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 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.
