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The simplest stochastic halo formation models assume that the traceless part of the shear field acts to increase the initial overdensity (or decrease the underdensity) that a protohalo (or protovoid) must have if it is to form by the present time. Equivalently, it is the difference between the overdensity and (the square root of the) shear that must be larger than a threshold value. To estimate the effect this has on halo abundances using the excursion set approach, we must solve for the first crossing distribution of a barrier of constant height by the random walks associated with the difference, which is now (even for Gaussian initial conditions) a non-Gaussian variate. The correlation properties of such non-Gaussian walks are inherited from those of the density and the shear, and, since they are independent processes, the solution is in fact remarkably simple. We show that this provides an easy way to understand why earlier heuristic arguments about the nature of the solution worked so well. In addition to modelling halos and voids, this potentially simplifies models of the abundance and spatial distribution of filaments and sheets in the cosmic web.
The excursion set approach provides a framework for predicting how the abundance of dark matter halos depends on the initial conditions. A key ingredient of this formalism comes from the physics of halo formation: the specification of a critical over
Insight into a number of interesting questions in cosmology can be obtained from the first crossing distributions of physically motivated barriers by random walks with correlated steps. We write the first crossing distribution as a formal series, ord
Recently, we provided a simple but accurate formula which closely approximates the first crossing distribution associated with random walks having correlated steps. The approximation is accurate for the wide range of barrier shapes of current interes
In excursion set theory the computation of the halo mass function is mapped into a first-passage time process in the presence of a barrier, which in the spherical collapse model is a constant and in the ellipsoidal collapse model is a fixed function
We provide a simple formula that accurately approximates the first crossing distribution of barriers having a wide variety of shapes, by random walks with a wide range of correlations between steps. Special cases of it are useful for estimating halo