Constraining cold dark matter halo merger rates using the coagulation equations

We place additional constraints on the three parameters of the dark matter halo merger rate function recently proposed by Parkinson, Cole & Helly by utilizing Smoluchowski’s coagulation equation, which must be obeyed by any binary merging process which conserves mass. We find that the constraints from Smoluchowski’s equation are degenerate, limiting to a thin plane in the three-dimensional parameter space. This constraint is consistent with those obtained from fitting to N-body measures of progenitor mass functions, and provides a better match to the evolution of the overall dark matter halo mass function, particularly for the most massive haloes. We demonstrate that the proposed merger rate function does not permit an exact solution of Smoluchowski’s equation and, therefore, the choice of parameters must reflect a compromise between fitting various parts of the mass function. The techniques described herein are applicable to more general merger rate functions, which may permit a more accurate solution of Smoluchowski’s equation. The current merger rate solutions are most probably sufficiently accurate for the vast majority of applications.