The excursion set theory of halo mass functions, halo clustering, and halo growth

I review the excursion set theory with particular attention toward applications to cold dark matter halo formation and growth, halo abundance, and halo clustering. After a brief introduction to notation and conventions, I begin by recounting the heuristic argument leading to the mass function of bound objects given by Press and Schechter. I then review the more formal derivation of the Press - Schechter halo mass function that makes use of excursion sets of the density field. The excursion set formalism is powerful and can be applied to numerous other problems. I review the excursion set formalism for describing both halo clustering and bias and the properties of void regions. As one of the most enduring legacies of the excursion set approach and one of its most common applications, I spend considerable time reviewing the excursion set theory of halo growth. This section of the review culminates with the description of two Monte Carlo methods for generating ensembles of halo mass accretion histories. In the last section, I emphasize that the standard excursion set approach is the result of several simplifying assumptions. Dropping these assumptions can lead to more faithful predictions and open excursion set theory to new applications. One such assumption is that the height of the barriers that de. ne collapsed objects is a constant function of scale. I illustrate the implementation of the excursion set approach for barriers of arbitrary shape. One such application is the now well- known improvement of the excursion set mass function derived from the moving barrier for ellipsoidal collapse. I also emphasize that the statement that halo accretion histories are independent of halo environment in the excursion set approach is not a general prediction of the theory. It is a simplifying assumption. I review the method for constructing correlated random walks of the density field in the more general case. I construct a simple toy model to illustrate that excursion set theory ( with a constant barrier height) makes a simple and general prediction for the relation between halo accretion histories and the large- scale environments of halos: regions of high density preferentially contain late- forming halos and conversely for regions of low density. I conclude with a brief discussion of the importance of this prediction relative to recent numerical studies of the environmental dependence of halo properties.