Peak-background split, renormalization, and galaxy clustering

We present a derivation of two-point correlations of general tracers in the peak-background split (PBS) framework by way of a rigorous definition of the PBS argument. Our expressions only depend on connected matter correlators and "renormalized" bias parameters with clear physical interpretation, and are independent of any coarse-graining scale. This result should be contrasted with the naive expression derived from a local bias expansion of the tracer number density with respect to the matter density perturbation δ_L coarse-grained on a scale R_L. In the latter case, the predicted tracer correlation function receives contributions of order âŸ̈δLnâŸ