Dark matter subhaloes in numerical simulations

We use cosmological Λ cold dark matter (CDM) numerical simulations to model the evolution of the substructure population in 16 dark matter haloes with resolutions of up to seven million particles within the virial radius. The combined substructure circular velocity distribution function (VDF) for hosts of 10¹¹ to  10¹⁴ M_⊙  at redshifts from zero to two or higher has a self-similar shape, is independent of host halo mass and redshift, and follows the relation  d n /d v = (1/8)( v _cmax/ v _cmax,host)⁻⁴  . Halo to halo variance in the VDF is a factor of roughly 2 to 4. At high redshifts, we find preliminary evidence for fewer large substructure haloes (subhaloes). Specific angular momenta are significantly lower for subhaloes nearer the host halo centre where tidal stripping is more effective. The radial distribution of subhaloes is marginally consistent with the mass profile for   r ≳ 0.3 r _vir  , where the possibility of artificial numerical disruption of subhaloes can be most reliably excluded by our convergence study, although a subhalo distribution that is shallower than the mass profile is favoured. Subhalo masses but not circular velocities decrease towards the host centre. Subhalo velocity dispersions hint at a positive velocity bias at small radii. There is a weak bias towards more circular orbits at lower redshift, especially at small radii. We additionally model a cluster in several power-law cosmologies of   P ∝ kⁿ   , and demonstrate that a steeper spectral index, n , results in significantly less substructure.     

Analytical approach to subhalo population in dark matter haloes

In the standard model of cosmic structure formation, dark matter haloes form by gravitational instability. The process is hierarchical: smaller systems collapse earlier, and later merge to form larger haloes. The galaxy clusters, hosted by the largest dark matter haloes, are at the top of this hierarchy and representing the largest as well as the last structures formed in the Universe, while the smaller and first haloes are those Earth-sized dark subhaloes that have been both predicted by theoretical considerations and found in numerical simulations, though there do not exist any observational hints of their existence. The probability that a halo of mass m at redshift z will be part of a larger halo of mass M at the present time can be described in the frame of the extended Press & Schecter theory making use of the progenitor (conditional) mass function. Using the progenitor mass function, we calculate analytically, at redshift zero, the distribution of subhaloes in mass, formation epoch and rarity of the peak of the density field at the formation epoch. That is done for a Milky Way size system, assuming both a spherical and an ellipsoidal collapse model. Our calculation assumes that small progenitors do not lose mass due to dynamical processes after entering the parent halo, and that they do not interact with other subhaloes. For a Λ cold dark matter power spectrum, we obtain a subhalo mass function  d n /d m   proportional to   m ^−α  with a model-independent  α∼ 2  . Assuming that the dark matter is a weakly interacting massive particle, the inferred distributions are used to test the feasibility of an indirect detection in the γ-ray energy band of such a population of subhaloes with a Gamma-ray Large Area Space Telescope like satellite.     

The clustering and host haloes of galaxy mergers at high redshift

High-resolution simulations of cosmological structure formation indicate that dark matter substructure in dense environments, such as groups and clusters, may survive for a long time. These dark matter subhaloes are the likely hosts of galaxies. We examine the small-scale spatial clustering of subhalo major mergers at high redshift using high-resolution N -body simulations of cosmological volumes. Recently merged, massive subhaloes exhibit enhanced clustering on scales  ∼100–300  h ⁻¹ kpc  , relative to all subhaloes of the same infall mass, for a short time after a major merger (<500 Myr). The small-scale clustering enhancement is smaller for lower mass subhaloes, which also show a deficit on scales just beyond the excess. Haloes hosting recent subhalo mergers tend to have more subhaloes; for massive subhaloes, the excess is stronger and it tends to increase for the most massive host haloes. The subhalo merger fraction is independent of halo mass for the scales we probe. In terms of satellite and central subhaloes, the merger increase in small-scale clustering for massive subhaloes arises from recently merged massive central subhaloes having an enhanced satellite population. Our mergers are defined via their parent infall mass ratios. Subhaloes experiencing major mass gains also exhibit a small-scale clustering enhancement, but these correspond to two-body interactions leading to two final subhaloes, rather than subhalo coalescence.     

The fate of substructures in cold dark matter haloes

We use the Millennium Simulation, a large, high-resolution N -body simulation of the evolution of structure in a Λ cold dark matter cosmology, to study the properties and fate of substructures within a large sample of dark matter haloes. We find that the subhalo mass function departs significantly from a power law at the high-mass end. We also find that the radial and angular distributions of substructures depend on subhalo mass. In particular, high-mass subhaloes tend to be less radially concentrated and to have angular distributions closer to the direction perpendicular to the spin of the host halo than their less massive counterparts. We find that mergers between subhaloes occur. These tend to be between substructures that were already dynamically associated before accretion into the main halo. For subhaloes larger than 0.001 times the mass of the host halo, it is more likely that the subhalo will merge with the central or main subhalo than with another subhalo larger than itself. For lower masses, subhalo–subhalo mergers become equally likely to mergers with the main subhalo. Our results have implications for the variation of galaxy properties with environment and for the treatment of mergers in galaxy formation models.     

The concentration–velocity dispersion relation in galaxy groups

Based on results from cold dark matter N -body simulations, we develop a dynamical model for the evolution of subhaloes within group-sized host haloes. Only subhaloes more massive than 5 × 10⁸ M_⊙ are considered, because they are massive enough to possibly host luminous galaxies. On their orbits within a growing host potential the subhaloes are subject to tidal stripping and dynamical friction. At the present time  ( z = 0)  , all model hosts have equal mass  ( M _vir= 3.9 × 10¹³ M_⊙)  but different concentrations associated with different formation times. We investigate the variation of subhalo (or satellite galaxy) velocity dispersion with host concentration and/or formation time. In agreement with the Jeans equation, the velocity dispersion of subhaloes increases with the host concentration. Between concentrations of ∼5 and ∼20, the subhalo velocity dispersions increase by a factor of ∼1.25. By applying a simplified tidal disruption criterion, that is, rejection of all subhaloes with a tidal truncation radius below 3  kpc at   z = 0  , the central velocity dispersion of the 'surviving' subhalo sample increases substantially for all concentrations. The enhanced central velocity dispersions in the surviving subhalo samples are caused by a lack of slow tangential motions. Additionally, we present a fitting formula for the anisotropy parameter which does not depend on concentration if the group-centric distances are scaled by r _s, the characteristic radius of the Navarro, Frenk & White profile. Since the expected loss of subhaloes and galaxies due to tidal disruption increases the velocity dispersion of surviving galaxies, the observed galaxy velocity dispersion can substantially overestimate the virial mass.     

Simulating subhaloes at high redshift: merger rates, counts and types

Galaxies are believed to be in one-to-one correspondence with simulated dark matter subhaloes. We use high-resolution N -body simulations of cosmological volumes to calculate the statistical properties of subhalo (galaxy) major mergers at high redshift ( z = 0.6–5). We measure the evolution of the galaxy merger rate, finding that it is much shallower than the merger rate of dark matter host haloes at   z > 2.5  , but roughly parallels that of haloes at   z < 1.6  . We also track the detailed merger histories of individual galaxies and measure the likelihood of multiple mergers per halo or subhalo. We examine satellite merger statistics in detail: 15–35 per cent of all recently merged galaxies are satellites, and satellites are twice as likely as centrals to have had a recent major merger. Finally, we show how the differing evolution of the merger rates of haloes and galaxies leads to the evolution of the average satellite occupation per halo, noting that for a fixed halo mass, the satellite halo occupation peaks at   z ∼ 2.5  .     

The signature of dark energy on the local Hubble flow

Using N -body simulations of flat, dark energy-dominated cosmologies, we show that galaxies around simulated binary systems resembling the Local Group (LG) have low peculiar velocities, in good agreement with observational data. We have compared results for LG-like systems selected from large, high-resolution simulations of three cosmologies: a ΛCDM model, a ΛWDM model with a 2-keV warm dark matter candidate, and a quintessence (QCDM) model with an equation-of-state parameter   w =−0.6  . The Hubble flow is significantly colder around LGs selected in a flat, Λ-dominated cosmology than around LGs in open or critical models, showing that a dark energy component manifests itself on the scales of nearby galaxies, cooling galaxy peculiar motions. Flows in the ΛWDM and QCDM models are marginally colder than in the ΛCDM one.
The results of our simulations have been compared to existing data and to a new data set of 28 nearby galaxies with robust distance measures (Cepheids and surface brightness fluctuations). The measured line-of-sight velocity dispersion is given by  σ_H= (88 ± 20  km s⁻¹) × ( R /7 Mpc)  . The best agreement with observations is found for LGs selected in the ΛCDM cosmology in environments with  −0.1 < δρ/ρ < 0.6  on scales of 7 Mpc, in agreement with existing observational estimates on the local matter density. These results provide new, independent evidence for the presence of dark energy on scales of a few megaparsecs, corroborating the evidence gathered from observations of distant objects and the early Universe.     

The formation and survival of discs in a ΛCDM universe

We study the formation of galaxies in a Λ cold dark matter (ΛCDM) universe using high-resolution hydrodynamical simulations with a multiphase treatment of gas, cooling and feedback, focusing on the formation of discs. Our simulations follow eight isolated haloes similar in mass to the Milky Way and extracted from a large cosmological simulation without restriction on spin parameter or merger history. This allows us to investigate how the final properties of the simulated galaxies correlate with the formation histories of their haloes. We find that, at   z = 0  , none of our galaxies contains a disc with more than 20 per cent of its total stellar mass. Four of the eight galaxies nevertheless have well-formed disc components, three have dominant spheroids and very small discs, and one is a spheroidal galaxy with no disc at all. The   z = 0  spheroids are made of old stars, while discs are younger and formed from the inside-out. Neither the existence of a disc at   z = 0  nor the final disc-to-total mass ratio seems to depend on the spin parameter of the halo. Discs are formed in haloes with spin parameters as low as 0.01 and as high as 0.05; galaxies with little or no disc component span the same range in spin parameter. Except for one of the simulated galaxies, all have significant discs at   z ≳ 2  , regardless of their   z = 0  morphologies. Major mergers and instabilities which arise when accreting cold gas is misaligned with the stellar disc trigger a transfer of mass from the discs to the spheroids. In some cases, discs are destroyed, while in others, they survive or reform. This suggests that the survival probability of discs depends on the particular formation history of each galaxy. A realistic ΛCDM model will clearly require weaker star formation at high redshift and later disc assembly than occurs in our models.     

The dynamics of subhaloes in warm dark matter models

We present a comparison of the properties of substructure haloes ( subhaloes ) orbiting within host haloes that form in cold dark matter (CDM) and warm dark matter (WDM) cosmologies. Our study focuses on selected properties of these subhaloes, namely their anisotropic spatial distribution within the hosts; the existence of a 'backsplash' population; the age–distance relation; the degree to which they suffer mass loss; and the distribution of relative (infall) velocities with respect to the hosts. We find that the number density of subhaloes in our WDM model is suppressed relative to that in the CDM model, as we would expect. Interestingly, our analysis reveals that backsplash subhaloes exist in both the WDM and CDM models. Indeed, there are no statistically significant differences between the spatial distributions of subhaloes in the CDM and WDM models. There is evidence that subhaloes in the WDM model suffer enhanced mass loss relative to their counterparts in the CDM model, reflecting their lower central densities. We note also a tendency for the (infall) velocities of subhaloes in the WDM model to be higher than in the CDM model. Nevertheless, we conclude that observational tests based on either the spatial distribution or the kinematics of the subhalo population are unlikely to help us to differentiate between the CDM model and our adopted WDM model.     

The role of minor mergers in the recent star formation history of early-type galaxies

We demonstrate that the large scatter in the ultraviolet (UV) colours of intermediate-mass early-type galaxies in the local Universe and the inferred low-level recent star formation (RSF) in these objects can be reproduced by minor mergers in the standard Λ cold dark matter (ΛCDM) cosmology. Numerical simulations of mergers with mass ratios ≤1:4, with reasonable assumptions for the ages, metallicities and dust properties of the merger progenitors, produce good agreement with the observed UV colours of the early-type population, if the infalling satellites are assumed to have (cold) gas fractions ≥20 per cent. Early-types that satisfy  ( NUV − r ) ≲ 3.8  are likely to have experienced mergers with mass ratios between 1:4 and 1:6 within the last ∼1.5 Gyr, while those that satisfy  3.8 < ( NUV − r ) < 5.5  are consistent with either recent mergers with mass ratios ≤1:6 or mergers with higher mass ratios that occurred more than ∼1.5 Gyr in the past. We demonstrate that the early-type colour–magnitude relations and colour distributions, in both the UV and optical spectral ranges, are consistent with the expected frequency of minor merging activity in the standard ΛCDM cosmology at low redshift. We present a strong plausibility argument for minor mergers to be the principal mechanism behind the large UV scatter and associated low-level RSF observed in early-type galaxies in the nearby Universe.     

Clustering of galaxy clusters in cold dark matter universes

We use very large cosmological N -body simulations to obtain accurate predictions for the two-point correlations and power spectra of mass-limited samples of galaxy clusters. We consider two currently popular cold dark matter (CDM) cosmogonies, a critical density model ( τ CDM) and a flat low density model with a cosmological constant (ΛCDM). Our simulations each use 10⁹ particles to follow the mass distribution within cubes of side 2  h ⁻¹ Gpc ( τ CDM) and 3  h ⁻¹ Gpc (ΛCDM) with a force resolution better than 10⁻⁴ of the cube side. We investigate how the predicted cluster correlations increase for samples of increasing mass and decreasing abundance. Very similar behaviour is found in the two cases. The correlation length increases from     for samples with mean separation     to     for samples with     The lower value here corresponds to τ CDM and the upper to ΛCDM. The power spectra of these cluster samples are accurately parallel to those of the mass over more than a decade in scale. Both correlation lengths and power spectrum biases can be predicted to better than 10 per cent using the simple model of Sheth, Mo & Tormen. This prediction requires only the linear mass power spectrum and has no adjustable parameters. We compare our predictions with published results for the automated plate measurement (APM) cluster sample. The observed variation of correlation length with richness agrees well with the models, particularly for ΛCDM. The observed power spectrum (for a cluster sample of mean separation     ) lies significantly above the predictions of both models.     

Power spectrum for the small-scale Universe

The first objects to arise in a cold dark matter (CDM) universe present a daunting challenge for models of structure formation. In the ultra small-scale limit, CDM structures form nearly simultaneously across a wide range of scales. Hierarchical clustering no longer provides a guiding principle for theoretical analyses and the computation time required to carry out credible simulations becomes prohibitively high. To gain insight into this problem, we perform high-resolution  ( N = 720³–1584³)  simulations of an Einstein–de Sitter cosmology where the initial power spectrum is   P ( k ) ∝ k ⁿ ,  with  −2.5 ≤ n ≤− 1  . Self-similar scaling is established for   n =−1  and −2 more convincingly than in previous, lower resolution simulations and for the first time, self-similar scaling is established for an   n =−2.25  simulation. However, finite box-size effects induce departures from self-similar scaling in our   n =−2.5  simulation. We compare our results with the predictions for the power spectrum from (one-loop) perturbation theory and demonstrate that the renormalization group approach suggested by McDonald improves perturbation theory's ability to predict the power spectrum in the quasi-linear regime. In the non-linear regime, our power spectra differ significantly from the widely used fitting formulae of Peacock & Dodds and Smith et al. and a new fitting formula is presented. Implications of our results for the stable clustering hypothesis versus halo model debate are discussed. Our power spectra are inconsistent with predictions of the stable clustering hypothesis in the high- k limit and lend credence to the halo model. Nevertheless, the fitting formula advocated in this paper is purely empirical and not derived from a specific formulation of the halo model.     

Resolving cosmic structure formation with the Millennium-II Simulation

We present the Millennium-II Simulation (MS-II), a very large N -body simulation of dark matter evolution in the concordance Λ cold dark matter (ΛCDM) cosmology. The MS-II assumes the same cosmological parameters and uses the same particle number and output data structure as the original Millennium Simulation (MS), but was carried out in a periodic cube one-fifth the size  (100  h ⁻¹ Mpc)  with five times better spatial resolution (a Plummer equivalent softening of  1.0  h ⁻¹ kpc  ) and with 125 times better mass resolution (a particle mass of  6.9 × 10⁶  h ⁻¹ M_⊙  ). By comparing results at MS and MS-II resolution, we demonstrate excellent convergence in dark matter statistics such as the halo mass function, the subhalo abundance distribution, the mass dependence of halo formation times, the linear and non-linear autocorrelations and power spectra, and halo assembly bias. Together, the two simulations provide precise results for such statistics over an unprecedented range of scales, from haloes similar to those hosting Local Group dwarf spheroidal galaxies to haloes corresponding to the richest galaxy clusters. The 'Milky Way' haloes of the Aquarius Project were selected from a lower resolution version of the MS-II and were then resimulated at much higher resolution. As a result, they are present in the MS-II along with thousands of other similar mass haloes. A comparison of their assembly histories in the MS-II and in resimulations of 1000 times better resolution shows detailed agreement over a factor of 100 in mass growth. We publicly release halo catalogues and assembly trees for the MS-II in the same format within the same archive as those already released for the MS.     

Voids in a ΛCDM universe

We study the formation and evolution of voids in the dark matter distribution using various simulations of the popular Λ cold dark matter cosmogony. We identify voids by requiring them to be regions of space with a mean overdensity of −0.8 or less – roughly the equivalent of using a spherical overdensity group finder for haloes. Each of the simulations contains thousands of voids. The distribution of void sizes in the different simulations shows good agreement when differences in particle and grid resolution are accounted for. Voids very clearly correspond to minima in the smoothed initial density field. Apart from a very weak dependence on the mass resolution, the rescaled mass profiles of voids in the different simulations agree remarkably well. We find a universal void mass profile of the form  ρ(< r )/ρ( r _eff) ∝ exp[( r / r _eff)^α]  , where r _eff is the effective radius of a void and  α∼ 2  . The mass function of haloes in voids is steeper than that of haloes that populate denser regions. In addition, the abundances of void haloes seem to evolve somewhat more strongly between redshifts ∼1 and 0 than the global abundances of haloes.     

Galactic halo cusp–core: tidal compression in mergers

We explain in simple terms how the build-up of dark haloes by merging compact satellites, as in the cold dark matter (CDM) cosmology, inevitably leads to an inner cusp of density profile  ρ∝ r ^−α  with  α≳ 1  , as seen in cosmological N -body simulations. A flatter halo core with  α < 1  exerts on the satellites tidal compression in all directions, which prevents the deposit of stripped satellite material in the core region. This makes the satellite orbits decay from the radius where  α∼ 1  to the halo centre with no local tidal mass transfer, and thus causes a rapid steepening of the inner profile to  α > 1  . These tidal effects, the resultant steepening of the profile to a cusp, and the stability of this cusp to tandem mergers with compact satellites are demonstrated using N -body simulations. The transition at  α∼ 1  is then addressed using toy models in the limiting cases of impulse and adiabatic approximations and using tidal radii for satellites on radial and circular orbits. In an associated paper, we address the subsequent slow convergence from either side to an asymptotic stable cusp with  α≳ 1  . Our analysis thus implies that an inner cusp is enforced when small haloes are typically more compact than larger haloes, as in the CDM scenario, such that enough satellite material makes it intact into the inner halo and is deposited there. We conclude that a necessary condition for maintaining a flat core, as indicated by observations, is that the inner regions of the CDM satellite haloes be puffed up by about 50 per cent such that when they merge into a larger halo they would be disrupted outside the halo core. This puffing up could be due to baryonic feedback processes in small haloes, which may be stimulated by the tidal compression in the halo cores.     

Constrained simulations of the local universe – II. The nature of the local Hubble flow

Using a suite of N -body simulations in different cold dark matter (CDM) scenarios, with cosmological constant (ΛCDM) and without (OCDM, SCDM), we study the Hubble flow (σ_H) in Local Volumes (LV) around Local Group (LG) like objects found in these simulations, and compare the numerical results with the most recent observations. We show that ΛCDM and OCDM models exhibit the same behaviour of σ_H. Hence, we demonstrate that the observed coldness of the Hubble flow is not likely to be a manifestation of the dark energy, contrary to previous claims. The coldness does not constitute a problem by itself but it poses a problem to the standard ΛCDM model only if the mean density within the LV is greater than twice the mean matter cosmic density. The lack of blueshifted galaxies in the LV, outside of the LG can be considered as another manifestation of the coldness of the flow. Finally, we show that the main dynamical parameter that affects the coldness of the flow is the relative isolation of the LG, and the absence of nearby Milky Way like objects within a distance of about  3 Mpc  .     

Dark matter halo profiles in scale-free cosmologies

We explore the dependence of the central logarithmic slope of dark matter halo density profiles α on the spectral index n of the linear matter power spectrum P ( k ) using cosmological N -body simulations of scale-free models [i.e. P ( k ) ∝ k ⁿ ]. These simulations are based on a set of clear, reproducible and physically motivated criteria that fix the appropriate starting and stopping times for runs, and allow one to compare haloes across models with different spectral indices and mass resolutions. For each of our simulations we identify samples of well-resolved haloes in dynamical equilibrium and we analyse their mass profiles. By parametrizing the mass profile using a 'generalized' Navarro, Frenk & White profile in which the central logarithmic slope α is allowed to vary while preserving the r ⁻³ asymptotic form at large radii, we obtain preferred central slopes for haloes in each of our models. There is a strong correlation between α and n , such that α becomes shallower as n becomes steeper. However, if we normalize our mass profiles by r ₋₂, the radius at which the logarithmic slope of the density profile is −2, we find that these differences are no longer present. This is apparent if we plot the maximum slope     as a function of r / r ₋₂– we find that the profiles are similar for haloes forming in different n models. This reflects the importance of concentration, and reveals that the concentrations of haloes forming in steep- n cosmologies tend to be smaller than those of haloes forming in shallow- n cosmologies. We conclude that there is no evidence for convergence to a unique central asymptotic slope, at least on the scales that we can resolve.     

Strong-lensing optical depths in a ΛCDM universe – II. The influence of the stellar mass in galaxies

We investigate how strong gravitational lensing in the concordance ΛCDM cosmology is affected by the stellar mass in galaxies. We extend our previous studies, based on ray tracing through the Millennium Simulation, by including the stellar components predicted by galaxy formation models. We find that the inclusion of these components greatly enhances the probability for strong lensing compared to a 'dark matter only' universe. The identification of the 'lenses' associated with strong-lensing events reveals that the stellar mass of galaxies (i) significantly enhances the strong-lensing cross-sections of group and cluster haloes and (ii) gives rise to strong lensing in smaller haloes, which would not produce noticeable effects in the absence of the stars. Even if we consider only image splittings ≳10 arcsec, the luminous matter can enhance the strong-lensing optical depths by up to a factor of 2.     

Topology and geometry of the CfA2 redshift survey

We analyse the redshift space topology and geometry of the nearby Universe by computing the Minkowski functionals of the Updated Zwicky Catalogue (UZC). The UZC contains the redshifts of almost 20 000 galaxies, is 96 per cent complete to the limiting magnitude m _Zw=15.5, and includes the Center for Astrophysics (CfA) Redshift Survey (CfA2). From the UZC we can extract volume-limited samples reaching a depth of 70  h ⁻¹ Mpc before sparse sampling dominates. We quantify the shape of the large-scale galaxy distribution by deriving measures of planarity and filamentarity from the Minkowski functionals. The nearby Universe shows a large degree of planarity and a small degree of filamentarity. This quantifies the sheet-like structure of the Great Wall, which dominates the northern region (CfA2N) of the UZC. We compare these results with redshift space mock catalogues constructed from high-resolution N -body simulations of two cold dark matter (CDM) models with either a decaying massive neutrino ( τ CDM) or a non-zero cosmological constant (ΛCDM). We use semi-analytic modelling to form and evolve galaxies in these dark matter‐only simulations. We are thus able, for the first time, to compile redshift space mock catalogues which contain galaxies, along with their observable properties, rather than dark matter particles alone. In both models the large-scale galaxy distribution is less coherent than the observed distribution, especially with regard to the large degree of planarity of the real survey. However, given the small volume of the region studied, this disagreement can still be a result of cosmic variance, as shown by the agreement between the ΛCDM model and the southern region of CfA2.