Matrix Formalism of Excursion Set Theory: A new approach to statistics of dark matter halo counting

Abstract

Excursion set theory (EST) is an analytical framework to study the large-scale structure of the Universe. EST introduces a procedure to calculate the number density of structures by relating the cosmological linear perturbation theory to the nonlinear structures in late time. In this work, we introduce a novel approach to reformulate the EST in matrix formalism. We propose that the matrix representation of EST will facilitate the calculations in this framework. The method is to discretize the two-dimensional plane of variance and density contrast of EST, where the trajectories for each point in the Universe lived there. The probability of having a density contrast in a chosen variance is represented by a probability ket. Naturally, the concept of the transition matrix pops up to define the trajectories. We also define the probability transition rate which is used to obtain the first up-crossing of trajectories and the number count of the structures. In this work we show that the discretization let us study the non-Markov processes by forcing them to look like a Wiener process. Also we discuss that the zero drift processes with Gaussian and also non-Gaussian initial conditions can be studied by this formalism. The continuous limit of the formalism is discussed, and the known Fokker-Planck dispersion equation is recovered. Finally we show that the probability of the most massive progenitors can be extracted in this framework.

18 pages, 13 figures, final version

References in corpus (4)

• The Excursion Set Theory of Halo Mass Functions, Halo Clustering, and Halo Growth
• Primordial non-gaussianity, statistics of collapsed objects, and the Integrated Sachs-Wolfe effect
• Dark Sky Simulations: Early Data Release
• Non-Gaussianity and Excursion Set Theory: Halo Bias

Cited by in corpus (10)

• Late time sky as a probe of steps and oscillations in primordial Universe
• Cosmological Filaments in the light of Excursion Set of Saddle Points
• Modified initial power spectrum and too big to fail problem
• Mass assembly history of dark matter halos in the light of $H_0$ tension
• Voids and Halos in Voids statistics as a probe of the Expansion History of the Universe
• Clustering of dark matter in the cosmic web as a probe of massive neutrinos
• 21-cm power spectrum and ionization bias as a probe of long-mode modulated non Gaussian sky
• Cosmic velocity, density and halo mass function: Insights from deep learning
• Luminosity Function of Galaxies in Voids: A Modification Inspired by Excursion Set Theory
• Addressing the too-big-to-fail problem and the void phenomenon through a modified initial power spectrum