On de Sitter Space, Scalar Fields and Inflation

TL;DR

This paper provides a comprehensive introduction to de Sitter space and inflation, covering geometric properties, scalar field quantization, and their relevance to cosmic microwave background observations.

Contribution

It offers a pedagogical overview with detailed derivations and visualizations of de Sitter space, and computes correlation functions relevant to cosmological observations.

Findings

Derived Killing vectors and visualized de Sitter geometry

Quantized scalar fields in the Bunch-Davies vacuum

Computed correlation functions linked to CMB anisotropies

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.