Geometrical Measures of Non-Gaussianity Generated by Single Field Models of Inflation. by Muhammad Junaid. Doctor of Philosophy

Transcription

1 Geometrical Measures of Non-Gaussianity Generated by Single Field Models of Inflation by Muhammad Junaid A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Physics University of Alberta c Muhammad Junaid, 015

2 Abstract In this thesis we have compiled the study of geometrical non-gaussianity generated by inflation along with necessary basics and background knowledge of inflationary universe. We effectively calculated the power spectrum and the bispectrum, as a measure of non-gaussianity, using the approach laid by Maldacena [1]. We developed a robust numerical technique to compute the bispectrum for different single field inflationary models that may even have some features in the inflationary potential. From the bispectrum, we evaluated the third order moments of scalar curvature perturbations in configuration space. We evaluate these moments analytically in the slow roll regime while we devised a numerical mechanism to calculated these moments even for non slow roll single field inflationary models with standard kinetic term that are minimally coupled to gravity. With help of these third order moments one can directly predict many non-gaussian and geometrical measures of three dimensional distributions as well as two dimensional CMB maps in the configuration space. Thus, we have devised a framework to calculate geometrical measures, for example Minkowski functionals or skeleton statistic, generated by different single field models of inflation. Finally, we also calculated these configuration space moments for the two dimensional projection maps on the sky. We subtracted the monopole contribution of the two dimensional perturbation field from these moments so that we can estimate observable geometrical non-gaussianity in CMB temperature maps generated by single field inflationary models. ii

3 Preface This dissertation is the result of work carried out in the Department of Physics, University of Alberta. Chapter one being an introduction to the subject of the thesis with background to the field of non-gaussianity and different geometrical measures. While chapters two and three contain basics of inflationary paradigm, primordial perturbation theory and non-gaussianity. In chapters 4, 5 and 6.1 we present our research work that we have published in []. In chapter four we presented a detailed calculation of cubic moments while chapter six is about geometrical statistics and precision non-gaussianity observables. In section 6., we have presented the two dimensional statistics of primordial fluctuations that we have recently calculated. These two dimensional moments and Minkowski functional calculation is our research work in progress with basic results presented in this section. iii

4 Acknowledgements In the name of Allah, the Most Gracious, the Ever Merciful. It is His kindness and help that made it possible to complete this thesis work. My deepest thanks and gratitude go to my supervisor Prof. Dmitri Pogosyan for his support, patience, guidance and providing me an excellent research environment. His flexible attitude always provided a space to work independently, and his professional skills allowed for timely guidance for me. I also appreciate all the facilities and the wonderful learning environment provided by the Department of Physics at the University of Alberta. Special thanks to Sarah Derr and Professors Valeri Frolov, Alexander Penin and Sharon Morsink for their guidance and helpful advices. I would love to acknowledge my beloved wife for her caring attitude and moral support in all aspects of my life. She has been wonderful life partner and has always been there for me, praying for me and encouraging me to move forward in my life. I am also thankful to all of my friends and colleagues, specially Asad and Asif, who were a great help. They were always there for us in times of need and they made our stay enjoyable in Canada with lots friend and family gatherings. We went for several trips to Banf and Jasper during our stay in Canada. We thank them for their wonderful company. iv

5 Table of Contents 1 Introduction 1 Inflationary Paradigm 6.1 Review of FRW Universe Einstein s Gravity and Friedmann equations Horizon and Flatness Problems of Friedmann Cosmology Inflation Basics Dynamics of Inflation and Scalar Field Models of Inflation Old Inflation New Inflation Chaotic Inflation Kinetic Inflation Multi-field and Hybrid Inflation Quantum Fluctuations During Inflation and Non-Gaussianity Cosmological Perturbations Theory Computational Strategy ADM Formalism Action Scalar Perturbations and Second Order Action (Linear Perturbations) Solution and Quantization of Mukhanov Equation Correlation Function and Power Spectrum Primordial Non-Gaussianity The Bispectrum and Measures of Non-Gaussianity Calculation of Third Order Action using ADM Formalism Quantization and Three Point Function v

6 4 Configuration Space Moments Cubic Moments in Single Field Slow Roll Models of Inflation Cubic Moments in Local Non-Gaussianity Numerical Analysis Case Study Models of Inflation Inflation with Quadratic m φ Potential Quartic or λφ 4 Inflation Step Potential Model of Inflation Resonance Model of Inflation Other Models with Large Non-Gaussianity Calculation of Power Spectrum Calculation of Bispectrum and Cesaro Sum Numerical Calculation of Moments Geometrical Statistics and Minkowski Functionals Three Dimensional Statistics Filling Factor Area of Isodensity Contours Euler Characteristic or Genus Two Dimensional Statistics and CMB Two Dimensional Cubic Moments Two Dimensional Minkowski Functionals Results and Future Studies 117 Bibliography 10 Appendices 17 A Reduced Planck Units 18 B Computations on HPC Cluster 130 C Constraint Evaluation in ADM 131 D Change in N, k and P ζ 133 vi

7 E Bessel Function and Spherical Harmonics 135 E.1 Cubic Moments in D with Monopole Subtraction vii

8 List of Tables 5.1 Moments in 3D for different Inflationary models Moments in D for different Inflationary models viii

9 List of Figures.1 Inflationary Paradigm in Comoving Scales Guth s Potential Coleman-Weinberg Potential Chaotic and Eternal Inflation ADM spatial Slices Slow roll parameter for the step potential Number of e-folds against physical time Inflaton field against number of e-folds Mode function u k Derivative of mode function u k Power spectrum for slow roll models Power spectrum for step potential Integrand of three point function Cesaro sum result f NL for quadratic inflation Error in f NL f NL for quartic inflation Equilateral triangle f NL for step potential Squeezed triangle f NL for step potentail f NL for resonance model Quadratic inflation moments as function of k max Quadratic inflation moments as function of k min Quartic inflation moments as function of k max Moments for the step potential model Moments as a function of step height c ix

10 5.1 Moments as a function of step width d Moments as a function of c and d Moments as a function of step position φ s Filling factor Area of isodensity contours as function of threshold Area of isodensity contours as function of filling factor Euler characteristic χ 3D as a function of threshold Euler characteristic χ 3D as a function of filling factor Euler characteristic χ 3D as a function of threshold Euler characteristic χ 3D as a function of filling factor CMB temperature map form Plank CMB power spectrum from Planck D moments as function of R for quadratic potential D moments as function of R for step potential Filling factor in D Length of isocontours as a function of threshold Length of isocontours as a function of filling factor Euler characteristic in D as a function of threshold Euler characteristic in D as a function of threshold for different R Euler characteristic in D as a function of filling factor Euler characteristic in D as a function of filling factor for step potential Euler characteristic in D as a function of filling factor for different R x

11 Chapter 1 Introduction Recent results from ground and balloon based experiments as well as from WMAP and Planck satellites have described the temperature anisotropies in the Cosmic Microwave Background(CMB) with high precision [3 13]. This primordial CMB is found close to being Gaussian, however, there are some features and anomalies that are unexplained as shown in [10] and references therein. The study of non-gaussian contributions in the cosmological perturbations is an ideal tool to study the inflationary dynamics. Inflation is initial accelerated expansion of the early universe [14 16]. Inflationary expansion by more than 65 e-folds is needed to explain the observed homogeneity and isotropy of the Universe. In widely used models the inflation is driven by the potential energy of a scalar inflaton field slowly rolling down the potential. Inflation also successfully describes the creation of small inhomogeneities, needed to seed observed structure in the Universe, as having been generated by quantum fluctuations in the inflaton field [17]. The quantum fluctuations got stretched and became imprinted on CMB maps and other observables on cosmological scales. Thus, inflation is able to explain why the universe is so homogeneous and isotropic but also what is the origin of the structures in the Universe [14 3]. Alternatives to inflation have been proposed but no other scenario is as simple and elegant as inflation produced by scalar field(s) [4 6]. 1

12 In single field inflationary models generated inhomogeneities are described via single scalar perturbation field ζ(x). Statistical properties of ζ(x) are the main observable signatures to distinguish inflationary models. We know that if the perturbations are exactly Gaussian then all odd n-point correlations functions vanish while all even n-point functions are related to the two point function. Thus, in the momentum space the Gaussian field ζ is completely described by the power spectrum P ζ (k). ζ k ζ k = (π) 3 δ 3 (k + k ) π k 3 P ζ(k) (1.1) where the δ 3 (k + k ) represents homogeneity and for isotropy the P ζ (k) is a function of magnitude k of the momenta. Inflation generally predicts that the power spectrum is nearly flat P ζ (k) k ns 1 const [16, 17, 19]. The fact that the observed scalar spectral index n s = ± [9] is close to, but not exactly unity is considered by many to support the existence of inflation in the early universe. However, many different kinds of inflationary models can be made compatible with observations of the power spectrum. Thus study of the non-gaussian signatures is important to reduce the degeneracy in inflationary models. Such signatures are contained in non-trivial higher order correlations starting with the cubic ones. Similar to the power spectrum, one can introduce the bispectrum B ζ (k 1, k, k 3 ) from the three point function as a measure of the non-gaussianity of the initial perturbations ζ k1 ζ k ζ k3 = (π) 3 δ 3 (k 1 + k + k 3 )B ζ (k 1, k, k 3 ), (1.) where bispectrum is also a function of magnitudes because of isotropy. This bispectrum carries much more information than the power spectrum as it contains three different length scales. while it has been concluded that for basic single field slow roll inflation with standard kinetic term the non-gaussian effects are small [1], there are many models of inflation that give large, potentially detectable, non-gaussianity [7 39]. Comparing non-gaussian predictions with observations will help us to constrain or rule out different inflationary models and

13 give us more insight about the physics of the early Universe [1, 40, 41]. Non-Gaussianity is direct measure of inflaton field interactions and can help us understand more about the inflationary dynamics [1, 40]. In this thesis we will focus on the study of non-gaussianity through geometrical measures that describe visual properties of the initial perturbations viewed as a random field. Examples of standard geometrical measures of random fields are Minkowski functionals [4,43], extrema statistics [44] and also more novel measures such as skeleton statistics [45, 46]. The first Minkowski function consist of fraction of above some threshold in an excursion set. The second Minkowski functional is area of isocontours above threshold while the last one is called Euler characteristic or genus that is a topological measure of isolated regions minus the number of holes above a threshold. The Euler characteristic or genus density of excursion sets as function of threshold is the most well known Minkowski functional. It has been shown that for mildly non-gaussian field such local geometrical characteristics can be expressed as a series of higher order moments of the perturbation field and it s derivatives [44, 47 51]. In particular the Euler characteristic [47, 48] is as follows χ 3D (ν) = ) 3 e ν [ H 3σ (π) (ν) ( ζ 3 + σ ( σ1 6σ 4 H 5(ν) 3 ζ ζ 4σ σ 1 H 3 (ν) 9 ) ] ( ζ) ζ H 4σ1 4 1 (ν) + O(σ ).(1.3) In the above expression σ = ζ, σ 1 = ( ζ), ν is the threshold in units of σ and H i (ν) are Hermite polynomials. The first term in the expansion denotes the Gaussian part that is proportional to H (ν) while other terms represent the first non-gaussian correction in σ. The above expression for Euler characteristic is the first order expansion in σ that contains three cubic moments ζ 3, ζ ζ and ( ζ) ζ that are in general needed to calculate different Minkowski functionals to this order. We have developed a robust mechanism to compute three dimensional cubic moments 3

14 as ζ 3, ζ ζ and ( ζ) ζ for single field models of inflation. This required careful study of the infrared divergences in these moments and designing numerical techniques to regularize them. This links the non-gaussianity generated by inflation to the geometrical observables such as Euler characteristics and other Minkowski functionals. We have applied our technique to predict non-gaussian Minkowski functionals in several popular models of inflation i.e. basic slow roll models with quadratic and quadric potentials that have small non- Gaussianity and the model with broken slow roll dynamics due to features in the potential that generate relatively large non-gaussianity [31, 3, 5]. Among the most direct observational datasets for studies of inflation and non-gaussianity are the CMB maps [7, 9 1]. CMB temperature fluctuations map is two dimensional (D) integral projection onto sphere of the 3D inflationary perturbation field ζ(x) modified by the transfer function and defined to have zero monopole contribution. Thus, we mapped the inflationary scalar perturbations on the D sky in plane parallel approximation. To calculate the D cubic moments for CMB maps we subtracted the background or monopole contribution to obtain observable quantities. Using the two dimensional cubic moments and variances with monopole contribution subtracted, we computed different Minkowski functional for two dimensional CMB maps. With help of these two dimensional Minkowski functionals one can get theoretical bounds on amount of non-gaussianity in the CMB statistics [53 55]. This thesis is organized into seven chapters. In chapter two, we review the the basics of inflationary cosmology. In chapter three we will review of quantum fluctuations in inflationary cosmology where we will describe the calculation of power spectrum and bispectrum of scalar perturbation. In chapter four we will study in detail the configuration space moments, specifically the cubic moments. In chapter five we will present our numerical technique for the calculation of three point function as well as cubic moments and briefly discuss different single field models of inflation with some features in the inflationary potential. In chapter 4

15 six, we will present the calculation of different geometrical Minkowski functionals in three dimensions and in two dimensional maps. In the last chapter we will conclude our results and discuss future studies. 5

16 Chapter Inflationary Paradigm It is now widely accepted that there was an early epoch in the history of the universe when the universe expanded exponentially (accelerated expansion to be more precise) and this period of cosmological expansion is called inflation. This inflationary expansion occurs when the energy density of the universe is dominated by vacuum energy while the Hubble parameter remains almost constant. The inflationary paradigm elegantly solves the flatness and the horizon problems of the standard Big Bang cosmology. Thus, inflation is responsible for the large-scale homogeneity and isotropy of the universe as well as being responsible for the small-scale perturbations that planted the seeds for the formation of galaxies. Now, the inflationary paradigm is able to predict the overall structure of the universe but little is known about the underlying physical mechanism that generates this inflationary expansion of the universe [1 3]..1 Review of FRW Universe Cosmology describes the structure and evolution of the universe on the largest scales. If we assume homogeneity and isotropy on large scales then one is led to the Friedmann-Robertson- Walker (FRW) metric for the spacetime of the universe ( ) dr ds = c dt + a (t) 1 kr + r (dθ + sin (θ)dφ ) (.1) 6

17 where a(t) is the scale factor and k is the curvature parameter that is +1 for positively curved, 0 for flat and 1 for negatively curved universe. An important quantity characterizing the FRW spacetime is the expansion rate in terms of Hubble parameter H H(t) = ȧ(t) a(t). (.) In these expressions t is the physical time and dots represent derivatives w.r.t the physical time. The causal structure of the universe can be determined by defining the FRW metric in conformal time τ. τ = c dt a(t) (.3) In terms of conformal time with coordinate transformations r = S k (χ) = sin χ, χ, sinh χ the metric can be written in a simple form ds = a (t) ( dτ + dχ + Sk(χ)(dθ + sin (θ)dφ ) ) sin χ k = +1 S k (χ) = χ k = 0 sinh χ k = 1 (.4) In conformal time the null geodesics of light(ds = 0) can be written as χ(τ) = ±τ that correspond to straight lines in radial direction at 45 o in the χ τ plane that make up the light cone. From this definition of null light cone we can define the maximum comoving distance(horizon) light can travel in some time from t i to t as χ p = τ τ i = t t i c dt a(t). (.5) From the above expression one can define the particle horizon d p = a(t)χ p as the physical distance light can travel during this time. The particle horizon is important in order to understand the causal structure of the universe and it will be fundamental to our discussion of inflation [56]. 7

18 . Einstein s Gravity and Friedmann equations The dynamics of space-time are governed by the Einstein equations of general relativity and in reduced Planck units( = c = 1, M pl written as = 8πG = 1, see Appendix A for detail) they are R µν 1 g µνr = T µν (.6) where R µν and R are the Ricci tenor and scalar, respectively, and T µ ν is the energy-momentum tensor and for a maximally symmetric space i.e. homogeneous and isotropic universe, ρ T µν = 0 p p 0 (.7) p where ρ is the matter energy density and p is the effective isotropic pressure. The Einstein equations using this energy-momentum tensor take the form of two ordinary differential equations known as the Friedmann equations (ȧ H = a ) = 1 3 ρ k a, (.8) Ḣ + H = ä a = 1 (ρ + 3p) (.9) 6 where the dots denote the derivatives with respect to physical time t. We can obtain a continuity equation of FRW cosmology from the above equation as follows dρ dt + 3H(ρ + p) = 0 (.10) This equation can also be rewritten in terms of equation of state parameter w d ln ρ d ln H = 3(1 + w), (.11) w = p ρ. (.1) 8

19 One can find the solution of this continuity equation and Friedmann equations as follows ρ(a) = ρ 0 a 3(1+w), (.13) { a(t) = a0 t 3(1+w), w 1 a 0 e Ht, w = 1. (.14) For a flat universe(k=0) dominated by non-relativistic matter(w = 0), radiation or relativistic matter(w = 1/3) and cosmological constant(w = 1) respectively. If more than one matter kind is present in the universe at the same time then energy density and pressure are the sum of all kinds of matter. ρ = i ρ i, p = i p i (.15) One can define relative energy density using the critical energy density ρ crit = 3H as follows Ω i = ρ i ρ crit, (.16) where subscript 0 denote quantities evaluated at present time t 0 and i signifies type of matter. Now, one can rewrite the Friedmann equation as H H 0 = i Ω i a 3(1+w i) + Ω k a (.17) with Ω k = k/(ah) being a curvature parameter. Ω i (t) + Ω k (t) = 1 (.18) i We will use this relation to quantify and explain the flatness problem..3 Horizon and Flatness Problems of Friedmann Cosmology The horizon problem is a problem of the standard hot Big Bang model of the universe which was identified in the late 1960s, primarily by Charles Misner. It states that different regions of the early universe were not in causal contact with each other because of the great 9

20 distances between them, but nevertheless they have the same temperature and other physical properties [1, ]. The comoving particle horizon in Friedmann universe is the distance a light ray can travel between time 0 and time t. τ = t 0 dt a(t) = a 0 d ln a 1 ah (.19) where (ah) 1 is called the comoving Hubble radius which in hot Big Bang model is a monotonically increasing function of time 1 ah = a 1 0 t 1+3w 3(1+w) (.0) and thus τ gets most of its contribution from recent times. Hence, the particle horizon is also an increasing function which implies that comoving scales entering the horizon today have been far outside the horizon at CMB decoupling. But, the near-homogeneity of the CMB tells us that the seen universe was in causal contact at the time of last-scattering. The flatness problem is an initial conditions fine-tuning problem within the Big Bang model of the universe. To quantify the problem we rewrite the Friedmann equation.18 as 1 Ω(t) = k (ah), Ω(t) = ρ(t) ρ crit (t), ρ crit (t) = 3H. (.1) Now, the Ω(t) = 1 point diverges away from unity if we have small curvature in the early universe in standard Big Bang cosmology. Because the factor 1 ah is growing function in Friedmann cosmology. Therefore, in standard Big Bang cosmology the near-flatness observed today, Ω(t 0 ) = 1, requires an extreme fine-tuning of close to 1 in the early universe. Thus, both these problems arise in the conventional cosmology because the comoving Hubble radius 1/(aH) is strictly increasing [1, 56]. 10

21 .4 Inflation Basics In order to solve the horizon and flatness Problems the concept of inflation naturally introduced. inflation is the initial period of the universe expansion when the universe expanded exponentially. Now, we know that the comoving particle horizon τ is a logarithmic integral of comoving Hubble radius 1/(aH) as seen in.19. The comoving Hubble radius measures distance particle can travel in one expansion time. This tells us that if particles are separated by distances larger than the comoving Hubble radius then they can not communicate right now. While, the comoving horizon is the distance across which no information can ever reach. Thus, τ can be larger than (ah) 1 right now, such that particles cannot communicate today but were in causal contact in earlier times. But this phenomenon can not happen in matter or radiation dominated regimes because in these regimes the comoving radius is an increasing function of time [, 56]. Figure.1: Comoving Hubble radius (ah) 1 in red plotted against conformal time τ. While blue curve shows a fixed comoving scale of density fluctuation [56]. To solve this problem we require an early phase in universe expansion when the comoving radius decreased so that τ gets most of its contribution from this earlier epoch of inflation. Now, if the comoving Hubble radius decreases sufficiently, the problems of hot big bang 11

22 cosmology are solved automatically in a very elegant fashion(see Fig..1). The Friedmann equations for shrinking Hubble radius leads to accelerated expansion as follows ( ) d 1 dt ah < 0 d a dt > 0 and thus from the second Friedmann equation eq. (.9) we get ρ + 3p < 0 p < ρ/3. (.) One can rewrite the second Friedmann equation in the following form and define a parameter ɛ such that ä a = H (1 ɛ), ɛ = Ḣ H (.3) an accelerating universe would correspond to ɛ < 1 that in turn corresponds to universe with negative pressure p < ρ/3. Hence, inflation is a stage of accelerated expansion of the universe when gravity acts as a repulsive force i.e. negative pressure..5 Dynamics of Inflation and Scalar Field Inflation is a peculiar phenomenon that drives the universe through an accelerated growth. This phenomenon can only be physically achieved if we have a negative pressure source and nearly constant positive energy density. The simplest models of inflation involve a single scalar field known as the inflaton field φ. Scalar fields or bosons are used to drive inflation because they can form condensates with positive vacuum energy density. Scalar fields are rather ubiquitous, all fundamental theories of particle physics such as Standard Model Higgs, Supersymmetry, Grand Unified Theories(GUTs) involve them. Historically, it is these extensions of Standard Model that led to the development of inflation that is driven 1

23 by scalar fields [4]. However, one can argue that inflation can be driven by something other than scalar fields. Vector fields or fermion condensates can also be used to drive inflation but they complicate things too much [0,] thus, in this thesis we will study inflation driven by scalar fields only. The dynamics of the scalar inflaton field φ is governed by the following simple action S = d 4 x ( ) 1 g gµν µ φ ν φ + V (φ), (.4) where V (φ) is the potential associated with inflation field. From this action one can find energy momentum tensor T µν T µν = µ φ ν φ + g µν ( 1 αφ α φ + V ). (.5) Now, for homogeneous and isotropic universe T µν = diag(ρ, p, p, p) from which we can find the energy-matter density ρ φ and isotropic pressure p φ of inflaton field ρ φ = 1 φ + V (φ) + 1 ( φ), p φ = 1 φ V (φ) 1 6 ( φ). (.6) where is a gradient operator. The Klein-Gordon (KG) equation for scalar inflaton field for homogeneous and isotropic universe can be written using the FRW metric as φ + 3H φ + V,φ = 0. (.7) The Friedmann equations for the scalar field inflation takes the following form ( ) 1 φ + V (φ), (.8) (V (φ) φ ) H = 1 3 ä a = 1 6 (ρ φ + 3p φ ) = 1 3 ( ɛ = Ḣ H = 1 φ H = 1 = H (1 ɛ), (.9) a φ ), (.30) ȧ 13

24 where ɛ is called the first slow roll parameter. The KG equation (.7) for inflaton field and the Friedmann equation (.8) for scale factor are together called the background equations of motion. Now, in order to achieve accelerated(or exponential) expansion of the universe one has to define the first slow roll condition as ɛ 1, φ V (φ). (.31) The universe goes through inflationary expansion phase only when the first slow roll condition holds. But, this accelerated expansion will only be sustained for a sufficiently long period of time if the second time derivative of φ is small too ( φ 3H φ, V (φ) ). Thus, one can define the second slow roll parameter η and a corresponding condition as follows η = ɛ ɛh 1 (.3) There are many other definitions of second slow roll parameter like η H η H = φ φh 1 η = ɛ η H. (.33) The parameters ɛ and η H are collectively called the Hubble slow roll parameters that are used more commonly in the studies of inflation. However, in this thesis we will use the first definition η = ɛ ɛh because this definition is more commonly used and is more helpful in the study of non-gaussianity. The slow roll conditions ɛ 1, η 1 ensure that the inflaton field rolls slowly down the potential and the universe inflates for significantly long period. These slow roll parameters depend on the potential of the inflaton field and the model of inflation. When we study inflation, we need to know how much the universe expanded during inflation. The number of e-folds gives us that measure of the universe expansion from start 14

25 to the end of inflation φ start tend N = ln a end = Hdt a start t start φend H φstart = φ dφ = dφ. (.34) ɛ φ end To solve the horizon and flatness problems one requires that N = ln a end a start 65, (.35) the total number of e-folds to be at least 65 [0,, 56]. In slow roll approximation the field slowly rolls down the potential from an initial field value φ start until it reaches the minimum where it oscillates and decay. Inflation ends when the slow-roll conditions break i.e. ɛ η 1 and this field value is denoted by φ end marking the end of inflation. The slow roll parameters can also be defined in terms of the inflationary potential V (φ). ɛ V = M pl ( ) V,φ 1 V η V = Mpl V,φφ V 1 (.36) where M pl = (8πG) 1 is the reduced Planck mass (see Appendix A). In terms of the above slow roll parameters the corresponding slow roll conditions can be defined as ɛ V, η V 1. In the slow-roll approximation the Hubble and potential slow-roll parameters are related as follows ɛ ɛ V, η H η V ɛ V (.37) η 4ɛ V η V (.38) The above relations are approximate, thus in our numerical study we will use the original definitions of slow roll parameters given in (.30) and (.3). It is important to mention here that in some inflationary models the slow roll conditions break down for short periods of time during inflation. These models of inflation give rise to interesting effects like large non-gaussianity [31, 41]. 15

26 .6 Models of Inflation The theory of inflation is now more than 35 years old, thus discussion about the history of inflation will gives us the basic insight into the theory of inflation. In early models of inflation, the main ingredient was the energy density of a scalar field that played the role of the constant vacuum energy and the other ingredient was cosmological phase transitions, due to first order or second order phase transitions that requires thermal equilibrium [4]. The earliest scenario of inflation was proposed by Alexei Starobinsky that considered quantum corrections to the theory of quantum gravity that led to the idea of inflation and gravitational waves from earlier universe. His model was rather complicated, and its goal was somewhat different from the goals of inflationary cosmology [4]..6.1 Old Inflation A much simple model of inflation now known as old inflation was proposed by Allan Guth in 1981 [14] that had clear physical motivation of inflation. This model was based on phase transitions in the thermal universe. In this inflationary scenario the universe undergoes an exponential expansion in a super cooled false vacuum state. False vacuum is a metastable state without any fields or particles but with large energy density (see Fig..). Thus, the energy density of the universe remains constant and the universe inflates exponentially. Now this false vacuum state decays quickly such that the vacuum bubble(small universe with different inflaton field value) formation is large and these bubbles collide and reheat the universe. But these bubbles created are smaller than the observed universe and hence bubble wall collisions make the universe very inhomogeneous. Hence this scenario was gracefully discarded due to large inhomogeneity and several bubbles with different vacuum energies [0, 4]. 16

27 Figure.: Guth s Potential for the theory of old inflation..6. New Inflation The first version of new inflationary scenario was introduced by Coleman-Weinberg theory of SU(5) symmetry breaking in Grand Unified Theory(GUT). They worked out the phase transition with symmetry breaking of SU(5) SU(3) SU() U(1) at finite temperatures. The effective potential for this theory can be written as ( (φ ) 4 ( ( ) φ V CW (φ) = V 0 ln 1 ) ) + 1 µ µ 4 4 (.39) for which the values V 0 and µ are parameters of SU(5) theory that depend on temperature and couplings. The inflation starts at φ = 0 that is a false vacuum where the symmetry is unbroken while the inflation ends when the field tunnels through the barrier and reaches the true vacuum as seen in Fig..3. The field then oscillates about true vacuum and decays into Higgs bosons and other Standard Model particles this causes the vacuum energy of inflation to reheat the universe. The problem with Coleman-Weinberg theory is that it requires a very flat plateau near φ = 0 that looks somewhat artificial. The inflaton field has small coupling constant so it cannot be in thermal equilibrium with other matter fields thus theory of phase transition will 17

28 Figure.3: Coleman-Weinberg potential of SU(5) theory. not work. The Coleman-Weinberg potential remains a popular model but it is incompatible with small amplitude of CMB fluctuations [4, 56]..6.3 Chaotic Inflation The old and new inflationary models fall in the broad category called the small field inflationary models. These models gave a solution to the problems of Big Bang cosmology but had short coming of their own. The assumption of state of thermal equilibrium that required large number of particles before the beginning of inflation was the biggest challenge. However, in 1983 the challenges of old and new inflation were resolved with the introduction of Chaotic inflation scenario [4, 5]. This model of inflation even works if there are no assumptions like thermal equilibrium in the early universe. Chaotic inflation can be achieved by simple polynomial potentials like V (φ) = kφ n with inflaton field rolling slowly down the side of a potential. Chaotic inflation is not just limited to polynomial potentials but can occur in any theory with potential that obeys slow-roll conditions eq..36. The non-minimal coupling to gravity, modified gravity and non standard kinetic term inflationary models also fall in this broad category of chaotic inflation. Chaotic inflation is categorized into large field models of 18

29 inflation. The chaotic, or some time also called eternal, inflation can occur without any preconditions as long as the field is placed high up on a potential like V (φ) = 1 m φ in Figure.4. This figure shows that the energies greater than Planck energy density, V (φ) > Mpl 4 (= 1) (see Appendix A), which corresponds to φ 1, are described by the space time foam. In m this region quantum fluctuations of space time are very large and can only be described in the framework of quantum gravity. At lower than Planck energies we can describe gravity as classical. In the region 1 m < φ < 1, the inflaton field φ quantum fluctuations are large m compared with the changes in the field due to rolling down the potential. Thus, in this period of inflation the inflaton field gets trapped because of the up and down quantum kicks of the inflaton field. Now, different regions of the universe get different quantum kicks thus during this regime some areas inflate more while some deflate hence it becomes a very chaotic and bubbling universe that gave this model it s name the Chaotic inflation. Now, if the field in some particular region get more downward kicks it reaches the region of slow roll inflationary stage 1 < φ < 1 m. In this region the quantum kicks of inflation are smaller than the rolling down of the field hence the inflaton field slowly rolls down the potential until the slow roll conditions break down, φ < 1, and the field starts oscillating at the bottom of the potential and reheat the universe as it decays. In this thesis we will focus our study on chaotic models of inflation with V (φ) = kφ n potentials and models with some feature in the potential that breaks the slow roll conditions and give rise to large non-gaussianity..6.4 Kinetic Inflation The kinetic inflation also falls in a broad category of chaotic inflation. Inflation is normally driven by standard kinetic term for the scalar field but we can even have non standard kinetic 19

30 Figure.4: Chaotic eternal inflation with V = 1 m φ potential [5]. term in high energy theories. The action for the kinetic inflation is given by S = 1 d 4 x g(r + P (X, φ)) (.40) where P (X, φ) is some function of the inflaton field and its derivatives [0]. For this kind theory the inflation can be driven by even steep potentials..6.5 Multi-field and Hybrid Inflation The simplest models of inflation involve just one scalar field. However, in supergravity and string theory there are many different scalar fields, so it does make sense to study models with several different scalar fields. If we allow more than one field to be dynamically relevant during inflation, then the possibilities for the inflationary dynamics expand dramatically; for instance one can have non-flat power spectrum. But the down side of multi-field inflation is that the theory loses a lot of its predictive power. The multi-field-models of supergravity are known as hybrid inflationary models since they fall in the regime of chaotic inflation [5, 57, 58]. 0

31 Chapter 3 Quantum Fluctuations During Inflation and Non-Gaussianity In this section we will review the quantum fluctuations of the metric and inflaton field φ as it rolls down the potential. These fluctuations lead to a local time delay in the time at which inflation ends, i.e. in different parts of the universe the inflation will end at slightly different times. Different parts of the universe therefore undergo slightly different evolutions that appear as density or temperature fluctuations. 3.1 Cosmological Perturbations Theory The CMB observations confirm that initially the universe was almost homogenous with small inhomogeneities in the metric and the matter fields. Therefore, perturbations can be naturally divided into homogeneous background X(t) with spatial perturbations δx(t, x). X(t, x) = X(t) + δx(t, x) (3.1) 1

32 Now, there are perturbations in the metric and as well as in the matter part of the Einstein equation. Where greek indices are µ = 0, 1,, 3 and latin ones are i = 1,, 3. g µν (t, x) = g µν (t) + δg µν (t, x), (3.) ds = (1 + A)dt ab i dx i dt + a [(1 D)δ ij + E ij ]dx i dx j. (3.3) B i = Bi V B,i, E ij = Eij S + Eij V + Eij T, (3.4) Eij S = ( i j 1 3 δ ij )E, Eij V = 1 (E i,j + E j,i ), (3.5) i Bi V = 0, i E i = 0, i Eij T = 0, Eii T = 0. (3.6) Perturbations are categorized into scalar(a, B, D and E), vector(b V i, E V i ) and tensor(e T ij) perturbations. In this thesis we will only study scalar perturbations, thus for scalar perturbation the metric can be written as ds = (1 + A)dt + ab,i dx i dt + a [(1 ψ)δ ij + E,ij ]dx i dx j, (3.7) ψ = D 1 3 E, (3.8) where ψ is called the curvature perturbation since the three dimensional Ricci scalar for the spatial metric is given by R (3) = 4a ψ. Now under the change of coordinates, with ξ µ = (ξ 0, ξ i ), the general metric, Eq. 3.3, transforms as follows [59] X µ X µ + ξ µ, δgµν = δg µν + L ξ g µν, (3.9) L ξ g µν = ξ λ λ g µν µ ξ λ g λν ν ξ λ g λµ, (3.10) Ã = A ξ 0 Hξ 0, (3.11) B i = B i + ξ i ξ 0,i, (3.1) D = D ξk,k + Hξ 0, H = a a, (3.13) where ξ 0 and ξ i are scalar and vector perturbations, respectively. Now, the scalar perturbation are the ones that are responsible for the structure formation of the universe. Thus, we

33 look at how the metric 3.7 transform using the scalar perturbation theory with ξ i = ξ,i à = A ξ 0 Hξ, (3.14) B = B + ξ + ξ 0, (3.15) Ẽ = E + ξ, ψ = ψ + Hξ 0. (3.16) The above transformations are called gauge transformations. For the matter part of the action the gauge transformation are T ν µ = T µ ν + δt ν µ, δρ = δρ ρ ξ 0, (3.17) δp = δp p ξ 0, ṽ i = v i + ξ i, (3.18) δφ = δφ φ ξ 0. (3.19) where v i = au i is fluid velocity with u ν u ν = 1. The above matter transformations can be expressed in scalar perturbations theory using v i = v,i and ξ i = ξ,i that leads to ṽ = v +ξ scalar transformation. The perturbation variables that do not change under gauge transformations are called gauge invariant. The minimal set of gauge invariant variables or physical degrees of freedom for the metric perturbations is two. For metric part we have two famous scalar gauge invariant variables called Bardeen potentials [60] Φ = A H(B E ) + (B E ) (3.0) Ψ = ψ H(B E ) (3.1) that are considered as physical space-time perturbations. In the Newtonian gauge, that is defined by B = E = 0, gives us A N = Φ and D N = ψ N = Ψ in terms of Bardeen potentials. Thus, the exact expressions for these gauge invariant perturbations will dependent on the choice of time coordinate ξ 0, as different gauge choices correspond to different hypersurfaces of constant time [59]. 3

34 Some important gauge invariant scalar perturbations are those that remain conserved on super horizon scales k > ah, for example, comoving curvature perturbation R = ψ +H(v B) and uniform density perturbation ζ = (ψ +H(δρ/ρ )). The perturbation variable R can be understood in the comoving gauge with v = B = 0 in which you move with the cosmic fluid and your constant time slices are orthogonal to the 4-velocity and R = ψ becomes curvature perturbation in these comoving slices [59]. Observers in the comoving picture will see themselves as freely falling, making the expansion seem isotropic, meaning that they do not measure any energy flux δt 0 i = (ρ + p)v i = 0 or v i = 0. On the other hand ζ acts as curvature perturbations with a negative sign in uniform density gauge δρ = 0. In the study of inflation that is driven by single scalar field, we have just only one degree of freedom in the matter part that is the inflaton field φ, for which we have δt 0 i = 0 δφ = 0. The uniform density perturbations can be defined in terms of the inflaton field as ζ = (ψ+h(δφ/φ )) [61]. In this thesis we will call ζ the comoving curvature perturbation [6]. Now if the perturbations are very small then linear perturbation theory approximates the Einstein equations to high accuracy. But, to study non-gaussianity we need to go beyond linear order in perturbation theory. 3. Computational Strategy We consider single scalar field models of inflation defined by the action with the inflaton field minimally coupled by gravity in units c = = 1, M pl = 8πG = 1 (see Appendix A) S = d 4 x ( 1 g R 1 ) gµν µ φ ν φ V (φ). (3.) In this section we will present the steps laid down by Maldacena to calculate the two-point and the three-point function in momentum space of the scalar curvature perturbations [1]. First, one writes the action for the inflaton field in Eq. 3. using the Arnowitt-Deser-Misner(ADM) formalism. Secondly, one expands the action to second order in perturbation theory for 4

35 calculation of two-point function and to third order for the calculation of three-point function. Thirdly, one quantizes the perturbations and imposes canonical commutation relations. Next, one defines the vacuum state by matching the mode function to Minkowski vacuum when the mode is deep inside the horizon that fixes the mode function completely. Following these steps one can find the power spectrum and the three-point function for scalar perturbations [1,56]. These steps are implemented in the sections to come. 3.3 ADM Formalism Action The ADM formalism is known as the Hamiltonian formalism of General Relativity. In this formalism the space-time manifold with metric g µν is sliced by three dimensional hypersurfaces Σ at constant time and the goal is to identify and describe dynamical quantities as they evolve from Σ(t) to Σ(t + dt). At a point on Σ we have a normal vector n µ with n µ n µ = 1 and a time direction vector t µ = (1, 0, 0, 0) (hereafter in this section, we will use greek indices as µ, ν = 0, 1,, 3 while latin indices as i, j = 1,, 3). From these vectors one can define N = t µ n µ, N µ = h µν t ν, h µν = g µν + n µ n ν, (3.3) where N and N µ are the lapse and shift functions and h µν is projection operator onto the hyper-surface Σ, h µν n ν = 0. The space part of this projection operator h ij is the induced three-metric of the hyper-surface Σ. In component representation n µ = ( N, 0 i ), N µ = (N i N i, N i ), n µ = (1/N, N i /N) N µ = (0, N i ), (3.4) These ADM quantities are related to one another by vector relationship (see Fig. 3.1) 5

36 Nn t N Figure 3.1: ADM vector relationship between n µ, t µ, N µ vectors which is given by t µ = Nn µ + N µ. t µ = Nn µ + N µ (3.5) To calculate the ADM metric g µν we find interval ds between point (t, x i ) on Σ(t) hypersurface and a point (t + dt, x + dx i ) on Σ(t + dt) hyper-surface. By Pythagorean theorem, this interval squared is the difference between the square of proper distance between two points projected onto the base hyper-surface Σ(t) and squared proper time interval between the hypersurfaces, ds = dl dτ. Noting that dτ = Ndt, dl = h ij (dx i + N i dt)(dx j + N j dt), (3.6) where dx i + N i dt gives the displacement to the normal projection of (t + dt, x + dx i ) point to the hyper-surface Σ(t) [63], one can write down the line element of the space-time as ds = N dt + h ij (dx i + N i dt)(dx j + N j dt) (3.7) and thus the metric components in the ADM formalism can be calculated as g 00 = N i N i N, g 0i = g i0 = N i, g ij = h ij, (3.8) 6

37 while inverse metric components and vectors are given by g 00 = N, g 0i = g i0 = N i /N, g ij = h ij N i N i /N. (3.9) Next, one has to introduce extrinsic curvature which is defined for three surface Σ embedded in four dimensional space time as K ab = h c a b n c = 1 L nh ab = 1 N (L th ab L N h ab ) = 1 N (ḣab a N b b N a ) (3.30) where a is the covariant derivative of the four metric g µν. In the above definition and in subsequent section we have used indices a, b, c,... for abstract index notation that is independent of the coordinate basis and just represent tensor type T (m, n) [64]. In this notation the dimensionality of tensors is concealed, however we know that the extrinsic curvature is three dimensional symmetric tensor. In order to write the action (Eq. 3.) in terms of the ADM parameters one has to calculate the gravitational part of the action, Ricci scalar, using the ADM formalism. For that we have to define the concept of exterior derivative D c [64] which is defined as D c T a 1...a i b 1...b j = h a 1 d 1...h a i d i h e 1 b 1...h e j b j h f c f T d 1...d i e 1...e j (3.31) where h a b act as projection operator. Using this definition and relation hab = g ab + n a n b we calculate D a D b ω c = h f ah g b hk c f (h d gh e k d ω e ), = h d bh e ch f a f d ω e + (h d bh k ck ak n e + h e ch g b K agn d ) d ω e, = h d bh e ch f a f d ω e + K ac K e b ω e + h e ck ab n d d ω e. (3.3) 7

38 From the above equation we find the 3D Reimanian tensor (3) R abcd tensor as D a D b ω c D b D a ω c = h d bh e ch f a( f d d f )ω e + (K ac Kb e K bc Ka)ω e e, (3) Rabcω d d = h d bh e ch f ar fde jω j + (K ac Kb e K bc Ka)ω e e, (3.33) (3) Rabc d = h f ah g b he cr fge d + (K bc Ka d K ac Kb d ), (3) R abcd = h e ah f b hg ch i dr efgi + (K bc K ad K ac K bd ) (3.34) for the Σ three-hyper-surface. Next we calculate Ricci scalar R abcd h ac h bd = R abcd (g ac + n a n c )(g bd + n b n d ), = R + R ac n a n c = G ac n a n c (3.35) where G ac = R ac 1 g acr hence R = (G ac n a n c R ac n a n c ), (3.36) G ac n a n c = R (3) + K K ac K ac, (3.37) R ac n a n c = Racbn c a n b = n a ( a c c a )n c, = a n a c n c a n c c n a = K K ac K ac, (3.38) R = R (3) K + K ac K ac. (3.39) where K ac is induced 3D extrinsic curvature and R (3) is intrinsic curvature of Σ. Next, we look at the inflaton field part of the action in Eq. 3. whose kinetic term is given by 1 gµν µ φ ν φ = g 00 φ + g 0i φ i φ + g ij i φ j φ = 1 N ( φ N i i φ) + h ij i φ j φ (3.40) 8

39 Lastly, we calculate the pre factor g in the action g = hn (3.41) Now from equations ( ) the action in Eq. (3.) take the following form S = 1 d 4 x hn ( R (3) + K ij K ij K + N ( φ ) N i i φ) h ij i φ j φ V (φ) (3.4) where K ij is the three dimensional extrinsic curvature of the spatial slice at constant t defined in Eq (3.30) and K = K i i is the trace of this extrinsic curvature tensor. This is the action of the ADM formalism for single filed models of inflation that we will use to calculate the two point and three point function of scalar perturbations. From the above action, the Lagrangian density can be simply written as follows L = hn ( R (3) + K ij K ij K + N ( φ ) N i i φ) h ij i φ j φ V (φ). (3.43) To calculate the Hamiltonian density H from the Lagrangian density we have H = π φ φ + π ij ḣ ij L (3.44) where π φ and π ij are the conjugate momenta defined as π φ = L φ = hn 1 ( φ N i i φ) (3.45) π ij = L = h(k ij h ij K). (3.46) ḣij For these conjugate momenta one can define the following relations π ij h ij = hk = π hk ij = π ij 1 hij π K ij π ij = h 1/ (π ij π ij π /) K ij K ij K = h 1 (π ij π ij π /) (3.47) 9

40 to calculate the Hamiltonian density H(π, h) as H = π ij (i N j) hn ( R (3) h ( 1 π ij π ij π / ) ) +N ( φ (N i i φ) ) + h ij i φ j φ V (φ) (3.48) = ( h N j i (h 1/ π ij ) + N (R (3) h ( 1 π ij π ij π / ) )) +N ( φ (N i i φ) ) + h ij i φ j φ V (φ). (3.49) Now, the important thing to note here is that N and N i are not dynamical variable of the theory but only constraint variables. Since these variables do not contribute to kinetic terms and there conjugate momenta are zero and these constraint variables give rise to constraint equations. The Hamiltonian constraint equations are as follows H N j = i (h 1/ π ij ) N 1 (N i i φ) j φ = 0, (3.50) H N = R(3) h 1 (π ij π ij π /) N ( φ (N i i φ) ) V (φ) = 0. (3.51) We will use ADM action (3.4) and these constraint equations in next sections to calculate the power spectrum of the scalar perturbations as well as the three point function and the primordial non-gaussianity. 3.4 Scalar Perturbations and Second Order Action (Linear Perturbations) To calculate the two point function or power spectrum we follow the steps laid down by Maldacena that were presented in section 3.. We choose the following gauge, the comoving gauge, for the dynamical fields φ and g ij δφ = 0, g ij = a (e ζ δ ij + t ij ), i t ij = t i i = 0, (3.5) where ζ is the comoving curvature perturbation at constant density hyper-surface δφ = 0. The ζ and t ij are the physical degrees of freedom for the theory. In this gauge the inflaton field 30

41 is unperturbed and all scalar degrees of freedom are parameterized by the metric fluctuation ζ(t, x) while the tensor perturbations are parameterized by t ij. This condition fixes the gauge completely at non zero momentum [1]. Geometrically, ζ is the measure of spatial curvature of constant-φ hyper-surfaces, R (3) = a e ζ (4 ζ + ( i ζ) ). An important property of ζ is that it remains constant outside the horizon i.e. k > ah. In ADM formalism the temporal part of the metric (g 00, g 0i ) are not dynamical variables of the theory and only act as constraint variables. We started by writing the action in Eq. 3. in the ADM formalism. To find power spectrum one has to expand the action to second order in the comoving gauge (Eq. 3.5) according to which the spatial derivative of field φ vanishes, thus action 3.4 can be written as S = 1 d 4 x hn ( ) R (3) + K ij K ij K + N φ V (φ) (3.53) The constraint equations 3.51 can be written in the comoving gauge in terms of extrinsic curvature i (K ij h ij K) = 0 R (3) (K ij K ij K ) N φ V (φ) = 0 (3.54) In order to solve these equation we follow [1] and expand N and N i = Ñi + ψ,i, with i Ñ i = 0, in powers of ζ as follows N = 1 + α 1 + α +... Ñ i = N (1) i + N () i +... ψ = ψ 1 + ψ +... (3.55) where α k, ψ k, N (k) i O(ζ k ). We plug these ansatz into the constraint equations and solve 31

42 them for N and N i = Ñi + ψ,i to first order in perturbation theory as follows N = 1 + ζ H, Ñ (1) i = 0, ψ = χ ζ H, χ = ɛa ζ (3.56) In order to calculate the effective action to the second or third order in perturbation one only needs to calculate N and N i to the first order. This is because the perturbations evaluated at second order, such as α, get multiplied by the constraints evaluated at first order O(ζ) that vanish. Similarly, the α 3 gets multiplied by constraint equation evaluated to zeroth order O(ζ 0 ) that also vanish [1, 65]. Details are given in Appendix C. Thus using the above results, the calculations are simplified and we expand the ADM action (3.53) to the second order by replacing the first order solutions of N and N i back into the action. We start by calculating the extrinsic curvature part of the action, for which we need the three-metric h ij and its derivatives in comoving gauge h ij = g ij = a e ζ δ ij ḣ ij = a e ζ ( ζ + H)δ ij h ij,k = a e ζ ( k ζ)δ ij h ij = a e ζ δ ij h = a 3 e 3ζ, g = a 3 e 3ζ N. (3.57) Using these above relations, the extrinsic curvature can be expanded as K ij = 1 N (ḣij i N j j N i ) = 1 (ḣij N i,j N j,i + N k (h ik,j + h jk,i h ij,k )) N = 1 ( a e ( ) ) ζ (HN + N i ζ,i )δ ij N i ζ,j N j ζ,i ψ,ij N K = K i i = 1 N ( 3HN + N i ζ,i e ζ ψ ) (3.58) 3

43 Thus, the extrinsic curvature part of action and R (3) can be written as g(kij K ij K ) = a 3 e ( 3ζ e 4ζ N ( ) 1 ψ,ij ψ,ii 4e 4ζ ζ,i ψ,j ψ,ij 6NH ), (3.59) R (3) = a e ζ ( 4 ζ + ( i ζ) ). (3.60) With help of the above relations we find the second order action as S () = 1 d 4 x (ae ζ (1 + ζ/h) ( 4 ζ ( ζ) ) + a 3 e 3ζ (1 + ζ/h) ( V (φ) 6H ) + a 3 e 3ζ (1 + ζ/h) ) 1 φ (3.61) The above action can further be simplified by the background equations of motion of the inflaton field V (φ) = 6H φ, Ḣ = φ /. (3.6) Expanding the action 3.61 to second order in scalar perturbations S () = 1 d 4 x (ae ζ (1 + ζ/h) ( 4 ζ ( ζ) ) + a 3 e 3ζ (1 + ζ/h) ( 1H φ ) ( + a 3 e 3ζ 1 ζ/h + ( ζ/h) ) ) φ = 1 d 4 x (ae ζ (1 + ζ/h) ( 4 ζ ( ζ) ) + a 3 e 3ζ (1 + ζ/h) ( 1H ) ( + a 3 e 3ζ + ( ζ/h) ) ) φ = 1 d 4 x ( ae ((1 ζ + ζ/h) ( 4 ζ ) ) + ( ζ) + a 3 e 3ζ ( ζ/h) ) φ (3.63) where we used Eq. 3.6 and integrated by parts in the last step. Now after some integrations by parts the action becomes S () = 1 ( d 4 x ( ae ζ ( ζ) φ ) ) + a 3 3ζ ζ e H H φ = 1 d 4 x φ ( ) a 3 H ζ a( ζ) ( ( ) ) = d 4 xa 3 φ ζ ζ H a ( ) = d 4 xa 3 ɛ ζ a ( ζ) (3.64) 33

44 After defining the Mukhanov variable v = zζ, z = a φ H = a ɛ (3.65) and transitioning to conformal time τ leads us to action for scalar perturbation v(τ, x) that is the Mukhanov variable S () = ) dτd 3 x (v + ( i v) + z z v. (3.66) We Fourier expand the v(τ, x) field and find the equation of motion for scalar perturbations d 3 k v(τ, x) = (π) v k(τ)e ik.x, (3.67) 3 ) v k + (k z v k = 0. (3.68) z This is called the Mukhanov-Sasaki equation [66] for the scalar quantum fluctuations v k = zζ k Solution and Quantization of Mukhanov Equation To find the solution of Mukhanov-Sasaki equation one can quantize the v field then define its commutation relations and the vacuum state. We start by quantizing the v field as d 3 k ˆv(τ, x) = (π) ˆv k(τ)e ik.x, 3 ˆv k (τ) = v k (τ)â k + v k(τ)â k. (3.69) Where the creation â k and annihilation â k operators satisfy the canonical commutation relations [â k, â k ] = δ 3 (k k ), [ˆv(x), ˆπ v (x )] = iδ 3 (x x ) (3.70) the above commutation relation holds when v k (τ) field is normalized as follows i (v kv k v k v k ) = 1. (3.71) 34

45 Next, one must choose a vacuum state â k 0 = 0 for the fluctuations at some early time. A natural length scale in curved spacetime is the radius of curvature while on scales much shorter than radius of curvature, the spacetime looks flat. In inflationary Universe the comoving curvature scale is (ah) 1 τ. Thus, in the remote past (τ ) the modes do not feel the curvature when kτ 1. In this limit the Mukhanov equation can be written as v k + k v k = 0, (3.7) and choosing the solution that asymptotically, τ, behave as v k e ikτ k. (3.73) corresponds to the choice of Minkowski vacuum for these short scale modes. One can make this choice for all modes of interest if we define the vacuum state sufficiently early. The Universe expanding in exactly exponential regime will have de Sitter space time. In de Sitter approximation z z a a = τ, (3.74) ( v k + k ) v τ k = 0. (3.75) one can define a de Sitter invariant vacuum by choosing the solution that satisfies the asymptotical behaviour in Eq. (3.73) ( v k (τ) = e ikτ 1 i ) k kτ. (3.76) The vacuum state that is defined by the above mode function is a preferred vacuum state in the de Sitter universe known as Bunch Davies vacuum. Form the above solution to the mode function v k, we will find the scalar power spectrum in the next subsection. 35

46 3.4. Correlation Function and Power Spectrum The power spectrum is given by the -point correlation function of the curvature perturbation ζ(x) ζ (x) = = d 3 k d 3 k ˆζk (π) 3 (π) 3 ˆζk e i(k k ).x (3.77) d 3 k k P ζ(k) (3.78) 3 We start by evaluating the -point function for v k in Fourier space as ˆv kˆv k = 0 ˆv kˆv k 0 = (π) 3 δ 3 (k k ) v(k), v(k) = τ k (1 + 3 k τ ). (3.79) To evaluate the -point function for ζ, that has mode function u k, by the relationship u k = v kz = H a φ v k or more precisely d 3 k ζ(τ, x) = (π) ˆζ 3 k (τ)e ik.x ˆζ k (τ) = u k (τ)â k + u k(τ)â k (3.80) Thus, the two point function for ˆζ k, the curvature perturbation, is evaluated as ˆζk ˆζk = (π) 3 δ 3 (k k ) ζ(k), ζ(k) = H φ H k 3 (1 + k τ ). (3.81) The above power spectrum ζ (k) = u k can be expressed in terms of dimensionless power spectrum P ζ (k) for scalar curvature perturbations ζ k as ζ(k) = π k 3 P ζ(k), P ζ (k) = k3 π u k = H H (π) φ (1 + k τ ). (3.8) 36

47 On super-horizon scale kτ 1, the power spectrum approaches a constant P ζ (k) = H (π) H φ = H 8π ɛ (3.83) where marks horizon crossing value of variables. Since ζ approaches a constant on superhorizon scales the spectrum at horizon crossing determines the future spectrum until a given fluctuation mode re-enters the horizon. 3.5 Primordial Non-Gaussianity Non-Gaussianity, i.e. the study of non-gaussian contributions to the correlations of cosmological fluctuations, is emerging as an important probe into the early universe. Being a direct measure of inflaton interactions, the primordial non-gaussianity will teach us a great deal about the inflationary dynamics. Now, if the perturbations ζ are drawn from a Gaussian distribution, then the power spectrum (or two-point correlation function) contains all the information about the perturbation field. However, for non-gaussian perturbations the higher-order correlation functions beyond the two-point function contain additional information about initial perturbations. Computing the leading non-gaussian effects requires expansion of the action in Eq. (3.4) to third order in perturbation theory to compute the three-point correlation function ζ(x 1 )ζ(x )ζ(x 3 ) The Bispectrum and Measures of Non-Gaussianity The Fourier equivalent of the scalar three-point function is called the bispectrum B ζ (k 1, k, k 3 ) ζ k1 ζ k ζ k3 = (π) 3 B ζ (k 1, k, k 3 )δ 3 (k 1 + k + k 3 ) (3.84) where the delta function, enforces momentum conservation, is a consequence of translational invariance of the background. In three point function, the presence of delta function of three momenta forces triangular condition k 1 +k +k 3 = 0 in the Fourier space and these triangles 37

48 can be of different sizes and shapes. Thus, the Bispectrum B ζ (k 1, k, k 3 ) is dependent on three different length scales that constitute triangles in the momentum space. The three point correlation is often described in terms of a local dimensionless nonlinearity parameter f NL. This non-linearity parameter f NL was first introduced as a measure of local non-gaussianity described by ζ(x) = ζ L (x) 3 5 f nl ( ζl (x) ζ L (x) ), (3.85) where factor 3 5 is a matter of convention. This factor comes from the relation Φ = 3 5 ζ during matter dominated era where Φ is the Bardeen potential in Newtonian gauge Calculation of Third Order Action using ADM Formalism To calculate the three point function one has to do the calculations beyond linear perturbations theory. This time we expand the action to next (third) order in perturbation theory since we are calculating the three point function. We follow the computational strategy of section 3. to calculate the Bispectrum. As mentioned previous section 3.4 and Appendix C that we only need to evaluate constraint equations to the first order in perturbations and then we plug these first order solutions back into the action. Thus, we start by expanding the action in Eq. 3.4 to third order by using the first order solution to constraint Eqs. (3.56). S (3) = 1 ( d 4 x ae ζ (1 + ζ/h) ( 4 ζ ( ζ) ) + a 3 e 3ζ (1 + ζ/h) ( V (φ) 6H ) + ( a 3 e 3ζ a 4 (1 ζ/h) ( ( i j ψ) ( ψ) ) 4a 4 i ζ i ψ ψ + (1 + ζ/h) )) 1 φ (3.86) 38

49 Then using the background equations of motion and integrate by parts in the last step we get S (3) = 1 ( d 4 x ae ζ (1 + ζ/h) ( 4 ζ ( ζ) ) + a 3 e 3ζ (1 + ζ/h) ( 1H φ ) = 1 +a 3 e (a 3ζ 4 (1 ζ/h) ( ( i j ψ) ( ψ) ) 4a 4 i ζ i ψ ψ ( + 1 ζ/h + ( ζ/h) ( ζ/h) ) ) ) 3 φ d 4 x ( ae ζ (1 + ζ/h) ( 4 ζ ( ζ) ) 1a 3 e 3ζ (1 + ζ/h)h +a 3 e (a 3ζ 4 (1 ζ/h) ( ( i j ψ) ( ψ) ) 4a 4 i ζ i ψ ψ ( + + ( ζ/h) ( ζ/h) ) ) ) 3 φ = 1 d 4 x (ae ζ (1 + ζ/h) ( 4 ζ ( ζ) ) ( + a 3 e 3ζ a 4 (1 ζ/h) ( ( i j ψ) ( ψ) ) 4a 4 i ζ i ψ ψ + (1 ζ/h) ζ )) φ /H (3.87) Now after some integrations by parts and dropping the total derivatives [65] we get S (3) = 1 ( d 4 x ae φ ζ H ( ζ) + a 3 e 3ζ (1 ζ H ) φ H ζ ( +a 1 e 3ζ (1 ζ H ) ( )) ψ,ij ψ,ij ( ψ) ) 4ζ,i ψ,i ψ = 1 ( d 4 x aɛζ( ζ) + a 3 (3ζ ζ )ɛ ζ H ( )) = 1 1 a 1 a (ζ,i + ζ,i H )ψ,iψ,jj + (3ζ,j ζ,j H )ψ,iψ,ij ( d 4 x aɛζ( ζ) + a 3 (3ζ ζ )ɛ ζ H ( (ζ,i + ζ,i H )ψ,i ψ 1 (3 ζ ζ H )ψ,i )) (3.88) 39

50 Now we perform integration by parts starting from the terms proportional to Kζ i ζi where K factors are multiplied by ζ i ζi. After integration by parts we get Kζ i ζi = ζ i d dt K where d dt acts on the K factors and we drop the total derivative terms. Next, we do the integration by parts of the following terms K ζ i = ζ i d dt K, K ζ = ζ d dt K, Kζ i ζ i = ζ i d K, K ζλ = dt ζ d dt (KΛ ) where K s are the coefficients of these terms and Λ = χ = ɛa ζ. We further integrate by parts the following terms ζ ζ ζ = ζ i i (ζ ζ), Kζ i ζ = ζ d dt (Kζ i ), ηaζ ζλ = ζ d dt (aηλ) d dt (ηa3 ζ ζ), ζζ i = ζ ζ here we will not drop one of the total time derivative term d dt (ηa3 ζ ζ) that will be important in our coming discussion. Finally after all these integration by parts we get S (3) = d 4 x (a 3 ɛ ζ ζ + aɛ ζ( ζ) aɛ ζ( i ζ)( i χ) + a3 ɛ ηζ ζ+ 1 a ɛ( iζ)( i χ) χ + 1 4a ɛ( ζ)( χ) + f(ζ) δl δζ ) (3.89) here the last term is variation of quadratic action which is given by δl δζ = d(a3 ɛ ζ ( ) ) d aɛ χ ζ = a + H χ ɛ ζ (3.90) dt dt f(ζ) = η 4 ζ + ζ ζ/h + 1 [ ( ζ) + ( 4a H i j ( i ζ j ζ)) ] + 1 [ ( ζ)( χ) + ( 4a i j ( i ζ j χ)) ], Λ = χ = ɛa ζ (3.91) H Now, the calculation of the three point function from the above action involves integration over the time variable. But, if we use the action in Eq to calculate the three point function, the integral over time does not converge to the end of inflation as pointed out by [68]. However, it was shown in [7,68] that this action can be converted into an equivalent form, that gives a convergent three point function, by adding a total derivative term that is given by d ( η ) dt ɛa3 ζ ζ (3.9) 40

51 which brings the action to the following form [7, 8]. S (3) = ( d 4 x a 3 ɛ ζ ζ + aɛ ζ( ζ) aɛ ζ( i ζ)( i χ) a 3 ηɛζ ζ a ɛηζ ζ + 1 a ɛ( iζ)( i χ) χ + 1 4a ɛ( ζ)( χ) + g(ζ) δl (3.93) δζ g(ζ) = ζ ζ/h + 1 [ ( ζ) + ( 4a H i j ( i ζ j ζ)) ] + 1 [ ( ζ)( χ) + ( 4a i j ( i ζ j χ)) ] (3.94) H Here the last term can be eliminated with a field redefinition ζ ζ n + g(ζ) because g(ζ) is only a function of derivatives of scalar perturbations ζ(t, x) that vanish outside the horizon. It has been shown in [30] that this last term can also be removed using to in-in formalism. The above third order action is an exact result without any slow roll approximations thus it is even valid for models that deviate from slow roll conditions. Another feature of this action is that it contains only first two slow roll parameters ɛ and η while it is independent of derivative terms like η Quantization and Three Point Function Finally, to calculate the 3-point function in momentum space we move to the interaction picture and write the Hamiltonian for the action in Eq. 3. as ) H(ζ) = H 0 (ζ) + H int (ζ) (3.95) where H 0 is the quadratic part of the Hamiltonian while H int represents all higher order terms in perturbation theory [1]. However, to calculate the 3-point function, the interaction Hamiltonian H int only has to be third order in perturbations. Thus H int is just equal to S (3) without the integral over time coordinate since the conjugate momenta vanish in the 41

52 ADM formalism. H int (τ) = d 3 x ( aɛ ζζ + aɛ ζ( ζ) ɛζ ( i ζ)( i χ) aɛηζζ a ɛηζ ζ + a ɛ( iζ)( i χ) χ + 1 ) 4a ɛ( ζ)( χ) (3.96) The three point function in the interaction vacuum at some time τ near the end of inflation is given by τ ζ k1 (τ)ζ k (τ)ζ k3 (τ) = i dτ a [ζ k1 (τ)ζ k (τ)ζ k3 (τ), H int (τ )] (3.97) τ o We can calculate this three point function analytically in the de Sitter limit where a = 1/(Hτ) and u k = ih (1 + 4ɛk 3 ikτ)e ikτ for a single inflation field with standard kinetic term given by action in Eq. (3.). We quantize the perturbation field ζ(x) and define the vacuum state. For this we expand the ζ(x) field into creation and annihilation operators and use ] commutation relations of scalar field [â k, â k = δ kk to get the following result ζ k1 ζ k ζ k3 = i(π) 3 τend ( dτδ 3 (k 1 + k + k 3 ) a ɛ u 1u u 3 + a ɛ(ɛ η)u 1u u 3 a ɛ(ɛk 1.k + ηk 3)u 1u u 3 + a ɛ3 u 1u u 3 k1 k.k 3 + a kk 3 ɛ3 u 1u ) 3 + distinct permutations j=1 u 3 k 1.k k + c.c. k 1.k k u j (τ end ). (3.98) where u 1 denotes complex conjugate of u k1 and c.c. is the complex conjugate of the whole expression [33]. This is the main formula for the three point function. This expression will be used in the sections to come for the exact numerical calculation of three point function in momentum space. The choice of the vacuum is specified by the choice of mode function u k selection. The three point function results can be summarized by each terms in the interaction Hamiltonian as follows. 4

53 Contribution from the ɛ(ɛ η)ζζ term ( τend 3 i dτ(π) 3 δ 3 (k 1 + k + k 3 )a ɛ(ɛ η) = H4 (ɛ η) 16ɛ (π) 3 δ 3 (k 1 + k + k 3 ) ( u du 1 dτ 3 j 1 k 3 j du 3 dτ j u j (τ end ) ) ) + two perm + c.c. ( k k3 K + k 1kk 3 K ) + sym (3.99) Contribution from the ɛ ζ( ζ) term i τend ( dτ(π) 3 δ 3 (k 1 + k + k 3 )a ɛ u i (τ end ) i (u 1u u 3k 1.k + two perm) + c.c. ( ) 16ɛ (π)3 δ 3 1 (k 1 + k + k 3 ) k 3 i i ( K + k 1k + k k 3 + k 3 k 1 + k ) 1k k 3 (k K K 1.k + sym) (3.100) = H4 Contribution from the ɛηζ ζ term τend ( ) i dτ(π) 3 δ 3 (k 1 + k + k 3 )a ɛη u i (τ end ) ( u 1u u 3k1 + two perm ) + c.c. ( ) = H4 η 3ɛ (π)3 δ 3 1 (k 1 + k + k 3 ) k 3 i i ( K + k 1k + k k 3 + k 3 k 1 + k ) 1k k 3 (k K K 1 + sym ) (3.101) i ) 43

54 Contribution from the ɛζ ( ζ)( χ) term τend ( i dτ(π) 3 δ 3 (k 1 + k + k 3 )a ɛ u i (τ end ) i ( ) u du du 3 k 1.k 1 + five perm + c.c. dτ dτ k ( ) = H4 16ɛ (π)3 δ 3 1 (k 1 + k + k 3 ) k 3 i i ( ( k1.k k3 + k ) ) 1 + k + sym K K ) (3.10) Contribution from order ɛ 3 term ( ) τend i dτ(π) 3 δ 3 (k 1 + k + k 3 ) a ɛ 3 u i (τ end ) i ( ) u du du 3 k 1.k 1 + five perm + c.c. (3.103) dτ dτ k ( ) τend i dτ(π) 3 δ 3 (k 1 + k + k 3 ) a ɛ 3 u i (τ end ) i ( ) u du du 3 k1k.k two perm + c.c. (3.104) dτ dτ kk 3 The integral relation for 3-point function given in Eq can be analytically evaluated in the slow roll limit. Ignoring ɛ 3 terms Eqs. ( ) and summing the leading order terms O(ɛ) in Eqs. ( ), gives us the following result that was first derived by Maldacena [1]. ( ) ζ k1 ζ k ζ k3 = (π) 7 δ 3 (k 1 + k + k 3 )(P ζ 1 k ) A k 3 i i (3.105) A = η ɛ ki 3 + ɛ k i kj + ɛ ki kj. 8 8 K (3.106) i where denotes the ɛ and η values at horizon crossing. This is a general result for single field slow roll inflation with standard kinetic term (c s = 1) with smooth potential. The quantity A is a convenient measure of non-gaussianity in the perturbation field. The relationship i j i>j 44

55 between A and bispectrum is given by B ζ (k 1, k, k 3 ) = (π) 4 (P ζ k ) A. (3.107) i k3 i The bispectrum and A are general measures of non-gaussianity however both these quantities are highly scale dependent. Thus, the three point correlation is often described in terms of a local dimensionless non-linearity parameter f NL. This non-linearity parameter f NL was first introduced as a measure of local non-gaussianity described by Eq For local non- Gaussianity, the quantity A is expressed via the non linearity parameter f nl as A local = 3 10 f nl ki 3. (3.108) Beyond the local model we can define a generalized f NL for general kind of non-gaussianity by the following equation, that also has the advantage of being nearly scale independent. i f NL 10B ζ(k 1, k, k 3 ) i k3 i 3(π) 4 (P ζ k ) ( i k3 i ) (3.109) Over the recent years f NL has become a widely used measure of non-gaussianity [40]. In this thesis we will be using this generalized definition of f NL instead of A or bispectrum that are both scale dependent quantities. The f NL depends on the shapes of three point function triangles and it will depend upon scale of triangles if there are features in the inflaton potential. 45

56 Chapter 4 Configuration Space Moments In this thesis we will study non-gaussianity through the higher order moments of the perturbation field in configuration space. Analysis of these moments provide a robust measures of non-gaussianity and has also become an important field of investigation [69, 70]. These moments can provide important information on the geometrical properties of the physical fields, e.g. CMB temperature fluctuations, and give us non-gaussianity observables such as extrema counts, genus and skeleton [50]. We will study how different inflationary models lead to different predictions for non-gaussian geometrical measures. We will calculate the cubic moments of scalar curvature perturbation ζ and it s derivatives in physical space. These rotation invariant cubic moments ζ 3, ζ ζ and ( ζ) ζ are needed to evaluate different Minkowski functional like genus. We will calculate these moments for single field slow roll models of inflation analytically in this chapter as base calculation. We will also calculate these moments for models with local non-gaussianity in this chapter. Then, we will calculate these moments numerically in the next chapter even for non slow roll models of inflation. 46

57 4.1 Cubic Moments in Single Field Slow Roll Models of Inflation To calculate the third order moments we have to take the inverse Fourier transform of three point function in momentum space ζ k1 ζ k ζ k3. These third order moments also contain derivatives of the perturbation field i ζ and ζ. The complete and independent set of moments that are needed to calculate the observables such Euler characteristic are given by ζ 3 (x) = ζ (x) ζ(x) = ( i ζ(x)) ζ(x) = dk 3 1 dkdk ζ (π) 9 k1 ζ k ζ k3 e i(k 1+k +k 3 ) x, (4.1) dk 3 1 dkdk ζ (π) 9 k1 ζ k ζ k3 k3e i(k 1+k +k 3 ) x, (4.) dk 3 1 dkdk ζ (π) 9 k1 ζ k ζ k3 k 1 k k3e i(k 1+k +k 3 ) x. (4.3) These moments are always scaled by the corresponding powers of variances σ and σ 1 in all physical observables like Minkowski functionals and Euler characteristics. We will calculate the above cubic order moments using Eqs. ( ). As a starting point we shall consider slow roll models where these moments can be calculated analytically for the case of flat power spectrum P ζ k const and then use the acquired insight to develop a general numerical procedure. For near flat spectra these integrals generally have both infrared and ultraviolet divergences, thus the correspondent moment is formally infinite. Ultraviolet divergences are regularized by the finite resolution of our measurements and should be studied in the context of a specific experiment. Infrared divergences, on the other hand, seem of more conceptual nature since they come from contribution of the modes much larger than the observed Universe, which potentially reflect also less understood physics. To study them we introduce infrared k min and ultraviolet k max cutoffs and consider asymptotic behaviour as z = k min /k max 0. This procedure is somewhat ambiguous as to the order of imposing the cutoff and integrating over the δ-function. This ambiguity does not, however, affect the coefficient of the leading infrared divergent term, which is, as we will see, what matters. It does affect subleading 47

58 terms and this freedom can be used to minimize them in the numerical calculations. We suggest the procedure to use the δ-function to eliminate in every term in the integral the least infrared divergent momentum and then integrate over the angles and the magnitudes of remaining two momenta in k min and k max limits. For instance, to calculate the ζ 3 (x), we substitute Eq into Eq. 4.1 to get ζ 3 (x) = dk 3 1 (π) 3 dk 3 dk 3 3 (π) 3 (π) ζ 3 k 1 ζ k ζ k3 e i(k 1+k +k 3 ) x (4.4) = dk 3 1 dk 3 dk3 3 (π) 3 (π) 3 (π) 3 (π)7 δ 3 (k 1 + k + k 3 )(P ζ k ) (4.5) ( ) 1 Ae i(k 1x+k +k 3 ) x i k 3 i A = η ɛ ki 3 + ɛ k i kj + ɛ 8 8 K i i j ki kj. (4.6) This calculation can be divided into three pieces corresponding to three terms in A. With first term being sum of cubes ki 3 ζ 3 (x) ( ) ( = dk 3 1 1dkdk δ 3 (k 1 + k + k 3 ) (P ζ k ) 1 (η ɛ 1 ) (π) k 3 j j 8 i = dk1dk 3 dk δ 3 (k 1 + k + k 3 )(η ɛ ) (P ζ ( ) k ) k k 3 + k3 3 (π) 8k1k 3 k = (η ɛ ) (P ζ ( ) k ) k dk 3 8(π) 1 dk k 3 + k 1 + k 3 k1k 3 k k 3 = 3 ( ) (η ɛ )(P ζ k ) ln kmax k min i>j k 3 i ) (4.7) = 3 (η ɛ )(P ζ k ) ln z, z = k min k max. (4.8) 48

59 The second term is sum of k i k j terms ζ 3 (x) = ( dk1dk 3 dk δ 3 (k 1 + k + k 3 ) (P ζ k ) (π) l 1 k 3 l ) ɛ ( 1 8 = dk1dk 3 dk δ 3 (k 1 + k + k 3 )ɛ (P ζ k ) (π) ( ) k1 k + k 1 k3 + k k1 + k k3 + k 3 k1 + k 3 k 8k1k 3 k = 3ɛ (P ζ ( ) k ) dk 3 (π) 1 dk 3 (k1 + k ) 8k1k 3 k k = 3 ( ) 4 ɛ (P ζ (k1 + k ) k ) dk 1 dk d cos θ k 1 k k 1 + k = 3 ( kmax k ( 1 4 ɛ (P ζ k ) dk dk ) k min k min k 1 k k 1 kmax k1 ( 1 + dk 1 dk + 1 ) ) k min k min k 1 k k While the third term is sum of k i k j K ( = 3 ɛ (P ζ (k min k max ) k ) + ln k max + k max k min terms given by ) k i kj i j ( ln k ) ) max. (4.9) k min ζ 3 (x) ( ) ( ) = dk 3 3 1dkdk δ 3 (k 1 + k + k 3 ) (P ζ k ) 1 ɛ 1 k (π) k 3 i kj l l K i>j = dk1dk 3 dk δ 3 (k 1 + k + k 3 )ɛ (P ζ ( ) k ) k 1 k + k1k 3 + kk 3 (π) Kk1k 3 k ( = 6ɛ (P ζ k k ) dk 1 dk d cos θ 1k (k 1 + k + k 1 + k )k1k 3 k 1 + k = 6ɛ (P ζ k ) ( kmax kmax + dk 1 k min k min dk k1 k k 1 k min dk 1 ln ((k 1 + k )/k ) k min dk ln ((k 1 + k )/k 1 ) k 1 = 6ɛ (P ζ k ) ( (1 ln 4) k min (k max + k min ) k max k min ( ln k max k max k max + k min + 1 ln ) ( kmax k min ) )). (4.10) ) 49

60 where in every terms in the integrations we eliminated the least infrared divergent momentum with help of the delta function δ 3 (k 1 + k + k 3 ) after that we integrated out the angles and momenta. Using this prescription we get the moment ζ 3 (x) by adding the above terms to get ζ 3 (x) = ζ 3 (x) + ζ 3 (x) + ζ 3 (x) 1 3 ( ) 3 = (P ζ k ) (η + ɛ ) ln (z) 3ɛ ln(z) + ɛ (15 4 ln()) + O(z ),. (4.11) This result for the ζ 3 (x) moment has ln (z) leading order divergence in z 0 limit. However, we will see that in the observed quantities like genus this moment will always be divided by the σ 4 ln (z) hence will give us finite result in z 0 limit. In the above calculation of ζ 3 (x) moment, we have used the regularization prescription to take out the least infrared divergent momentum in every term with help of the delta function. But, if we use some other prescription by removing the momenta that is not least infrared divergent with help of the delta function the leading order term, 3 (η + ɛ ) ln (z), will remain the same however all next to leading order terms will get modified and their values will increase in magnitude. In the next two moments ζ ζ and ( ζ) ζ has a derivative operator of the form D(ζ k1, ζ k ) ζ k3, where is Laplace operator and D is a differential operator of k 1 and k. Hence, our integrand has the following form D(k 1, k )k 3A ijk (4.1) where D(k 1, k )k 3 is differential operator and A ijk is the three point function in momentum space, i.e. ζ ki ζ kj ζ kk, that has all permutations of i, j, k momenta indices. Namely D(k 1, k )k 3(A 13 + A 13 + A 13 + A 31 + A 31 + A 31 ). (4.13) Now, let the last index k in A ijk correspond to least infrared divergent momenta i.e. k 3. 50

61 Thus, we need to relabel the terms to make this index 3, we have D(k 1, k )k 3(A 13 + A 13 ) + D(k 1, k 3 )k (A 13 + A 13 ) + D(k 3, k )k 1(A 13 + A 13 ) (4.14) where in the second term with changed indexes 3, 3 and in the third term 3 1, 1 3. Thus we have (D(k 1, k )k 3 + D(k 1, k 3 )k + D(k 3, k )k 1)(A 13 + A 13 ) (4.15) which is a general result. In our case ζ ζ moment has D(k 1, k ) = 1, ( ζ) ζ has D(k 1, k ) = k 1 k. Using the above mentioned procedure to evaluate the momentum space integrals, we now calculate the ζ (x) ζ(x) moment with D(k 1, k ) = 1 by substituting Eq into Eq. 4.. Again dividing the calculation into three parts as follows ζ (x) ζ(x) = 1 = dk 3 1 dk 3 1 dk 3 dk3δ 3 3 (k 1 + k + k 3 ) (P ζ k ) (π) (η ɛ ) ( ) ( ) k3 1 1 k k 3 i 3 i i 8 i dk 3 dk3δ 3 3 (k 1 + k + k 3 )(η ɛ ) (P ζ k ) (π) ( ) ( ) k1 + k + k3 k3 3 8k1k 3 k ( dk1 3 dk 3 k1 + k + 1 k1k 3 3 k 1 k 3 = (η ɛ ) (P ζ k ) 8(π) = η ɛ (P ζ k 4 ) dk 1 dk d cos θ ( k 1 + k + k 1 k cos θ k 1 k + k 1 k + k = (η ɛ )(P ζ k ) ( k max k min ( ) ) kmax ln k min k 1 ) + 1 ) k1k 3 = (ɛ η )(P ζ k ) k max ( 1 z ) ln(z). (4.16) 51

62 The second term gives ζ (x) ζ(x) = = dk 3 1 k 3 = ɛ (P ζ k ) 4(π) dk 3 dk3δ 3 3 (k 1 + k + k 3 ) (P ζ k ) (π) ɛ ( ) ( ) 1 1 k i kj 8 i k 3 i i j dk1dk 3 dk δ 3 (k 1 + k + k 3 )ɛ (P ζ k ) (π) ( ) ( ) k1 + k + k3 k 1 k3 + k k3 8k1k 3 k ( k dk1 3 dk k + k 1 k k 3 k 1 k 3 ) = ɛ (P ζ k ) k max ( ln(z) z + O(z ) + k 3 + k 3 + k ) k 3 k 1 k and the third term gives ζ (x) ζ(x) = dk dk 3 dk3δ 3 3 (k 1 + k + k 3 ) (P ζ k ) (π) ɛ ( ) ( ) k3 1 1 k k 3 i kj i i K i>j = dk1 3 dk 3 dk3δ 3 3 (k 1 + k + k 3 )ɛ (P ζ k ) (π) ( ) ( ) k1 + k + k3 k1k 3 + kk 3 Kk1k 3 k = dk1 3 dk 3 dk3δ 3 3 (k 1 + k + k 3 ) (P ζ k ) (π) ɛ ( k1 k k3k + k ) 3 k 1 k3k k 1 k k 3 K ( π = ɛ (P ζ k ) kmax ) 3 ln() 18 ln(z) + z + O(z ) Adding the above three terms gives us the expression for the moment ζ (x) ζ(x) ( = (P ζ k ) kmax (η + ɛ ) ln(z) ɛ ( π ln() ) ) + O(z ) 6 k 3 (4.17) (4.18) (4.19) 5

63 This moment ζ (x) ζ(x) is to leading order ln(z) divergent in the z 0 limit. However, in all observables this moment will be divided by σ σ 1 ln(z) giving us finite result in z 0 limit. However, in the calculation of this moment if we do not use the prescription of eliminating the least divergent momenta with help of the delta function, then next to leading order terms can change significantly. The third moment ( ζ(x)) ζ(x), with D(k 1, k ) = k 1 k, can be calculated by substituting Eq into Eq. 4.3 splitting the calculation into three parts ( ζ(x)) ζ(x) 1 = = dk1 3 dk 3 dk3δ 3 3 (k 1 + k + k 3 ) (P ζ k ) (π) (η ɛ ) ( ) ( ) 1 1 k k 3 i 3 k i i 8 3k 1 k i dk1 3 dk 3 dk3δ 3 3 (k 1 + k + k 3 )(η ɛ ) (P ζ k ) (π) ( ) k 3 3 (k ) 8k 1k 1k 3 k k 3 + kk 1 k 3 + k3k 1 k = (η ɛ ) (P ζ k ) 8(π) dk k 1 + k 4 k 3 1k 3 dk 3 ( k1 k 3 + k 1 + k k 1 k 3 + k k 3 1 = (η ɛ ) (P ( ζ k 6 ) kmaxk min kmax 4 kmin 4 1 k 1 k + k 1 + k k1k 3 ) = (η ɛ ) ( (P ζ k 6 ) kmax 4 1 z + z 4). (4.0) ) 53

64 The second terms gives ( ζ(x)) ζ(x) = dk 3 1 dk 3 dk3δ 3 3 (k 1 + k + k 3 ) (P ζ k ) (π) ɛ ( ) ( ) 1 k i kj 1 k 8 k 3k 3 1 k i j i i = dk1 3 dk 3 dk3δ 3 3 (k 1 + k + k 3 )ɛ (P ζ k ) (π) ( ) k1 k3 + k k3 (k ) 8k 1k 1k 3 k k 3 + kk 1 k 3 + k3k 1 k = ɛ (P ζ ( k ) dk 3 4(π) 1 dk 3 k1 + k ) k1k 3 3 k 1 k 3 k1k 3 ( = ɛ (P ζ k ) kmax 4 3 ) z 3 + O(z3 ). (4.1) The third terms gives ( ζ(x)) ζ(x) = dk dk 3 dk3δ 3 3 (k 1 + k + k 3 ) (P ζ k ) (π) ɛ ( ) ( ) 1 ki kj 1 k K k 3k 3 1 k i>j i i = dk1 3 dk 3 dk3δ 3 3 (k 1 + k + k 3 )ɛ (P ζ k ) (π) ( ) k 1 k (k ) Kk 1k 1k 3 k k 3 + kk 1 k 3 + k3k 1 k = ɛ (P ζ k ) kmax 4 ( ) 1 ln(4) + O(z 3 ). (4.) 3 Thus, adding the above three terms give us ( ζ(x)) ζ(x) ( ( ) ) 1 4 = (P ζ k ) kmax 4 6 η + ɛ 57 ln() + O(z ) (4.3) This moment is finite moment in the z 0 limit but in every physical observable it will be divided by σ 4 1 k 4 max that will solve the ultraviolet divergence. However, if we do not follow 54

65 the regularization prescription of elimination of least infrared momenta by the delta function then this moment changes the most and will not give us the correct answer. As this moment is not sensitive to infrared cutoff thus next the leading order correction contribute a lot to the moment. As briefly discussed before, the first two moments, Eqs(4.11 and 4.19), are divergent in z 0 limit, ζ 3 (x) ln (z) while ζ (x) ζ(x) ln(z). The modified regularization sequence retains the coefficients in the divergent terms but changes the constant ones. Here we note that the last moment with the most derivatives ( ζ(x)) ζ(x) is finite as z 0 and thus remains sensitive to regularization details. Now, if we look at the expression for Euler characteristic (Eq. 1.3) or any other observables, the configuration space moments are always normalized [50] by the variances σ and σ 1 given in Eqs σ = ζ (x) ( ) = P ζ k ln kmax σ 1 = ( ζ(x)) = P ζ k k min k max k min (4.4). (4.5) Thus, even though the moments by themselves are divergent, the observed quantities are finite. Normalized or scaled by these variances the moments to leading order in slow roll are given by S 3 ζ3 (x) σ 4 3 (η + ɛ ), (4.6) T 3 ζ (x) ζ(x) (η + ɛ ), (4.7) σ σ1 U 3 ( ζ(x)) ζ(x) ( (η + ɛ 8 ln() 57 )). (4.8) σ Note that the numerical coefficient in front of ɛ in U 3 is also equal to two with better than % accuracy, thus all three normalized moments are determined by η + ɛ. 55

66 4. Cubic Moments in Local Non-Gaussianity Next we want to find these cubic moments for the local non-gaussianity, using Eq. (3.108), that can easily be derived using the method described in previous section as follows ζ k1 ζ k ζ k3 local = (π) 7 δ 3 (k 1 + k + k 3 )(P ζ 3 k ) 10 f nl i k3 i i k3 i Now it can be seen that the above expression is similar the the sum of cubes k 3 i terms with a different prefactor 3 10 f nl. Hence the three moments for local non-gaussianity can be easily deduced from Eqs. (4.8, 4.16, 4.0) as follows ζ 3 (x) local = 18 5 f nl(p ζ k ) ln z, (4.9) ζ (x) ζ(x) local = 1 5 f ( nl(p ζ k ) k max 1 z ln(z), (4.30) ( i ζ) (x) ζ(x) local = 5 f ( nl(p ζ k ) kmax 4 1 z + z 4), (4.31) ζ 3 (x) local = 18 σ 4 5 f nl, (4.3) ζ (x) ζ(x) local σ σ 1 = 4 5 f nl, (4.33) ( i ζ(x)) ζ(x) local σ 4 1 = 8 5 f nl. (4.34) From the above moments, we have found that the local non-gaussianity parameter for single field slow roll inflation is f nl = 5 1 (η + ɛ ) which is independent of the shape and scale of the triangles. These expression for cubic moments for local non-gaussianity can also be derived straight from the original definition of non linearity parameter, given in Eq. 3.85, in configuration space. This will work as a verification test for our calculations of momentum space integrals. We can rewrite the definition of f nl as ζ(x) = ζ L (x) 3 5 f nl ( ζ L (x) σ ), (4.35) 56

67 where ζ L (x) is the linear or Gaussian part of the perturbations and σ = ζl (x). Now the ζ 3 (x) moment can be expanded to first order in f nl as ζ 3 (x) = ζl(x) f ( nl ζ L (x) σ ) ζl(x) (4.36) = f ( nl 3σ 4 σ 4) = 18 5 f nlσ 4 (4.37) where ζ 3 L (x) = 0 as it is a Gaussian part and all odd power vanish while ζ4 L (x) = 3σ4. Thus, we find the same result as we got from the other method in Eq. (4.3). The next moment ζ L (x) ζ L(x) can be calculated as follows ζ L (x) ζ L (x) = ( ζl(x) 3 ) 5 f nlζ L (x)(ζl(x) σ ) (4.38) ( ζ L (x) 3 5 f ( nl ζl (x) ζ L (x) ζl,i(x) )) = 3 5 f nl ( ζl (x)(ζ L(x) σ ) ζ L (x) + ζl(x) ζ 3 L (x) + ζl(x)ζ L,i(x) ) = 3 ( 5 f nl ζl(x) ζ 3 L (x) + σ σ1 ) σ ζ L (x) ζ L (x) = 6 ( 5 f nl (3 ζl(x) ) ζ L (x) ζ L (x) ) + σ σ1 = 4 5 f nlσ σ 1. (4.39) where σ 1 = ζ L,i (x) and using ζ 3 L (x) ζ L(x) = 3 ζ L (x) ζ L(x) ζ L (x), ζ L (x) ζ L (x) = 57

68 σ 1 identities. The third moment can be similarly calculated as ζ L,i (x) ζ L (x) = ( ζl,i(x) 4 3 ) 5 f nlζl,i(x)ζ L (x) (4.40) ( ζ L (x) 3 5 f ( nl ζl (x) ζ L (x) ζl,i(x) )) = 3 5 f nl ( ζ L (x)ζ L,i(x) ζ L (x) + ζ L (x)ζl,i(x) ζ L (x) + ζl,i(x)ζ L,i(x) ) = 6 ( 5 f nl 3 ζl,i(x) ζ L (x) ζ L (x) + 5 ) 3 σ4 1 = 6 ( 5 f nl 3σ ) 3 σ4 1 = 8 5 f nlσ 4 1 (4.41) where we have used the the identity ζl,i (x)ζ L,i (x) = 5 3 σ4 1 and we get the same result as Eq Thus, we see that both methods, the momentum space integration method and the configuration space expectation value method give the same results. That show that our prescription for regularization in momentum space integrals by taking out least divergent momenta by the delta function works very well and gives us correct results. However if we do not follow this prescription we do not get the correct results especially for the U 3 moment. 58

69 Chapter 5 Numerical Analysis This chapter consists of case study models of inflation with which we will check our numerical calculations. For these models of inflation we will calculate the power spectrum and bispectrum and the generalized f NL. Then we will numerically calculate the different cubic moments for some models of inflation. 5.1 Case Study Models of Inflation We shall consider few models of inflation that consist of single inflaton field minimally coupled to gravity with standard kinetic term. The models we will study can be divided into two classes one are standard slow roll models of inflation. While the other class of models consist of models that break slow roll conditions. The standard quadratic m φ potential inflation and the λφ 4 inflation models fall in the first category of slow roll models. The quadratic inflation with a sharp step in the potential and the resonance model of inflation are the models that break slow roll conditions. However, all these models are chaotic models of inflation with base model being the quadratic inflation model. Finally we will briefly discuss some other model of inflation that may give rise to large non-gaussianity. 59

70 5.1.1 Inflation with Quadratic m φ Potential Single Field with V (φ) = 1 m φ, or quadratic, potential inflation is known as chaotic inflation in large field models of inflation. The dynamics of inflation can be solved exactly in the slow roll limit for 1 m φ potential. Therefore, we will consider it as our case study in this section. The equation for the background i.e. the equations for the scale factor a(t) and the scalar inflaton field φ(t). H = 1 ( ) 1 3 φ + V (φ) φ + 3H φ + V,φ = 0 (5.1) In the slow roll approximation, the potential dominates over the kinetic term and φ V,φ thus, the above equations takes the following form H = ȧ a = mφ 6 (5.) 3H φ + V,φ = 0 (5.3) solving these equation for φ we get φ = m φ 3H = 3 m φ(t) = φ i + φ, φ = mt (5.4) 3 where φ i marks the starting value of the scalar field. At this stage do not know the value of φ i but we can find the final value φ at which the slow roll condition breaks φ V (φ) 4 φ = 3 thus in terms of equation for φ(t) can now be written in terms of φ as follows ( ) φ(t) = φ + 3 mt = + mt 3 (5.5) (5.6) 60

71 where t identify the time from the start of inflation t i to the variable time t. Similarly we can solve for the scale factor from Eq. 5. da a = m 6 φ(t)dt a(t) = a 0 e ( mt/3+m t /6). (5.7) Now in in order to solve the problems of hot big bang model we need at least 70 e-folds of inflationary expansion. 3 m t m t = 70, 4 t = t t i =, m 844 φ i = 3 ( + m t) = 3 (5.8) Thus, we have found that in order solve the flatness and horizon problem we need to start at least with the above value φ i to ensure 70 e-folds of inflation. Next we can find the slow roll parameters and the number of e-folds N as follows N(φ) = Hdt = φ φ ɛ = 1 φ H = φ η = ɛ ɛh = 4 φ dφ = φ ɛ (5.9) from where we can see that value of η is twice that of ɛ. Now inflation ends when scalar field drops down to the value φ or when the slow roll approximation breaks. Next we calculate the scalar perturbations v k from the Mukhanov equation v k + ) (k z v k = 0, (5.10) z 61

72 in the slow roll limit, with z = a ɛ and ignoring O(ɛ ) terms, we find that z z a H ( ɛ + 3 ) η, ah 1 + ɛ τ z z n 1/4 v k +,, n 3 τ + 3ɛ + 1 η, (5.11) ( ) k n 1/4 v τ k = 0, v k = xy(x), x = kτ, ẏ = dy dx, ÿ + 1 ( ) xẏ + 1 n y = 0 (5.1) x thus the above equation, Bessel Equation, has solutions in terms of Hankel function of first H 1 n and second H n kind given by v k = xy = α kτh 1 n(kτ) + β kτh n(kτ), (5.13) where n is order of Hankel function. Now the values of α and β can be found from the Bunch Davies vacuum solution 3.73 in the limit kτ using lim x H1, n (x) = πx e ix e iπ(n 1/)/. (5.14) Thus, after ignoring the constant phase factors we get the value for α = π 4k we can write the complete solution for v k in the slow roll limit as v k = and β = 0 and π 4 τh1 n(kτ). (5.15) To calculate the power spectrum for u k = v k /z and using z = ɛ H τ 1 n we get P ζ (k) = k3 π u k = k3 H 16πɛ τ n H 1 n(kτ). (5.16) 6

73 To find the super-horizon limit kτ 0 of the power spectrum we use Thus, the scalar spectral index is found as lim x 0 H1 n(x) = i ( x ) n π Γ(n) (5.17) ) n (5.18) P ζ (k) = k3 H 16π 3 ɛ τ n Γ (n) ( kτ = n Γ (n)h 16π 3 ɛ k 3 n (5.19) = H 8π ɛ k 3 n (5.0) n s 1 = 3 n ɛ η, (5.1) that it non-zero for the slow roll inflation hence the spectrum is not flat. On similar ground one can calculate the power spectrum in the de Sitter limit as follows u k = ihe ikτ 4ɛk 3 (1 + ikτ) (5.) P ζ = H 8π ɛ = m φ 4 96π (5.3) P ζ m N 6π (5.4) from which we can calculate or estimate the mass m of the inflaton field for the quadratic inflation Quartic or λφ 4 Inflation Single field model with λφ 4 potential is another example of chaotic or slow roll models of inflation. We will work out different non-gaussian observables for this model as well. The potential is given by V (φ) = λ 4 φ4 (5.5) 63

74 where λ is coupling constant. After that one can calculate the Hubble parameter and the slow roll parameters as follows λ H = 1 φ, (5.6) λ φ = φ, (5.7) 3 ɛ = η = 8 φ, (5.8) N(φ) = 1 8 (φ φ end). (5.9) with, φ end =. Now one can calculate power spectrum as follows P ζ = H 8π ɛ (5.30) = λφ6 3π (5.31) 16λN 3. π (5.3) Hence, we can find the approximate the value of λ from the above equation for power spectrum Step Potential Model of Inflation In single field models of inflation significant non-gaussianity can arise from different kinds of non trivial potentials, for example adding some features to quadratic potential. One such model consist of localized breaking of slow roll conditions. This can be achieved be adding a step in the quadratic potential as proposed by [5, 71 73] V step (φ) = 1 ( m φ 1 + c tanh φ φ ) s. (5.33) d where c is the height of the step and d the width of the step centred at φ s. This model has been used to improve the fit between LCDM and observed power spectrum and we have used the best fit parameters for the step potential [73]. When inflation rolls down through 64

75 Figure 5.1: Slow roll parameters ɛ and η plotted against the inflaton field φ with step located at φ s = 15.86M pl. this step it goes through a sharp acceleration in inflationary dynamics. The three point correlation function is proportional to ɛ and η parameters and deviation from slow roll gives rise to large interaction of modes near the step. It important to mention here that ɛ does not change much, as can be seen in Fig. 5.1a, by the step potential however η 7c3/ ɛd > 1 is more sensitive to step as observed Fig. 5.1b. This breaking of slow roll η > 1 gives rise to large non-gaussianity and hance large values to the generalized f NL 7c3/ ɛd parameter [31]. Since, the three point function integral Eq gets most of its contribution form near the horizon crossing part of the modes. Thus, due to the step in the potential, the modes that cross the horizon near the step get a sharp kick that gives rise to large non-gaussianity. The physical size to step φ d = 0.01M pl leads to the step size in k space from Eq. D.4 is given by that we incorporated in our numerics as k = Resonance Model of Inflation k = k 0φ 0 φ (5.34) The other kind of models that can give rise to large non-gaussianity are models with global features in the potential like small oscillation on top of quadratic potential also know as 65

76 Resonance model [3] V res (φ) = 1 m φ (1 + c sin(φ/λ)). (5.35) where c is the amplitude of oscillation and Λ is the frequency. In this model the non-gaussianity is generated in the sub-horizon scales when modes are deep inside the horizon and they resonate, or interfere constructively, with the physical frequency ω This model introduces a new scale Λ into the quadratic potential as there are small ripples in the entire plane of the potential. These ripples in the potential causes oscillation in k space in the generalized f NL with amplitude φ πλ. 5c/4 Λ.5 φ 0.5 (see Fig. 5.15) while the power spectrum remains almost flat with tiny ripples [3]. This kind of mechanism may be realized in brane inflation with duality cascade [74, 75]. The physical scale of oscillations Λ = requires k = that is need to resolve the oscillations Other Models with Large Non-Gaussianity The kinetic inflation with non-standard kinetic energy term also gives large non-gaussianity in the limit small speed of sound c s limit. The non-gaussianity parameter in this model is ( ) proportional to f NL O + O ( λ Σ). Thus in the limit cs 1 and λ/σ 1 gives large 1 c s non-gaussianity. Where c s is speed of sound while Σ = H ɛ c s and parameter λ consist of third derivative of P (X, φ) of Kinetic inflation given in Eq..40 [33]. The kinetic inflation is a very broad class of inflation because of large parameter space with models like DBI inflation. In this thesis however, we will not study this class of inflationary models. The non-attractor initial conditions can also be enhance non-gaussianity if the inflaton trajectory is initially displaced from its attractor, slow-roll solution. Any deviation from the attractor solution of the inflaton generically generates a component of non-bunch-davies vacuum [9, 3]. Thus, models that consist of non-standard vacuum other than the Bunch- Davies vacuum may give rise to large non-gaussianity as well [38, 39]. 66

77 5. Calculation of Power Spectrum To calculate the power spectrum of scalar perturbations we need to solve the background equations of motion Eqs. 5.1 and the Mukhanov equation Eq written in terms of conformal time. The scalar perturbations are governed by the Mukhanov equation 3.68 that can be elegantly written in conformal time parametrization. a = a 6 ( φ + 4a V (φ) ) φ = a a φ a V,φ (5.36) ) v k = (k z v k z where primes donate derivatives with respect to conformal time. Numerically, it is more convenient to work out the differential equation for u k rather than v k since we finally require u k = v k /z to calculate the non-gaussianity parameters A and f NL. Thus, we convert the Mukhanov equation to perturbation equation for u k. u k = k u k z z u k (5.37) These are coupled differential equation with first two 5.36 representing the background while the third equation 5.37, for u k, representing the quantum fluctuations. We have solved these equations using the Runge-Kutta method of order four. The initial condition for solving these differential equations are given by the following equations a(τ 0 ) = 1, φ(τ 0 ) = 16.77, a (τ 0 ) = a 0 6 V (φ) + V (φ) + V (φ) /3, φ (τ 0 ) = a 0 V (φ) + V (φ) + V (φ) /3, 1 k v k (τ 0 ) = k, v k(τ 0 ) = i, τ a 0 0 = a 0(1 ɛ 0 ). (5.38) 67

78 100 N=ln(a) 10 1 N=ln(a) e-05 1e t Figure 5.: Number of e-folds N = ln(a) for quadratic potential plotted against physical time t = τ dτ a(τ ) with t plotted in logarithmic units i.e. t = 10 x with x shown on the x-axis. We chose initial value of inflaton field φ 0 such that universe expands for 70 e-folds. The mode functions originate deep inside the horizon that correspond to, our choice of vacuum, the Bunch-Davies vacuum. Using these initial conditions we numerically solved the differential equations using Runge-Kutta method of order four and the results are displayed below(figs ). The above equations of motion are for single field inflation with standard kinetic term with any potential V (φ). However in this thesis we will specifically study quadratic inflation V (φ) = 1 m φ as our base model to check our numerics. We will also study quartic inflation and other models that have features added to the quadratic model that are known as the step and resonance models, details of which are given in the next section. The first two plots(fig. 5., 5.3) show the solutions to the background equations of motion. Figure 5. shows scale factor a(τ) or the number of e-folds from which it clear that it behaves linearly with respect to physical time in a log-log plot. Thus, it is found that a t before the end of inflation, at t = 10 7, while it flattens out when inflation ends when inflaton field 68

79 10 Inflaton Field 1 Inlfaton Field e-05 1e Figure 5.3: Absolute value of inflaton field φ plotted against the Number of e-folds for quadratic potential. ln(a) starts to oscillates about the minimum. The linear behaviour of scale factor in physical time corresponds to a τ 1 in conformal time. Thus, numerical computation in conformal time τ is a challenging task as most of the dynamics occur close to a pole at τ = 0 thus one has to decrease dτ in your numerics as you approach this pole. The second plot(fig. 5.3) shows the solution to KG equation of the inflaton field. Here one can see that the inflaton field drops down the potential and then starts to oscillate around the minimum of the potential with decaying amplitude in post inflationary stage. Next two Figures(5.4, 5.5) show the mode function for scalar perturbations u k and it s derivative u k that is also needed in calculation of Bispectrum. Here it is evident that the mode oscillates inside the horizon and then becomes constant as the mode crosses horizon at around 6.5 e-folds. While, the derivative of the mode function u k approaches to zero as the mode crosses horizon depicting the stability of our numerics. After solving for the u k and background equations of motion we now move towards the calculation of power spectrum. The calculation dimensionless power spectrum is pretty 69

80 100 u k 10 1 u k Figure 5.4: Scalar Perturbations mode function u k against the number of e-folds for k = 0.005M pl for quadratic potential. ln(a) u k 1 u k e-05 1e-06 1e-07 1e-08 1e-09 1e ln(a) Figure 5.5: Derivative of Scalar Perturbations mode function u k against the number of e-folds for k = 0.005M pl for quadratic potential. 70

81 straight forward as P ζ (k) = k3 π u k. (5.39) The observational constraints from Planck and WMAP satellites [9] are given by P k = A s ( k k 0 ) ns 1 (5.40) where A s = ± 0.05 and n s = ± with pivot at k 0 = 0.05Mpc 1 = 3.5M 1 pl. Using these Planck result one can find the mass m = M pl of inflaton field for the quadratic inflation and coupling λ = for the quartic inflation. We have calculated the dimensionless power spectrum according to Eq Figure 5.6 shows the power spectrum for the quadratic m φ and λφ 4 models of inflation. The power spectrum is mildly dependent on k for the quadratic and quartic potentials with d ln P ζ d ln k = ɛ η. This can be viewed in Fig. 5.6 that for m φ potential P k drops slowly when compared with λφ 4 potential. On the other hand for the step potential, due to the breaking of the slow roll condition because of a sharp step in the potential, we see oscillating power spectrum near the step but as we move away from the step it follows the same behaviour of the quadratic potential (Fig. 5.7). 5.3 Calculation of Bispectrum and Cesaro Sum The starting point for the calculation of three point function is to solve the background and Mukhanov equations in Eqs. (5.36, 5.37) determine ɛ, η, u k and u k. After numerically solving the background equations and the equation for scalar perturbations, we insert these solutions 71

82 3.e-09 3e-09 m Potential 4 Potential.8e-09.6e-09 P k.4e-09.e-09 e e k Figure 5.6: Power spectrum of scalar perturbations P ζ (k) against momenta k for quadratic potential with m = M pl and λφ 4 potential with λ = e e-09 Quadratic Potential Step Potential 4e e-09 P k 3e-09.5e-09 e e-09 1e-09 5e k Figure 5.7: Power spectrum of scalar perturbations P ζ (k) against momenta k for quadratic potential with m = M pl and step potential with c = 0.00, d = 0.0M pl and φ s = 15.86M pl. 7

83 back into Eq to calculate the three point function integral. ζ k1 ζ k ζ k3 = δ 3 (k 1 + k + k 3 ) j=1 τend 3 ( I(τ) = i(π) 3 u j (τ end ) a ɛ u 1u dτ I(τ) (5.41) u 3 k 1.k k a ɛ(ɛk 1.k + ηk3)u 1u u 3 + a ɛ3 u 1u u 3 k1 k.k 3 kk 3 + a ɛ3 u 1u u 3 k 1.k k + a ɛ(ɛ η)u 1u u 3 ) + c.c. + distinct permutations. (5.4) Thus, the three point function calculation is numerically a challenging task as it involves integrations that arise from equations 5.41 and 5.4. The integrand I(τ) consist of three factors of u k or u k multiplied by the background factors of a, ɛ and η. The scalar perturbation function u k oscillates before horizon crossing at τ, while after horizon it freezes out. Thus, the integration consists of two parts, before horizon crossing (BHC) part and after horizon crossing (AHC) part τend dτi(τ) = τ dτi(τ) + τend τ dτi(τ) (5.43) where τ is the horizon crossing point of the largest k mode in the three point correlation function and I(τ) is the integrand of the 3-point function given in Eq that contains background factors and product of three oscillating mode functions. The BHC and AHC parts of integration present different numerical challenges as the first has growing oscillations, as τ approaches negative infinity, while for the AHC part we have to regularize the three point function by adding a total derivative term(eq. 3.9) in the action. Without adding this term in the action, the AHC part of the integral is divergent as one of the term a 3 ɛ ηζ ζ in the initial action grows as the scale factor [7, 68]. The contribution to the integral that arises from before horizon crossing poses significant technical challenges. In conformal time the initial big bang singularity is pushed back in conformal time to τ. Thus, the scalar perturbations start deep inside the horizon and 73

84 Integrand Integrand 1e-05 1e-06 1e-07 1e-08 1e-09 1e ln(a) Figure 5.8: Integrand I of three point function for an equilateral triangle with k = 0.005M pl plotted against the number of e-folds. keep oscillating till horizon crossing point τ of the largest k mode in the 3-point function. Now, there are different methods to numerically evaluate an oscillating integral over an infinite range. If we cutoff this infinite integral to some finite value, due to large oscillations this induces an spurious contribution of O(1). Numerically it was shown that these kind of integrals can be evaluated by introducing an arbitrary damping factor into the integrand but this damping factor needs to be chosen carefully [31]. Other techniques, such as boundary regularizaion, for evaluating such integrals are even more complex [7, 3]. We have developed a different numerical technique, which is numerically more robust and elegant, using the Cesaro resummation of improper series. For oscillating integrand I(τ), the following expressions gives the definition of Cesaro integration τ ( 1 τ ) τ dτi(τ) lim dτ dτ I(τ ) τ τ τ τ τ. (5.44) This gives a specific definition to the improper integral on the left hand side whereas the right hand side is an average over the partial integrals that give convergent result for a wide range of improper integrals [76]. However, we extended this method further and we defined a 74

85 higher order Cesaro integral, with one additional average, to further improve the convergence as lim τ 1 τ τ τ τ dτ 1 τ τ τ τ dτ ( τ τ dτ I(τ ) ). (5.45) In our numerical program we have used this Cesaro integral with additional average Eq. (5.45) for faster convergence Equilateral Squeezed 3-point Function Integral Figure 5.9: Three point correlation function, or generalized f NL, integral plotted against the number of e-folds N = ln(a) for Equilateral(k 1 = k = k 3 = 0.005M pl ) and Squeezed(k 1 = 0.005M pl, k = k 3 = 0.05M pl ) triangles with horizon crossings at 6.5 and 8 e-folds. The generalized f NL is just the sum of the two asymptote(plateau) values at small and large e-folds corresponding to the two integrals in Eq ln(a) This method quickly gives convergent results without introducing any artificial damping factors. This can be seen in Fig. 5.9 which shows the three-point function integral result plotted against the number of e-folds for an equilateral triangle and squeezed triangle cases. In this figure horizon crossings occur τ that corresponds to e-folds values of 6.5 and 8 for equilateral and squeezed triangle. This Fig 5.9 describe two different integration regimes BHC τ < τ and AHC τ > τ. In BHC regime, we integrate in backward direction from the horizon crossing points using the Cesaro Integral Eq Our technique converges very 75

86 quickly as can be seen that the integral plateaus as we go 5-6 e-folds before horizon crossing points. In the AHC regime, we integrate in the forward direction that also plateaus soon after horizon crossing. Thus, the three point function integral, or generalized f NL, is just the sum the two asymptote(plateau) values in the before and after horizon crossing regimes for each kind of triangle. To test our procedure, we have calculated the three point function numerically for 1 m φ potential and compared it with the corresponding analytical results given by Eq The numerical results when compared the analytical results are plotted in Fig. 5.10(a, b) for equilateral and squeezed triangles. As can be seen that our numerics follows very closely the analytical results for the quadratic potential. The Fig shows that our numerical technique is accurate to below one percent error for equilateral and squeezed triangles. In the extreme squeezed triangle limit k 5 in Fig the errors are somewhat large as a compromise was made in the calculation due to the time and memory constraints. We have also calculated the three point function and generalized f NL for the λφ 4 model of inflation. The results are shown in Fig. 5.1 and since it also a slow roll model of inflation the f NL is small like the quadratic potential model. But the values for f NL are larger than those for quadratic inflation. In Fig we present the calculation of f NL for equilateral triangle for the step potential model with parameters c = 0.00, d = 0.0M pl and φ s = 15.86M pl. The mode function u k that exits horizon at φ = φ s corresponds to k s 0.01M pl. It is noted that peaks in f NL plot for the step potential are of O(10) in amplitude that is about 500 times the f NL of quadratic potential. However, if we move away from the step k s, k > 0.5M pl and k < 0.005M pl, the value of f NL for step potential tend to approach values for the quadratic potential. These large value for the f NL parameter are still within the observational constraints from Planck satellite [1]. 76

87 Analytical Numerical f NL f NL k 1 =k k Analytical Numerical Figure 5.10: Numerical calculation of f NL for an equilateral triangle case(k = k 1 = k = k 3 ) alongside analytical result (Fig. a) for quadratic potential. Second plot show results for a squeezed triangle k 3 = 0.005M pl while k 1 = k are on the x-axis (Fig. b) for quadratic potential. 77

88 .5 Squeezed Equilateral Percentage error in f NL k Figure 5.11: Percentage error in calculation of f NL for an equilateral triangle case when compared with analytical results plotted against k = k 1 = k = k 3. While for squeezed triangle percentage error is plotted against k = k = k 3 while fixed k 1 = 0.005M pl for quadratic potential m Potential 4 Potential f NL k Figure 5.1: f NL for the λφ 4 model plotted against k = k 1 = k = k 3 for equilateral triangle case in comparison with quadratic potential model. 78

89 Quadratic Potential Step Potential 4 f NL k Figure 5.13: f NL for the step potential model plotted against k = k 1 = k = k 3 for equilateral triangle case with step height and width given by c = 0.00 and d = 0.0M pl respectively while step is located at φ s = 15.86M pl..5 Step Potential Quadratic Potential f NL k 1 =k Figure 5.14: f NL for the step potential model plotted against k 1 = k and k 3 = 0.005M pl for squeezed triangles with step height and width given by c = 0.00 and d = 0.0M pl respectively while step is located at φ s = 15.86M pl. 79

90 Fig shows our computational technique works well for resonance model of inflation as well Quadratic Potential Resonance Potential 5 f NL k 0.01 Figure 5.15: f NL for resonance model plotted against k = k 1 = k = k 3 for equilateral triangle case for c = and Numerical Calculation of Moments The calculation of cubic moments involves taking inverse Fourier transform of momentum space three point function as can seen in Eqs. ( ). To calculate moments numerically we have to numerically integrate Eqs. ( ) but using the exact result for three point function in momentum space in Eq The numerical calculation of momentum space integrals with finite momentum space cutoff is unavoidable. The analytical results in Chapter 4 guide us to the following numerical procedure for the general single field models of inflation. We calculate the moments using the exact result of 3-point function Eq. 3.98, with the triangularity condition k 1 +k +k 3 = 0 applied to the least divergent momentum as described by Eqs. (4.11,4.15). To numerically obtain z 0 result we integrate the remaining two momenta in finite range [k min,k max ], then vary z = k min k max and find the asymptotic limit 80

91 S 3 T 3 U Moments k max Figure 5.16: Moments plotted against k max with fixed k min = 0.005M pl for quadratic potential. (plateau value) for the scaled moments S 3, T 3 and U 3. Procedure we follow is that we fix k min = 0.005M pl that corresponds to the mode that inflates 64 e-folds after horizon crossing that is roughly the largest scales of the observed universe(see Eq. A.7 for detail) and vary k max or z. As seen in Fig. 5.16, for quadratic potential the numerical calculation of S 3, T 3 and U 3 gives very stable result already by k max 0.1M pl. This fast convergence reflects smallness of the constant terms in Eqs and 4.19, 1.6ɛ and 0.8ɛ correspondingly. Slight drop in values of moments in Fig is a numerical artifact as higher values of k max > M pl require very fine resolution k in momenta integrations. Another important test for the numerical calculation of moments was in which we fix the range kmax k min = 100 and we vary k min such that k max = 100k min. The moments are plotted against k min in Fig which shows that moments are very stable with respect to k min as well. The increase in magnitude of moments comes from the increase in values of slow roll parameters as seen is Eqs. ( ). This slight increase in values of the moments is also because of the small decrease in value of power spectrum P k and the variances σ and σ 1 81

92 S 3 T 3 U Moments k min Figure 5.17: Moments plotted against k min with k max = 100k min for quadratic potential. as function of k min. But it is important to note here that the moments vary by roughly 5 percent in magnitude that is our estimate for the percentage error in moments calculation. Thus, we see that the infrared divergences for near flat spectra can be dealt with very efficiently during numerical momenta integration our devised mechanism. It is sufficient to integrate over just a two orders of wave-numbers to obtain a good approximation to the asymptotic values. The quadratic potential gives the values S 3 = , T 3 = and U 3 = with at most 5% error in the calculation of these scaled moments. Like the quadratic potential one can also calculate these moments for the other slow roll model of λφ 4. In this model the moments plateau quickly in range k min = 0.005M pl and k max = 0.5M pl as seen in Fig The values for the moments are S 3 = , T 3 = 0.1 and U 3 = with almost 5% errors. It can be seen that in this model the moments come out to be roughly twice the quadratic potential model. For potential with features such as step potential, the answer for the moments depends on the range of k-integration in relation to the modes that give additional contribution to non-gaussian signal. For the step potential, as Fig demonstrates, the support for 8

93 S 3 T 3 U Moments k max Figure 5.18: Moments plotted against k max with fixed k min = 0.005M pl for λφ 4 potential with λ = correction to the f NL is finite, encompassing the range from k 0.01M pl to k 0.M pl for the parameters used there. Fig shows the change in the moments if we fix k min = 0.005M pl but vary k max. The asymptotic behaviour can be understood from the following consideration. Let us write S step 3 (k min, k max ) = S quad ζ σ 4 (k min, k max ), (5.46) and correspondingly for the other moments, where ζ 3 signifies additional contribution to the momentum integral above the baseline quadratic values. In such a split the first term, S quad 3 is practically independent on the k min or k max, as we discussed above. The second correction term depends on the integration range. As soon as this range encompasses the support for the extra non-gaussian contribution, it starts decreasing as the correspondent powers of the variances, i.e. as 1/σ 4 ln (k max /k min ) for S 3, 1/(σ σ 1) k max ln 1 (k max /k min ) for T 3 and 1/σ 4 1 k 4 max for U 3. This explains the quick convergence of T 3 and U 3 moments to their baseline values as k max > 1M pl and slow logarithmic decrease of S 3 exhibited in Fig In cosmological applications the k min to k max range depends on observational setup or 83

94 4 3 S 3 T 3 U 3 Moments k max Figure 5.19: Moments plotted against k max with fixed k min = 0.005M pl for step potential with c = 0.00, d = 0.0M pl and φ s = 15.86M pl. analysis choice. For large-scale structure studies the ratio of largest observable scale 10Gpc or k min = 0.005M pl to the smallest presently mildly non-linear scale 10Mpc is of order of a thousand, which corresponds to k max 5M pl. This is the value that we adopt in the following sample calculations. For the step potential we plot the moments against the parameters c and d as displayed in Figs. 5.0 and 5.1. The limit of the quadratic potential is reached as c 0 or d. It can be seen that the magnitude of S 3 moment increases significantly as the height of the step c increases (Fig 5.0) or the width d decreases (Fig. 5.1). At the same time, the moments T 3 and U 3 remain practically unchanged from their small values in quadratic potential limit, as expected from our analysis of k max behaviour. The first moment S 3 is plotted against c and d in Fig. 5. which shows that sharper the step, smaller width and higher step, the higher the moment. In Fig. 5.3 the moments are plotted against the step location φ s in range φ s = (15.9, 15.5). This range of φ s corresponds to modes with range k = (0.01M pl, 1M pl ) that cross the horizon at step location. Thus if the step occurs at k = 1M pl or φ s = 15.5M pl 84

95 10 S 3 T 3 U 3 1 Moments Figure 5.0: Absolute values of the moments for the step potential plotted against parameter c for fixed d = 0.01M pl. c 10 S 3 T 3 U 3 1 Moments d Figure 5.1: Absolute values of the moments for the step potential plotted against parameter d for fixed c =

96 < Ζ3 > Σ d c 0.00 hζ 3 i Figure 5.: Moment S3 = σ4 for the step potential plotted against parameters c and d showing that this moment peeks when the step is sharpest. the moments obtain high values because of the step support region near the upper integration limit kmax = 5Mpl. Inflationary Model 1 m φ Potential λφ4 Potential Stepc=0.01, d=0.01mpl Stepc=0.00, d=0.0mpl S T U Table 5.1: Moments for three models of inflation calculated with integration limits kmin = 0.005Mpl and kmax = 1Mpl for quadratic and quartic potential and kmax = 5Mpl for step potential. Note these values of moments are accurate with in 5 percent error bars. 86

97 S 3 T 3 U 3 Moments s Figure 5.3: Moments for the step potential plotted against step location φ s for fixed d = 0.01M pl and c =

98 Chapter 6 Geometrical Statistics and Minkowski Functionals 6.1 Three Dimensional Statistics Many geometrical statistics, including Minkowski functionals, of the mildly non-gaussian fields can be expressed as series expansion in higher-order moments of the field and its derivatives with powers of σ controlling the expansion order [48]. First order non-gaussian corrections are linear in σ and defined by cubic normalized moments that we have studied earlier. For the models with nearly flat power spectrum that we consider, we adopt the value of the variance σ P ζ ln(k max /k min ) with P ζ (k 0 ) = suggested by Planck data [9] and k max /k min = Filling Factor The simplest geometrical statistic or Minkowski functional is the filling factor f V = dζp (ζ), νσ i.e. the fraction of volume above the threshold ν. Its moment expansion gives f V (ν) = 1 Erfc ( ν ) + e ν σh (ν) S 3 π 6 + O(σ ). (6.1) The first non-gaussian correction only depends on S 3 multiplied by Hermite polynomial of order two. Thus, linear in σ non-gaussian part of f V (ν) has a global shape that is independent 88

99 of any model while its magnitude will depend on the magnitude of moment S 3. Thus for the step potential, the filling factor can be as large as 500 times the f V for quadratic potential. In Fig. 6.1 a single curve shows the non-gaussian part of the filling factor both for the model with quadratic potential with f V correction of order O(10 7 ) (as labeled on the left vertical axis) and for the step potential for which it is O(10 5 ) (the right vertical axis) f V Ν Ν Figure 6.1: Non gaussian part of filling factor f V as the function of threshold ν. Values on the left vertical axis are for quadratic potential while values on the right axis are for step potential with c = 0.01 and d = 0.01M pl. There are several advantages to use the value of the filling factor f V instead of ν as a variable in which to express all other statistics. Indeed, the fraction of volume occupied by a data set is often available even from the limited data, whereas specifying ν requires prior knowledge of the variance σ which may not be easily obtainable. In some cases non-gaussian analysis itself gives more robust way to determine the variance. Following [77] we introduce the effective threshold ν f Erfc 1 (f V ) be used as an observable alternative to ν. To the first order correction in σ we have the relation ν = ν f + σ S 3 6 H (ν f ). (6.) 89

100 6.1. Area of Isodensity Contours Another Minkowski functional in 3D is the area (per unit volume) of isodensity contours N 3 (ν). Up to first non-gaussian contribution, this quantity is expressed in terms of third order moments as N 3 (ν) e ν ( σ 1 S3 [1 + σ 3πσ 6 H 3(ν) + T ) ] 3 H 1(ν), (6.3) depending on moments S 3 and T 3. In Fig. 6. we show the non-gaussian N 3 (ν) for quadratic and step potential. Besides vastly different amplitudes, it exhibits different shape that distinguishes the two models. For quadratic potential H 3 (ν) contribution is small and N 3 has just two measurable extrema with small secondary ones near the edges, while due to the large value of S 3 the H 3 (ν) part is prominent for the step potential, giving rise to four distinct extrema, as seen in Fig N3 Ν Figure 6.: Non gaussian part of area of isodensity contours N 3 (ν) for quadratic potential in black with values on left vertical axis as a function of threshold ν. Similarly, N 3 (ν) for step potential, for c = 0.01 and d = 0.01M pl, is show in red(lighter colour) with values on right vertical axis as a function of threshold ν. 90

101 As a function of ν f, i.e of the filling factor, the area of isodensity contours is ν f ( N 3 (ν f ) e σ 1 [1 σ S 3 + T ) ] 3 H 1 (ν f ), (6.4) 3πσ 4 which demonstrates the general outcome of eliminating the highest order Hermite polynomial term when switching from ν to ν f. Because of that, in Fig. 6.3, the area of isodensity contours N 3 (ν f ) has a model independent shape that is proportional to e ν f H 1 (ν f ). The N 3 (ν f ) for the step potential has values of order 10 3 on right axis while for quadratic potential it has values of order 10 5 on the left axis N3 Ν f Figure 6.3: Non gaussian part of area of isodensity contours N 3 (ν f ) for quadratic potential in black with values on left vertical axis as a function of filling factor ν f. Similarly, N 3 (ν f ) for step potential, for c = 0.01 and d = 0.01M pl, in red(lighter colour) with values on right vertical axis as a function of filling factor ν f Euler Characteristic or Genus An important Minkowski functional in cosmology is the Euler characteristic (or genus) that is used to characterize the topology of the isocontours of random fields. The excursion set is part of the space where field value exceeds the threshold. In D this area will have, in 91

102 general, some isolated island, and large continents with inner lakes. The definition generally used by cosmologists is that the genus is the number of holes minus the number of isolated regions above a threshold ν in a random field (thus, for one isolated region, the genus is just the number of holes minus one), while Euler characteristic of the excursion set is just minus genus, see Eq χ 3D (ν) 1 (π) ( ) [ 3 ( σ1 e ν S3 H (ν) + σ 3σ 6 H 5(ν) 3T 3 4 H 3(ν) 9U ) ] 3 4 H 1(ν) (6.5) Euler characteristic density can be also considered as a function of filling-factor-deduced threshold ν f rather than the threshold ν. χ 3D (ν f ) 1 (π) ( σ1 3σ ) 3 e ν f / [H (ν f ) ( σ S ) ( 4 T 3 H 3 (ν f ) σ S ) 4 U 3 H 1 (ν f ) ] (6.6) We have calculated Euler characteristic χ 3D of the 3D perturbation field ζ for two models of inflation, the quadratic inflation and step potential model. For quadratic potential, the non-gaussian part of χ 3D as a function of threshold ν is plotted in Fig showing the small amplitude of the non-gaussianity of order 10 7 and the presence of all three H 1, H 3, H 5 harmonics. Euler characteristic as a function of ν f is given in Fig For the quadratic inflation S 3 + 3T 3 / is notably smaller than S 3 + 9U 3 / , hence the result is dominated by the H 1 term that has only one zero crossing at origin as can be seen in Fig While the O(10 7 ) non-gaussian correction to Euler characteristic for the quadratic potential is expectingly small and is hardly observable, for the step potential, it will have the magnitude 10 to 500 times larger. Fig. 6.6 shows χ 3D (ν) curves for two different sets of c and d parameters, with magnitude of the effect differing by an order of magnitude, namely O(10 5 ) for c = and d = 0.0M pl and O(10 4 ) for c = 0.01 and d = 0.01M pl. At the same time the shapes of the Euler characteristic curves are very similar dominated by S 3 H 5 (ν) term. 9

103 Χ 3 D Ν Figure 6.4: Non gaussian part of Euler characteristic χ 3D as a function of threshold ν for the quadratic potential. The shaded region shows the error bars coming from the error in calculation of moments that are roughly 5 percent. Ν Χ 3 D Ν f Figure 6.5: Non gaussian part of genus as a function of filling factor ν f for the quadratic potential. The shaded region shows the error bars coming from the error in calculation of moments that are roughly 5 percent. 93

104 Χ3 D Ν Ν Figure 6.6: Non gaussian part of genus as a function of threshold ν for the step potential for two parameter sets (c = , d = 0.0M pl in black on left axis) and (c = 0.01, d = 0.01M pl in red, lighter colour, on right axis). This can be see explicitly also when measuring Euler characteristic as a function of ν f as shown in Fig With S 3 dominant for all the values of step parameters c and d studied, the non-gaussian part of Euler characteristic exhibits a universal form χ 3D σs 3 (H 1 (ν f ) + H 3 (ν f )) that is a distinguishing signature of this particular model of non- Gaussianity. 6. Two Dimensional Statistics and CMB The CMB maps are two dimensional maps of the last scattering surface of primordial photons that traversed the Universe largely free after they decoupled from matter 375,000 years after the end of inflation. The CMB is fundamental in the study of physical cosmology as it is the oldest observable in the universe. This signal is observed as a faint thermal radiation background of.75k [13] that is almost the same in every direction in the sky with tiny temperature and polarization fluctuations. The accidental discovery of CMB in 1964 by Arno Penzias and Robert Wilson was accepted as landmark test for Big Bang model of the 94

105 Χ3 D Ν f Figure 6.7: Non gaussian part of genus as a function of filling factor ν f for the step potential for two parameter sets (c = , d = 0.0M pl in black on left axis) and (c = 0.01, d = 0.01M pl in red, lighter colour, on right axis). universe. The tiny temperature fluctuations in the CMB mark the imprints of inflation in the early universe. Adiabatic scalar mode of these fluctuations are generated by the quantum fluctuations of initial perturbation field ζ. Thus, we will study how these quantum perturbations get imposed on the CMB maps as seen in Fig In two dimensional map of the sky, the temperature fluctuations can be related to ζ via k-mode integral T T (θ, ϕ) = d 3 k (π) 3 ζ kθ(k, ˆk ˆn, τ rec, τ 0 ) (6.7) where the transfer function Θ(k, ˆk ˆn, τ rec, τ 0 ) describes how observed temperature of the photons coming from the direction ˆn = (θ, φ) was formed from earlier times to the moment of recombination τ rec and subsequently modified to the present moment of observation τ 0. As a function on a sky sphere, T (θ, ϕ) can be expanded in terms of spherical harmonics T as follows T T (θ, ϕ) = lm a lm Y lm (θ, ϕ) (6.8) 95

106 Figure 6.8: CMB temperature map form Plank 015 [13]. where alm expansion coefficients are random quantities1 Z d3 k = ζk dωn Θ(k, k n, τrec, τ0 )Ylm (n ) (π)3 Z d3 k ζk Θl (k, τrec, τ0 ) (4π) Ylm (k ). = (π)3 Z alm (6.9) The variance these coefficients defines the power spectrum of the temperature fluctuations halm a l0 m0 i where ζ (k) = Z 3 0 d3 k dk = hζk ζk 0 i Θl (k)θl0 (k 0 )(4π) Ylm (k )Yl0 m0 (k 0 ) 3 (π) (π)3 Z d3 k (k)θl (k, τrec, τ0 )(4π) Ylm (k )Yl0 m0 (k ) = (π)3 ζ Z = k dk ζ (k)θl (k, τrec, τ0 )δmm0 δll0 = Cl δmm0 δll0 π π P (k) k3 ζ Z (6.10) is the scalar power spectrum, defined in Eq. (3.8), for ζ perturba- tions. We used the identities in Eqs. (E.3-E.8) in above derivation. Thus Cl can be written 1 In this derivation we have used the definitions given in Eqs. E.8-E.10 that different from the standard definition that contains a factor il as used in []. Due to this definition in Eq. E.8 an extra factor of il appear in Eq