An Application of Wiener Hermite Expansion to Non-linear Evolution of
  Dark Matter
  We apply the Wiener Hermite (WH) expansion to the non-linear evolution of
Large-Scale Structure, and obtain an approximate expression for the matter
power spectrum in full order of the expansion. This method allows us to expand
any random function in terms of an orthonormal bases in space of random
functions in such a way that the first order of the expansion expresses the
Gaussian distribution, and others are the deviation from the Gaussianity. It is
proved that the Wiener Hermite expansion is mathematically equivalent with the
$\Gamma$-expansion approach in the Renormalized Perturbation Theory (RPT).
While an exponential behavior in the high-$k$ limit has been proved for the
mass density and velocity fluctuations of Dark Matter in the RPT, we reprove
the behavior in the context of the Wiener Hermite expansion using the result of
Standard Perturbation Theory (SPT). We propose a new approximate expression for
the matter power spectrum which interpolates the low-$k$ expression
corresponding to 1-loop level in SPT and the high-$k$ expression obtained by
taking a high-$k$ limit of the WH expansion. The validity of our prescription
is specifically verified by comparing with the 2-loop solutions of the SPT. The
proposed power spectrum agrees with the result of $N$-body simulation with the
accuracy better than 1% or 2% in a range of the BAO scales, where the wave
number is about $k= 0.2 \sim 0.4$ $h{\rm Mpc^{-1}}$ at $z=0.5-3.0$. This
accuracy is comparable to or slightly less than the ones in the Closure Theory
whose the fractional difference from N body result is within 1%. A merit of our
method is that the computational time is very short because only single and
double integrals are involved in our solution.
