THESIS. Primordial Curvature Perturbations in Inflationary Universe. Shuichiro Yokoyama

Transcription

1 THESIS Primordial Curvature Perturbations in Inflationary Universe Shuichiro Yokoyama Department of Physics. Kyoto University January, 2008

2 Abstract We study the spectrum of primordial curvature perturbations generated during inflation. By virtue of current precise observations, inflation paradigm, meaning accelerating expansion driven by a scalar field (inflaton) in the early universe, has been widely recognized as an elegant mechanism generating the primordial curvature perturbations which are seeds of the large scale structure and Cosmic Microwave Background (CMB) anisotropies. On the theoretical side, in constructing the realistic inflation model based on the particle physics, a lot of models have been predicted and it seems natural to consider complicated models, for example, multi-inflaton models. In order to identify necessary conditions for viable models of inflation among various proposed possibilities, any simple and sufficiently accurate formula for the spectrum is useful, especially if the formula is applicable to a wide class of models. First, following a brief review of the standard formulation of the power spectrum of the primordial curvature perturbations, we present a new formulation of the power spectrum of curvature perturbations generated during inflation, using the fact that the Wronskian of the scalar field perturbation equation is constant. We deal with the general multi-component scalar field, and show that our new formalism which gives a method to evaluate the final amplitude of curvature perturbations systematically and economically. The advantage of the new method is that one only has to solve a single mode of the scalar field perturbation equation backward in time from the end of inflation to the stage at which the perturbation is within the Hubble horizon, at which the initial values of the scalar field perturbations are given. Second, we analyze the non-gaussianity of the primordial curvature perturbations generated during inflation. If the perturbations satisfy the pure Gaussian statistics, then the bispectrum of the perturbation, which is related to three point-correlation functions, is exactly zero. If, however, the perturbations have the deviation from pure Gaussian, the bispectrum has non-zero value. Recently, this non-gaussianity (non-linearity) of the primordial perturbations also has been a focus of constant attention all over the world. The main reason for attracting much attention is that meaningful measurement of this quantity, which brings us valuable information about the dynamics of inflation if detected, will become observationally available in the near future. Following a brief review of the primordial non-gaussianity, we give a concise formula for the non-gaussianity of the primordial curvature perturbations generated on super-horizon scales in general multi-scalar inflation models. This formula reduces the problem to calculate the non-linear parameter f NL, which is related to the amplitude of the bispectrum, to solving only two vector quantities. Using this formula, we show that even in multi-scalar inflation the non-linear parameter f NL is suppressed by the slow-roll parameters under the slow-roll conditions. We also study the generation of non-gaussianity for the double inflation models as a example in which the slow-roll conditions are temporarily violated after horizon exit, and we show that the non-linear parameter f NL for such models is suppressed by the slow-roll parameters evaluated at the time of horizon exit. We also discuss the possibility of generating the large non-gaussianity during inflation. 1

3 Contents 1 Introduction 5 2 Overview of the inflationary universe; Background dynamics 8 3 The power spectrum of primordial curvature perturbations in single-scalar inflation Cosmological linear perturbation theory Gauge-invariant variables The Einstein equations for the gauge-invariant variables The evolution of curvature perturbations in standard single-scalar slow-roll inflation Quantization The power spectrum of curvature perturbation in the standard single slow-roll inflation Specific models in the slow-roll approximation General slow-roll formula in the single-scalar inflation Formulation Power spectrum The long-wavelength approximation Formulation Example - Starobinsky model Curvature perturbations in multi-scalar inflation Adiabatic and iso-curvature perturbations in Inflation Adiabatic field and iso-curvature field Evolution equations for the adiabatic perturbation and iso-curvature perturbation δn formalism Basic idea Detailed studies about the δn formalism Wronskian formulation of the power spectrum of the primordial curvature perturbations Use of Wronskian Extended general slow-roll formula Single field case

4 4.4 Appendix: δn = δn in general? Non-Gaussianity of the primordial curvature perturbations The non-gaussianity vs the bispectrum The non-linear parameter f NL Can we detect the primordial non-gaussianity? δn formalism extended to non-linear regime Non-linear δn formalism Power Spectrum and bispectrum The non-linear parameter in the δn formalism The non-linearity on sub-horizon scales due to the coupling of the field fluctuations δφ δφ The three-point correlation function of δφ in single slow-roll inflation ADM formalism Quadratic terms in the action Cubic terms in the action The expression for f NL in the single slow-roll inflation fnl sub in multi-scalar inflation The generic Lagrangian case Background Power spectrum The non-linear parameter f NL for the generic Lagrangian The non-linearity of the curvature perturbations on super-horizon scales Single field inflation Curvaton scenario Multi-scalar inflation Formulation Non-linearity generated during slow-roll phase in multi-scalar inflation Double inflation model with large mass ratio Hybrid type and brief discussions Appendix: Single field slow-roll case Appendix: Constancy of δn Appendix: Specific expression Appendix: Extension to the general field space metric Appendix: Spectrum of the waterfall field Appendix: The trispectrum of the curvature perturbations Summary and discussion 113 A New δn formalism for the curvature perturbation 117 A.1 δn A.2 Background equations A.2.1 Basic background equations A.2.2 N φ a and N φa

5 A.3 Perturbations A.3.1 δφ on flat slices A.3.2 Wronskian A.3.3 δn A.4 Multi-field general slow-roll power spectrum A.4.1 Zeroth order A.4.2 First order A.4.3 Inverse formula A.5 Appendix: Covariant partial derivatives with respect to (φ, φ) and (φ, π) A.6 Appendix: Properties of n 0 (x), z (x), z (x), W (x) and s(x) A.7 Appendix: Connection with single field formula A.8 Appendix: Configuration space formula B Second order perturbation theory 138 B.1 Definitions B.2 The second order perturbation equations on super-horizon scales B.2.1 Gauge fixing B.2.2 The Einstein tensor and the energy momentum tensor in the uniform N gauge at the second order B.2.3 The evolution equations in the uniform N gauge

6 Chapter 1 Introduction Inflation [1,2] provides an elegant mechanism to solve shortcomings of the standard Big Bang Model, for example, the flatness and horizon problems. Moreover, in the inflationary universe, primordial curvature perturbations, which seed cosmic microwave background (CMB) temperature anisotropies and the structure formation of the universe, are generated from vacuum fluctuations of the scalar field, so-called inflaton. One can test various models of inflation, by comparing the theoretical predictions for the spectrum of the primordial curvature perturbations with observations. Theoretically, in the simplest single-field slow-roll inflation models, the spectrum of the primordial curvature perturbations is almost Gaussian and almost scale-invariant according to the cosmological perturbation theory [3 7]. On the observational side, thanks to recent advances in technologies, an era of precision cosmology has commenced, and we can determine, or at least constrain, possible models/theories of the early universe by cosmological observations. Notably, the observations of CMB temperature anisotropies remarkably make progress starting with COBE satellite [8] and the analysis of the first and third year WMAP data [9 11] strongly indicates that the universe is spatially flat, that the primordial perturbation is adiabatic and Gaussian, and that the spectrum is almost scale-invariant. By and large, these support the now-standard picture that our universe experienced an inflationary phase at its early stage. Furthermore, the PLANCK satellite [12], to be launched within a couple of years from now, will provide us with more detailed information about not only the CMB temperature spectrum but also the polarization spectrum [12 14]. On the current theoretical side, in making realistic models of inflation based on supersymmetry and supergravity theories [15, 16], it seems more natural to consider models composed of a multi-component scalar field, in which the slow-roll conditions are also possibly violated. Obviously, the calculation of the perturbations in the simplest single slow-roll case is not sufficient to evaluate the spectrum of the curvature perturbations generated in such models. In order to identify necessary conditions for viable models of inflation among various proposed possibilities, any simple and sufficiently accurate formula for the spectrum is useful, especially if the formula is applicable to a wide class of models. Furthermore, recently, the non-linearity (non-gaussianity) of the primordial curvature perturbations also has been a focus of constant attention by many authors. The main reason for attracting much attention is that meaningful measurement of this quantity, which brings us valuable information about the dynamics of inflation if detected, will become observationally available in the near future. In order to parameterize the amount of non-gaussianity of 5

7 primordial perturbations, commonly used is a non-linear parameter, f NL, which is related to the bispectrum of the curvature perturbation [17]. In the single field slow-roll inflation, it is theoretically predicted that f NL is suppressed by slow-roll parameters to undetectable level. But, for example, in the curvaton scenario, it is predicted by many authors that there is a possibility of large non-gaussianity enough to be detectable by future experiments, such as PLANCK [12], which is expected to detect the non-linear parameter if f NL 5 [17]. In the curvaton scenario, primordial curvature perturbations are sourced by iso-curvature perturbations related to the vacuum fluctuations of a light scalar field (other than inflaton), called curvaton, which is an energetically subdominant component during inflation. As the energy density of the universe drops after inflation, the fraction of this component becomes significant. Then, through the process that the curvaton decays into radiation after inflation, the curvaton iso-curvature perturbations are converted into curvature (adiabatic) perturbations. Curvaton scenario predicts nearly scale invariant spectrum as in the case of the standard inflation scenario, but a large value of f NL is possible in this scenario. In order to evaluate the non-linearity of the primordial curvature perturbations, we have to solve full set of the second order perturbation equations in general. It is much more uphill task. Thus, any simple and sufficiently accurate formula even for the non-linearity of the primordial perturbations is useful, especially if the formula is applicable to a wide class of models which include multi-component inflation. On against this background, we formulate the power spectrum of the primordial curvature perturbations generated during inflation in a form as general as possible and also we give the formula for the non-linear parameter f NL. The plan of this thesis is as follows; In chapter 2, we overview homogeneous background dynamics of the inflationary universe. In chapter 3, we review the basic linear cosmological perturbation theory and the formulation of the power spectrum of the curvature perturbations generated during single scalar inflation. In chapter 4, we present the useful formulation of the power spectrum of the curvature perturbations generated during multi-scalar inflation. Especially, in section 4.3, the original contribution is based on Ref. [18]. In chapter 5, we introduce the non-linear parameter in order to parameterize the non- Gaussianity of the primordial curvature perturbations generated during inflation and briefly review the non-linear δn formalism, which is a powerful tool to calculate the non-linear parameter. In chapter 6, we review the non-gaussianity generated on sub-horizon scales during inflation due to the non-linear coupling of the field perturbations. In chapter 7, we analyze the non-gaussianity generated on super-horizon scales due to the non-linear coupling between the field perturbations and the curvature perturbations. Especially, in section 7.3, the original contribution is based on Ref. [19, 20]. In chpater 8, we draw the summary and discussion of this thesis. In keeping with conventional unit in cosmology, we set the speed of light c, the Planck constant and the Boltzmann constant k B to one throughout this thesis. 6

8 Figure 1.1: (From This image is a Internal Linear Combination Map of the WMAP three year data, where the foreground contamination is subtracted Angular Scale l(l+1)cl/2 [ K 2 ] Multipole moment l Figure 1.2: (From The binned WMAP three-year angular power spectrum (in black) from l = The grey shadow region represents a cosmic variance. The red line is the angular power spectrum predicted for the best fit three-year WMAP only Λ-CDM model. ( The resolution of COBE satellite is corresponding to l 30.) 7

9 Chapter 2 Overview of the inflationary universe; Background dynamics Modern standard big bang model of the universe is based on the premise that the Universe is homogeneous and isotropic on large scales. In the big bang model, the universe is expanding and is very hot back to very early epoch. One of the evidences of the hot big bang is known as the theory of nucleosynthesis [21, 22] which explain the primordial abundance of the light elements, hydrogen, helium, etc. The theoretical value of the abundance is amazingly consistent with current observations. Another evidence is the discovery of cosmic microwave background (CMB) radiation coming from the whole sky. This shows just that the Universe was a heat bath back to very early epoch and the isotropy of the temperature of CMB has confirmed the assumption that the universe is isotropic on large scales. From these evidences the big bang model of the Universe seems successful, but there remain some questions about the initial conditions required for the big bang model; Why is our universe so flat, so homogeneous and so isotropic? The historical motivation for inflation was to solve these shortcomings of the standard big bang model. Inflation is a stage of accelerated expansion of the early universe. This is the general definition of inflation. If the universe goes through the accelerated expansion at an early stage, the whole observable universe can be produced from a very small homogeneous and flat domain. That is, inflation gives the initial conditions needed for the big bang universe. However, ordinary matter or radiation cannot drive the accelerated expansion of the universe. In the inflationary stage, gravity needs to become repulsive. As is well known, from the Einstein equation, the dynamics of the flat homogeneous universe is represented by the Friedmann equations; (ȧ ) 2 H 2 = 1 2 ρ, (2.1) a 3M Pl ä a = 1 2 (ρ +3P ), (2.2) 6M Pl where a is the scale factor, H is the Hubble parameter, a dot represents the derivative in terms of cosmic time, M 2 Pl =(8πG) 1 is the reduced Planck mass squared, ρ represents the total energy density of the universe and P the pressure. From Eq. (2.2), when the universe 8

10 is dominated by the material whose equation of state satisfies P < ρ/3, the accelerating universe is realized, i.e., ä > 0. A natural candidate of matter with negative pressure necessary to drive inflation is a scalar field, so-called inflaton. Of course, such scalar field has not been detected by experiments so far. However, scalar fields play an important role in modern particle physics, e.g, superstring theory. A homogeneous classical scalar field φ = φ(t) is characterized by the energy density ρ(φ) and the pressure P (φ) ; ρ(φ) = 1 2 φ 2 + V (φ), (2.3) P (φ) = 1 2 φ 2 V (φ), (2.4) where V (φ) is the potential of the scalar field. From the Klein-Gordon equation, the equation of motion for the homogeneous scalar field is given by φ +3H φ + V φ =0, (2.5) where V φ = dv/dφ. In the scalar field dominant universe, the Friedmann equations (2.1) and (2.2) become H 2 = 1 ( ) 1 2 3M Pl 2 φ 2 + V (φ), (2.6) Ḣ = 1 2M Pl 2 φ 2. (2.7) From the forms of the energy density and the pressure, inflation is realized when 1 φ 2 V 2 (Fig. 2.1). For analyzing the dynamics of the inflation, the slow-roll approximation is well known as a standard one. Let us define two parameters, so-called slow-roll parameters, as ɛ M Pl 2 2 ( Vφ V ) 2, η M Pl 2 V φφ V, (2.8) where V φφ = d 2 V/dφ 2. The conditions ɛ 1 and η 1 are satisfied in the slow-roll approximation. Under these conditions, the homogeneous background equations (2.5) and (2.6) are reduced to 3H φ V φ, (2.9) H M V. (2.10) Pl Under the slow-roll conditions, slow-roll parameter ɛ can be rewritten in terms of Hubble parameter as ɛ M Pl 2 2 φ 2 H = Ḣ 2 H = ä +1. (2.11) 2 ah2 From this expression, ɛ<1 is equivalent to the accelerated expansion, ä>0, and ɛ>1 is the decelerated expansion, ä<0. In the slow-roll limit, the Hubble parameter becomes 9

11 V(þ) slow-roll inflation reheating þ Figure 2.1: This picture represents a naive picture of inflation. The slow-roll inflation is induced by the scalar-field which has very flat potential. After inflation, the reheating era is needed for the successful inflationary scenario. constant. Then, the universe expands as if dominated by the cosmological constant, that is, the universe behaves as de Sitter universe, expanding exponentially. η can be also rewritten as η V φφ H 2. (2.12) Since V φφ represents the effective mass squared of scalar field, m 2 eff, η 1 means that m 2 eff H2, that is, the mass of scalar field is much smaller than the Hubble parameter during slow-roll inflation. In addition, let us define a parameter important for later discussion, N, which represents the amount of inflation, as N ln a(t end) a(t) = tend t H(t )dt, (2.13) where t end represents a time when inflation ends. This parameter, N, is commonly called the e-folding number. To solve the initial condition problems of big bang universe, the e-folding number need to be typically more than about sixty (Fig. 2.2). To realize the standard big bang universe after the inflation, some process is needed. Generally, this process is called reheating, in which the inflaton decays into some particles and the thermalization of the decay products occurs (Fig. 2.1). Thus, modern standard cosmology consist of hot big bang and inflation. 10

12 log (physical scale) cosmological scale ~ a(t)/k Hubble scale ~ 1/H(t) N ~ 60 t end log a(t) Inflationary phase Big bang universe Figure 2.2: This figure shows the behavior of a cosmological comoving scale 1/k and the physical Hubble scale 1/H (horizon scale). The Hubble scale stays almost constant in time during inflation, whereas, the physical scale a(t)/k evolves. Thus, a certain scale which is smaller than the horizon scale at initial time can exit the horizon scale in the inflationary phase. After the inflation, the Hubble parameter starts to evolve and the scales which are larger than the horizon scale at the end of inflation reenter the horizon. The e-folding number between the time when the cosmological scale exit the horizon scale and the time when inflation ends needs to be about sixty to solve the flatness and horizon problems. 11

13 Chapter 3 The power spectrum of primordial curvature perturbations in single-scalar inflation Inflationary cosmology provides more than the mechanism for solving the initial condition problems of the big bang model as was shown in the previous chapter. The results of CMB observation by COBE [8] showed the existence of the CMB temperature anisotropies, which have the amplitude In order to explain temperature anisotropies and also the structure formation of the universe, primordial density perturbations need to be generated. In the inflationary universe, this primordial density perturbations are generated from vacuum fluctuations of the scalar field. Current detailed analysis of CMB anisotropies indicates that the primordial perturbations is almost scale-invariant, almost adiabatic and almost Gaussian [9 11]. Such observational characters of the spectrum of the primordial perturbations give us the various information about the inflation dynamics, for example, the slow-roll parameters ɛ and η. On the theoretical side, many authors have recently proposed realistic models of inflation based on recent progress of particle physics. It seems natural to consider more complicated models, e.g., low energy inflation models and multi-inflaton models. In order to identify necessary conditions for viable models of inflation among various proposed possibilities, any simple and sufficiently accurate formula for the spectrum is useful, especially if the formula is applicable to a wide class of models. In this chapter, we would like to show the formulations of the power spectrum of curvature perturbations generated from various single-scalar inflation models, following mainly Ref. [4]. In order to evaluate the power spectrum, we have to solve the perturbation equations based on general relativity. This chapter is organized as follows. In section 3.1, we review the first order cosmological perturbation theory, which is necessary to discuss the metric perturbations and the matter fluctuations simultaneously. In section 3.2, we show the standard formula of the power spectrum of the curvature perturbation for single slowroll inflation models. In sections 3.3 and 3.4, we would like to introduce two useful formulae aiming at generalizing the standard single slow-roll formula. Hereinafter, we set M 2 Pl =1. 12

14 3.1 Cosmological linear perturbation theory In order to study the evolutions of fluctuations in the universe, we have to consider such fluctuations based on general relativity. Cosmological perturbation theory [4, 5] is the standard tool for studying the power spectrum of the primordial perturbation generated during inflation. Evaluating the power spectrum of primordial density perturbation at the leading order, we only have to consider the perturbations to the first order. In this section, we briefly review the cosmological perturbation theory at the linear order, following Ref. [4] Gauge-invariant variables The perturbations can be classified into three types based on their behavior under the transformation of space-coordinates; scalar type, vector type and tensor type. At the first order, the perturbation equations are decomposed into groups of equations each of which contains only components of one type [4]. Since the density perturbation which we want to evaluate here is obviously scalar type, we, hereinafter, consider only the scalar type perturbations. Incidentally, the tensor type is called gravitational wave perturbation and the vector type is frequently called rotational perturbation. Where only scalar perturbations are concerned, the perturbed metric is generally expressed as ds 2 = a 2 { (1+2AY )dτ 2 2BY j dτdx j +[(1+2H L Y )δ ij +2H T Y ij ] dx i dx j}, (3.1) where τ dt/a(t) is called conformal time, Y is the spatial scalar harmonic function with the eigenvalue k 2 and Y j k 1 j Y, ( Y ij k 2 i j + 1 ) 3 δ ij Y. (3.2) Here, we omit the indices to distinguish different eigenvalues. The formulation in this section does not make much difference between for the single scalar case and for the multi-component case. The difference between two cases become pronounced when one try to evaluate the power spectrum of the density perturbation after inflation ends. Thus, in this section, we consider a D-component scalar field whose action is given by S matter = d 4 x [ ] 1 g 2 gμν h IJ μ φ I ν φ J + V (φ), (3.3) I,J =1, 2,,D, where g μν represents the spacetime metric and h IJ is the field space metric. For simplicity, we assume h IJ = δ IJ in this section, but the extension to a general field space metric h IJ is straightforward. We also decompose the scalar fields to the background and perturbations as φ I = φ(τ) I + δφ(x μ ) I Y, (3.4) 13

15 where a tilde represents the perturbed quantity. In general relativistic perturbation theories, the variables representing perturbations introduced in Eqs. (3.1) and (3.4) (A, B, H L,H T,δφ I ), change their values under a coordinate transformation in the perturbed world. This transformation is the so-called a gauge transformation. Let us consider infinitesimal gauge transformations, x μ x μ with x μ = x μ + ξ μ, (3.5) where ξ μ are regarded as quantities of the same order as the perturbation variables. The perturbations of any tensor X defined as δx X X transform as δx = δx + ξ X. (3.6) Gauge transformations are classified into the scalar type and vector type. There is no gauge transformation of the tensor type. We can also discuss the gauge transformation of the perturbation variables for each type independently. Under the gauge scalar type transformation (ξ 0,ξ i )=(TY,LY i ), at the first order, the change of the perturbation variables is given by Ā = A + T + a T, a (3.7) B = B L kt, (3.8) H L = H L + k 3 L + a T, (3.9) a H T = H T kl, (3.10) δφ = δφ I + φ I T, (3.11) δρ = δρ + ρ T, (3.12) where a prime represents the derivative in terms of conformal time, τ, and δρ = ρ ρ represents the density perturbation. From these equations which represent the change of perturbation variables, we can construct the gauge-invariant variables. Generally, we can construct various gauge-invariant variables. Here, we introduce some gauge-invariants which will be useful in the later discussion. At first, we define the intrinsic curvature perturbation as R = H L H T, (3.13) which appears in the perturbation of the spatial scalar curvature, δ s R,as δ s R =4 k2 RY. (3.14) a2 Using this intrinsic curvature perturbation R, which itself is not gauge-invariant, we construct a gauge-invariant variable as ζ = R H δρ, (3.15) ρ 14

16 where H a /a is the comoving Hubble parameter. This gauge-invariant variable ζ is called the curvature perturbation on the uniform density hypersurface. We also construct the density perturbation on flat hypersurface as δρ f = δρ ρ H R = H ζ, (3.16) ρ and similarly the field perturbation on flat hypersurface, which is often called Sasaki- Mukhanov variable [6, 7], as From Eq. (3.17), such a variable as δφ I f = δφ I φi H R. (3.17) R c = R H φ 2 φ Iδφ I = H φ 2 φ Iδφ I f, (3.18) is also gauge-invariant, where φ I = δ IJ φ J, φ 2 = δ IJ φ I φ J. This variable R c is called the curvature perturbation on comoving hypersurface, δt 0 i = 0. Especially, in the single field case R c is given by R c = R H δφ. (3.19) φ The relation between the curvature perturbation on the uniform energy density hypersurface, ζ, and the curvature perturbation on the comoving hypersurface is given by [23] with ζ = R c H ρ δρ c, (3.20) δρ c δρ +3Hφ Iδφ I, (3.21) which is also gauge-invariant and represents the perturbation of the energy density on comoving hypersurface. In general, these gauge slicings disagree with each other. In the following discussions, we use these gauge-invariant variables as the situation demands. In the next subsection, we derive the evolution equations for gauge-invariant variables, δφ I f and R c, from the perturbed Klein-Gordon equation and Einstein equations in terms of the gauge-invariant variables The Einstein equations for the gauge-invariant variables From the perturbed Klein-Gordon equation, we obtain the evolution equation for the perturbation of the scalar field, which is given by δφ I +2Hδφ I + k 2 δφ I + a 2 V I Jδφ J +2a 2 V I A φ I A + φ I (3R kσ g )=0, (3.22) 15

17 where V I = δ IJ V J δ IJ V/ φ J and we have introduced a perturbation variable kσ g H T kb which represents the shear perturbation. We decompose the Einstein tensor G μν and the energy-momentum tensor T μν into the background quantities and the perturbations as G μν = G μν + δg μν, T μν = T μν + δt μν, (3.23) Then, the perturbed Einstein equations to the first order are given by δg 0 0 = δt 0 0 : 3H 2 A H(3R kσ g ) k 2 R = 1 ( 2 φ I δφ I a 2 V I δφ I + Aφ I φ I ), (3.24) 2M Pl δg 0 i = δt 0 i : trace part of δg i j = δt i j : HA R = 1 2 φ I δφ I, (3.25) HA 1 3 [ ] d dη +2H (3R kσ g ) k2 (A + R) 3 = 1 ) ( a 2 VA+ φ I δφ I a 2 V I δφ I 2, (3.26) traceless part of δg i j = δt i j : kσ g +2Hkσ g k 2 (A + R) =0, (3.27) where φ I = δ IJ φ J. From these equations, we obtain the evolution equation for a gauge-invariant variable δφ I f as δφ I f +2Hδφ I f + k 2 δφ I f + a 2 VJ I δφ J f = 1 ( ) d a 2 a 2 dη H φi φ J δ JK δφ K f. (3.28) This is a basic evolution equation of the gauge-invariant perturbation in multi-component inflation. In the single field case, the evolution equation for the curvature perturbation on comoving hypersurface R c is also useful, which is given by R c +2 z z R c + k 2 R c =0, (3.29) where z = aφ /H. Before the end of this section, let us consider the relation between the curvature perturbation on uniform energy density hypersurface, ζ, and that on the comoving hypersurface, R c. The relation between ζ and R c is given by Eq. (3.20). From the perturbed Einstein 16

18 equations, Eqs. (3.24) and (3.25), we have the gauge-invariant generalization of the Poisson equation as with k 2 a 2 Ψ= 1 2 δρ c, (3.30) Ψ R+ Hkσ g, (3.31) which is often called the gravitational potential in the Newtonian gauge. From this equation, we can find ζ = R c + O(k 2 ). (3.32) That is, in the long wavelength limit k 2 0, the uniform energy density hypersurface becomes identical to the comoving one. In the next section, we will consider the evolution of the perturbation in the single-scalar inflation and show the standard formula of power spectrum of the primordial perturbation. 17

19 3.2 The evolution of curvature perturbations in standard single-scalar slow-roll inflation In order to evaluate the amplitude of the primordial fluctuations which seeds CMB temperature anisotropies, it is convenience to use the curvature perturbation on comoving hypersurface as a gauge-invariant variable. In the single-scalar inflation models, the evolution equation for the curvature perturbation on comoving hypersurface is given by Eq. (3.29). On large scales k 2 0, this equation is reduced to R c +2 z z R c =0, (3.33) where there exists a solution with R c constant in time, which corresponds to the growing mode 1. That is, basically, we do not need to evolve R c on super-horizon scales. This fact makes the problem easy. Thus, in this section, we review some formulations of the power spectrum of the curvature perturbations in the single-scalar inflation. First, we mention about the quantization of the field perturbations in subsection 3.2.1, following Ref. [24, 25]. In subsection 3.2.2, we show the most standard formula for the power spectrum of the curvature perturbations in the single slow-roll inflation, following Ref. [24, 25]. In subsection 3.2.3, we show theoretical implications for several specific inflation models using the standard formula Quantization In this subsection, we briefly review the quantization of the fluctuation of the inflaton, following [22, 24, 25]. The action of single-scalar inflation is given by S = M Pl 2 d 4 x gr d 4 x ( ) 1 g 2 2 gμν μ φ ν φ + V (φ), (3.34) where R represents the Ricci scalar. The first order evolution equations for the perturbations are given by the second order action, which is given by S 2 = 1 ] d 4 x [(u ) 2 ( i u) 2 + z 2 z u2, (3.35) for the scalar-type perturbation with u zr c = aδφ f, (3.36) where z = aφ /H. The Fourier expansion of the Heisenberg operator of u is given by û(τ,x i d 3 k [ )= u (2π) 3/2 k (τ)â k e ik ix i + u k(τ)â k e ik ix i], (3.37) 1 Of course, Eq. (3.33) also has a decaying mode solution. Since this decaying mode is proportional to a 3, this solution rapidly decay during inflationary phase. Hence, neglecting the decaying mode may be good approximation. If there is no iso-curvature mode, this behavior of the decaying mode solution is satisfied for any matter. 18

20 where u k is a mode function of ˆϕ, â k and â k represent the creation and annihilation operators, respectively. The commutation relation and a vacuum state 0 are given by ] [â k, â k = δ 3 (k k ), [â k, â k ]=0, (3.38) â k 0 =0. (3.39) The equation of motion for the Fourier mode u k is given by ) u k + (k 2 z u k =0, (3.40) z and the normalization condition for u k (τ) u ku k u k u k = i, (3.41) are satisfied. Well before horizon exit k 2 H 2, Eq. (3.40) is reduced to u k + k 2 u k 0. (3.42) This evolution equation is equivalent to that for the free field in flat spacetime. naturally the positive frequency mode, u k is initially given by Hence u k (τ) 1 2k e ikτ, (3.43) satisfying the normalization condition (3.41). On the other hand, well after horizon exit k 2 H 2, Eq. (3.40) is reduced to u k z z u k =0, (3.44) and a growing mode solution 2 is given by u k (τ) z. (3.45) Since u k = zr ck, on the super-horizon scales R ck stays constant in time neglecting the decaying mode. The power spectrum of R c, P Rc (k) is defined as R ck R c k 2π2 k P 3 R c (k)δ 3 (k + k ), (3.46) 2 Eq. (3.44) is also rewritten as [ 1 d a 2 d ( uk ) ] =0. z dτ dτ z We can easily find that this equation also have a decaying mode solution ; dτ u k z z. 2 This solution is corresponding to R c dτ z 2. 19

21 where R ck is given by the Fourier expansion of R c as d 3 k R c = (2π) R cke ik x. (3.47) 3/2 Then, in terms of u k we have P Rc (k) = k3 2π lim u k 2. (3.48) 2 τ 0 z In the next subsection, we obtain the amplitude of the curvature perturbations in the standard single slow-roll inflation. In such cases, the formulation is quite simple The power spectrum of curvature perturbation in the standard single slow-roll inflation Power spectrum In the slow-roll limit, i.e., the both slow-roll parameters ɛ, η defined by Eq. (2.8) are small, φ and H become almost constant. Hence, da τ = a 2 H 1 H, z = aφ H 1 φ τ H, 2 z z a a 2τ 2, (3.49) in the slow-roll limit. From these equations, the equation of motion for u k in the slow-roll limit can be rewritten as ( u k + k 2 2 ) u τ 2 k =0. (3.50) With the initial condition given by Eq. (3.43), the solution of Eq. (3.50) becomes u k (τ) = 1 ( 1 i ) e ikτ. (3.51) 2k kτ On the superhorizon scales kτ 0, the growing mode dominates to give u k (τ) τ 1. (3.52) Then, the power spectrum P Rc (3.48) becomes k 3 u k 2 ( ) H 2 2 P Rc (k) = lim kτ 0 2π 2 z 2 2π φ, (3.53) a=k/h where the last expression is evaluated at the time when the scale of interest (a/k) exit the horizon scale (1/H). Here, we impose the fact that R c = u k /z stays constant in time on super-horizon scales (kτ 1) for any single-component matter and also in the slow-roll limit u k /z stays constant because of the constancy of φ/h 2. However, if we consider the case in which the slow-roll conditions are violated, φ/h 2 stays constant no longer. In the section 3.4, we will discuss the power spectrum for such cases. 20

22 log (physical scale) scale of the perturbation ~ a(t)/k R c ' const: super-horizon scales Hubble scale ~ 1/H(t) sub-horizon scales k=ah horizon crossing time t end log a(t) Inflationary phase Big bang universe Figure 3.1: This figure shows the evolution of curvature perturbations in the single-scalar inflation. The curvature perturbations are generated from the quantum fluctuations of the inflaton on sub-horizon scales. On super-horizon scales, the curvature perturbations on comoving hypersurface stay constant in time. 21

23 The spectral index and the running of the spectral index Important observational quantities are not only the amplitude of the primordial curvature perturbations but also the spectral index which represents the scale dependence of the power spectrum, given by n s 1 d ln P R c (k). (3.54) d ln k We also define the running of the spectral index, which represents the scale dependence of the spectral index, by α dn s d ln k. (3.55) The standard formula for the power spectrum (3.53) is expressed as the value evaluated at the time when k = ah. We have d ln k = d ln(a k H k ) dn k = H k dt k, (3.56) where the suffix k represents the value evaluated at the horizon crossing time, k = ah, and N represents the e-folding number given by Eq. (2.13). Thus, the spectral index of the power spectrum (3.53) is given by and n s 1= d ln P R c d ln k ( φ H 2 ) 2 d Hdt ( H 2 φ ) 2 = 4 Ḣ H 2 2 φ H φ = 6ɛ k +2η k, (3.57) α = dn s d ln k 24ɛ2 k 8ɛ kη k 2 V φv φφφ a=k/h. (3.58) V 2 Thus, if the spectral index and the running of the spectral index may be found from observations, the slow-roll parameters at the horizon crossing time can be determined using these standard formulae. That is, we can measure the derivatives of potential of the inflaton at the horizon exit from the CMB observations!! Tensor Perturbations As we mentioned in the previous section 3.1, the perturbation can be classified into three types; scalar, vector and tensor perturbations. In general, since the vector perturbation decays during inflation, it can be neglected [4]. On the other hand, the amplitude of the tensor perturbations also stays constant in time on large scales, as well as the scalar perturbations [24, 25]. We can directly detect the primordial tensor perturbations as background gravitational waves, but the gravitational waves have not been detected yet. It is well known that the tensor perturbations also affect the CMB temperature anisotropies. The polarization of the temperature anisotropies are classified into two modes; E-mode and B-mode. The 22

24 tensor perturbations generate the B-mode polarization of the temperature anisotropies. The scalar perturbation can generate the E-mode polarization, but cannot generate the B-mode polarization at the linear level. Then, the B-mode polarization of the CMB temperature anisotropies are mainly generated by the tensor perturbations. Thus, the future observation of CMB B-mode polarization [12 14] will tell the amplitude of primordial tensor perturbations. In general, the power spectrum of the tensor perturbations generated from inflation is given by [24,25] P T (k) = ( ) 2 H 2π a=k/h. (3.59) That is, the amplitude of the primordial tensor perturbations generated from inflation is given by the Hubble scale at the horizon crossing time, in other words, the energy scale of inflation. Thus, if the primordial tensor perturbation is detected in the future experiments, we can measure the energy scale of inflation directly!! As the cosmological parameter related to the primordial tensor perturbation, the ratio between the amplitude of the primordial tensor perturbation and that of the primordial curvature perturbation is commonly used, which is defined by r P T P Rc, (3.60) and is called tensor-to-scalar ratio. Substituting the expression (3.53) and (3.59) to Eq. (3.60), we obtain the expression of the tensor-to-scalar ratio for the standard single-scalar model as r =2ɛ. (3.61) Implication for the inflation models from current observations Here, we briefly review the current observational results of the primordial curvature perturbations, following [8 11] and restore the reduced Planck mass M Pl. The WMAP team parameterized the model of the universe in terms of 15 parameters as shown in Table 3.1 (quoted from Ref. [11]). In this 15 cosmological parameters, Δ 2 R,n s,α,r, related with the primordial curvature perturbations generated during inflation are called inflationary parameters. The analysis of these inflationary parameters give us constraints on the inflation models. From the amplitude of the CMB temperature anisotropies, which is related to the inflationary parameters Δ 2 R P R c, the normalization of the power spectrum of curvature perturbations is given by P 1/2 R c (3.62) From the standard formula (3.53) and using the slow-roll background equations (2.9) and (2.10), we have V 1/4 ɛ 1/ M Pl GeV. (3.63) 23

25 Parameter ω b ω c f ν N ν Ω k Ω DE w Δ 2 R n s α r τ A SZ b sdss z s Description Baryon density Cold dark matter density Massive neutrino fraction Effective number of relativistic neutrino species Spatial curvature Dark energy density Dark energy equation of state Amplitude of curvature perturbations R Scalar spectral index at 0.002/Mpc Running in scalar spectral index Ratio of the amplitude of tensor fluctuations to scalar potential fluctuations at k=0.002/mpc Reionization optical depth SZ marginalization factor Galaxy bias factor for SDSS sample Weak lensing source redshift Table 3.1: (From Ref. [11]) 15 cosmological parameters in WMAP three year analysis. Indeed, in the WMAP three year analysis, they also use other parameters for convenience, but such parameters are functions of these 15 parameters. Thus, it is right in thinking that the basic cosmological parameters are these 15 parameters. 24

26 Since the dependence of the slow-roll parameter ɛ is weak with a small power index 1/4, we can find that if the standard slow-roll inflation has occurred in the early universe the energy scale of the inflation should be about GUT scale GeV. The COBE satellite does not have the resolution enough to analyze the scale dependence of the power spectrum. The WMAP, which has enough resolution, indicates that the marginalized value for the spectral index is n s =0.961 ± 0.017, (3.64) at the 68% confidence level. From the standard expression of the spectral index (3.54), we find that this result n s 1 1 indicates that the slow-roll parameters at the horizon crossing time are much smaller than unity. That is, the slow-roll approximation seems to be valid around the horizon crossing time. The marginalized value for the running spectral index is α = , (3.65) at the 68% confidence level. Thus, we also find that the higher derivatives of the potential of scalar field in terms of the scalar field are also suppressed. As for the tensor-to-scalar ratio, r, we have not detected tensor perturbations until now. The current observations, therefore, gives only upper bound on r. Fig. 3.2 shows the twodimensional marginalized contours for inflationary parameters (r, n s ) predicted by monomial potential models, V (φ) φ n. From this figure, it seems that the λφ 4 model is not preferred at 68% confidence level. In this way, the formulae for the power spectrum of curvature perturbations help us to extract information about the dynamics of inflation from observations Specific models in the slow-roll approximation In the last part of the previous section, we have shown the implication for the inflation models from current observations. In this subsection, we briefly show the theoretical predictions for the cosmological parameters in various inflation models, using the standard slow-roll formula and also restore the reduced Planck mass M Pl, here. Chaotic Inflation ; V (φ) m 2 φ 2, λφ 4 Let us consider the most simplest models, so-called chaotic inflation, whose potential is given by V = 1 2 m2 φ 2. (3.66) For this model, the slow-roll parameters defined by Eq. (2.8) are given by ɛ = M Pl 2 ( ) 2 Vφ = 2M Pl 2, (3.67) 2 V φ 2 η = M Pl 2 V φφ V 25 = 2M 2 Pl = ɛ. (3.68) φ 2

27 WMAP WMAP + SDSS r WMAP + 2dF WMAP + CBI + VSA r n s n s Figure 3.2: (From Ref. [11]) Joint two-dimensional marginalized contours (68% and 95% confidence levels) for inflationary parameters (r 0.002, n s ). r =16ɛ(= 8r) represents a tensor-to-scalar ratio at k = 0.002/M pc. In this figure, they assume a power-law primordial power spectrum, dn s /d ln k = 0. (Upper left) WMAP only. (Upper right) WMAP+SDSS. (Lower left) WMAP+2dFGRS. (Lower right) WMAP+CBI+VSA. The dashed and solid lines show the range of values predicted for monomial inflaton models with 50 and 60 e-folding number of inflation, respectively. The open and filled circles show the predictions of m 2 φ 2 and λφ 4 models for 50 and 60 e-folding number of inflation. The rectangle denotes the scale-invariant Harrison-Zel dovich-peebles (HZ) spectrum (n s =1,r= 0). From this figure, it is noted that the current data prefers the m 2 φ 2 model over both the HZ spectrum and the λφ 4 model. 26

28 Therefore in this model the slow-roll conditions, ɛ, η 1, are corresponding to φ 2 M Pl 2. (3.69) In order to realize the slow-roll inflation, we have to take the initial value of the scalar field to be φ M Pl. We also need to get the 60 e-folding number. In this model the e-folding number defined by Eq. (2.13) is written as N = tend t 1 M Pl 2 Hdt φend φend φ H φ dφ V V φ dφ = 1 2M Pl 2 φ φdφ φ φ end φ 2 2 4M = 1, (3.70) Pl 2ɛ where t = t represents the time when our cosmological scale exits the horizon during inflation, and φ end represents the value of the scalar field at the end of inflation when ɛ becomes unity for the first time. From Eq. (3.67), we have 1=ɛ end 2M Pl 2 φ 2 end, φ end M Pl 2. (3.71) In order to realize N 60, the initial value of the scalar field φ should be set to φ 20M Pl. (3.72) When we consider the extension to the general chaotic inflation models whose potential is given by the general power of the scalar field, V (φ) φ n, the expressions of the slow-roll parameters and the e-folding number are ɛ = M Pl 2 2 η = M Pl 2 N n 2 φ, 2 (3.73) n(n 1) 2(n 1) = ɛ, φ 2 n (3.74) φ2 2 2nM = n, (3.75) Pl 4ɛ where in the expression for N, we have neglected the contribution of φ end.in Fig. 3.2, we have shown the implication for the chaotic inflation models from the relation between the spectral index n s and the tensor-to-scalar ratio r. In such chaotic inflation models, we need to set the initial value of the scalar field to the much larger than M Pl. During the initial phase, φ M Pl, the potential energy and also the Hubble parameter are quite large. The Hubble parameter affects the equation of motion for the scalar field through an effective friction and the large Hubble parameter means the strong friction force. Thus, even if the scalar field is fast-rolling initially, the slow-roll solution is an attractor. That is, in the chaotic inflation model, an appropriate initial condition for the 27

29 þç þ Figure 3.3: The inflationary attractor behavior of the chaotic model V (φ) =m 2 φ 2 /2 scalar field is dynamically selected, as long as the initial value of the field is sufficiently large. Let us consider the implications for chaotic models from the observations. As we mentioned before, from the amplitude of the primordial curvature perturbations we can determine the energy scale of inflation for the single slow-roll inflation as V 1/4 ɛ 1/ M Pl GeV. For the chaotic inflation model whose potential is V (φ) =m 2 φ 2 /2, this condition with the aid of Eq. (3.70) fixes the mass of scalar field, m, as m 10 5 M Pl. (3.76) N And, in this model, the theoretical predictions for the spectral index and tensor-to-scalar ratio are n s 1= 6ɛ +2η = 4ɛ = 2 N < 0, (3.77) r =2ɛ = 1 N. (3.78) For the model whose potential is V (φ) =λφ 4 /4, we can also obtain the necessary condition for λ as and also, we have λ 10 14, (3.79) n s 1= 3ɛ = 3 N < 0, (3.80) r =2ɛ = 2 N. (3.81) 28