ansatz that states "the fraction of trajectories with a first upcrossing of the barrier at is equal to the mass fraction at time that is incorporated in halos with masses ". Consequently, this ansatz ensures that each trajectory will upcross the barrier given some arbitrarily large , and as a result it guarantees that each mass element will ultimately become part of a halo. Furthermore, the fraction of mass that originated from collapsed objects of a specific mass at an earlier time can be used to determine average number of progenitors at time within the mass interval that have merged to produce a halo of at time . This is accomplished by considering a spherical region of mass with a corresponding mass variance and linear overdensity , where is the linear growth rate that is normalized to unity at time and is the critical overdensity at which the initial spherical region has collapsed to form a virialized object. Mathematically, the progenitor mass function is expressed as:where and is the Press-Schechter multiplicity function that describes the fraction of mass associated with halos in a range . Various comparisons of the progenitor mass function with numerical simulations have concluded that good agreement between theory and simulation is obtained only when is small, otherwise the mass fraction in high mass progenitors is significantly underestimated, which can be attributed to the crude assumptions such as assuming a perfectly spherical collapse model and using a linear density field as opposed to a non-linear density field to characterize collapsed structures. Nevertheless, the utility of the EPS formalism is that it provides a computationally friendly approach for determining properties of dark matter halos. Halo merger rate Another utility of the EPS formalism is that it can be used to determine the rate at which a halo of initial mass M merges with a halo with mass between M and M+ΔM. This rate is given by where , . In general the change in mass, , is the sum of a multitude of minor mergers. Nevertheless, given an infinitesimally small time interval it is reasonable to consider the change in mass to be due to a single merger events in which transitions to . Galactic cannibalism (minor mergers) Throughout their lifespan, satellite galaxies orbiting in the dark matter halo experience dynamical friction and consequently descend deeper into the gravitational potential of their host as a result of orbital