An Application of Wiener Hermite Expansion to Non-linear Evolution of Dark Matter

TL;DR

This paper introduces a Wiener Hermite expansion-based method to approximate the non-linear evolution of dark matter's matter power spectrum, achieving high accuracy with reduced computational effort.

Contribution

It develops a new approximation for the matter power spectrum using Wiener Hermite expansion, bridging low- and high-k regimes with high accuracy and computational efficiency.

Findings

Achieves better than 2% accuracy compared to N-body simulations.

Validates the method against 2-loop Standard Perturbation Theory results.

Computationally efficient with only single and double integrals involved.

Abstract

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.