PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

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CHAPTER 4. BI-SPECTRA ASSOCIATED WITH LOCAL AND NON-LOCAL FEATURES 1e-15 0.1 1e-16 0.01 1e-17 0.001 1e-18 0.0001 k 6 |Gn(k)| 1e-19 1e-11 f eq NL 1e-05 1 1e-12 1e-13 0.1 1e-14 0.01 1e-15 1e-16 1e-05 0.0001 0.001 0.01 k 0.001 1e-05 0.0001 0.001 0.01 k Figure 4.4: The quantities k 6 times the absolute values of the non-zero contributions in the power law case, viz.G 1 +G 3 ,G 2 andG 5 +G 6 , obtained numerically, have been plotted on the left for two different values of γ (γ = −2.02 on top and γ = −2.25 below), as solid curves with the same choice of colors to represent the different quantities as in the previous two figures. Note that we have followed the same color scheme to represent the differential quantities as in the previous two figures. The dots on these curves are the spectral shape arrived at from the analytical arguments, with amplitudes chosen to match the numerical results at a specific wavenumber. The dots of a different color on the solid purple curves represents G 5 + G 6 obtained from its relation to G 1 + G 3 discussed in the text. The plots on the right are the non-Gaussianity parameter f eq associated with NL the different contributions, arrived at using the numerical code. Note that, as indicated by the analytical arguments, the quantityf eq corresponding to all the contributions turns NL out to be strictly scale invariant for both values ofγ. above analytically. Let us now turn to the Starobinsky model. As we have discussed earlier, in this case, the change in the slope causes a brief period of fast roll which leads to sharp features in the scalar power spectrum (as we had illustrated in Figure 4.1). It was known that, for certain range of parameters, one could evaluate the scalar power spectrum analytically 78
4.3. THE NUMERICAL COMPUTATION OF THE SCALAR BI-SPECTRUM in the Starobinsky model, which matches the actual, numerically computed spectrum exceptionally well [71, 54]. Interestingly, it has been recently shown that, in the equilateral limit, the model allows the analytic evaluation of the scalar bi-spectrum too (see Ref. [54]; in this context, also see Refs. [55]) In Figure 4.5, we have plotted the numerical as well as the analytical results for the functionsG 1 +G 3 ,G 2 ,G 4 +G 7 , andG 5 +G 6 for the Starobinsky model. We have plotted for parameters of the model for which the analytical results are considered to be a good approximation [54]. It is evident from the figure that the numerical results match the analytical ones very well. Importantly, the agreement proves to be excellent in the case of the dominant contribution G 4 +G 7 . A couple of points concerning concerning the numerical results in the case of the Starobinsky model (both in Figure 4.1 wherein we have plotted the power spectrum as well as in Figure 4.5 above containing the bi-spectrum) require some clarification. The derivatives of the potential (4.3) evidently contain discontinuity. These discontinuities needs to be smoothened in order for the problem to be numerically tractable. The spectra and the bi-spectra in the Starobinsky model we have illustrated have been computed with a suitable smoothing of the discontinuity, while at the same time retaining a sufficient level of sharpness so that they closely correspond to the analytical results that have been arrived at [54, 55]. 79
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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
Page 51 and 52: 2.1. THE INFLATIONARY MODELS OF INT
Page 53 and 54: 2.2. METHODOLOGY, DATASETS, AND PRI
Page 55 and 56: 2.2. METHODOLOGY, DATASETS, AND PRI
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
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 101: 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 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
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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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