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 nablaᵢ nablaⱼ phi and nablaᵢ vⱼ, where phi is the potential and v 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 sᵢj sⱼi, where sᵢj=nablaᵢ nablaⱼ ^-2 - 1/3 deltaKᵢj delta. The other multiplies sᵢj tⱼi, where tᵢj=nablaᵢ nablaⱼ nabla^-2 - 1/3 deltaKᵢj (theta-delta), with theta=- (a H dlnD/dlna) ^-1 nablaᵢ vᵢ. (There are other, observationally equivalent, ways to write the two terms, e. g. , using theta-delta instead of sᵢj sⱼi. ) We show how short-range (non-gravitational) non-locality can be included through a controlled series of higher derivative terms, starting with R² nabla² delta, where R is the scale of non-locality (this term will be a small correction as long as k² R² 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.