On the environmental dependence of halo formation

ABSTRACT A generic prediction of hierarchical gravitationalclustering models is that the distribu-tion of halo formation times should depend relatively strongly on halo mass, massivehaloes forming more recently, and depend only weakly, if at all, on the large scaleenvironment of the haloes. We present a novel test of this assumption which uses thestatistics of weighted or ‘marked’ correlations, which prove to be particularly well-suited to detecting and quantifying weak correlations with environment. We ﬁnd thatclose pairs of haloes form at slightly higher redshifts than do more widely separatedhalo pairs, suggesting that haloes in dense regions form at slightly earlier times thando haloes of the same mass in less dense regions. The environmental trends we ﬁndare useful for models which relate the properties of galaxies to the formation historiesof the haloes which surround them.Key words: galaxies: clustering – cosmology: theory – dark matter. 1 INTRODUCTIONThe excursion set model of hierarchical clustering (Epstein1983; Bond et al. 1991) has been remarkably successful. Itprovides useful analytic approximations for the abundanceof haloes of mass m at time t (Bond et al. 1991; Sheth,Mo & Tormen 2001), for the conditional mass function ofm haloes at t which are later (at T > t) in more massivehaloes M > m (Bond et al. 1991; Lacey & Cole 1993; Sheth& Tormen 2002), for the abundance of haloes as a functionof the larger scale environment (Mo & White 1996; Lemson& Kauﬀmann 1999; Sheth & Tormen 2002) for the distribu-tion of halo formation times (Lacey & Cole 1993) and masses(Nusser & Sheth 1999; Sheth & Tormen 2004). Here, forma-tion is typically deﬁned as that time when the most massiveprogenitor contains at least half the ﬁnal mass.In the simplest and most used approximation, this ap-proach ignores most correlations between diﬀerent spatialscales. In this approximation, the approach predicts thatthere should be no correlation between halo formation andthe large scale environment in which the halo sits (White1996). This is because, in the model, formation refers toa smaller mass than the ﬁnal virial mass, and hence toa smaller spatial scale than that associated with the La-grangian radius of an object, whereas the larger scale envi-ronment, by deﬁnition, refers to scales which are larger thanthat of the halo.Lemson & Kauﬀmann (1999) presented evidence frommeasurements in numerical simulations of clustering thathalo formation times were indeed independent of environ-ment. They interpreted this as evidence that the excursionset neglect of correlations was justiﬁed. (Lemson & Kauﬀ-mann also presented evidence that a number of other physi-cal properties of haloes were also independent of environ-ment, and this evidence has been used to justify an as-sumption which enormously simpliﬁes semi-analytic mod-els of galaxy formation: that the properties of galaxies aredetermined by the haloes in which they form, and not bythe surrounding larger-scale environment.) Their conclusionis somewhat surprising for the following reason. It is quitewell established that the ratio of massive to low mass haloesis larger in dense regions, and that the excursion set modelis able to quantify this dependence quite well (see the refer-ences given earlier). It is also well established that, on aver-age, low mass haloes form at higher redshifts (see referencesgiven earlier). Together, these suggest that if one averagesover the entire range of halo masses in any given region,then the mean formation time in dense regions should beshifted to lower redshifts, simply because these regions con-tain more massive haloes which, on average, form later. InFigure 4 of their paper, Lemson & Kauﬀmann averaged overthe entire range of halo masses accessible to them in theirsimulations, and found no dependence of formation time onenvironement; at face value, this is inconsistent with thesimplest excursion set prediction! The main goal of this paper is to repeat the test for envi-ronmental eﬀects on halo formation. Section 2 shows that asimple plot of formation time versus local density does notshow strong trends, suggesting that the excursion set ap-proximation is rather accurate. But then, Section 3 presents