Exact Analytic Solution for Non-Linear Density Fluctuation in a LCDM Universe

TL;DR

This paper derives an exact third-order analytic solution for matter density fluctuations in a LCDM universe, clarifying relativistic effects and initial conditions, which is crucial for understanding galaxy bias.

Contribution

It provides the first exact third-order relativistic solution for density fluctuations in a LCDM universe, including explicit time dependence and gauge comparisons.

Findings

Exact third-order solution derived

Comparison with previous gauge and Newtonian results

Fourier kernels for relativistic effects provided

Abstract

We derive the exact third-order analytic solution of the matter density fluctuation in the proper-time hypersurface in a $\Lambda$CDM universe, accounting for the explicit time-dependence and clarifying the relation to the initial condition. Furthermore, we compare our analytic solution to the previous calculation in the comoving gauge, and to the standard Newtonian perturbation theory by providing Fourier kernels for the relativistic effects. Our results provide an essential ingredient for a complete description of galaxy bias in the relativistic context.