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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 overdensity threshold (barrier) which protohalos must exceed if they are to form bound virialized halos at a later time. Another ingredient is statistical, as it requires the specification of the appropriate statistical ensemble over which to average when making predictions. The excursion set approach explicitly averages over all initial positions, thus implicitly assuming that the appropriate ensemble is that associated with randomly chosen positions in space, rather than special positions such as peaks of the initial density field. Since halos are known to collapse around special positions, it is not clear that the physical and statistical assumptions which underlie the excursion set approach are self-consistent. We argue that they are at least for low mass halos, and illustrate by comparing our excursion set predictions with numerical data from the DEUS simulations.
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
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
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. Eq
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
The excursion set approach uses the statistics of the density field smoothed on a wide range of scales, to gain insight into a number of interesting processes in nonlinear structure formation, such as cluster assembly, merging and clustering. The app