Sufficient observables for large scale structure in galaxy surveys

Beyond the linear regime, the power spectrum and higher order moments of the matter field no longer capture all cosmological information encoded in density fluctuations. While non-linear transforms have been proposed to extract this information lost to traditional methods, up to now, the way to generalize these techniques to discrete processes was unclear; ad hoc extensions had some success. We pointed out in Carron and Szapudi (2013) that the logarithmic transform approximates extremely well the optimal "sufficient statistics", observables that extract all information from the (continuous) matter field. Building on these results, we generalize optimal transforms to discrete galaxy fields. We focus our calculations on the Poisson sampling of an underlying lognormal density field. We solve and test the one-point case in detail, and sketch out the sufficient observables for the multi-point case. Moreover, we present an accurate approximation to the sufficient observables in terms of the mean and spectrum of a non-linearly transformed field. We find that the corresponding optimal non-linear transformation is directly related to the maximum a posteriori Bayesian reconstruction of the underlying continuous field with a lognormal prior as put forward in Kitaura et al (2010). Thus simple recipes for realizing the sufficient observables can be built on previously proposed algorithms that have been successfully implemented and tested in simulations.