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Peak-Background Split, Renormalization, and Galaxy Clustering

Fabian Schmidt, Donghui Jeong, Vincent Desjacques

TL;DR

The article develops a rigorous PBS-based, renormalized bias formalism for galaxy/tracer clustering, expressing tracer two-point functions in terms of connected matter correlators and $R_L$-independent PBS biases $b_N$. It shows how curvature biases ($ abla^2 ilde ho$) cure smoothing-induced $R_L$-dependence, producing a scale-dependent bias $ abla^2$-type term that aligns with peak-model results. For non-Gaussian initial conditions, the authors introduce bivariate PBS biases $b_{NM}$ to absorb $R_L$-dependent pieces and obtain a tree-level, convergent expansion whose leading NG effect is captured by $b_{01}$ (and its generalizations) depending on the NG shape. The framework unifies Gaussian and non-Gaussian cases, links PBS biases to universal mass functions, and provides a physically meaningful, scale-aware description that can incorporate various tracer-formation models. This approach enhances the interpretability and predictive power of large-scale structure bias, with clear pathways to testing against simulations and extending to higher-point statistics.

Abstract

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 <δ_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 ~ 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.

Peak-Background Split, Renormalization, and Galaxy Clustering

TL;DR

The article develops a rigorous PBS-based, renormalized bias formalism for galaxy/tracer clustering, expressing tracer two-point functions in terms of connected matter correlators and -independent PBS biases . It shows how curvature biases () cure smoothing-induced -dependence, producing a scale-dependent bias -type term that aligns with peak-model results. For non-Gaussian initial conditions, the authors introduce bivariate PBS biases to absorb -dependent pieces and obtain a tree-level, convergent expansion whose leading NG effect is captured by (and its generalizations) depending on the NG shape. The framework unifies Gaussian and non-Gaussian cases, links PBS biases to universal mass functions, and provides a physically meaningful, scale-aware description that can incorporate various tracer-formation models. This approach enhances the interpretability and predictive power of large-scale structure bias, with clear pathways to testing against simulations and extending to higher-point statistics.

Abstract

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 <δ_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 ~ 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.

Paper Structure

This paper contains 20 sections, 116 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Peak-background split and tracer correlations
  3. Correlations
  4. PBS and bias parameters
  5. PBS biases for universal mass functions
  6. Application to regions above threshold
  7. Exact calculation for a Gaussian density field
  8. Peak-background split calculation
  9. Smoothed correlation function and curvature bias
  10. Connection with the peak model
  11. Non-Gaussian case
  12. Primordial non-Gaussianity of the local type
  13. Correlations
  14. Bivariate PBS bias parameters
  15. Non-local non-Gaussianity
  16. ...and 5 more sections

Figures (1)

  • 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. (\ref{['eq:dsdef']}) in Sec. \ref{['sec:NG']}; 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. (\ref{['eq:Fp']})], where the explicit dependence on $\mathbf{x}$ encodes the scatter around the mean relation with $\delta_L$, which is assumed to be uncorrelated with $\delta_L$.