tokens: (e.g. Dalal et al. 2008 ; Desjacques Sheth 2010 ). Combination of the peaks approach plus N -body simulations led to further insights, including that haloes in dense regions that do not correspond to high peaks in the initial conditions accrete more slowly than haloes that do correspond to high peaks (Ludlow Porciani 2011 ). Faster supercomputers and better codes have led to improved N -body simulations, in which the haloes and subhaloes that host galaxies, groups and clusters are resolved. Such simulations have permitted more detailed analyses of the correlations of halo properties with each other and with the halo environments measured in various ways. One of the first papers to do this was Lemson Kauffmann ( 1999 ), which used Ω 0 = 1 τCDM and Ω 0 = 0.3 ΛCDM simulations and concluded that only the halo mass distribution varies as a function of environment, with more high-mass haloes in denser environments, in reasonable agreement with the analytic calculation by Mo White ( 1996 ) based on extended Press–Schechter theory and the spherical top-hat model. While subsequent N -body calculations confirmed this density effect on the halo mass function, they implied that environmental density also affects halo major merging rates (Gottlöber, Klypin Kravtsov 2001 ) and other halo properties. Bullock et al. ( 2001 ) found that haloes in dense environments tend to have higher concentrations than isolated haloes. Sheth Tormen ( 2004 ), using the same simulations as Lemson Kauffmann ( 1999 ), found evidence that low-mass haloes form somewhat earlier in dense environments. Avila-Reese et al. ( 1999 ) found that dark matter haloes that are isolated or in intermediate-density environments have outer density profiles r −β with β ≈ 2.8 ± 0.5 while haloes in denser regions have a wider range of β up to ∼5. Avila-Reese et al. ( 2005 ), again using ΛCDM simulations that were very small by modern standards, nevertheless found that haloes in dense regions had lower spin parameters and higher concentrations and were less prolate than haloes in lower density environments, with the differences arising mostly at low redshifts z 1 from phenomena such as tidal stripping in dense environments. In this paper, we confirm and expand on these results. Maulbetsch et al. ( 2007 ) used a 512 3 particle ΛCDM simulation in a (50 h −1 Mpc) 3 volume to study the mass accretion history of galaxy-mass haloes in different environments, and found that haloes of the same final mass accreted their mass earlier in denser environments, and also accreted a significantly higher fraction of their mass in major mergers. They suggested that this could help to explain the galaxy density–morphology relation and that early-type galaxies are more common in dense environments. They also found that ∼80 per cent of haloes in higher density environments are not accreting (i.e. have ), while this fraction is only ∼20 per cent in low-density environments. They defined the environmental density on a
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