PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

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CHAPTER 1. INTRODUCTION 1.3 The generation and imprints of non-Gaussianities So far, we have discussed the generation of correlations during inflation and the comparison of the theoretical predictions with the observational evidence at the level of the two point function, viz. the primordial and the CMB angular power spectra. Over the last decade, there has been frenetic activity towards gaining a theoretical understanding of the amplitude and the form of the three and the four point functions that are produced during inflation. In particular, a large fraction of these investigations have been dedicated to analyzing inflationary and post-inflationary dynamics that lead to deviations from Gaussianity (see Refs. [49, 50, 51, 52, 53, 54, 55]; for earlier efforts in this direction, see Refs. [56]). At the same time, there has also been a constant effort to arrive at increasingly tighter constraints on the actual extent of primordial non-Gaussianities observed in the CMB and the LSS data [57, 58]. It is now commonly recognized that non-Gaussianities, if detected, can play a significant role in helping us discriminate between the various inflationary as well as the post-inflationary scenarios. In a linear theory described by a quadratic action, when one works with vacuum initial states, the correlations prove to be Gaussian in nature. It is the non-linearities in the action governing the variable of interest that leads to deviations from Gaussianities in the correlation functions. The simplest signature of non-Gaussianity is a non-zero three point function, which, in a given theory, will be determined by the cubic terms in the action. Then, evidently, the first step towards evaluating the three point correlation functions generated in an inflationary model would be to arrive at the action describing the scalar and tensor perturbations at the next higher, i.e. the cubic, order [49, 51]. Having obtained such an action, one can make use of the standard procedures of quantum field theory to arrive at the corresponding scalar and tensor bi-spectra, viz. the three point functions in Fourier space, as well as cross-correlations between the scalars and the tensors, corresponding to the cubic order action. Based on these arguments, there now exists a standard formalism, originally due to Maldacena [49], to evaluate the bi-spectrum generated during inflation (for more details, see Chapter 4). The scalar bi-spectrum, say, B S (k 1 ,k 2 ,k 3 ), associated with the modes ˆR k of the curvature perturbation is defined as 〈 ˆR k1 ˆRk2 ˆRk3 〉 = (2π) 3 B S (k 1 ,k 2 ,k 3 ) δ (3) (k 1 +k 2 +k 3 ). (1.26) Ideally, one would have liked to compute the CMB angular bi-spectrum corresponding to the primordial bi-spectrum generated during inflation and compare with the observa- 18
1.3. THE GENERATION AND IMPRINTS OF NON-GAUSSIANITIES tions, as is done in the case of the power spectra. However, some of the issues towards carrying out such comparison still remain to be resolved (in this context, see, for instance, Ref. [59]). Further, efficient computational tools that will be required for such an analysis are yet to be thoroughly developed. As a result, for convenience in characterizing the observations on the one hand and the theoretical models on the other, a dimensionless parameter f NL is often introduced to reflect the amplitude of the deviations from Gaussianity in the curvature perturbation through the relation [57] R = R G − 3f NL 5 ( R 2 G −〈 R 2 G〉) , (1.27) where R G denotes the Gaussian quantity, and the factor of 3/5 arises due to the relation between the Bardeen potential and the curvature perturbation during the matter dominated epoch. Upon making use of the corresponding relation between R and R G in Fourier space and the Wick’s theorem, which applies to Gaussian distributions, one obtains that [49, 51, 52] 〈 ˆR k1 ˆRk2 ˆRk3 〉 = − 3f NL 10 (2π)4 (2π) −3/2 1 δ (3) (k k1 3 1 +k 2 +k 3 ) k3 2 k3 3 × [ k1 3 P S (k 2 ) P S (k 3 )+two permutations ] . (1.28) This expression can then be utilized to arrive at the following relation between the non- Gaussianity parameterf NL (k 1 ,k 2 ,k 3 ) and the scalar bi-spectrum B S (k 1 ,k 2 ,k 3 ) [53, 54]: f NL (k 1 ,k 2 ,k 3 ) = − 10 3 (2π)−4 (2π) 9/2 k 3 1 k3 2 k3 3 B S (k 1,k 2 ,k 3 ) × [ k 3 1 P S (k 2) P S (k 3 )+two permutations ] −1 , (1.29) which suggests that the non-Gaussianity parameter is, in fact, a suitable ratio of the scalar bi-spectrum to the corresponding power spectrum. Note that the presence of the delta-function in the definition (1.26) of the bi-spectrum B S (k 1 ,k 2 ,k 3 ) implies that the wavevectors k 1 , k 2 and k 3 have to constitute a triangle. The present observational constraints on the non-Gaussianity parameter f NL (k 1 ,k 2 ,k 3 ) are often quoted in the equilateral (i.e. when k 1 = k 2 = k 3 ) and the squeezed (or the local) limits (i.e. when, say, k 1 = k 2 ≫ k 3 ). In the equilateral limit, analysis of the WMAP seven-year data indicates that f NL = 26 ± 140, whereas in the squeezed limit one has f NL = 32 ± 21, with the errors denoting the 1-σ deviations from the mean values [19, 57, 58]. It is worth mentioning that, while a Gaussian distribution lies within 2-σ, the mean values seem to suggest relatively large levels of non-Gaussiantiy. For instance, slow 19
Page 1: Primordial features and non-Gaussia
Page 5: Declaration This thesis is a presen
Page 9 and 10: Acknowledgments According to a popu
Page 11 and 12: Abstract Currently, inflation is th
Page 13 and 14: ABSTRACT tensors prove to be small
Page 15 and 16: ABSTRACT the best fit values arrive
Page 17 and 18: Contents 1 Introduction 1 1.1 The c
Page 19: CONTENTS 6 Effects of primordial fe
Page 23 and 24: List of Figures 1.1 A schematic tim
Page 25 and 26: Chapter 1 Introduction At a first g
Page 27 and 28: 2.725K. It is also found to be high
Page 29 and 30: 1.1. THE CONCORDANT, BACKGROUND COS
Page 31 and 32: 1.2. THE INFLATIONARY PARADIGM 1.2
Page 33 and 34: 1.2. THE INFLATIONARY PARADIGM log
Page 35 and 36: 1.2. THE INFLATIONARY PARADIGM wher
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Page 39 and 40: 1.2. THE INFLATIONARY PARADIGM The
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Page 45 and 46: 1.5. THE FORMATION OF THE LARGE SCA
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Page 49 and 50: Chapter 2 Generation of localized f
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Page 53 and 54: 2.2. METHODOLOGY, DATASETS, AND PRI
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Page 57 and 58: 2.3. BOUNDS ON THE PARAMETERS 2.3 B
Page 59 and 60: 2.3. BOUNDS ON THE PARAMETERS Model
Page 61 and 62: 2.4. THE SCALAR AND THE CMB ANGULAR
Page 63 and 64: 2.4. THE SCALAR AND THE CMB ANGULAR
Page 65 and 66: 2.5. DISCUSSION by the canonical sc
Page 67 and 68: Chapter 3 Non-local features in the
Page 69 and 70: 3.1. MODELS AND METHODOLOGY on the
Page 71 and 72: 3.1. MODELS AND METHODOLOGY values
Page 73 and 74: 3.2. RESULTS Datasets WMAP-7 WMAP-7
Page 75 and 76: 3.2. RESULTS 0.2 0.15 0.1 ∆ χ 2
Page 77 and 78: 3.2. RESULTS 7000 150 6000 100 50 l
Page 79 and 80: 3.3. DISCUSSION WMAP as well as the
Page 81 and 82: Chapter 4 Bi-spectra associated wit
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Page 85 and 86: 4.1. MODELS OF INTEREST AND POWER S
Page 87 and 88: 4.2. THE SCALAR BI-SPECTRUM IN THE
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4.3. THE NUMERICAL COMPUTATION OF T
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4.4. RESULTS IN THE EQUILATERAL LIM
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4.4. RESULTS IN THE EQUILATERAL LIM
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Chapter 5 The scalar bi-spectrum du
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5.1. BEHAVIOR OF BACKGROUND AND PER
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5.2. THE CONTRIBUTIONS TO THE BI-SP
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5.3. DISCUSSION significant during
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Chapter 6 Effects of primordial fea
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6.1. MODELS AND COMPARISON WITH THE
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6.2. FROM THE PRIMORDIAL SPECTRUM T
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6.3. THE NUMERICAL METHODS: ESSENTI
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6.4. RESULTS Model Quadratic Quadra
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6.4. RESULTS 6e-09 Quadratic potent
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6.4. RESULTS 30 4 20 Change in R Ch
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6.5. DISCUSSION wherein one works w
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Chapter 7 Imprints of primordial no
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7.1. FORMALISM and the LSS studies.
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7.1. FORMALISM only mildly non-line
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7.1. FORMALISM If the bi-spectrum c
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7.2. RESULTS 0.12 0.13 2e3 , 2.2e3
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7.3. CONCLUSIONS 7.3 Conclusions To
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Chapter 8 Summary and outlook In th
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8.2. OUTLOOK factorily [69, 70]. Ra
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Bibliography [1] See,http://magnum.
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BIBLIOGRAPHY [27] See,http://cfht.h
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BIBLIOGRAPHY [55] F. Arroja, A. E.
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BIBLIOGRAPHY [77] R. K. Jain, P. Ch
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BIBLIOGRAPHY [103] R. Allahverdi, J
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BIBLIOGRAPHY [132] S. Sasaki, Publ.
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PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

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