On the assembly history of dark matter haloes

We study the mass assembly history (MAH) of dark matter haloes. We compare MAHs obtained using (i) merger trees constructed with the extended Press–Schechter (EPS) formalism, (ii) numerical simulations and (iii) the Lagrangian perturbation code pinocchio . We show that the pinocchio MAHs are in excellent agreement with those obtained using numerical simulations, while the EPS formalism predicts MAHs that occur too late. pinocchio , which is much less CPU intensive than N -body simulation, can be run on a simple personal computer, and does not require any labour intensive post-simulation analysis, therefore provides a unique and powerful tool to investigate the growth history of dark matter haloes. Using a suite of 55 pinocchio simulations, with 256³ particles each, we study the MAHs of 12 924 cold dark matter (CDM) haloes in a ΛCDM concordance cosmology. This is by far the largest set of haloes used for any such analysis. For each MAH we derive four different formation redshifts, which characterize different epochs during the assembly history of a dark matter halo. We show that haloes less massive than the characteristic non-linear mass scale establish their potential wells much before they acquire most of their mass. The time when a halo reaches its maximum virial velocity roughly divides its mass assembly into two phases, a fast-accretion phase which is dominated by major mergers, and a slow-accretion phase dominated by minor mergers. Each halo experiences about 3 ± 2 major mergers since its main progenitor had a mass equal to 1 per cent of the final halo mass. This major merger statistic is found to be virtually independent of halo mass. However, the average redshift at which these major mergers occur is strongly mass dependent, with more massive haloes experiencing their major mergers later.     

The build-up of haloes within Press–Schechter theory

Modelling the build-up of haloes is important for linking the formation of galaxies with cosmological models. A simple model of halo growth is provided by Press–Schechter (PS) theory, where the initial field of density fluctuations is smoothed using spherically symmetric filters centred on a given position to obtain information about the likelihood of later collapse on varying scales. In this paper the predicted halo mass growth is compared for three filter shapes: Gaussian, top-hat and sharp k -space. Preliminary work is also presented analysing the build-up of haloes within numerical simulations using a friends-of-friends group finder. The best-fit to the simulation mass function was obtained using PS theory with a top-hat filter. By comparing both the backwards conditional mass function, which gives the distribution of halo progenitors, and the distribution of halo mergers in time, the build-up of haloes in the simulations is shown to be better fitted by PS theory with a sharp k -space filter. This strengthens previous work, which also found the build-up of haloes in simulations to be well matched to PS theory with a sharp k -space filter by providing a direct comparison of different filters and by extending the statistical tools used to analyse halo mass growth. The usefulness of this work is illustrated by showing that the cosmological evolution in the proportion of haloes that have undergone recent merger is predicted to be independent of mass and power spectrum and to only depend upon cosmology. Recent results from observations of field galaxies are shown to match the evolution expected, but are not sufficiently accurate to distinguish usefully between cosmological parameters.     

Extended Press–Schechter theory and the density profiles of dark matter haloes

An inside–out model for the formation of haloes in a hierarchical clustering scenario is studied. The method combines the picture of the spherical infall model and a modification of the extended Press–Schechter theory. The mass accretion rate of a halo is defined to be the rate of its mass increase due to minor mergers. The accreted mass is deposited at the outer shells without changing the density profile of the halo inside its current virial radius. We applied the method to a flat Λ-cold dark matter universe. The resulting density profiles are compared with analytical models proposed in the literature, and a very good agreement is found. A trend is found of the inner density profile to become steeper for larger halo mass, which also results from recent N -body simulations. Additionally, present-day concentrations as well as their time evolution are derived and it is shown that they reproduce the results of large cosmological N -body simulations.     

Dual black holes in merger remnants – I. Linking accretion to dynamics

We study the orbital evolution and accretion history of massive black hole (MBH) pairs in rotationally supported circumnuclear discs up to the point where MBHs form binary systems. Our simulations have high resolution in mass and space which, for the first time, makes it feasible to follow the orbital decay of a MBH either counter- or corotating with respect to the circumnuclear disc. We show that a moving MBH on an initially counter-rotating orbit experiences an 'orbital angular momentum flip' due to the gas-dynamical friction, i.e. it starts to corotate with the disc before a MBH binary forms. We stress that this effect can only be captured in very high resolution simulations. Given the extremely large number of gas particles used, the dynamical range is sufficiently large to resolve the Bondi–Hoyle–Lyttleton radii of individual MBHs. As a consequence, we are able to link the accretion processes to the orbital evolution of the MBH pairs. We predict that the accretion rate is significantly suppressed and extremely variable when the MBH is moving on a retrograde orbit. It is only after the orbital angular momentum flip has taken place that the secondary rapidly 'lights up' at which point both MBHs can accrete near the Eddington rate for a few Myr. The separation of the double nucleus is expected to be around ≲10 pc at this stage. We show that the accretion rate can be highly variable also when the MBH is corotating with the disc (albeit to a lesser extent) provided that its orbit is eccentric. Our results have significant consequences for the expected number of observable double active galactic nuclei at separations of ≲100 pc.     

Modelling the cosmological co-evolution of supermassive black holes and galaxies – I. BH scaling relations and the AGN luminosity function

We model the cosmological co-evolution of galaxies and their central supermassive black holes (BHs) within a semi-analytical framework developed on the outputs of the Millennium Simulation. This model, described in detail by Croton et al. and De Lucia and Blaizot, introduces a 'radio mode' feedback from active galactic nuclei (AGN) at the centre of X-ray emitting atmospheres in galaxy groups and clusters. Thanks to this mechanism, the model can simultaneously explain: (i) the low observed mass dropout rate in cooling flows; (ii) the exponential cut-off in the bright end of the galaxy luminosity function and (iii) the bulge-dominated morphologies and old stellar ages of the most massive galaxies in clusters. This paper is the first of a series in which we investigate how well this model can also reproduce the physical properties of BHs and AGN. Here we analyse the scaling relations, the fundamental plane and the mass function of BHs, and compare them with the most recent observational data. Moreover, we extend the semi-analytic model to follow the evolution of the BH mass accretion and its conversion into radiation, and compare the derived AGN bolometric luminosity function with the observed one. While we find for the most part a very good agreement between predicted and observed BH properties, the semi-analytic model underestimates the number density of luminous AGN at high redshifts, independently of the adopted Eddington factor and accretion efficiency. However, an agreement with the observations is possible within the framework of our model, provided it is assumed that the cold gas fraction accreted by BHs at high redshifts is larger than at low redshifts.     

Testing tidal-torque theory – I. Spin amplitude and direction

We evaluate the success of linear tidal-torque theory (TTT) in predicting galactic-halo spin using a cosmological N -body simulation with thousands of well-resolved haloes. The protohaloes are identified by tracing today's haloes back to the initial conditions. The TTT predictions for the protohaloes match, on average, the spin amplitudes of the virialized haloes of today, if linear growth is assumed until ∼ t ₀/3, or  55–70  per cent of the halo effective turn-around time. This makes it a useful qualitative tool for understanding certain average properties of galaxies, such as total spin and angular momentum distribution within haloes, but with a random scatter of the order of the signal itself. Non-linear changes in spin direction cause a mean error of ∼50° in the TTT prediction at t ₀, such that the linear spatial correlations of spins on scales ≥1  h ⁻¹ Mpc are significantly weakened by non-linear effects. This questions the usefulness of TTT for predicting intrinsic alignments in the context of gravitational lensing. We find that the standard approximations made in TTT, including a second-order expansion of the Zel'dovich potential and a smoothing of the tidal field, provide close-to-optimal results.     

Merger rates of dark matter haloes

We derive analytic merger rates for dark matter haloes within the framework of the extended Press–Schechter (EPS) formalism. These rates become self-consistent within EPS once we realize that the typical merger in the limit of a small time-step involves more than two progenitors, contrary to the assumption of binary mergers adopted in earlier studies. We present a general method for computing merger rates that span the range of solutions permitted by the EPS conditional mass function, and focus on a specific solution that attempts to match the merger rates in N -body simulations. The corrected EPS merger rates are more accurate than the earlier estimates of Lacey & Cole by ∼20 per cent for major mergers and by up to a factor of ∼3 for minor mergers of mass ratio 1:10⁴. Based on the revised merger rates, we provide a new algorithm for constructing Monte Carlo EPS merger trees, which could be useful in semi-analytic modelling. We provide analytic expressions and plot numerical results for several quantities that are very useful in studies of galaxy formation. This includes (i) the rate of mergers of a given mass ratio per given final halo, (ii) the fraction of mass added by mergers to a halo and (iii) the rate of mergers per given main progenitor. The creation and destruction rates of haloes serve for a self-consistency check. Our method for computing merger rates can be applied to conditional mass functions beyond EPS, such as those obtained by the ellipsoidal collapse model or extracted from N -body simulations.     

Evolution of the mass function of dark matter haloes

We use a high-resolution ΛCDM numerical simulation to calculate the mass function of dark matter haloes down to the scale of dwarf galaxies, back to a redshift of 15, in a  50 h ⁻¹ Mpc  volume containing 80 million particles. Our low-redshift results allow us to probe low-σ density fluctuations significantly beyond the range of previous cosmological simulations. The Sheth & Tormen mass function provides an excellent match to all of our data except for redshifts of 10 and higher, where it overpredicts halo numbers increasingly with redshift, reaching roughly 50 per cent for the  10¹⁰–10¹¹ M_⊙  haloes sampled at redshift 15. Our results confirm previous findings that the simulated halo mass function can be described solely by the variance of the mass distribution, and thus has no explicit redshift dependence. We provide an empirical fit to our data that corrects for the overprediction of extremely rare objects by the Sheth & Tormen mass function. This overprediction has implications for studies that use the number densities of similarly rare objects as cosmological probes. For example, the number density of high-redshift  ( z ≃ 6) QSOs  , which are thought to be hosted by haloes at 5σ peaks in the fluctuation field, are likely to be overpredicted by at least a factor of 50 per cent. We test the sensitivity of our results to force accuracy, starting redshift and halo-finding algorithm.