The cold dark matter model has become the leading theoretical picture for the formation of structure in the Universe. This model, together with the theory of cosmic inflation, makes a clear prediction for the initial conditions for structure formation and predicts that structures grow hierarchically through gravitational instability. Testing this model requires that the precise measurements delivered by galaxy surveys can be compared to robust and equally precise theoretical calculations. Here we present a simulation of the growth of dark matter structure using 2,160(3) particles, following them from redshift z = 127 to the present in a cube-shaped region 2.230 billion lightyears on a side. In postprocessing, we also follow the formation and evolution of the galaxies and quasars. We show that baryon-induced features in the initial conditions of the Universe are reflected in distorted form in the low-redshift galaxy distribution, an effect that can be used to constrain the nature of dark energy with future generations of observational surveys of galaxies.

We simulate the growth of galaxies and their central supermassive black holes by implementing a suite of semi‐analytic models on the output of the Millennium Run, a very large simulation of the concordance Λ cold dark matter cosmogony. Our procedures follow the detailed assembly history of each object and are able to track the evolution of all galaxies more massive than the Small Magellanic Cloud throughout a volume comparable to that of large modern redshift surveys. In this first paper we supplement previous treatments of the growth and activity of central black holes with a new model for ‘radio’ feedback from those active galactic nuclei that lie at the centre of a quasi‐static X‐ray‐emitting atmosphere in a galaxy group or cluster. We show that for energetically and observationally plausible parameters such a model can simultaneously explain: (i) the low observed mass drop‐out rate in cooling flows; (ii) the exponential cut‐off at the bright end of the galaxy luminosity function; and (iii) the fact that the most massive galaxies tend to be bulge‐dominated systems in clusters and to contain systematically older stars than lower mass galaxies. This success occurs because static hot atmospheres form only in the most massive structures, and radio feedback (in contrast, for example, to supernova or starburst feedback) can suppress further cooling and star formation without itself requiring star formation. We discuss possible physical models that might explain the accretion rate scalings required for our phenomenological ‘radio mode’ model to be successful.

We use the Millennium Simulation (MS) to study the statistics of Λ cold dark matter (ΛCDM) halo concentrations at z= 0. Our results confirm that the average halo concentration declines monotonically with mass; the concentration–mass relation is well fitted by a power law over three decades in mass, up to the most massive objects that form in a ΛCDM universe (∼ 1015 h−1 M⊙). This is in clear disagreement with the predictions of the model proposed by Bullock et al. for these rare objects, and agrees better with the original predictions of Navarro, Frenk & White. The large volume surveyed, together with the unprecedented numerical resolution of the MS, allows us to estimate with confidence the distribution of concentrations and, consequently, the abundance of systems with unusual properties. About one in a hundred cluster haloes (M200≳ 3 × 1014 h−1 M⊙) have concentrations exceeding c200= 7.5, a result that may be useful in interpreting the likelihood of unusually strong massive gravitational lenses, such as Abell 1689, in the ΛCDM cosmogony. A similar fraction of about 1 per cent of galaxy‐sized haloes (M200∼ 1012 h−1 M⊙) have c200 < 4.5 and this could be relevant to models that attempt to reconcile the ΛCDM cosmology with rotation curves of low surface brightness galaxies by appealing to haloes of unexpectedly low concentration. We find that halo concentrations are independent of spin once haloes manifestly out of equilibrium have been removed from the sample. Compared to their relaxed brethren, the concentrations of out‐of‐equilibrium haloes tend to be lower and have more scatter, while their spins tend to be higher. A number of previously noted trends within the halo population are induced primarily by these properties of unrelaxed systems. Finally, we compare the result of predicting halo concentrations using the mass assembly history of the main progenitor with predictions based on simple arguments regarding the assembly time of all progenitors. The latter are typically as good or better than the former, suggesting that halo concentration depends not only on the evolutionary path of a halo's main progenitor, but on how and when all of its constituents collapsed to form non‐linear objects.

We investigate the subhalo populations of dark matter haloes in the concordance Λ cold dark matter (ΛCDM) cosmology. We use a large cosmological simulation and a variety of high‐resolution resimulations of individual cluster and galaxy haloes to study the systematics of subhalo populations over ranges of 1000 in halo mass and 1000 in the ratio of subhalo to parent halo mass. The subhalo populations of different haloes are not scaled copies of each other, but vary systematically with halo properties. On average, the amount of substructure increases with halo mass. At fixed mass, it decreases with halo concentration and with halo formation redshift. These trends are comparable in size to the scatter in subhalo abundance between similar haloes. Averaged over all haloes of given mass, the abundance of low‐mass subhaloes per unit parent halo mass is independent of parent mass. It is very similar to the abundance per unit mass of low‐mass haloes in the Universe as a whole, once differing boundary definitions for subhaloes and haloes are accounted for. The radial distribution of subhaloes within their parent haloes is substantially less centrally concentrated than that of the dark matter. It varies at most weakly with the mass (or concentration) of the parent halo and not at all with subhalo mass. It does depend on the criteria used to define the subhalo population considered. About 90 per cent of present‐day subhaloes were accreted after z= 1 and about 70 per cent after z= 0.5. Only about 8 per cent of the total mass of all haloes accreted at z= 1 survives as bound subhaloes at z= 0. For haloes accreted at z= 2, the survival mass fraction is just 2 per cent. Subhaloes seen near the centre of their parent typically were accreted earlier and retain less of their original mass than those seen near the edge. These strong systematics mean that comparison with galaxies in real clusters is only possible if the formation of the luminous component is modelled appropriately.