Towards optimal estimation of the galaxy power spectrum

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 generalize 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.