Reconstructing Nonlinear Stochastic Bias from Velocity Space Distortions

We propose a strategy to measure the dark matter power spectrum using minimal assumptions about the galaxy distribution and the galaxy-dark matter cross-correlations. We argue that on large scales the central limit theorem generically assures Gaussianity of each smoothed density field, but not coherence. Asymptotically, the only surviving parameters on a given scale are galaxy variance, bias b=^. 6/ and the galaxy-dark matter correlation coefficient r. These can all be determined by measuring the quadrupole and octupole velocity distortions in the power spectrum. Measuring them simultaneously may restore consistency between all determinations independent of galaxy type. The leading deviations from Gaussianity are conveniently parameterized by an Edgeworth expansion. In the mildly non-linear regime, two additional parameters describe the full picture: the skewness parameter s and non-linear bias b₂. They can both be determined from the measured skewness combined with second order perturbation theory or from an N-body simulation. By measuring the redshift distortion of the skewness, one can measure the density parameter with minimal assumptions about the galaxy formation process. This formalism also provides a convenient parametrization to quantify statistical galaxy formation properties.