Coagulation and branching process models of gravitational clustering

Smoluchowski's coagulation equation describes the growth of clustering in a system in which clusters coalesce irreversibly by binary mergers. If the rate with which two objects merge is proportional to the sum of their masses, and the process starts with all clusters having the same mass initially (monodisperse initial conditions), then the growth of clustering in the system can be derived from a Poisson Galton-Watson branching process. The branching process is consistent with, and provides a more detailed description of gravitational clustering from Poisson initial conditions than, the excursion set, Press-Schechter approach. The correspondence between the branching process and additive coagulation schemes is used to efficiently simulate realizations of the merger history tree. These results are easily extended to describe clustering from white-noise Gaussian initial conditions. Extending this approach to describe clustering from Gaussian fields with arbitrary power spectra is more complicated. To illustrate this, Smoluchowski's additive coagulation equation with arbitrary discrete initial conditions is solved. The result is used to argue that if Smoluchowski's equation is to describe the growth of clustering from arbitrary Gaussian fluctuation fields, and if it is to be consistent with the excursion set results, then the merger rule must depend on the power spectrum. However, with the exception of the white-noise case, the excursion set approach is formally inconsistent with the binary merger assumption, although, for all power spectra of current interest, binary mergers remain a reasonable approximation. This means that, except for the white-noise case, an analogue of the Smoluchowski coagulation approach cannot provide a description of clustering that is exactly equivalent to the excursion set description. Nevertheless, since binary mergers remain a good approximation, Smoluchowski's equation, with a merger rate that can be determined from the excursion set approach, so that it is a calculable function of the power spectrum, can provide a useful approximate description of the growth of clustering.