Delta N Formalism in Cosmological Perturbation Theory

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

• Publisher: WSPC
• eBook ISBN: 9789813238770
• Topic: Physical Sciences
• Subtopic: Astronomy & Astrophysics

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 fluc

CHAPTER 1

Introduction

Thanks to tremendous progresses in observational cosmology during about past two decades, see e.g. Refs. [1–7], the theory of primordial inflation has become a widely accepted, almost standard model of the early universe. Inflation [8–10] was originally proposed to solve a couple of fundamental problems associated with the standard Big Bang cosmology, namely the flatness problem that the universe is spatially so flat while the Einstein equation alone naturally suggests a highly curved universe, and the horizon problem that the universe appears so homogeneous and isotropic on large scales while there were potentially an extremely large number of causally disconnected regidons in the early universe over such large scales. Through various studies, it has become clear that qualitatively inflation can be regarded as a solution to these problems, but not in the sense of quantitative, testable predictions.

It has turned out that among predictions of inflation, observationally the most important one is the generation of cosmological perturbations from quantum vacuum fluctuations that seed all large scale structures and inhomogeneities in the universe. During inflation, the quantum vacuum fluctuations of a scalar field which drives inflation, the inflaton, or those of any light scalar fields, on very small scales are stretched to very large scales due to the accelerated expansion of the universe and behave as classical fluctuations when the wavelength exceeds the Hubble horizon size. One of them or a combination of them gives rise to the curvature perturbation, namely, perturbations in the spatial curvature of the universe around the homogeneous and isotropic background Friedmann–Lemaître–Robertson–Walker (FLRW) universe. The curvature perturbation leaves a distinct imprint on the temperature anisotropy of the cosmic microwave background (CMB), and eventually develops into seed density perturbations for the large scale structure (LSS) of the universe such as galaxies and clusters of galaxies.

In this situation, it is very important to quantify theoretical predictions of various inflationary models in detail, and test them against the observed CMB anisotropy data and against various observations of the large scale structure of the universe. For this purpose, the relativistic cosmological perturbation theory plays the central role. In particular, in the leading order, linear perturbation theory has turned out to be extremely powerful. Nevertheless, in the last decade much attention has been paid to the non-linear nature encoded in the primordial perturbations since it is now within the scope of observations. In particular, such non-linearities generate deviations from the Gaussian statistics, that is, non-Gaussianities even if primordial fluctuations originate from vacuum fluctuations with the Gaussian statistics [11]. As a result, any detection of primordial non-Gaussianities through the statistics of CMB anisotropy or distributions of LSS may give a valuable clue to decipher the detailed mechanism of primordial inflation, e.g. non-linear interaction among fields.

The simplest class of inflationary models is the single-field slow-roll model, in which a single scalar field, the inflaton field ϕ, is responsible for both driving inflation and generating the curvature perturbations. One of the most remarkable fact of primordial fluctuations from single-field inflation is that one of the modes of the spatial curvature perturbation on the velocity-orthogonal hypersurface (misleadingly but commonly dubbed as “comoving” hypersurface), namely the hypersurface on which the scalar field is homogeneous, is conserved after the mode exits the Hubble horizon [12, 13].

To be specific, the spatial curvature on the comoving hypersurface at linear order may be expressed as

where R_c is the comoving curvature perturbation and a is the cosmological scale factor. It is known that R_c satisfies the following second order differential equation,

in which the quantity z is related to the inflaton field ϕ via z ≡ a(ϕ′/H) = a(

/H). Here ′ denotes a derivative with respect to the conformal time, dη = a⁻¹dt while t represents the cosmic time and H = a′/a is the conformal Hubble parameter related to the standard Hubble parameter H via H = aH. On sufficiently large scales, one can safely neglect the last term in Eq. (1.2) which involves spatial gradients and hence R′_c ∼ 0 becomes one of the solutions of this equation. This conserved mode is called the adiabatic growing mode (for a reason we will explain later).

Furthermore, under the slow-roll approximation, z essentially behaves as the scale factor a and then the other solution gives R′_c ∝ z⁻² ∼ a⁻² which soon becomes negligible once the mode is outside the Hubble horizon scale during inflation. So the comoving curvature perturbation is indeed conserved on superhorizon scales.¹ This fact leads to another important consequence. In principle, there are two dynamical degrees of freedom in single-field inflation since the scalar field satisfies a second order differential equation, and the two physical degrees of freedom are imprinted in the field value ϕ and its time derivative

at a given initial time. However, on large scales, one of these modes decays out and only one dynamical degree of freedom survives. Reflecting this fact, one finds the coincidence of several time slicings on large scales.

One can always consider a slicing where the energy density of the universe is spatially homogeneous, which is called the uniform energy density slicing. Since the energy density ρ of the scalar field is given by the sum of the potential and kinetic energies, its value can be determined by specifying both ϕ and

. However, it turns out that the energy density may be regarded as a function of ϕ alone due to the reduction of the two degrees of freedom to a single degree of freedom. Hence on the comoving slice on which ϕ is uniform, one notices that the energy density ρ is also uniform, which implies the coincidence of the two slicings. Moreover, the Friedmann equation, or more precisely the Hamiltonian constraint equation which includes perturbations, equates the energy density with the Hubble expansion rate H on large scales. Then one also finds the coincidence of the above two slicings with the uniform Hubble (expansion rate) slicing.

To summarize, we have

on large (superhorizon) scales. Based on this fact, one notices that the conservation of the curvature perturbation is not a special property of the comoving slicing. Even for other slicings, such as the uniform energy density or uniform Hubble slicings, the curvature perturbation may be also conserved. This is a rather generic physical consequence of single-field slow-roll inflation.

On the other hand, the conservation of the curvature perturbation no longer holds in multi-field inflation models [14, 15]. Even under the slow-roll approximation under which all the decaying modes may be neglected, there still remain multiple dynamical degrees of freedom. These dynamical degrees of freedom of multi-scalar fields affect the evolution of the curvature perturbation even on superhorizon scales. Those effects appear on the right-hand side of Eq. (1.2) as a source term. This may be seen by writing down the energy conservation law of a perfect fluid on uniform density slicing. On superhorizon scales, it reads

where R_e is the curvature perturbation on uniform density slices,

is the non-adiabatic pressure perturbation, and we have neglected spatial gradients by adopting the large-scale approximation. It is clear that the adiabatic pressure perturbation vanishes on the uniform energy density slicing because the adiabatic pressure means P = P(ρ). So any non-vanishing pressure perturbation on the uniform density slicing is non-adiabatic, and it seeds the curvature perturbation.

In multi-field inflation, δP_nad may be a complicated combination of the multi-field components, whose evolution equations may be coupled to each other. Thus, although one may be able to find the time evolution of δP_nad and then that of the curvature perturbation by carefully solving the perturbation equations, it may not be an easy task.

The δN formalism is invented to save this situation [16–18]. It is based on the fact that on superhorizon scales each Hubble horizon size region evolves mutually independently because they are not causally connected to each other [15, 19–21]. It is essentially equivalent to the leading order approximation in spatial gradient expansion, and it is often called the separate universe approach. It turns out that the comoving curvature perturbation can be computed by solving the equations for the homogeneous background with slightly different initial conditions, without resorting to the complicated perturbation equations.

In this monograph, we review the δN formalism in depth. It is organized as follows. In Chapter 2 we formulate the δN formalism at linear and non-linear order. First the δN formalism is introduced at linear order in cosmological perturbation theory. The derivation and analysis of the linear δN formalism are quite useful because one will be able to clearly understand the geometrical meaning of the δN formalism, and explicitly observe the correspondence between a set of background homogeneous equations/solutions and those...

Table of contents

• Cover
• Halftitle
• Series Editors
• Title
• Copyright
• Contents
• Acknowledgements
• Chapter 1 Introduction
• Chapter 2 Basic formulation of δN formalism
• Chapter 3 Application of δN formalism: Warm-up studies
• Chapter 4 Application of δN formalism: Multi-brid inflation
• Chapter 5 Application of δN formalism: Non-attractor inflation
• Chapter 6 Application of δN formalism: Inflation with local features
• Appendix A δN for general cs in non-attractor background
• Appendix B Variance of δχ fluctuations
• Appendix C Correlation functions of δχ2
• Appendix D Bispectrum with localized feature
• Appendix E δN up to Δχ4
• Bibliography