Delta N Formalism In Cosmological Perturbation Theory

by Ali Akbar Abolhasani, Hassan Firouzjahi, Atsushi Naruko, Misao Sasaki

Overview

Early Universe cosmology is an active area of research and cosmic inflation is a pillar of modern cosmology. Among predictions of inflation, observationally the most important one is the generation of cosmological perturbations from quantum vacuum fluctuations that source all inhomogeneous structures in the Universe, not to mention the large-scale structures such as clusters of galaxies.Cosmological perturbation theory is the basic tool to study the perturbations generated from inflation. There are a few different approaches to primordial cosmological perturbations. In the conventional approach one perturbs the field equations and after quantizing the perturbations by the use of the corresponding action, one calculates the power spectrum of cosmological observables. This approach extends to higher order perturbations such as bispectrum etc., but the analysis becomes increasingly difficult.The delta N formalism, the topic of this book, is an alternative approach. The novelty of this approach is that, under the condition that the scale of interest is very large so that the spatial derivatives may be ignored in the dynamics, it can be applied to all orders in perturbation theory and has a rigorous foundation in general relativity. Thanks to the fact that one can evaluate perturbations with only the knowledge of background solutions, it is proved to be much easier than the conventional approach in evaluating higher order effects in many cases.

Product Details

ISBN-13:	9789813238756
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	01/30/2019
Pages:	184
Product dimensions:	6.00(w) x 9.00(h) x 0.44(d)

Table of Contents

Chapter 1 Introduction 01

Chapter 2 Basic formulation of δN formalism 09

2.1 Preliminary 09

2.1.1 Linear perturbation theory 09

2.1.2 Single field slow-roll inflation 15

2.2 δN formalism in linear perturbation theory 20

2.2.1 δN formalism in slow-roll inflation 20

2.2.2 δN formalism in multi-field inflation beyond slow-roll 29

2.3 Non-linear δN formalism 40

2.3.1 The Einstein equations 40

2.3.2 Gradient expansion 42

2.3.3 Leading order in gradient expansion 45

2.3.4 Curvature perturbation and non-linear δN formula 48

2.4 Statistical quantities 52

2.4.1 Power spectrum in cosmological perturbation theory 52

2.4.2 Power spectrum and spectral index in δN formalism 55

2.4.3 Non-Gaussianities 58

2.4.4 Implementation of δN formalism for non-Gaussianity 62

Chapter 3 Application of δN formalism: Warm-up studies 67

3.1 A specific model: Chaotic inflation 67

3.2 Curvaton model 69

3.2.1 Curvature perturbation in curvaton scenario 72

3.2.2 Spectrum and bispectrurn in curvaton scenario 74

Chapter 4 Application of δN formalism: Multi-brid inflation 79

4.1 The exact soluble class 79

4.2 Multi-brid inflation 84

4.3 The power spectrum and the bispectrum in multi-brid scenario 90

Chapter 5 Application of δN formalism: Non-attractor inflation 95

5.1 Motivation for non-attractor inflation 95

5.2 Non-attractor background 97

5.3 Power spectrum for non-attractor background 103

5.4 δN Formalism in non-attractor backgrounds 107

5.4.1 The case with c_s = 1 108

Chapter 6 Application of δN formalism: Inflation with local features 115

6.1 Motivation 115

6.2 The model 117

6.2.1 Dynamics of inflaton 119

6.2.2 Dynamics of waterfall field 122

6.3 δN formalism in models with localized feature 128

6.4 Power spectrum with localized feature 132

6.4.1 Contribution of inflaton to power spectrum 133

6.4.2 Contribution of the waterfall field to power spectrum 133

6.4.3 Total curvature perturbation power spectrum 134

Appendix A δN for general c_s in non-attractor background 139

Appendix B Variance of δΧ fluctuations 145

Appendix C Correlation functions of δΧ² 149

Appendix D Bispectrum with localized feature 153

D.1 Dynamically generated non-Gaussianities 154

D.2 Bispectrum from intrinsic non-Gaussianity 159

D.3 Total f_NL parameter 160

Appendix E δN up to ΔΧ⁴ 165

Bibliography 167