The distribution of ejected subhaloes and its implication for halo assembly bias

Using a high-resolution cosmological N -body simulation, we identify the ejected population of subhaloes, which are haloes at redshift   z = 0  but were once contained in more massive 'host' haloes at high redshifts. The fraction of the ejected subhaloes in the total halo population of the same mass ranges from 9 to 4 per cent for halo masses from  ∼10¹¹  to  ∼10¹²  h ⁻¹ M_⊙  . Most of the ejected subhaloes are distributed within four times the virial radius of their hosts. These ejected subhaloes have distinct velocity distribution around their hosts in comparison to normal haloes. The number of subhaloes ejected from a host of given mass increases with the assembly redshift of the host. Ejected subhaloes in general reside in high-density regions, and have a much higher bias parameter than normal haloes of the same mass. They also have earlier assembly times, so that they contribute to the assembly bias of dark matter haloes seen in cosmological simulations. However, the assembly bias is not dominated by the ejected population, indicating that large-scale environmental effects on normal haloes are the main source for the assembly bias.     

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 graininess of dark matter haloes

We use the recently completed one billion particle Via Lactea II Λ cold dark matter simulation to investigate local properties like density, mean velocity, velocity dispersion, anisotropy, orientation and shape of the velocity dispersion ellipsoid, as well as the structure in velocity space of dark matter haloes. We show that at the same radial distance from the halo centre, these properties can deviate by orders of magnitude from the canonical, spherically averaged values, a variation that can only be partly explained by triaxiality and the presence of subhaloes. The mass density appears smooth in the central relaxed regions but spans four orders of magnitude in the outskirts, both because of the presence of subhaloes as well as of underdense regions and holes in the matter distribution. In the inner regions, the local velocity dispersion ellipsoid is aligned with the shape ellipsoid of the halo. This is not true in the outer parts where the orientation becomes more isotropic. The clumpy structure in local velocity space of the outer halo cannot be well described by a smooth multivariate normal distribution. Via Lactea II also shows the presence of cold streams made visible by their high 6D phase space density. Generally, the structure of dark matter haloes shows a high degree of graininess in phase space that cannot be described by a smooth distribution function.     

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 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.     

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.     

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  .     

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.     

Feedback and the formation of dwarf galaxy stellar haloes

Stellar population studies show that low-mass galaxies in all environments exhibit stellar haloes that are older and more spherically distributed than the main body of the galaxy. In some cases, there is a significant intermediate age component that extends beyond the young disc. We examine a suite of Smoothed Particle Hydrodynamic simulations and find that elevated early star formation activity combined with supernova feedback can produce an extended stellar distribution that resembles these haloes for model galaxies ranging from   v ₂₀₀= 15  to 35 km s⁻¹, without the need for accretion of subhaloes.     

The image separation distribution of strong lenses: halo versus subhalo populations

We present a halo model prediction of the image separation distribution of strong lenses. Our model takes into account the subhalo population, which has been ignored in previous studies, as well as the conventional halo population. Haloes and subhaloes are linked to central and satellite galaxies by adopting a universal scaling relation between masses of (sub)haloes and luminosities of galaxies. Our model predicts that 10–20 per cent of lenses should be caused by the subhalo population. The fraction of lensing by satellite galaxies (subhaloes) peaks at ∼1 arcsec and decreases rapidly with increasing image separations. We compute fractions of lenses which lie in groups and clusters and find them to be ∼14 and ∼4 per cent, respectively; nearly half of such lenses are expected to be produced by satellite galaxies, rather than central parts of haloes. We also study mass distributions of lensing haloes and find that, even at image separations of ∼3 arcsec, the deviation of lens mass distributions from isothermal profiles is large; at or beyond ∼3 arcsec, image separations are enhanced significantly by surrounding haloes. Our model prediction agrees reasonably well with observed image separation distributions from galaxy to cluster scales.     

Subhaloes in scale-free cosmologies

We explore the dependence of the subhalo mass function on the spectral index n of the linear matter power spectrum using scale-free Einstein-de Sitter simulations with   n =−1  and −2.5. We carefully consider finite volume effects that may call into question previous simulations of   n < −2  power spectra. Subhaloes are found using a 6D friends-of-friends algorithm in all haloes originating from high-σ peaks. For   n =−1  , we find that the cumulative subhalo mass function is independent of the parameters used in the subhalo finding algorithm and is consistent with the subhalo mass function found in Λ cold dark matter (ΛCDM) simulations. In particular, the subhalo mass function is well fit by a power-law with an index of  α=−0.9  , that is the mass function has roughly equal mass in subhaloes per logarithmic interval in subhalo mass. Conversely, for   n =−2.5  , the algorithm parameters affect the subhalo mass function since subhaloes are more triaxial with less well-defined boundaries. We find that the index α is generally larger with  α≳−0.75  . We infer that although the subhalo mass function appears to be independent of n so long as   n ≳−2  , it begins to flatten as   n →−3  . Thus, the common practice of using  α≈−1.0  may greatly overestimate the number of subhaloes at the smallest scales in the CDM hierarchy.     

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.     

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 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 haloes of merger remnants

We perform collisionless N -body simulations of 1:1 galaxy mergers, using models which include a galaxy halo, disc and bulge, focusing on the behaviour of the halo component. The galaxy models are constructed without recourse to a Maxwellian approximation. We investigate the effect of varying the galaxies' orientation, their mutual orbit and the initial velocity anisotropy or cusp strength of the haloes upon the remnant halo density profiles and shape, as well as on the kinematics. We observe that the halo density profile (determined as a spherical average, an approximation we find appropriate) is exceptionally robust in mergers, and that the velocity anisotropy of our remnant haloes is nearly independent of the orbits or initial anisotropy of the haloes. The remnants follow the halo anisotropy – local density slope (β–γ) relation suggested by Hansen & Moore in the inner parts of the halo, but β is systematically lower than this relation predicts in the outer parts. Remnant halo axis ratios are strongly dependent on the initial parameters of the haloes and on their orbits. We also find that the remnant haloes are significantly less spherical than those described in studies of simulations which include gas cooling.     

Star formation and chemical evolution of damped Lyman α systems

In this paper, we investigate the star formation and chemical evolution of damped Lyman α systems (DLAs) based on the disc galaxy formation model developed by Mo, Mao & White. We propose that the DLAs are the central galaxies of less-massive dark haloes present at redshifts z ∼3, and they should inhabit haloes of moderately low circular velocity. The empirical Schmidt law of star formation rates, and closed box model of chemical evolution that an approximation known as instantaneous recycling is assumed, are adopted. In our models, when the predicted distribution of metallicity for DLAs is calculated, two cases are considered. One is that, using the closed-box model, empirical Schmidt law and star formation time, the distribution of metallicity can be directly calculated. The other is that, when the simple gravitational instability of a thin isothermal gas disc as first discussed by Toomre is considered, the star formation occurs only in the region where the surface density of gas satisfies the critical value, not everywhere of a gas disc. In this case, we first obtain the region where the star formation can occur by assuming that the disc has a flat rotation curve and rotational velocity is equal to the circular velocity of the surrounding dark matter halo, and then calculate the metallicity distribution as in case one. We assume that star formation in each DLA lasts for a period of 1 Gyr from redshifts z =3. There is only one output parameter in our models, i.e. the stellar yield, which relates to the time of star formation history and is obtained by normalizing the predicted distribution of metallicity to the mean value of 1/13 Z_⊙ as presented by Pettini et al.. The predicted metallicity distribution is consistent with the current (rather limited) observational data. A random distribution of galactic discs is taken into account.     

Tracing the nature of dark energy with galaxy distribution

Dynamical dark energy (DE) is a viable alternative to the cosmological constant. Constructing tests to discriminate between Λ and dynamical DE models is difficult, however, because the differences are not large. In this paper we explore tests based on the galaxy mass function, the void probability function (VPF), and the number of galaxy clusters. At high z , the number density of clusters shows large differences between DE models, but geometrical factors reduce the differences substantially. We find that detecting a model dependence in the cluster redshift distribution is a significant challenge. We show that the galaxy redshift distribution is potentially a more sensitive characteristic. We do this by populating dark matter haloes in N -body simulations with galaxies using well-tested halo occupation distributions. We also estimate the VPF and find that samples with the same angular surface density of galaxies, in different models, exhibition almost model-independent VPF which therefore cannot be used as a test for DE. Once again, geometry and cosmic evolution compensate each other. By comparing VPFs for samples with fixed galaxy mass limits, we find measurable differences.     

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.     

Non-Gaussian cosmic microwave background temperature fluctuations from peculiar velocities of clusters

We use numerical simulations of a (480 Mpc  h ⁻¹)³ volume to show that the distribution of peak heights in maps of the temperature fluctuations from the kinematic and thermal Sunyaev–Zeldovich (SZ) effects will be highly non-Gaussian, and very different from the peak-height distribution of a Gaussian random field. We then show that it is a good approximation to assume that each peak in either SZ effect is associated with one and only one dark matter halo. This allows us to use our knowledge of the properties of haloes to estimate the peak-height distributions. At fixed optical depth, the distribution of peak heights resulting from the kinematic effect is Gaussian, with a width that is approximately proportional to the optical depth; the non-Gaussianity comes from summing over a range of optical depths. The optical depth is an increasing function of halo mass and the distribution of halo speeds is Gaussian, with a dispersion that is approximately independent of halo mass. This means that observations of the kinematic effect can be used to put constraints on how the abundance of massive clusters evolves, and on the evolution of cluster velocities. The non-Gaussianity of the thermal effect, on the other hand, comes primarily from the fact that, on average, the effect is larger in more massive haloes, and the distribution of halo masses is highly non-Gaussian. We also show that because haloes of the same mass may have a range of density and velocity dispersion profiles, the relation between halo mass and the amplitude of the thermal effect is not deterministic, but has some scatter.