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 ₋ coarse-grained on a scale R₋. In the latter case, the predicted tracer correlation function receives contributions of order ⟨₋^n⟩ at each perturbative order n, whereas, in our formalism, these are absorbed in the PBS bias parameters at all orders. Further, this approach naturally predicts both a scale-dependent bias k^2 such as found for peaks of the density field, and the scale-dependent bias induced by primordial non-Gaussianity in the initial conditions. The only assumption made about the tracers is that their abundance at a given position depends solely on the matter distribution within a finite region around that position.