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

Understand

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.