The Excursion set approach: Stratonovich approximation and Cholesky decomposition

TL;DR

This paper introduces efficient analytic and numerical methods, including the Stratonovich approximation and a Cholesky-decomposition algorithm, for calculating the first crossing distribution in the excursion set approach, relevant for cosmic structure formation.

Contribution

It presents a novel, faster Cholesky-decomposition algorithm and applies the Stratonovich approximation for accurate estimation of first crossing distributions in cosmology.

Findings

The Cholesky-based algorithm is significantly faster than existing methods.

The Stratonovich approximation accurately estimates the first crossing distribution.

The methods are validated with Monte-Carlo simulations.

Abstract

The excursion set approach is a framework for estimating how the number density of nonlinear structures in the cosmic web depends on the expansion history of the universe and the nature of gravity. A key part of the approach is the estimation of the first crossing distribution of a suitably chosen barrier by random walks having correlated steps: The shape of the barrier is determined by the physics of nonlinear collapse, and the correlations between steps by the nature of the initial density fluctuation field. We describe analytic and numerical methods for calculating such first up-crossing distributions. While the exact solution can be written formally as an infinite series, we show how to approximate it efficiently using the Stratonovich approximation. We demonstrate its accuracy using Monte-Carlo realizations of the walks, which we generate using a novel Cholesky-decomposition based algorithm, which is significantly faster than the algorithm that is currently in the literature.