The merger and accretion probabilities of dark matter halos have so far only been calculated for an infinitesimal time interval. This means that a Monte-Carlo simulation with very small time steps is necessary to find the merger history of a parent halo. In this paper we use the random walk formalism to find the merger and accretion probabilities of halos for a finite time interval. Specifically, we find the number density of halos at an early redshift that will become part of a halo with a specified final mass at a later redshift, given that they underwent $n$ major mergers, $n=0,1,2,...$ . We reduce the problem into an integral equation which we then solve numerically. To ensure the consistency of our formalism we compare the results with Monte-Carlo simulations and find very good agreement. Though we have done our calculation assuming a flat barrier, the more general case can easily be handled using our method. This derivation of finite time merger and accretion probabilities can be used to make more efficient merger trees or implemented directly into analytical models of structure formation and evolution.
Download