Towards optimal estimation of the galaxy power spectrum

TL;DR

This paper develops a generalized, optimal method for estimating the galaxy power spectrum that accounts for galaxy formation physics and survey limitations, ensuring unbiased results for cosmological analysis.

Contribution

It extends the Feldman et al. (1994) framework to include galaxy-halo relationships, bias, and selection effects, providing a comprehensive optimal estimation scheme.

Findings

The optimal weights depend on halo bias and occupation moments, not directly on galaxy luminosity.

The scheme guarantees unbiased reconstruction of the matter power spectrum if the model assumptions hold.

Practical implementation challenges are discussed, guiding future observational applications.

Abstract

The galaxy power spectrum encodes a wealth of information about cosmology and the matter fluctuations. Its unbiased and optimal estimation is therefore of great importance. In this paper we generalise the framework of Feldman et al. (1994) to take into account the fact that galaxies are not simply a Poisson sampling of the underlying dark matter distribution. Besides finite survey-volume effects and flux-limits, our optimal estimation scheme incorporates several of the key tenets of galaxy formation: galaxies form and reside exclusively in dark matter haloes; a given dark matter halo may host several galaxies of various luminosities; galaxies inherit part of their large-scale bias from their host halo. Under these broad assumptions, we prove that the optimal weights "do not" explicitly depend on galaxy luminosity, other than through defining the maximum survey volume and effective galaxy density at a given position. Instead, they depend on the bias associated with the host halo; the first and second factorial moments of the halo occupation distribution; a selection function, which gives the fraction of galaxies that can be observed in a halo of mass M at position r in the survey; and an effective number density of galaxies. If one wishes to reconstruct the matter power spectrum, then, provided the model is correct, this scheme provides the only unbiased estimator. The practical challenges with implementing this approach are also discussed.