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

We derive the exact third-order analytic solution of the matter density fluctuation in the proper-time hypersurface in a $Λ$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.