PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

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CHAPTER 5. THE SCALAR BI-SPECTRUM DURING PREHEATING whereγ(b,x) is the incomplete Gamma function (see, for example, Refs. [115, 116]), while the quantity C is a dimensionless constant of integration. Then, the relation between the incomplete Gamma and the Gamma functions allows us to express the function K(t) as follows: K(t) ≃ t e 4a 2 e ∫ t [ d¯t 6 cos 2 (m¯t+∆) 5 ( tē t ) 1/3 ( ) ( ) 2 5 +e 2i∆ (−2imt e ) −5/3 tē Γ 3 t ( ) 4/3 ( ) ( ) 2 −e 2i(m¯t+∆) (−2imt e ) −1 tē 5 +e −2i∆ (2imt e ) −5/3 tē Γ t 3 t ( ) 4/3 ( ) ] 2 −e −2i(m¯t+∆) (2imt e ) −1 tē tē +C t t ≃ 3t4/3 e 10a 2 e ∫ t d¯t ¯t −1/3 +··· . (5.23) cos 2 (m¯t+∆) In arriving at the final equality, we have used the asymptotic property of the incomplete Gamma function [115, 116] and have retained only the dominant term in inverse power ofmt. The final expression above can be integrated by parts to arrive at ∫ t K(t) ≃ 3t4/3 e t −1/3 tan(mt+∆)+ t4/3 e d¯t ¯t −4/3 tan(m¯t+∆). (5.24) 10ma 2 e 10ma 2 e The second term containing the integral in this expression is of the order of the other terms that we have already neglected and, hence, it too can be ignored. As a result, the growing mode of the curvature perturbation can be written as [ ] f k ≃ A k 1− 3 k 2 t 4/3 e 10 a 2 e mt tan(mt+∆) 1/3 [ = A k 1− 1 ( ) ] 2 k H 5 aH m tan(mt+∆) , (5.25) in perfect agreement with the result that has been obtained recently in the literature [64]. It is evident from the above expression that the evolution of the curvature perturbation will contain sharp spikes during preheating, a feature that is clearly visible in Figure 5.3 wherein we have plotted the above analytical expression as well as the corresponding numerical result (in this context, also see Figure 4 in the first reference in Refs. [63], where the spikes are also clearly visible). It is important that we make a couple of remarks concerning the appearance of the spikes in the evolution of the curvature perturbation. Firstly, as the spikes are encountered both analytically and numerically, evidently, they are not artifacts of the adopted 96
5.1. BEHAVIOR OF BACKGROUND AND PERTURBATIONS DURING PREHEATING Figure 5.3: The behavior of the curvature perturbation during preheating. The blue curve denotes the numerical result, while the dashed red curve represents the analytical solution (5.25). We have chosen to work with a very small scale mode k that leaves the Hubble radius at about two e-folds before the end of inflation. We have made use of the same value of ∆ as in the previous two figures and we have fixed A k [cf. Eq. (5.25)] by choosing it to be the numerical value of the curvature perturbation at a suitable time close to the end of inflation. It is clear that the agreement between the analytical and the numerical results is quite good. 97
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Primordial features and non-Gaussia
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Declaration This thesis is a presen
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Acknowledgments According to a popu
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Abstract Currently, inflation is th
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ABSTRACT tensors prove to be small
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ABSTRACT the best fit values arrive
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Contents 1 Introduction 1 1.1 The c
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CONTENTS 6 Effects of primordial fe
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List of Figures 1.1 A schematic tim
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Chapter 1 Introduction At a first g
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2.725K. It is also found to be high
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1.1. THE CONCORDANT, BACKGROUND COS
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1.2. THE INFLATIONARY PARADIGM 1.2
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1.2. THE INFLATIONARY PARADIGM log
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1.2. THE INFLATIONARY PARADIGM wher
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1.2. THE INFLATIONARY PARADIGM wher
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1.2. THE INFLATIONARY PARADIGM The
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1.2. THE INFLATIONARY PARADIGM yet
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1.3. THE GENERATION AND IMPRINTS OF
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1.5. THE FORMATION OF THE LARGE SCA
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1.7. NOTATIONS AND CONVENTIONS resu
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Chapter 2 Generation of localized f
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2.1. THE INFLATIONARY MODELS OF INT
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2.2. METHODOLOGY, DATASETS, AND PRI
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2.2. METHODOLOGY, DATASETS, AND PRI
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2.3. BOUNDS ON THE PARAMETERS 2.3 B
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2.3. BOUNDS ON THE PARAMETERS Model
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2.4. THE SCALAR AND THE CMB ANGULAR
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2.4. THE SCALAR AND THE CMB ANGULAR
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2.5. DISCUSSION by the canonical sc
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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
Page 83 and 84: 4.1. MODELS OF INTEREST AND POWER S
Page 85 and 86: 4.1. MODELS OF INTEREST AND POWER S
Page 87 and 88: 4.2. THE SCALAR BI-SPECTRUM IN THE
Page 89 and 90: 4.3. THE NUMERICAL COMPUTATION OF T
Page 105 and 106: 4.4. RESULTS IN THE EQUILATERAL LIM
Page 107 and 108: 4.4. RESULTS IN THE EQUILATERAL LIM
Page 109 and 110: Chapter 5 The scalar bi-spectrum du
Page 111 and 112: 5.1. BEHAVIOR OF BACKGROUND AND PER
Page 119: 5.1. BEHAVIOR OF BACKGROUND AND PER
Page 123 and 124: 5.2. THE CONTRIBUTIONS TO THE BI-SP
Page 129 and 130: 5.3. DISCUSSION significant during
Page 131 and 132: Chapter 6 Effects of primordial fea
Page 133 and 134: 6.1. MODELS AND COMPARISON WITH THE
Page 135 and 136: 6.2. FROM THE PRIMORDIAL SPECTRUM T
Page 137 and 138: 6.3. THE NUMERICAL METHODS: ESSENTI
Page 139 and 140: 6.4. RESULTS Model Quadratic Quadra
Page 141 and 142: 6.4. RESULTS 6e-09 Quadratic potent
Page 143 and 144: 6.4. RESULTS 30 4 20 Change in R Ch
Page 145 and 146: 6.5. DISCUSSION wherein one works w
Page 147 and 148: Chapter 7 Imprints of primordial no
Page 149 and 150: 7.1. FORMALISM and the LSS studies.
Page 151 and 152: 7.1. FORMALISM only mildly non-line
Page 153 and 154: 7.1. FORMALISM If the bi-spectrum c
Page 155 and 156: 7.2. RESULTS 0.12 0.13 2e3 , 2.2e3
Page 157 and 158: 7.3. CONCLUSIONS 7.3 Conclusions To
Page 159 and 160: Chapter 8 Summary and outlook In th
Page 161 and 162: 8.2. OUTLOOK factorily [69, 70]. Ra
Page 163 and 164: Bibliography [1] See,http://magnum.
Page 165 and 166: BIBLIOGRAPHY [27] See,http://cfht.h
Page 167 and 168: BIBLIOGRAPHY [55] F. Arroja, A. E.
Page 169 and 170: BIBLIOGRAPHY [77] R. K. Jain, P. Ch
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BIBLIOGRAPHY [103] R. Allahverdi, J
Page 173 and 174:
BIBLIOGRAPHY [132] S. Sasaki, Publ.
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PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

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