On de Sitter Space, Scalar Fields and Inflation

TL;DR

The paper provides a concise, pedagogical overview of de Sitter space, its embedding geometry, and the role of its symmetries in cosmology and inflation. It integrates the classical geometry with quantum field theory in de Sitter, deriving the Bunch–Davies vacuum, two-point functions, and the in-in computation of the inflaton bispectrum within slow-roll inflation. By connecting the quantum fluctuations to the CMB, the work explains how horizon-crossing dynamics and the nearly scale-invariant power spectrum seed the observed Gaussian temperature anisotropies, while outlining the small non-Gaussian signatures expected from cubic interactions. The discussion highlights the significance of the dS symmetry, planar/global patches, and ADM formalism for understanding early-universe physics and motivates future directions including stochastic/multi-field inflation and holographic approaches to de Sitter space.

Abstract

An introductory, self-contained overview, with pedagogical figures, is provided to acquaint readers unfamiliar with the subjects with key aspects of de Sitter space and inflation. The connection between de Sitter space and cosmology is reviewed. The embedding of a hyperboloid surface in higher-dimensional Minkowski space is analysed, and the Killing vectors of de Sitter space are derived in both global and planar coordinate systems; their integral curves are visualised on the de Sitter Penrose diagram, illustrating the static patch and cosmological horizon. The inflaton scalar field is quantised in the Bunch-Davies vacuum within the slow-roll regime. The two- and three-point correlation functions are computed using the ADM decomposition and the in-in formalism. These computations are related to the observed small temperature anisotropies of the cosmic microwave background.

Figures (12)

• Figure 1: The cosmic inventory based on Eqn. \ref{['rho']}. The densities $\rho_\Lambda$ and $\rho_c$ are respectively defined as follows: $\rho_\Lambda = \Lambda/8\pi G_\text{N}$ and $\rho_c=3H_0^{2}/8\pi G_\text{N}$. The subscripts M and R refers to matter and radiation respectively. The subscript $x$ in $\rho_x$ denotes R, M or $\Lambda$. In the early universe, radiation made the dominant contribution to the total energy density of the universe, followed by a long period in which matter dominated. $a_\text{eq}$ refers to the value of $a$ at which the energy densities of radiation and matter became equal. To the right hand side, a period of accelerated expansion dominated by $\Lambda$. More details can be found in reference dodelson2020modern for example.
• Figure 2: Full–sky map of CMB temperature anisotropies (Mollweide projection; dipole removed). Colours show fluctuations $\Delta T$ about the mean CMB temperature $T_0 = 2.72548 \pm 0.00057~\mathrm{K}$ (the uncertainty refers to the monopole) mather1994measurementfixsen1996cosmicfixsen2009temperature. The anisotropy amplitude is at the level $\Delta T/T_0 \sim \text{few}\times10^{-5}$ (tens of $\mu$K)smoot1992structure. The picture is captured from the European Space Agency website.
• Figure 3: Schematic slow-roll inflationbaumann2022cosmology. The potential energy is a function of the inflaton field. The homogeneous background slowly rolls along the shallow part of the potential while the slow-roll parameters satisfy $\epsilon,|\eta|\ll1$ (blue shading), leading to accelerated expansion. Inflation ends when $\epsilon\simeq1$, typically close to the minimum of the curve, after which the field oscillates and reheats the Universe. Quantum fluctuations generated during the roll (suggested by the gradient shading) source curvature perturbations that later seed structure formation. More details can be found in reference baumann2022cosmology, and inflation is discussed further in section \ref{['sec5']}.
• Figure 4: Three-dimensional (one for time, $X^{0}$, and two for space) de Sitter spacetime. To the right, constant time slices, circles, are shown. The lower part (or branch) corresponds to $X^0=-\sqrt{\left(X^1\right)^2+\left(X^2\right)^2-\ell^2}$, and the upper part corresponds to $X^0=\sqrt{\left(X^1\right)^2+\left(X^2\right)^2-\ell^2}$. More details can be found in reference hawking2023large.
• Figure 5: The Penrose diagram for dS$_4$. The metric of Eqn. \ref{['de Sitter Penrose']} is included for clarity, and the shadow is included to remind of the higher-dimensional character of the figure. Each point in the "square", except for when $\psi =0$ or $\pi$, represents a two-sphere. Time slices (Cauchy surfaces) are three-spheres with their poles being on opposite edges of the square. $i^+$ and $i^-$ respectively refer to the future and past time-like infinities which occur at finite values for $T$, the conformal time, as shown on the left-hand side $-$ an example of a time-like trajectory is included. $\mathcal{I}^+$ and $\mathcal{I}^-$ refer to the future and past light-like infinities. A light ray, propagating at 45$^{\circ}$ angle, starts at $\mathcal{I}^-$ and ends at $\mathcal{I}^+$. Once it reaches the south pole, it is re-introduced at the corresponding north pole of $S^3$. Note that in this conformal diagram, $i^+$ (the endpoint of time-like trajectories) and $\mathcal{I}^+$ (the endpoint of light-like trajectories) may appear to coincide at the top, which is a feature of this compactified representation of the de Sitter space; the same note applies to $i^-$ and $\mathcal{I}^-$ at the bottom. More details on the causal structure of de Sitter space can be found in reference hawking2023large, and details on Penrose diagrams in references penrose1963asymptoticpenrose2011republication.