Density profiles and substructure of dark matter halos: converging results at ultra-high numerical resolution

TL;DR

This study probes the convergence of dark matter halo structure in ultra-high-resolution N-body simulations of a Virgo-scale cluster. By increasing mass and force resolution by nearly an order of magnitude, the authors show the cluster density cusp approaches a slope of $-1.5$, consistent with Moore et al., and demonstrate that subhalo velocity and mass distributions (VDF and MDF) remain largely invariant over several Gyr and across environments for halos with $v_{circ}\gtrsim 100\,\mathrm{km\,s^{-1}}$. They find subhalos are spatially anti-biased relative to the mass, with a mild positive velocity bias in the core, and that overmerging is not a major issue for the bulk of substructure; tidal mass loss reduces $v_{circ}$ by about 20% over 5 Gyr. Tracing cluster progenitors from high redshift shows the central cD-like object forms early (roughly $z\sim3$ to $z\sim1$) through mergers, consistent with hierarchical structure formation. Collectively, the results indicate robust convergence of key structural and substructure properties, enabling precise tests of CDM predictions against cluster observations and guiding future high-resolution studies of galaxy-scale halos as well as clusters.

Abstract

Can N-body simulations reliably determine the structural properties of dark matter halos? Focussing on a Virgo-sized galaxy cluster, we increase the resolution of current ``high resolution simulations'' by almost an order of magnitude to examine the convergence of the important physical quantities. We have 4 million particles within the cluster and force resolution 0.5 kpc/h (0.05% of the virial radius). The central density profile has a logarithmic slope of -1.5, as found in lower resolution studies of the same halo, indicating that the profile has converged to the ``physical'' limit down to scales of a few kpc. Also the abundance of substructure is consistent with that derived from lower resolution runs; on the scales explored, the mass and circular velocity functions are close to power laws of exponents ~ -1.9 and -4. Overmerging appears to be globally unimportant for suhalos with circular velocities > 100 km/s. We can trace most of the cluster progenitors from z=3 to the present; the central object (the dark matter analog of a cD galaxy)is assembled between z=3 and 1 from the merging of a dozen halos with v_circ \sim 300 km/s. The mean circular velocity of the subhalos decreases by ~ 20% over 5 billion years, due to tidal mass loss. The velocity dispersions of halos and dark matter globally agree within 10%, but the halos are spatially anti-biased, and, in the very central region of the cluster, they show positive velocity bias; however, this effect appears to depend on numerical resolution.

Figures (13)

• Figure 1: Maps of the cluster's density as seen in run LORES (upper panel) and HIRES, which has 8 times better mass resolution. The change in the appearance between the two runs is mainly due to HIRES's ability of resolving further down the substructure mass function.
• Figure 2: The density profile of the cluster measured in three runs with increasing resolutions (from triangles to squares to circles). In the best run, the cluster contains over 4 million particles and the force resolution is $0.05$% of the cluster's virial radius. The curves are an NFW profile (lower curve) and a fit with the profile of Moore et al (1999a), which rises more steeply ($\propto r^{-1.5}$) at the center than the NFW profile ($\propto r^{-1}$). With increasing resolution, the cluster's profile continues to approach M99a's curve $i.e.$ this appears to be the asymptotic profile in the limit of infinite resolution. The vertical bars mark the radii at which the measured profiles are no longer affected by finite numerical resolution.
• Figure 3: The typical density profile of a substructure halo with $v_{circ}\sim 200\, {{\rm km}}\,{{\rm s}}^{-1}$ (and clustercentric distance $\sim 0.5\,R_{200}$) in the two runs, compared with the profile of an isolated halo (simulated with much higher resolution). Increasing the resolution brings the profiles of substructure halos closer to those of their isolated counterparts.
• Figure 4: Velocity distribution function (VDF) of the substructure halos, at two redshifts (lower panel: the final epoch, $z=0$; upper panel: the young cluster at $z=0.5$, when it is $\sim 1$ billion years old). The figure shows the effect of increasing the resolution by a factor of 8 in mass (from LORES, dashed curve, to HIRES, solid curve). The errorbars represent (1-$\sigma$) Poisson errors on the counts. The curves agree well where we expect both runs to be close to completeness (roughly $v_{circ}>100\, {{\rm km}}\,{{\rm s}}^{-1}$). The fall off at the low velocity end is caused by finite numerical resolution.
• Figure 5: The VDF for the cluster's substructure halos at $z=0$ (solid line) compared with that of the halos at $z=0.5$ contained within the same physical volume (dashed), using in both cases $R_{200}|_{z=0}$ as limiting distance (main panel). The inset shows the VDFs obtained considering only the halos within the cluster's virial radius at $z=0$ (dashed) and $z=0.5$ (dotted) and and measuring $v_{circ}$ in units of the cluster's "circular velocity" at the respective epoch. In both cases, there are no significant changes in the shape and amplitude of the VDF during the lifetime of the cluster. The two dotted curves in the main panel show the effect of changing environment inside and around the evolved cluster ($z=0$); they are for inner subhalos ($R/R_{200} < 0.5$; upper curve) and "peripheral" halos ($1 < R/R_{200} < 2.5$; lower). Within and around the cluster the shape of the VDF is very similar.