On de Sitter Space, Scalar Fields and Inflation

Abstract: An introductory, self-contained overview with pedagogical figures is provided to acquaint readers unfamiliar with the subject 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 patches and cosmological horizons. 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.