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Stochastic Nonlinear Galaxy Biasing

Avishai Dekel, Ofer Lahav

TL;DR

Stochastic Nonlinear Galaxy Biasing reframes galaxy bias as a probabilistic relation P(g|δ) rather than a simple linear mapping, introducing three local parameters that separate mean bias, nonlinearity, and scatter. The authors derive how these properties affect two-point and three-point statistics, redshift-space distortions, and cosmological inferences (notably β = f(Ω)/b), showing that stochasticity mainly restricts the usable scales in redshift analyses while nonlinearity often yields modest corrections. They connect theory to simulations and observations, outlining methods to constrain b(δ) from PDFs and counts-in-cells and discussing the implications for interpreting diverse β measurements. Overall, the framework clarifies why β estimates vary and highlights the potential of redshift surveys and weak lensing to tighten constraints on galaxy biasing across cosmic time.

Abstract

We propose a general formalism for galaxy biasing and apply it to methods for measuring cosmological parameters, such as regression of light versus mass, the analysis of redshift distortions, measures involving skewness and the cosmic virial theorem. The common linear and deterministic relation g=b*d between the density fluctuation fields of galaxies g and mass d is replaced by the conditional distribution P(g|d) of these random fields, smoothed at a given scale at a given time. The nonlinearity is characterized by the conditional mean <g|d>=b(d)*d, while the local scatter is represented by the conditional variance s_b^2(d) and higher moments. The scatter arises from hidden factors affecting galaxy formation and from shot noise unless it has been properly removed. For applications involving second-order local moments, the biasing is defined by three natural parameters: the slope b_h of the regression of g on d, a nonlinearity b_t, and a scatter s_b. The ratio of variances b_v^2 and the correlation coefficient r mix these parameters. The nonlinearity and the scatter lead to underestimates of order b_t^2/b_h^2 and s_b^2/b_h^2 in the different estimators of beta (=Omega^0.6/b_h). The nonlinear effects are typically smaller. Local stochasticity affects the redshift-distortion analysis only by limiting the useful range of scales, especially for power spectra. In this range, for linear stochastic biasing, the analysis reduces to Kaiser's formula for b_h (not b_v), independent of the scatter. The distortion analysis is affected by nonlinear properties of biasing but in a weak way. Estimates of the nontrivial features of the biasing scheme are made based on simulations and toy models, and strategies for measuring them are discussed. They may partly explain the range of estimates for beta.

Stochastic Nonlinear Galaxy Biasing

TL;DR

Stochastic Nonlinear Galaxy Biasing reframes galaxy bias as a probabilistic relation P(g|δ) rather than a simple linear mapping, introducing three local parameters that separate mean bias, nonlinearity, and scatter. The authors derive how these properties affect two-point and three-point statistics, redshift-space distortions, and cosmological inferences (notably β = f(Ω)/b), showing that stochasticity mainly restricts the usable scales in redshift analyses while nonlinearity often yields modest corrections. They connect theory to simulations and observations, outlining methods to constrain b(δ) from PDFs and counts-in-cells and discussing the implications for interpreting diverse β measurements. Overall, the framework clarifies why β estimates vary and highlights the potential of redshift surveys and weak lensing to tighten constraints on galaxy biasing across cosmic time.

Abstract

We propose a general formalism for galaxy biasing and apply it to methods for measuring cosmological parameters, such as regression of light versus mass, the analysis of redshift distortions, measures involving skewness and the cosmic virial theorem. The common linear and deterministic relation g=b*d between the density fluctuation fields of galaxies g and mass d is replaced by the conditional distribution P(g|d) of these random fields, smoothed at a given scale at a given time. The nonlinearity is characterized by the conditional mean <g|d>=b(d)*d, while the local scatter is represented by the conditional variance s_b^2(d) and higher moments. The scatter arises from hidden factors affecting galaxy formation and from shot noise unless it has been properly removed. For applications involving second-order local moments, the biasing is defined by three natural parameters: the slope b_h of the regression of g on d, a nonlinearity b_t, and a scatter s_b. The ratio of variances b_v^2 and the correlation coefficient r mix these parameters. The nonlinearity and the scatter lead to underestimates of order b_t^2/b_h^2 and s_b^2/b_h^2 in the different estimators of beta (=Omega^0.6/b_h). The nonlinear effects are typically smaller. Local stochasticity affects the redshift-distortion analysis only by limiting the useful range of scales, especially for power spectra. In this range, for linear stochastic biasing, the analysis reduces to Kaiser's formula for b_h (not b_v), independent of the scatter. The distortion analysis is affected by nonlinear properties of biasing but in a weak way. Estimates of the nontrivial features of the biasing scheme are made based on simulations and toy models, and strategies for measuring them are discussed. They may partly explain the range of estimates for beta.

Paper Structure

This paper contains 19 sections, 59 equations, 1 figure.

Table of Contents

  1. INTRODUCTION
  2. LOCAL MOMENTS
  3. Conditional Distribution
  4. Conditional Mean -- Nonlinearity
  5. Conditional Variance -- Stochasticity
  6. Variances and Linear Regression
  7. TWO-POINT CORRELATIONS
  8. REDSHIFT DISTORTIONS
  9. General Linear Redshift Distortions at Non-zero Lag
  10. Distortions for Linear, Stochastic and Local Biasing
  11. SKEWNESS AND THREE-POINT CORRELATIONS
  12. Skewness
  13. Cosmic Virial Theorem and Energy Equation
  14. CONSTRAINTS ON THE BIASING RELATION
  15. Preliminary Results from Simulations
  16. ...and 4 more sections

Figures (1)

  • Figure 1: Biasing of galactic halos versus mass in a cosmological $N$-body simulation, demonstrating nonlinearity and stochasticity. The conditional mean (solid curve) and scatter (error bars) are marked. The fields smoothed with a top-hat window of radius $8\,h^$ ^-1${\rm Mpc}$ are plotted at the points of a uniform grid. Left: at the time when $\sigma_8=0.6$ (e.g., $z=0$). Right: at an earlier time when $\sigma_8=0.3$ (e.g., $z=1$). Based on Somerville et al. (1998).