Dark matter haloes are biased tracers of the underlying dark matter distribution. We use a simple model to provide a relation between the abundance of dark matter haloes and their spatial distribution on large scales. Our model shows that knowledge of the unconditional mass function alone is sufficient to provide an accurate estimate of the large-scale bias factor. We then use the mass function measured in numerical simulations of SCDM, OCDM and LambdaCDM to compute this bias. Comparison with these simulations shows that this simple way of estimating the bias relation and its evolution is accurate for less massive haloes as well as massive ones. In particular, we show that haloes that are less/more massive than typical M_* haloes at the time they form are more/less strongly clustered than is predicted by formulae based on the standard Press-Schechter mass function

A B S T R A C TWe simulate the assembly of a massive rich cluster and the formation of its constituent galaxies in a flat, low-density universe. Our most accurate model follows the collapse, the star formation history and the orbital motion of all galaxies more luminous than the Fornax dwarf spheroidal, while dark halo structure is tracked consistently throughout the cluster for all galaxies more luminous than the SMC. Within its virial radius this model contains about 2 Â 10 7 dark matter particles and almost 5000 distinct dynamically resolved galaxies. Simulations of this same cluster at a variety of resolutions allow us to check explicitly for numerical convergence both of the dark matter structures produced by our new parallel N-body and substructure identification codes, and of the galaxy populations produced by the phenomenological models we use to follow cooling, star formation, feedback and stellar aging. This baryonic modelling is tuned so that our simulations reproduce the observed properties of isolated spirals outside clusters. Without further parameter adjustment our simulations then produce a luminosity function, a mass-to-light ratio, luminosity, number and velocity dispersion profiles, and a morphology -radius relation which are similar to those observed in real clusters. In particular, since our simulations follow galaxy merging explicitly, we can demonstrate that it accounts quantitatively for the observed cluster population of bulges and elliptical galaxies.

The Press-Schechter, excursion set approach allows one to make predictions about the shape and evolution of the mass function of bound objects. The approach combines the assumption that objects collapse spherically with the assumption that the initial density fluctuations were Gaussian and small. The predicted mass function is reasonably accurate, although it has fewer high-mass and more low-mass objects than are seen in simulations of hierarchical clustering. We show that the discrepancy between theory and simulation can be reduced substantially if bound structures are assumed to form from an ellipsoidal, rather than a spherical, collapse. In the original, standard, spherical model, a region collapses if the initial density within it exceeds a threshold value, δsc. This value is independent of the initial size of the region, and since the mass of the collapsed object is related to its initial size, this means that δsc is independent of final mass. In the ellipsoidal model, the collapse of a region depends on the surrounding shear field, as well as on its initial overdensity. In Gaussian random fields, the distribution of these quantities depends on the size of the region considered. Since the mass of a region is related to its initial size, there is a relation between the density threshold value required for collapse and the mass of the final object. We provide a fitting function to this δec(m) relation which simplifies the inclusion of ellipsoidal dynamics in the excursion set approach. We discuss the relation between the excursion set predictions and the halo distribution in high-resolution N-body simulations, and use our new formulation of the approach to show that our simple parametrization of the ellipsoidal collapse model represents an improvement on the spherical model on an object-by-object basis. Finally, we show that the associated statistical predictions, the mass function and the large-scale halo-to-mass bias relation, are also more accurate than the standard predictions

The excursion set approach allows one to estimate the abundance and spatial distribution of virialized dark matter haloes efficiently and accurately. The predictions of this approach depend on how the non-linear processes of collapse and virialization are modelled. We present simple analytic approximations that allow us to compare the excursion set predictions associated with spherical and ellipsoidal collapse. In particular, we present formulae for the universal unconditional mass function of bound objects and the conditional mass function which describes the mass function of the progenitors of haloes in a given mass range today. We show that the ellipsoidal collapse based moving barrier model provides a better description of what we measure in the numerical simulations than the spherical collapse based constant barrier model, although the agreement between model and simulations is better at large lookback times. Our results for the conditional mass function can be used to compute accurate approximations to the local-density mass function, which quantifies the tendency for massive haloes to populate denser regions than less massive haloes. This happens because low-density regions can be thought of as being collapsed haloes viewed at large lookback times, whereas high-density regions are collapsed haloes viewed at small lookback times. Although we have applied our analytic formulae only to two simple barrier shapes, we show that they are, in fact, accurate for a wide variety of moving barriers. We suggest how they can be used to study the case in which the initial dark matter distribution is not completely cold