An improved model for the formation times of dark matter haloes

A dark matter halo is said to have formed when at least half its mass has been assembled into a single progenitor. With this definition, it is possible to derive a simple but useful analytic estimate of the distribution of halo formation times. The standard estimate of this distribution depends on the shape of the conditional mass function - the distribution of progenitor masses of a halo as a function of time. If the spherical collapse model is used to estimate the progenitor mass function, then the formation times one infers systematically underestimate those seen in numerical simulations of hierarchical gravitational clustering. We provide estimates of halo formation which may be related to an ellipsoidal collapse model. These estimates provide a substantially better description of the simulations. We also provide an alternative derivation of the formation time distribution which is based on the assumption that haloes increase their mass through binary mergers only. Our results are useful for models which relate halo structure to halo formation.