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Clustering of dark matter tracers: renormalizing the bias parameters

Patrick McDonald

TL;DR

Addresses how perturbation theory can model large-scale clustering of dark matter tracers via a Taylor expansion of the tracer density. Introduces renormalization-inspired redefinitions that absorb problematic higher-order terms and remove explicit smoothing-scale dependencies, preserving the linear bias on very large scales. Derives a compact galaxy power spectrum $P_g(k)$ depending on a renormalized linear bias $b_1$, a single higher-order parameter $ ilde{b}_2$, and a shot-noise term $N$, with the third-order bias eliminated as an independent parameter; treats the three-point function within the same renormalized framework and finds only modest BAO modifications. The approach improves convergence and offers a practical, extendable framework for interpreting large-scale structure surveys, with straightforward extensions to redshift-space distortions and multi-tracer analyses.

Abstract

A commonly used perturbative method for computing large-scale clustering of tracers of mass density, like galaxies, is to model the tracer density field as a Taylor series in the local smoothed mass density fluctuations, possibly adding a stochastic component. I suggest a set of parameter redefinitions, eliminating problematic perturbative correction terms, that should represent a modest improvement, at least, to this method. As presented here, my method can be used to compute the power spectrum and bispectrum to 4th order in initial density perturbations, and higher order extensions should be straightforward. While the model is technically unchanged at this order, just reparameterized, the renormalized model is more elegant, and should have better convergence behavior, for three reasons: First, in the usual approach the effects of beyond-linear-order bias parameters can be seen at asymptotically large scales, while after renormalization the linear model is preserved in the large-scale limit, i.e., the effects of higher order bias parameters are restricted to relatively high k. Second, while the standard approach includes smoothing to suppress large perturbative correction terms, resulting in dependence on the arbitrary cutoff scale, no cutoff-sensitive terms appear explicitly after my redefinitions (and, relatedly, my correction terms are less sensitive to high-k, non-linear, power). Third, the 3rd order bias parameter disappears entirely, so my model has one fewer free parameter than usual (this parameter was redundant at the order considered). This model predicts a small modification of the baryonic acoustic oscillation (BAO) signal, in real space, supporting the robustness of BAO as a probe of dark energy, and providing a complete perturbative description over the relevant range of scales.

Clustering of dark matter tracers: renormalizing the bias parameters

TL;DR

Addresses how perturbation theory can model large-scale clustering of dark matter tracers via a Taylor expansion of the tracer density. Introduces renormalization-inspired redefinitions that absorb problematic higher-order terms and remove explicit smoothing-scale dependencies, preserving the linear bias on very large scales. Derives a compact galaxy power spectrum depending on a renormalized linear bias , a single higher-order parameter , and a shot-noise term , with the third-order bias eliminated as an independent parameter; treats the three-point function within the same renormalized framework and finds only modest BAO modifications. The approach improves convergence and offers a practical, extendable framework for interpreting large-scale structure surveys, with straightforward extensions to redshift-space distortions and multi-tracer analyses.

Abstract

A commonly used perturbative method for computing large-scale clustering of tracers of mass density, like galaxies, is to model the tracer density field as a Taylor series in the local smoothed mass density fluctuations, possibly adding a stochastic component. I suggest a set of parameter redefinitions, eliminating problematic perturbative correction terms, that should represent a modest improvement, at least, to this method. As presented here, my method can be used to compute the power spectrum and bispectrum to 4th order in initial density perturbations, and higher order extensions should be straightforward. While the model is technically unchanged at this order, just reparameterized, the renormalized model is more elegant, and should have better convergence behavior, for three reasons: First, in the usual approach the effects of beyond-linear-order bias parameters can be seen at asymptotically large scales, while after renormalization the linear model is preserved in the large-scale limit, i.e., the effects of higher order bias parameters are restricted to relatively high k. Second, while the standard approach includes smoothing to suppress large perturbative correction terms, resulting in dependence on the arbitrary cutoff scale, no cutoff-sensitive terms appear explicitly after my redefinitions (and, relatedly, my correction terms are less sensitive to high-k, non-linear, power). Third, the 3rd order bias parameter disappears entirely, so my model has one fewer free parameter than usual (this parameter was redundant at the order considered). This model predicts a small modification of the baryonic acoustic oscillation (BAO) signal, in real space, supporting the robustness of BAO as a probe of dark energy, and providing a complete perturbative description over the relevant range of scales.

Paper Structure

This paper contains 4 sections, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Calculation
  3. Three-point function
  4. Discussion

Figures (2)

  • Figure 1: Thick lines show $(P_g(k)-N)/ b_1^2 P_{\rm mass}(k)$ for $\tilde{b}_2=0.6$ (black), $\tilde{b}_2=1.3$ (green), and $\tilde{b}_2=-0.15$ (red). For comparison, the thinner, blue, wiggly lines show the ratio of mass power in a realistic model to the power in a similar model with no baryonic acoustic oscillations (solid is non-linear power, dotted is linear).
  • Figure 2: $(P_g(k)-N)/ b_1^2 P_{\rm mass}(k)$ for $\tilde{b}_2=0.6$. The solid curve is my standard calculation, where the RG-corrected (non-linear) mass power spectrum has been used to compute the bias terms. To show the lack of sensitivity to small scales, for the dashed curve the usual linear mass power, which becomes dramatically different from the RG power with increasing $k$, was used to compute the bias terms. The dotted curve shows the effect of a $2 \, h^{-1} \, {\rm Mpc}$ rms Gaussian smoothing applied to the result of the standard calculation.