Lagrangian bias in the local bias model

TL;DR

This paper proves that in the local bias model with Gaussian initial conditions, the halo-mass correlation is linearly proportional to the mass auto-correlation on all scales, simplifying the understanding of halo bias.

Contribution

It establishes exact relations showing that local Lagrangian bias leads to linear proportionality between halo-mass cross-correlation and mass auto-correlation, with no higher order terms.

Findings

Cross-correlation is linearly proportional to auto-correlation on all scales.

Auto-correlation of biased tracers can be expressed as a Taylor series in mass auto-correlation.

The linear bias factor for auto-correlation is the square of that for cross-correlation.

Abstract

It is often assumed that the halo-patch fluctuation field can be written as a Taylor series in the initial Lagrangian dark matter density fluctuation field. We show that if this Lagrangian bias is local, and the initial conditions are Gaussian, then the two-point cross-correlation between halos and mass should be linearly proportional to the mass-mass auto-correlation function. This statement is exact and valid on all scales; there are no higher order contributions, e.g., from terms proportional to products or convolutions of two-point functions, which one might have thought would appear upon truncating the Taylor series of the halo bias function. In addition, the auto-correlation function of locally biased tracers can be written as a Taylor series in the auto-correlation function of the mass; there are no terms involving, e.g., derivatives or convolutions. Moreover, although the leading order coefficient, the linear bias factor of the auto-correlation function is just the square of that for the cross-correlation, it is the same as that obtained from expanding the mean number of halos as a function of the local density only in the large-scale limit. In principle, these relations allow simple tests of whether or not halo bias is indeed local in Lagrangian space. We discuss why things are more complicated in practice. We also discuss our results in light of recent work on the renormalizability of halo bias, demonstrating that it is better to renormalize than not. We use the Lognormal model to illustrate many of our findings.