A nonlinear and biased view of dark matters

Studies of the peculiar velocities of galaxies and clusters of galaxies suggest that much of the matter of the universe is dark. The spatial distribution of light, the galaxies, is most probably not the same as the distribution of the dark matter. I present a very biased view of how these two distributions are related. In this view, galaxies form in gravita-tionally bound dark matter haloes, and the formation of galaxies in one halo is independent of what happens in other, more distant haloes. Thus, galaxies are biased tracers of the dark halo distribution, and the haloes themselves are biased tracers of the underlying dark matter distribution. The spatial distribution of haloes at any given time is closely related to the way in which they were assembled from smaller haloes which existed earlier. 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 is stochastic. This stochastic biasing can be described simply and analytically, even though it is, in general, nonlinear, scale-dependent and time-dependent. At any time, the halo-to-mass bias depends only on the halo mass: on large scales, less massive haloes are less strongly clustered than massive ones. This mass dependence depends on how haloes formed: if they formed from an ellipsoidal collapse, then the model describes the number and spatial distribution of haloes more accurately than if they formed from a spherical collapse. A simple model for the galaxy distribution follows easily from this halo-to-mass bias model if one assumes that the number of galaxies per halo is a deterministic function of halo mass. 1. Biasing and the forest of merger history trees It is thought that the large scale structures which we see today grew gravitation-ally from smaller initial uctuations; initially denser regions grew denser, while less dense regions became still less dense. 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 to construct the nal object. Not all objects of a given mass need have formed in the same way: there is a forest of diierent possible merger history trees. Sheth (1996b) showed …