Recovering the nonlinear density field from the galaxy distribution with a Poisson-Lognormal filter

TL;DR

This paper introduces a Poisson-Lognormal filter for reconstructing the nonlinear matter density field from galaxy data, demonstrating its effectiveness and superiority over previous methods in simulations.

Contribution

The authors derive a new lognormal filter as a maximum a posteriori solution for nonlinear density reconstruction, tested with efficient implementation and compared to existing methods.

Findings

Good agreement with dark matter field even at high overdensities

Superior correlation coefficients and smaller Euclidean distances compared to previous methods

Effective recovery of the positive tail of the density distribution down to ~2 Mpc/h scales

Abstract

We present a general expression for a lognormal filter given an arbitrary nonlinear galaxy bias. We derive this filter as the maximum a posteriori solution assuming a lognormal prior distribution for the matter field with a given mean field and modeling the observed galaxy distribution by a Poissonian process. We have performed a three-dimensional implementation of this filter with a very efficient Newton-Krylov inversion scheme. Furthermore, we have tested it with a dark matter N-body simulation assuming a unit galaxy bias relation and compared the results with previous density field estimators like the inverse weighting scheme and Wiener filtering. Our results show good agreement with the underlying dark matter field for overdensities even above delta~1000 which exceeds by one order of magnitude the regime in which the lognormal is expected to be valid. The reason is that for our filter the lognormal assumption enters as a prior distribution function, but the maximum a posteriori solution is also conditioned on the data. We find that the lognormal filter is superior to the previous filtering schemes in terms of higher correlation coefficients and smaller Euclidean distances to the underlying matter field. We also show how it is able to recover the positive tail of the matter density field distribution for a unit bias relation down to scales of about >~2 Mpc/h.