A Universal Density Profile from Hierarchical Clustering

TL;DR

This study tests whether dark matter halos formed through hierarchical clustering share a universal equilibrium density profile independent of mass, initial power spectrum, and cosmology. Using zoom-in N-body simulations across eight cosmologies, the authors show that a two-parameter profile fits halos over two decades in radius, and that the characteristic density δ_c scales with the mean cosmic density at the halo's assembly time, captured by a Press-Schechter-based collapse redshift z_coll. They demonstrate tight correlations between halo mass, concentration, and Vmax, and explain apparent discrepancies in prior work as arising from selection effects and numerical resolution. The results provide a practical analytic framework to predict halo structure in any hierarchical model and suggest observational tests that could constrain cosmological parameters through halo density profiles.

Abstract

We use high-resolution N-body simulations to study the equilibrium density profiles of dark matter halos in hierarchically clustering universes. We find that all such profiles have the same shape, independent of halo mass, of initial density fluctuation spectrum, and of the values of the cosmological parameters. Spherically averaged equilibrium profiles are well fit over two decades in radius by a simple formula originally proposed to describe the structure of galaxy clusters in a cold dark matter universe. In any particular cosmology the two scale parameters of the fit, the halo mass and its characteristic density, are strongly correlated. Low-mass halos are significantly denser than more massive systems, a correlation which reflects the higher collapse redshift of small halos. The characteristic density of an equilibrium halo is proportional to the density of the universe at the time it was assembled. A suitable definition of this assembly time allows the same proportionality constant to be used for all the cosmologies that we have tested. We compare our results to previous work on halo density profiles and show that there is good agreement. We also provide a step-by-step analytic procedure, based on the Press-Schechter formalism, which allows accurate equilibrium profiles to be calculated as a function of mass in any hierarchical model.

Figures (13)

• Figure 1: Particle plots illustrating the time evolution of halos of different mass in an $\Omega_0=1$, $n=-1$ cosmology. Box sizes of each column are chosen so as to include approximately the same number of particles. At $z_0=0$ the box size corresponds to about $6 \times r_{200}$. Time runs from top to bottom. Each snapshot is chosen so that $M_{\star}$ increases by a factor of $4$ between each row. Low mass halos assemble earlier than their more massive counterparts. This is true for every cosmological scenario in our series.
• Figure 2: Density profiles of one of the most and one of the least massive halos in each series. In each panel the low-mass system is represented by the leftmost curve. In the SCDM and CDM$\Lambda$ models radii are given in kpc (scale at the top) and densities are in units of $10^{10} M_{\odot}$/kpc$^3$. In all other panels units are arbitrary. The density parameter, $\Omega_0$, and the value of the spectral index, $n$ is given in each panel. Solid lines are fits to the density profiles using eq. (1). The arrows indicate the value of the gravitational softening. The virial radius of each system is in all cases two orders of magnitude larger than the gravitational softening.
• Figure 3: The fits to the density profiles of Figure 2, scaled to the virial radius, $r_{200}$, of each system and to the critical density of the universe at $z=0$. Solid and dashed lines correspond to the low- and high-mass systems, respectively. Note that low-mass systems are denser than high-mass systems near the center, indicating that the characteristic density of a halo increases as the halo mass decreases.
• Figure 4: The circular velocity profiles of the halos shown in Figure 2. Radii are in units of the virial radius and circular speeds are normalized to the value at the virial radius. The thin solid line shows the data from the simulations. All curves have the same shape: they rise near the center until they reach a maximum and then decline at the outer edge. Low mass systems have higher maximum circular velocities in these scaled units because of their higher central concentrations. Dashed lines are fits using eq.(3). The dotted lines are the fit to the low-mass halo in each panel using a Hernquist profile. Note that this model fits rather well the inner regions of the halos, but underestimates the circular velocity near the virial radius.
• Figure 5: The correlation between the mass of a halo and its characteristic density. Masses are given in units of the nonlinear mass scale $M_{\star}$ (see text for a definition). Densities are relative to the critical value. Three curves are shown in each panel for different values of the parameter $f$ (see eq. 5). The fits are normalized to intersect at $M_{200}=M_{*}$ in the case $\Omega=1$. This normalization is then used for the low-density models ($\Omega_0 <1$). Note that for $f=0.01$ this procedure results in good fits to the results of the simulations in all cases.