The Excursion set approach: Stratonovich approximation and Cholesky decomposition

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.

arXiv comment6 pages, 2 figures, Accepted for publication in MNRAS

References in corpus (10)

• Halo Models of Large Scale Structure
• Ellipsoidal collapse and an improved model for the number and spatial distribution of dark matter haloes
• A hierarchy of voids: Much ado about nothing
• Excursion set peaks: a self-consistent model of dark halo abundances and clustering
• Halo abundances and counts-in-cells: The excursion set approach with correlated steps
• Level Crossing Analysis of the Stock Markets
• First Passage Time Densities in Non-Markovian Models with Subthreshold Oscillations
• The importance of stepping up in the excursion set approach
• The excursion set approach in non-Gaussian random fields
• On the Markovian assumption in the excursion set approach: the approximation of Markov Velocities

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