as a slow-roll condition. We will make this assumption, where additionally we will assume since ϕ 2 is small, ϕ is as well more accurately ϕ H ϕ. With this approximation, equations 2.63 and 2.64 show us that: P V ϕ ρ 2.68 Thus, if the universe is dominated by a particular scalar field s potential energy; i.e. we have V ϕ ϕ 2, and that same scalar field rolls slowly down its potential, then we obtain precisely the condition required for a de Sitter stage, and hence inflation will ensue. Thus, a slow-roll approximation ensures inflation. We have derived the equation of motion, 2.52, which for a homogeneous scalar field simplifies to, ϕ + 3H ϕ + V ϕ = where we have defined V ϕ dv/dϕ = V/ ϕ. Under the approximation 2.68, Friedmann equation 2.28 becomes again neglecting K, 10 See Section 4.4 for more details on the relation between the scalar field and the energy density. 25

29 while the new equation of motion is: H 2 8πG N V ϕ H ϕ V ϕ 2.71 These conditions on the potential are referred to as flatness conditions, and are sufficient signs of inflation. They then can be manipulated to obtain additional slow-roll conditions, 3H ϕ V = H 2 ϕ 2 V 2 = V 2 H 2 ϕ2 V = V 2 V H H ϕ V = 3H d ϕ dϕ V = 3H d ϕ dt V dt dϕ = 3H 2 ϕ H ϕv = V H where in the last step we recall our assumption ϕ H ϕ; if the second to last line s equality must approximately hold, H 2 must be far greater than V in order to offset the fact that ϕ is much less than H ϕ. These slow-roll conditions may be conveniently expressed in terms of the so-called slow-roll parameters ɛ and η, which are defined by: ɛ M 2 P 2 V 2, η M 2 V P V V 2.74 Here we use the reduced Planck mass M 2 P 1/8πG N 11. Additionally, one may define a second order slow-roll parameter, ξ 2, given by: ξ 2 V V V The conditions 2.72 and 2.73 can be re-represented in terms of ɛ and η using 2.70 as follows, V 2 V H2 = V 2 2 3V 1 = 1 = ɛ V H2 8πG N V 2 11 Note that with this definition, κ 2 is the reciprocal of the reduced Planck mass; i.e. κ 2 = 1/M 2 P. 26

30 and: V H 2 = V 8πG N 3 V = 3M 2 P V V 1 = η Thus the slow-roll conditions, which, fundamentally, are the requirements for a successful period of inflation, are encapsulated in the smallness of the slow-roll parameters ɛ and η. 2.7 Conformal Time In this and future sections we will often use the definition of conformal time, τ, rather than that of the ordinary cosmic time, t. The two are related by: dτ dt at From this definition, it follows that the conformal time τ is always given by: τt = dt at With the introduction of the new conformal time variable τ, we may introduce the conformal Hubble rate H a, where primes denote derivatives with respect to a conformal time while dots are still exclusively representative of derivatives with respect to cosmic time. H and H are simply related by, H = a a 2, H = ȧ 2.80 which gives the extremely useful and important relation: ah = H 2.81 This last equality shows that H is also the comoving Hubble rate, for it incorporates the scale factor at; it is thus also referred to by this name. The second thing one can do with this new variable is rewrite the spatially flat FRW metric, given by equation 2.2, in terms of conformal time. Through the relation dτ = dt/a, we see dt 2 = a 2 τdτ 2. With these changes, the final conformal metric we will often work with becomes: ] ds 2 g µν xdx µ dx ν = a 2 τdτ 2 + a 2 τ [dr 2 + r 2 dω 2 This metric has the matrix form,

31 [ ] a 2 τ 0 g µν x = 0 a 2 τγ ij while the contravariant metric tensor is then given by: g µν x = [ aτ 2 0 ] 0 aτ 2 γ ij One may immediately see the use of the conformal time variable from the form of the metric in 2.82: it is the metric for an ordinary, flat universe, mod an overall scale factor aτ. The metric is then sometimes referred to as conformally flat. The introduction of the conformal time variable into the metric gives rise to new Christoffel symbols, as well as new components to the Einstein and energy-momentum tensors. Since the latter are in fact tensors, they transform simply when changing between conformal and cosmic time. In particular, when looking at equations that equate two tensors like Einstein s equations, changing between cosmic and conformal time requires no other work than to simply interchange t and τ. The Christoffel symbols are not tensors, and will not transform in such an easy way. The relation between conformal time and cosmic time differs when distinguishing between pure and quasi de Sitter stages. Recall in a pure de Sitter stage, the Hubble rate H is constant in time, from which it followed that: at = e Ht From the definition of conformal time, 2.79, we then see that: τt = dt at = dte Ht = 1 H e Ht From our expression for at and τt, we then see that: τt = 1 1, or at = ath τth 2.85 This is the relation between at and τt in a pure de Sitter stage. We now consider what these expressions look like in a quasi de Sitter stage. During this stage, as before mentioned, the Hubble rate is not entirely constant, but varies slightly by the slow-roll parameter ɛ; to see this, from 2.70, taking a total derivative with respect to ϕ gives 12, 12 We have just introduced primes as derivatives with respect to τ, however, a prime on the potential still denotes, and always will, a total derivative with respect to the field itself. 28

32 where, using 2.71: dh dϕ = dh dt With this 2.86 becomes: 2H dh dϕ V 3M 2 P dt 3H dϕ Ḣ = 3HḢ V V H2 Ḣ V V 3M 2 P Multiplying both sides by V, and dividing by expression 2.70 squared, we arrive at: 6Ḣ V H 3M P V This, upon using the definition of ɛ in 2.74, shows that the Hubble rate varies slightly in time as: Ḣ ɛh Note that this result is valid for a quasi de Sitter stage, because in a pure de Sitter stage, if P = ρ, then we must have ϕ = 0, and the scalar field has no kinetic energy; it rolls infinitely slowly down its potential it does not move at all!. This then implies that V = 0 from 2.69, and hence ɛ = 0. In other words, during a pure de Sitter stage the slow-roll parameters vanish. To put this in terms of conformal time, we simply make the appropriate change of variables: H H 2 ɛh This result stresses a very important point, namely, that while the time derivative of the Hubble rate is negligible to first order in cosmic time, in conformal time it is not negligible; only the combination H H 2 is first order in the slow-roll parameters. Now, in order to derive the first order version of 2.85, we follow the same analysis as before, retaining expressions to only first order in ɛ: Ḣ = ɛh 2 = Ḣ H 2 = ɛ = 1 H = ɛt = H = 1 ɛt 29

33 From the definition of H we then have: ȧ a = 1 ɛt = lna = 1 ɛ lnt = a = t 1 ɛ From 2.79 we then have: τ = t 1 1 ɛ dt = 1 + ɛ 1t 1 ɛ +1 = 1 ɛ 1 1 ɛ t = 1 ɛt ɛ t ɛ ɛt = 1 ah1 ɛ Thus we find that in a quasi de Sitter universe: τt = 1 ath1 ɛ, or aτ = 1 τth1 ɛ

34 3. Scalar Field Fluctuations Now that we have seen how important and fundamental a scalar field can be to inflation, let us look at the basic dynamics of just such a field in a realistic light, by considering the effect of quantum fluctuations. In this chapter we would like to consider a scalar field ϕ, with a potential V ϕ, during both a pure and quasi de Sitter stage, and introduce the idea of perturbations. As a step of improved accuracy from our background equations, in order to compare theoretical results with observations, we must take into account deviations from pure homogeneity and isotropy. The study of inflation relies heavily on the evolution of first and second order perturbations of various quantities. We will look at the first order Klein-Gordon equation for the quantum fluctuations of the scalar field, and derive the exact solution, as well as analyze the solution in particular limits. Finally, to make a connection with observation, we will introduce the idea of the power spectrum, which will be used later in Chapter Fluctuations In A Pure de Sitter Space-time We will consider fluctuations or perturbations to the scalar field in a pure de Sitter spacetime, in which we recall both ρ and H are constants in time. To start, we perturb the scalar field, resulting in a split between the homogeneous and perturbed parts, denoted as: For convenience we will redefine the scalar field as: ϕτ, x = ϕτ + δϕτ, x 3.1 δ ϕ aδϕ 3.2 Following the standard practice of second quantization, we develop creation and annihilation operators a k and a k, re-expressing δ ϕτ, x as: δ ϕτ, x = d 3 k [ ] u 2π 3/2 k τa k e i k x + u kτa k e i k x The operators satisfy the simple commutation relations 13 : 3.3 [a k, a k ] = 0, and [a k, a k ] = δ3 k k Note that the a k operators of δϕ do not satisfy these same commutation relations; the factor of a difference is what is important. 31