A marked correlation function analysis of halo formation times in the Millennium Simulation

We study the environmental dependence of the formation epoch of dark matter haloes in the Millennium Simulation: a ten billion particle N-body simulation of standard Lambda cold dark matter cosmology. A sensitive test of this dependence - the marked correlation function - reveals highly significant evidence that haloes of a given mass form earlier in denser regions. We define a marked cross-correlation function, which helps quantify how this effect depends upon the choice of the halo population used to define the environment. The mean halo formation redshift as a function of the local overdensity in dark matter is also well determined, and we see an especially clear dependence for galaxy-sized haloes. This contradicts one of the basic predictions of the excursion set model of structure formation, even though we see that this theory predicts other features of the distribution of halo formation epochs rather well. It also invalidates an assumption usually employed in the popular halo, or halo occupation distribution, models of galaxy clustering, namely that the distribution of halo properties is a function of halo mass but not of halo environment.