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.     

Scaling up tides in numerical models of galaxy and halo formation

The purpose of this article is to show that when dynamically cold, dissipationless self-gravitating systems collapse, their evolution is a strong function of the symmetry in the initial distribution. We explore with a set of pressureless homogeneous fluids the time evolution of ellipsoidal distributions and map the depth of potential achieved during relaxation as function of initial ellipsoid axis ratios. We then perform a series of N -body numerical simulations and contrast their evolution with the fluid solutions. We verify an analytic relation between collapse factor and particle number N in spherical symmetry, such that  ∝ N ^1/3  . We sought a similar relation for axisymmetric configurations, and found an empirical scaling relation such that  ∝ N ^1/6  in these cases. We then show that when mass distributions do not respect spherical or axial symmetry, the ensuing gravitational collapse deepens with increasing particle number N but only slowly: 86 per cent of triaxial configurations may collapse by a factor of no more than 40 as   N →∞  . For   N ≈10⁵  and larger, violent relaxation develops fully under the Lin–Mestel–Shu instability such that numerical N -body solutions now resolve the different initial morphologies adequately.     

An analytic model for the epoch of halo creation

In this paper we describe the Bayesian link between the cosmological mass function and the distribution of times at which isolated haloes of a given mass exist. By assuming that clumps of dark matter undergo monotonic growth on the time-scales of interest, this distribution of times is also the distribution of 'creation' times of the haloes. This monotonic growth is an inevitable aspect of gravitational instability. The spherical top-hat collapse model is used to estimate the rate at which clumps of dark matter collapse. This gives the prior for the creation time given no information about halo mass. Applying Bayes' theorem then allows any mass function to be converted into a distribution of times at which haloes of a given mass are created. This general result covers both Gaussian and non-Gaussian models. We also demonstrate how the mass function and the creation time distribution can be combined to give a joint density function, and discuss the relation between the time distribution of major merger events and the formula calculated. Finally, we determine the creation time of haloes within three N -body simulations, and compare the link between the mass function and creation rate with the analytic theory.     

Microlensing by halo MACHOs with a spatially varying mass function

The main aim of microlensing experiments is to evaluate the mean mass of massive compact halo objects (MACHOs) and the mass fraction of the Galactic halo made by this type of dark matter. Statistical analysis shows that by considering a Dirac-Delta mass function (MF) for the MACHOs, their mean mass is about that of a white dwarf star. This result is, however, in discrepancy with other observations such as those of non-observed expected white dwarfs in the Galactic halo which give rise to metal abundance, polluting the interstellar medium by their evolution. Here we use the hypothesis of the spatially varying MF of MACHOs, proposed by Kerins and Evans to interpret microlensing events. In this model, massive lenses with a lower population contribute to the microlensing events more frequently than do dominant brown dwarfs. This effect causes the mean mass of the observed lenses to be larger than the mean mass of all the lenses. A likelihood analysis is performed to find the best parameters of the spatially varying MF that are compatible with the duration distribution of Large Magellanic Cloud microlensing candidates of the MACHO experiment.     

On the origin of cold dark matter halo density profiles

N -body simulations predict that cold dark matter (CDM) halo-assembly occurs in two phases: (i) a fast-accretion phase with a rapidly deepening potential well; and (ii) a slow-accretion phase characterized by a gentle addition of mass to the outer halo with little change in the inner potential well. We demonstrate, using one-dimensional simulations, that this two-phase accretion leads to CDM haloes of the Navarro, Frenk & White (NFW) form and provides physical insight into the properties of the mass-accretion history that influence the final profile. Assuming that the velocities of CDM particles are effectively isotropized by fluctuations in the gravitational potential during the fast-accretion phase, we show that gravitational collapse in this phase leads to an inner profile  ρ( r ) ∝ r ⁻¹  . Slow accretion on to an established potential well leads to an outer profile with  ρ( r ) ∝ r ⁻³  . The concentration of a halo is determined by the fraction of mass that is accreted during the fast-accretion phase. Using an ensemble of realistic mass-accretion histories, we show that the model predictions of the dependence of halo concentration on halo formation time and, hence, the dependence of halo concentration on halo mass, and the distribution of halo concentrations all match those found in cosmological N -body simulations. Using a simple analytic model that captures much of the important physics, we show that the inner   r ⁻¹  profile of CDM haloes is a natural result of hierarchical mass assembly with an initial phase of rapid accretion.     

Numerical modelling of the vertical structure and dark halo parameters in disc galaxies

The non‐linear dynamics of bending instability and vertical structure of a galactic stellar disc embedded into a spherical halo are studied with N‐body numerical modelling. Development of the bending instability in stellar galactic disc is considered as the main factor that increases the disc thickness. Correlation between the disc vertical scale height and the halo‐to‐disc mass ratio is predicted from the simulations. The method of assessment of the spherical‐to‐disc mass ratio for edge‐on spiral galaxies with a small bulge is considered. Modelling of eight edge‐on galaxies: NGC 891, NGC 4738, NGC 5170, UGC 6080, UGC 7321, UGC 8286, UGC 9422 and UGC 9556 is performed. Parameters of stellar discs, dark haloes and bulges are estimated. The lower limit of the dark‐to‐luminous mass ratio in our galaxies is of the order of one within the limits of their stellar discs. The dark haloes dominate by mass in the galaxies with very thin stellar discs (NGC 5170, UGC 7321 and UGC 8286) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inferring dark halo structure from observed scaling laws of late-type galaxies and LSBs

We re-examine the Fall & Efstathiou scenario for galaxy formation, including the dark halo gravitational reaction to the formation of the baryon disc, as well as continuous variations in the intrinsic halo density profile. The recently published rotation curves of low surface brightness (LSB) and dwarf galaxies together with previously known scaling relations provide sufficient information on the present-day structure of late-type disc galaxies to invert the problem. By requiring that the models reproduce all the observational restrictions we can fully constrain the initial conditions of galaxy formation, with a minimum of assumptions, in particular without the need to specify a cold dark matter (CDM) halo profile. This allows one to solve for all the initial conditions, in terms of the halo density profile, the baryon fraction and the total angular momentum. We find that a unique initial halo shape is sufficient to accurately reproduce the rotation curves of both LSB and normal late-type spiral galaxies. This unique halo profile differs substantially from that found in standard CDM models. A galactic baryon fraction of 0.065 is found. The initial value of the dimensionless angular momentum is seen to be the principal discriminator between the galaxy classes we examine. The present-day scalings between structural parameters are seen to originate in the initial conditions.     

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.     

Dark matter halo response to the disc growth

We consider the sensitivity of the circular-orbit adiabatic contraction approximation to the baryon condensation rate and the orbital structure of dark matter haloes in the Λ cold dark matter (ΛCDM) paradigm. Using one-dimensional hydrodynamic simulations including the dark matter halo mass accretion history and gas cooling, we demonstrate that the adiabatic approximation is approximately valid even though haloes and discs may assemble simultaneously. We further demonstrate the validity of the simple approximation for ΛCDM haloes with isotropic velocity distributions using three-dimensional N -body simulations. This result is easily understood: an isotropic velocity distribution in a cuspy halo requires more circular orbits than radial orbits. Conversely, the approximation is poor in the extreme case of a radial orbit halo. It overestimates the response of a core dark matter halo, where radial orbit fraction is larger. Because no astronomically relevant models are dominated by low angular momentum orbits in the vicinity of the disc and the growth time-scale is never shorter than a dynamical time, we conclude that the adiabatic contraction approximation is useful in modelling the response of dark matter haloes to the growth of a disc.     

Tidal effects and the environment dependence of halo assembly

We explore a possible origin for the puzzling anti-correlation between the formation epoch of galactic dark-matter haloes and their environment density. This correlation has been revealed from cosmological N -body simulations and is in conflict with the extended Press–Schechter model of halo clustering. Using similar simulations, we first quantify the straightforward association of an early formation epoch with a reduced mass-growth rate at late times. We then find that a primary driver of suppressed growth, by accretion and mergers, is tidal effects dominated by a neighbouring massive halo. The tidal effects range from a slowdown of the assembly of haloes due to the shear along the large-scale filaments that feed the massive halo to actual mass loss in haloes that pass through the massive halo. Using the restricted three-body problem, we show that haloes are prone to tidal mass loss within 1.5 virial radii of a larger halo. Our results suggest that the dependence of the formation epoch on environment density is a secondary effect induced by the enhanced density of haloes in filaments near massive haloes where the tides are strong. Our measures of assembly rate are particularly correlated with the tidal field at high redshifts   z ∼ 1  .