Joachim Stadel

Dark Matter Substructure within Galactic Halos

We use numerical simulations to examine the substructure within galactic and cluster mass halos that form within a hierarchical universe. Clusters are easily reproduced with a steep mass spectrum of thousands of substructure clumps that closely matches the observations. However, the survival of dark matter substructure also occurs on galactic scales, leading to the remarkable result that galaxy halos appear as scaled versions of galaxy clusters. The model predicts that the virialized extent of the Milky Ways halo should contain about 500 satellites with circular velocities larger than the Draco and Ursa Minor systems, i.e., bound masses 108 M☉ and tidally limited sizes 1 kpc. The substructure clumps are on orbits that take a large fraction of them through the stellar disk, leading to significant resonant and impulsive heating. Their abundance and singular density profiles have important implications for the existence of old thin disks, cold stellar streams, gravitational lensing, and indirect/direct detection experiments.

Cold collapse and the core catastrophe

We show that a universe dominated by cold dark matter fails to reproduce the rotation curves of dark matter dominated galaxies, one of the key problems that it was designed to resolve. We perform numerical simulations of the formation of dark matter haloes, each containing ≳106 particles and resolved to 0.003 times the virial radius, allowing an accurate comparison with rotation curve data. A good fit to both Galactic and cluster-sized haloes can be achieved using the density profile ρ(r)∝[(rrs)1.5(1+(rrs)1.5)]−1, where rs is a scale radius. This profile has a steeper asymptotic slope, ρ(r)∝r−1.5, and a sharper turn-over than found by lower resolution studies. The central structure of relaxed haloes that form within a hierarchical universe has a remarkably small scatter. We compare the results with a sample of dark matter dominated, low surface brightness (LSB) galaxies with circular velocities in the range 100–300xa0kmxa0s−1. The rotation curves of discs within cold dark matter haloes rise too steeply to match these data, which require a constant mass density in the central regions. The effects of Ωmass and Λ cannot reconcile the cold dark matter (CDM) model with data – even if we leave the concentration as a free parameter, we are unable to reproduce the observations with such a steep central density profile. It is important to confirm these results using stellar rather than Hxa0i rotation curves for LSB galaxies. We test the effects of introducing a cut-off in the power spectrum that may occur in a universe dominated by warm dark matter. In this case, haloes form by a monolithic collapse but the final density profile barely changes, demonstrating that the merger history does not play a role in determining the halo structure.

Resolving the Structure of Cold Dark Matter Halos

We examine the effects of mass resolution and force softening on the density profiles of cold dark matter halos that form within cosmological N-body simulations. As we increase the mass and force resolution, we resolve progenitor halos that collapse at higher redshifts and have very high densities. At our highest resolution we have nearly 3×106 particles within the virial radius, which is several orders of magnitude more than previously used, and we can resolve more than 1000 surviving dark matter halos within this single virialized system. The halo profiles become steeper in the central regions, and we may not have achieved convergence to a unique slope within the inner 10% of the virialized region. Results from two very high resolution halo simulations yield steep inner density profiles, ρ(r)~r−1.4. The abundance and properties of arcs formed within this potential will be different from calculations based on lower resolution simulations. The kinematics of disks within such a steep potential may prove problematic for the cold dark matter model when compared with the observed properties of halos on galactic scales.

The inner structure of ΛCDM haloes - I. A numerical convergence study

We present a comprehensive set of convergence tests which explore the role of various numerical parameters on the equilibrium structure of a simulated dark matter halo. We report results obtained with two independent, state-of-the-art, multi-stepping, parallel N–body codes: PKDGRAV and GADGET. We find that convergent mass profiles can be obtained for suitable choices of the gravitational softening, timestep, force accuracy, initial redshift, and particle number. For softenings chosen so that particle discreteness effects are negligible, convergence in the circular velocity is obtained at radii where the following conditions are satisfied: (i) the timestep is much shorter than the local orbital timescale; (ii) accelerations do not exceed a characteristic acceleration imprinted by the gravitational softening; and (iii) enough particles are enclosed so that the collisional relaxation timescale is longer than the age of the universe. Convergence also requires sufficiently high initial redshift and accurate force computations. Poor spatial, time, or force resolution leads generally to systems with artificially low central density, but may also result in the formation of artificially dense central cusps. We have explored several adaptive time-stepping choices and obtained best results when individual timesteps are chosen according to the local acceleration and the gravitational softening (�ti / (ǫ/ai) 1/2 ), although further experimentation may yield better and more efficient criteria. The most stringent requirement for c is typically that imposed on the particle number by the collisional relaxation criterion, which implies that in order to estimate accurate circular velocities at radii where the density contrast may reach � 10 6 , the region must enclose of order 3000 particles (or more than a few

Density Profiles and Substructure of Dark Matter Halos: Converging Results at Ultra-High Numerical Resolution

Can dissipationless N-body simulations be used to reliably determine the structural and substructure properties of dark matter halos? A large simulation of a galaxy cluster in a cold dark matter universe is used to increase the force and mass resolution of current high-resolution simulations by almost an order of magnitude to examine the convergence of the important physical quantities. The cluster contains ~5 million particles within the final virial radius, Rvir 2 Mpc (with H0 = 50 km s-1 Mpc-1), and is simulated using a force resolution of 1.0 kpc (≡0.05% of Rvir); the final virial mass is 4.3 × 1014 M☉, equivalent to a circular velocity of vcirc ≡ (GM/R)1/2 1000 km s-1 at the virial radius. The central density profile has a logarithmic slope of -1.5, identical to lower resolution studies of the same halo, indicating that the profiles measured from simulations of this resolution have converged to the physical limit down to scales of a few kpc (~0.2% of Rvir). In addition, the abundance and properties of substructure are consistent with those derived from lower resolution runs; from small to large galaxy scales (vcirc > 100 km s-1, m > 1011 M☉), the circular velocity function and the mass function of substructures can be approximated by power laws with slopes of ~-4 and ~-2, respectively. At the current resolution, overmerging (a numerical effect that leads to structureless virialized halos in low-resolution N-body simulations) seems to be globally unimportant for substructure halos with circular velocities of vcirc > 100 km s-1 (~10% of the clusters vcirc). We can identify subhalos orbiting in the very central region of the cluster (R 100 kpc), and we can trace most of the cluster progenitors from high redshift to the present. The object at the cluster center (the dark matter analog of a cD galaxy) is assembled between z = 3 and z = 1 from the merging of a dozen halos with vcirc 300 km s-1. Tidal stripping and halo-halo collisions decrease the mean circular velocity of the substructure halos by ≈20% over a 5 billion yr period. We use the sample of 2000 substructure halos to explore the possibility of biases using galactic tracers in clusters: the velocity dispersions of the halos globally agree with the dark matter within 10%, but the halos are spatially antibiased, and in the very central region of the cluster (R/Rvir < 0.3) they show positive velocity bias (bv ≡ σv3D,halos/σv3D,DM 1.2-1.3); however, this effect appears to depend on numerical resolution.

Dark matter haloes within clusters

We examine the properties of dark matter haloes within a rich galaxy cluster using a high-resolution simulation that captures the cosmological context of a cold dark matter universe. The mass and force resolution permit the resolution of 150 haloes with circular velocities larger than 80 kmxa0s−1 within the cluster virial radius of 2 Mpc (with Hubble constant H0xa0=xa050xa0kmxa0s−1xa0Mpc−1). This enables an unprecedented study of the statistical properties of a large sample of dark matter haloes evolving in a dense environment. The cumulative fraction of mass attached to these haloes varies from close to zero per cent at 200 kpc to 13 per cent at the virial radius. Even at this resolution the overmerging problem persists; haloes that pass within 100–200 kpc of the cluster centre are tidally disrupted. Additional substructure is lost at earlier epochs within the massive progenitor haloes. The median ratio of apocentric to pericentric radii is 6:1, so that the orbital distribution is close to isotropic, circular orbits are rare and radial orbits are common. The orbits of haloes are unbiased with respect to both position within the cluster and the orbits of the smooth dark matter background, and no velocity bias is detected. The tidal radii of surviving haloes are generally well-fitted using the simple analytic prediction applied to their orbital pericentres. Haloes within clusters have higher concentrations than those in the field. Within the cluster, halo density profiles can be modified by tidal forces and individual encounters with other haloes that cause significant mass loss —‘galaxy harassment’. Mergers between haloes do not occur inside the cluster virial radius.

On the survival and destruction of spiral galaxies in clusters

We follow the evolution of disk galaxies within a cluster that forms hierarchically in a cold dark matter N-body simulation. At a redshift z=0.5 we select several dark matter halos that have quiet merger histories and are about to enter the newly forming cluster environment. The halos are replaced with equilibrium high resolution model spirals that are constructed to represent examples of low surface brightness (LSB) and high surface brightness (HSB) galaxies. Varying the disk and halo structural parameters reveals that the response of a spiral galaxy to tidal encounters depends primarily on the potential depth of the mass distribution and the disk scale length. LSB galaxies, characterised by slowly rising rotation curves and large scale lengths, evolve dramatically under the influence of rapid encounters with substructure and strong tidal shocks from the global cluster potential --- galaxy harassment. We find that up to 90% of their stars are tidally stripped and congregate in large diffuse tails that trace the orbital path of the galaxy and form the diffuse intra-cluster light. The bound stellar remnants closely resemble the dwarf spheroidals (dEs) that populate nearby clusters. HSB galaxies are stable to the chaos of cluster formation and tidal encounters. These disks lie well within the tidally limited dark matter halos and their potentials are more concentrated. Although very few stars are stripped, the scale height of the disks increases substantially and no spiral features remain, therefore we speculate that these galaxies would be identified as S0 galaxies in present day clusters.

Tidal stirring and the origin of dwarf spheroidals in the local group

N-body + smoothed particle hydrodynamics (SPH) simulations are used to study the evolution of dwarf irregular galaxies (dIrrs) entering the dark matter halo of the Milky Way or M31 on plunging orbits. We propose a new dynamical mechanism driving the evolution of gas-rich, rotationally supported dIrrs, mostly found at the outskirts of the Local Group (LG), into gas-free, pressure-supported dwarf spheroidals (dSphs) or dwarf ellipticals (dEs), observed to cluster around the two giant spirals. The initial model galaxies are exponential disks embedded in massive dark matter halos and reproduce nearby dIrrs. Repeated tidal shocks at the pericenters of their orbits partially strip their halos and disks and trigger dynamical instabilities that dramatically reshape their stellar components. After only 2-3 orbits low surface brightness dIrrs are transformed into dSphs while high surface brightness dIrrs evolve into dEs. This evolutionary mechanism naturally leads to the morphology-density relation observed for LG dwarfs. Dwarfs surrounded by very dense dark matter halos, such as the dIrr GR8, are turned into Draco or Ursa Minor, the faintest and most dark matter dominated among LG dSphs. If disks include a gaseous component, this is both tidally stripped and consumed in periodic bursts of star formation. The resulting star formation histories are in good qualitative agreement with those derived using Hubble Space Telescope (HST) color-magnitude diagrams for local dSphs.

Substructure in Dark Halos: Orbital Eccentricities and Dynamical Friction

The virialized regions of galaxies and clusters contain significant amounts of substructure; clusters have hundreds to thousands of galaxies, and satellite systems and globular clusters orbit the halos of individual galaxies. These orbits can decay owing to dynamical friction. Depending on their orbits and their masses, the substructures either merge, are disrupted, or survive to the present day. We examine the distributions of eccentricities of orbits within mass distributions similar to those we see for galaxies and clusters. A comprehensive understanding of these orbital properties is essential to calculate the rates of physical processes relevant to the formation and evolution of galaxies and clusters. We derive the orbital eccentricity distributions for a number of spherical potentials. These distributions depend strongly on the velocity anisotropy, but only slightly on the shape of the potential. The eccentricity distributions in the case of an isotropic distribution function are strongly skewed toward high eccentricities, with a median value of typically ~0.6, corresponding to an apocenter-to-pericenter ratio of 4.0. We also present high-resolution N-body simulations of the orbital decay of satellite systems on eccentric orbits in an isothermal halo. The dynamical friction timescales are found to decrease with increasing orbital eccentricity because of the dominating deceleration at the orbits pericenter. The orbital eccentricity stays remarkably constant throughout the decay; although the eccentricity decreases near pericenter, it increases again near apocenter, such that there is no net circularization. We briefly discuss several applications for our derived distributions of orbital eccentricities and the resulting decay rates from dynamical friction. We compare the theoretical eccentricity distributions to those of globular clusters and galactic satellites for which all six phase-space coordinates (and therewith their orbits) have been determined. We find that the globular clusters are consistent with a close-to-isotropic velocity distribution, and they show large orbital eccentricities because of this (not in spite of this, as has been previously asserted). In addition, we find that the limited data on the Galactic system of satellites appears to be different and warrants further investigation as a clue to the formation and evolution of our Milky Way and its halo substructure.

The seeds of rich galaxy clusters in the universe

The discovery of a population of young galaxies at a redshift when the Universe was about a tenth of its current age has shed new light on the question of when and how galaxies formed. Within the context of popular models, this is the population of primeval galaxies that built themselves up to the size of present-day galaxies through the process of repeated mergers called hierarchical clustering. But the recent detection of a large concentration of these primeval galaxies appears to be incompatible with hierarchical clustering models, which generally predict that clusters of this size are fully formed later in time. Here we use a combination of theoretical techniques — semi-analytic modelling and n-body simulations — to show that such large concentrations should be quite common in a universe dominated by cold dark matter, and that they are the progenitors of the rich galaxy clusters seen today. We predict the clustering properties of primeval galaxies which should, when compared with data that will be collected in the near future, test our current understanding of galaxy formation within the framework of a universe dominated by cold dark matter.