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Stochastic Bias from Non-Gaussian Initial Conditions

Daniel Baumann, Simone Ferraro, Daniel Green, Kendrick M. Smith

TL;DR

The paper demonstrates that multi-source inflation can produce a stochastic form of scale-dependent halo bias, signaling boosted collapsed four-point functions relative to the square of squeezed three-point functions. It derives general, model-independent formulas for both the non-stochastic and stochastic halo bias in terms of N-point cumulants of the curvature perturbation, using barrier crossing and peak-background split methods that are proven equivalent. Through explicit examples—τ_NL, g_NL, and quasi-single-field inflation—it shows how stochastic bias arises and scales with k, often enhancing detectability, and discusses how measurements of stochasticity can probe hidden-sector non-Gaussianity and the number of light degrees of freedom during inflation. The results provide a framework for using large-scale halo clustering as a diagnostic of early-universe physics, including the squeezed and collapsed limits of primordial correlators.

Abstract

In this article, we show that a stochastic form of scale-dependent halo bias arises in multi-source inflationary models, where multiple fields determine the initial curvature perturbation. We derive this effect for general non-Gaussian initial conditions and study various examples, such as curvaton models and quasi-single field inflation. We present a general formula for both the stochastic and the non-stochastic parts of the halo bias, in terms of the N-point cumulants of the curvature perturbation at the end of inflation. At lowest order, the stochasticity arises if the collapsed limit of the four-point function is boosted relative to the square of the three-point function in the squeezed limit. We derive all our results in two ways, using the barrier crossing formalism and the peak-background split method. In a companion paper, we prove that these two approaches are mathematically equivalent.

Stochastic Bias from Non-Gaussian Initial Conditions

TL;DR

The paper demonstrates that multi-source inflation can produce a stochastic form of scale-dependent halo bias, signaling boosted collapsed four-point functions relative to the square of squeezed three-point functions. It derives general, model-independent formulas for both the non-stochastic and stochastic halo bias in terms of N-point cumulants of the curvature perturbation, using barrier crossing and peak-background split methods that are proven equivalent. Through explicit examples—τ_NL, g_NL, and quasi-single-field inflation—it shows how stochastic bias arises and scales with k, often enhancing detectability, and discusses how measurements of stochasticity can probe hidden-sector non-Gaussianity and the number of light degrees of freedom during inflation. The results provide a framework for using large-scale halo clustering as a diagnostic of early-universe physics, including the squeezed and collapsed limits of primordial correlators.

Abstract

In this article, we show that a stochastic form of scale-dependent halo bias arises in multi-source inflationary models, where multiple fields determine the initial curvature perturbation. We derive this effect for general non-Gaussian initial conditions and study various examples, such as curvaton models and quasi-single field inflation. We present a general formula for both the stochastic and the non-stochastic parts of the halo bias, in terms of the N-point cumulants of the curvature perturbation at the end of inflation. At lowest order, the stochasticity arises if the collapsed limit of the four-point function is boosted relative to the square of the three-point function in the squeezed limit. We derive all our results in two ways, using the barrier crossing formalism and the peak-background split method. In a companion paper, we prove that these two approaches are mathematically equivalent.

Paper Structure

This paper contains 23 sections, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Stochastic Bias
  3. Predictions from Barrier Crossing
  4. Definitions and Notation
  5. Edgeworth Expansions
  6. Barrier Crossing
  7. Matter-Halo Correlations
  8. Halo-Halo Correlations
  9. Stochastic Halo Bias
  10. Examples
  11. $\tau_{\rm NL}$ Cosmology
  12. Peak-Background Split
  13. Barrier Crossing
  14. $g_{\rm NL}$ Cosmology
  15. Peak-Background Split
  16. ...and 8 more sections

Figures (2)

  • Figure 1: The squeezed limit of the three-point function, $k_1 \to 0$, gives the dominant contribution to the scale-dependent halo bias. A stochastic form of scale-dependent halo bias arises if the four-point function is large in the collapsed limit, $k_{12} \equiv |{\boldsymbol{k}}_1+{\boldsymbol{k}}_2| \to 0$.
  • Figure 2: Numerical evaluation of eqs. (\ref{['equ:Sigma']}) and (\ref{['equ:fs']}). For $\Delta \gtrsim 1.0$, the cumulant $f_{\hat{1},2}$ depends significantly on the halo mass scale $M$. This is in contrast to local non-Gaussianity, which corresponds to the limit $\Delta \to 0$.