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 $\Lambda$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.
