A Survey through Conformal Time

TL;DR

This paper provides a clear, pedagogical analysis of conformal time in flat FRW universes, exploring its relation to cosmic time, geodesics, and curvature in both 1+1 and 3+1 dimensions.

Contribution

It offers a detailed, elementary exposition of conformal time's role in cosmology, including explicit formulas and extensions to higher dimensions.

Findings

Distinct conformal-time behaviors for radiation, matter, and de Sitter universes.

Explicit relations among cosmic time, conformal time, and the scale factor.

General affine-parameter formalism for conformal metrics.

Abstract

We revisit conformal time $\eta$ in a spatially flat Friedmann--Robertson--Walker universe and use a $1+1$-dimensional setting as a technically transparent pedagogical arena. Our purpose is to clarify the relation among cosmic time $t$, conformal time $\eta$, and the scale factor $a(t)$, and then to follow how this relation governs the geodesics of freely moving particles and the curvature of the corresponding manifold. The radiation-dominated, matter-dominated, and exact vacuum-only de Sitter cases are treated separately, because each of them produces a distinct conformal-time dependence and therefore a distinct geodesic structure. We then write the affine-parameter formalism in a form that is genuinely general for any spatially flat conformal metric, and we record the straightforward extension to the spatially flat $3+1$ case. The presentation remains elementary in spirit, but the notation, the curvature formulas, and the de Sitter interpretation are kept explicit.