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 ${\delta }_L$ coarse-grained on a scale $R_L$. In the latter case, the predicted tracer correlation function receives contributions of order $⟨{\delta }_L^n⟩$ 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 $\propto k^2$ 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.

Figure 1

Sketch of the separation of the density field (blue, thin line) into large-scale part ${\delta }_L$ (red, thick line) and a small-scale part ${\delta }_s$ [Eq. (86) in Sec. 4; thin black line below], via an arbitrary coarse-graining scale $R_L$. The tracer density coarse-grained on scale $R_L$ (circles) is described by the function $F_{h,L}({\delta }_L;\mathbf{x})$ [Eq. (6)], where the explicit dependence on $\mathbf{x}$ encodes the scatter around the mean relation with ${\delta }_L$, which is assumed to be uncorrelated with ${\delta }_L$.