Stochastic biasing and the forest of merger history trees

Gravitationally bound dark matter haloes are biased tracers of the underlying dark matter distribution. I summarize a model in which the problem of describing the spatial distribution of haloes at any given time is closely related to the problem of describing the way in which dark matter haloes are assembled from smaller haloes which formed at earlier times. In the model, if not all haloes formed similarly, that is, if there is some scatter among diierent halo merger histories, then the bias between the halo and dark matter distributions will be stochastic. This stochastic biasing can be described analytically, if the initial uctuations were Gaussian distributed and small. 1. The forest of merger history trees It is commonly believed that the large scale structures that we see today grew gravitationally from smaller initial uctuations; initially denser regions grew denser, while less dense regions became still less dense, a sort of cosmic capitalism ! In this model, structure grows hierarchically: objects which virialize at late times form by the merging of smaller ones which themselves formed at earlier times, from mergers of still smaller objects which existed even earlier. Associated with any object is a merger history tree which describes the order in which smaller pieces were assembled as the larger object was constructed. Absent any other information about the initial building blocks and the subsequent assembly process, it is clear that not all objects need have formed in the same way; objects of the same nal mass may well have had diierent formation histories. Thus, associated with objects of a given mass is a forest of diierent possible merger history trees. Can one determine if this forest is comprised of trees of primarily one type, or is one type of tree as likely as another? Sheth (1996) showed that the statistics of the initial uctuation eld can be used to provide a detailed answer to this question. The logic behind his argument is illustrated below. Fig. 1 is a schematic plot of the merger history of an object. Time increases with increasing height, so that the object was initially made up of sixteen small objects. The gure serves two purposes. The rst is to show how the initial spatial distribution of the mass which makes up an object determines the merger history of the object. This is illustrated by the fact that the merger history tree has two main branches. …