http://arxiv.org/abs/1401.8177
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 approach treats the curve defined by the overdensity fluctuation field when changing the smoothing scale as a random walk. Most implementations of the approach then assume that, at least to a first approximation, the walks have uncorrelated steps, so that the walk heights are a Markov process. This assumption is known to be inaccurate: smoothing filters that are most easily related to the physics of structure formation generically yield walks whose steps are correlated with one another. We develop models in which it is the steps, rather than the walk heights, that are a Markov process. In such models, which we call Markov Velocity processes, each step correlates only with the previous one. We show that TopHat smoothing of a power law power spectrum with index n = -2 gives Markov Velocities, but using suitably truncated smoothing filters always produces walks with Markov Velocities for any power spectrum. We also describe a Markov Velocity model whose statistics are very close to those of TopHat smoothed LCDM walks, and show how to use the Markov property to gain insight into merger rates and assembly histories, as well as to generate fast Monte Carlo realizations of the walks. Our analysis illustrates that Markovian Velocity walks generically exhibit a simple but realistic form of Assembly bias, so they can be useful to construct more realistic merger history trees.
Mon, 3 Feb 14
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