On the Equivalence of Barrier Crossing, Peak-Background Split, and Local Biasing

TL;DR

This paper proves the mathematical equivalence of three major formalisms used to predict dark matter halo clustering, unifying previously observed case-by-case agreements into a general theoretical framework.

Contribution

It provides a simple, rigorous proof demonstrating that barrier crossing, peak-background split, and local biasing are fundamentally equivalent in predicting halo clustering.

Findings

Mathematical proof of equivalence of the three formalisms

Unification of different halo clustering prediction methods

Clarification of theoretical foundations in cosmology

Abstract

Several, apparently distinct, formalisms exist in the literature for predicting the clustering of dark matter halos. It has been noticed on a case-by-case basis that the predictions of these different methods agree in specific examples, but there is no general proof that they are equivalent. In this paper, we give a simple proof of the mathematical equivalence of barrier crossing, peak-background split, and local biasing.