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Publication# Conditional probabilities in the excursion set theory: generic barriers and non-Gaussian initial conditions

Abstract

The excursion set theory, where density perturbations evolve stochastically with the smoothing scale, provides a method for computing the dark matter halo mass function. The computation of the mass function is mapped into the so-called first-passage time problem in the presence of a moving barrier. The excursion set theory is also a powerful formalism to study other properties of dark matter haloes such as halo bias, accretion rate, formation time, merging rate and the formation history of haloes. This is achieved by computing conditional probabilities with non-trivial initial conditions, and the conditional two-barrier first-crossing rate. In this paper we use the path integral formulation of the excursion set theory to calculate analytically these conditional probabilities in the presence of a generic moving barrier, including the one describing the ellipsoidal collapse, and for both Gaussian and non-Gaussian initial conditions. While most of our analysis associated with Gaussian initial conditions assumes Markovianity (top-hat in momentum space smoothing, rather than generic filters), the non-Markovianity of the random walks induced by non-Gaussianity is consistently accounted for. We compute, for a generic barrier, the first two scale-independent halo bias parameters, the conditional mass function and the halo formation time probability, including the effects of non-Gaussianities. We also provide the expression for the two-constant-barrier first-crossing rate when non-Markovian effects are induced by a top-hat filter function in real space.

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In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occurring with some sort of relationship with another event A. In this event, the event B can be analyzed by a conditional probability with respect to A. If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A under the condition B", is usually written as P(AB) or occasionally P_B(A).

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