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 $\sigma$, bias $b=\Omega^{.6}/\beta$ 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 $\beta$ 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 $b_2$. 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 $\Omega$ with minimal assumptions about the galaxy formation process. This formalism also provides a convenient parametrization to quantify statistical galaxy formation properties.