Living in a Non-Flat Universe: Theoretical Formalism

TL;DR

This paper develops a comprehensive, gauge-invariant theoretical framework for understanding cosmological probes in a non-flat universe, which is essential for analyzing upcoming large-scale survey data and constraining spatial curvature.

Contribution

It provides the first complete gauge-invariant formalism for light propagation and cosmological probes in a non-flat universe, including relativistic effects, vector, and tensor contributions.

Findings

Derived gauge-invariant expressions for cosmological probes in a non-flat universe.

Accounted for relativistic, vector, and tensor effects in light propagation.

Verified the formalism's consistency and compared with previous work.

Abstract

Recent analysis of the Planck measurements opened a possibility that we live in a non-flat universe. Given the renewed interest in non-zero spatial curvature, here we re-visit the light propagation in a non-flat universe and provide the gauge-invariant expressions for the cosmological probes: the luminosity distance, galaxy clustering, weak gravitational lensing, and cosmic microwave background anisotropies. With the positional dependence of the spatial metric, the light propagation in a non-flat universe is much more complicated than in a flat universe. Accounting for all the relativistic effects and including the vector and tensor contributions, we derive the expressions for the cosmological probes and explicitly verify their gauge invariance. We compare our results to previous work in a non-flat universe, if present, but this work represents the first comprehensive investigation of the cosmological probes in a non-flat universe. Our theoretical formalism in a non-flat universe will play a crucial role in constraining the spatial curvature in the upcoming large-scale surveys.