Clustering of dark matter tracers: generalizing bias for the coming era of precision LSS

On very large scales, density fluctuations in the Universe are small, suggesting a perturbative model for large-scale clustering of galaxies (or other dark matter tracers), in which the galaxy density is written as a Taylor series in the local mass density, delta, with the unknown coefficients in the series treated as free bias parameters. We extend this model to include dependence of the galaxy density on the local values of del(i)del(j phi) and del(ivj), where phi is the potential and nu is the peculiar velocity. We show that only two new free parameters are needed to model the power spectrum and bispectrum up to 4th order in the initial density perturbations, once symmetry considerations and equivalences between possible terms are accounted for. One of the new parameters is a bias multiplying s(ij)s(ji), where s(ij) = [del(i)del(j)del(-2)- 1/3 delta(K)(ij)]delta. The other multiplies s(ij)t(ji), where t(ij) = [del(i)del(j)del(-2)- 1/3 delta(K)(ij)] (theta-delta), with theta = -(alpha H d ln D/d ln alpha)(-1) del . nu. (There are other, observationally equivalent, ways to write the two terms, e.g., using theta-delta instead of s(ij)s(ji).) We show how short-range (non-gravitational) non-locality can be included through a controlled series of higher derivative terms, starting with R-2 del(2)delta, where R is the scale of non-locality (this term will be a small correction as long as k(2)R(2) is small, where k is the observed wavenumber). We suggest that there will be much more information in future huge redshift surveys in the range of scales where beyond-linear perturbation theory is both necessary and sufficient than in the fully linear regime.