The Diversity and Similarity of Simulated Cold Dark Matter Halos

TL;DR

This study leverages the Aquarius project to resolve outstanding questions about the structure of ΛCDM halos with unprecedented numerical resolution. It demonstrates that halo density profiles are not strictly universal and are better described by the Einasto form with a halo-dependent shape parameter $α$, while the inner cusp is shallower than $-1$, ruling out steeper claims such as $ρ∝ r^{-1.2}$. Remarkably, the pseudo-phase-space density $ρ/σ^3$ follows an almost universal power law $∝ r^{-1.875}$ across halos, even as the individual density and velocity profiles vary, indicating partial universality captured by dynamical similarity in phase-space. Together, these results provide a robust benchmark for theoretical models and observational tests in a baryon-free context and establish a baseline against which baryonic modifications in real galaxies can be judged.

Abstract

We study the mass, velocity dispersion, and anisotropy profiles of $Λ$CDM halos using a suite of N-body simulations of unprecedented numerical resolution (the {\it Aquarius Project}). Our analysis confirms a number of results claimed by earlier work, and clarifies a few issues where conflicting claims may be found in the recent literature. The spherically-averaged density profile becomes progressively shallower inwards and, at the innermost resolved radius, the logarithmic slope is $γ\equiv -$d$\lnρ/$d$\ln r \simlt 1$. Asymptotic inner slopes as steep as the recently claimed $ρ\propto r^{-1.2}$ are clearly ruled out. The radial dependence of $γ$ is well approximated by a power-law, $γ\propto r^α$ (the Einasto profile). The shape parameter, $α$, varies slightly but significantly from halo to halo, implying that the mass profiles of $Λ$CDM halos are not strictly universal: different halos cannot, in general, be rescaled to look identical. Departures from similarity are also seen in velocity dispersion profiles and correlate with those in density profiles so as to preserve a power-law form for the spherically averaged pseudo-phase-space density, $ρ/σ^3\propto r^{-1.875}$. Our conclusions are reliable down to radii below 0.4% of the virial radius, providing well-defined predictions for halo structure when baryonic effects are neglected, and thus an instructive theoretical template against which the modifications induced by the baryonic components of real galaxies can be judged.

Figures (13)

• Figure 1: Spherically-averaged density (left) and circular velocity (right) profiles for the Aq-A halo simulation series. Different colours correspond to different resolution runs, as labeled in the figure. The density profile is multiplied by $r^2$ in order to emphasize small deviations. The bumps in the outer regions may be traced to the presence of substructure and unrelaxed tidal debris. Profiles are shown from $\sim 3 r_{200}$ down to the "convergence radius", $r_{\rm conv}^{(1)}$, corresponding to the radius where the relaxation time, $t_{\rm relax}$, is of the order of the age of the Universe. The thick portion of each profile indicates the region $r>r_{\rm conv}^{(7)}$ where $t_{\rm relax}$ is more than 7 times the age of the universe and where stricter convergence is achieved. Outside $r_{\rm conv}^{(7)}$ circular velocity estimates converge to better than $2.5\%$ (see Fig. \ref{['FigConvVc']}). The dot-dashed line shows an Einasto profile with $\alpha=0.17$ matched at ($r_{-2}$,$\rho_{-2}$), the peak in the $r^2\rho$ profile. This provides an excellent fit to the structure of the inner regions of the halo, as shown by the residuals plotted in the bottom panels. Arrows indicate the softening length $h_s$ of each simulation.
• Figure 2: Top panel: Fractional deviations in the circular velocity profile of the Aq-A convergence series versus the (enclosed) relaxation time, $t_{\rm relax}$, expressed in units of the circular orbit period at the virial radius, $t_{\rm circ}(r_{200})$. Deviations are measured relative to the highest resolution halo, Aq-A-1. Note that departures from convergence for all simulations are similar when expressed this way, indicating that $t_{\rm relax}$ is the main parameter determining convergence. Solid circles mark the location of the convergence criterion proposed by P03. Note that $V_c$ estimates converge there to about 10%. A stricter convergence criterion, e.g., $2.5\%$ convergence in $V_c$, is achieved at larger radii, where $t_{\rm relax}\sim 7 \, t_{\rm circ}(r_{200})$ (right vertical line). Bottom panel: Relaxation time versus radius for all five Aq-A simulations. Arrows indicate $h_s=2.8\, \epsilon_G$, the lengthscale where pairwise interactions become Newtonian.
• Figure 3: Left: Spherically-averaged density profiles of all level-2 Aquarius halos. Density estimates have been multiplied by $r^2$ in order to emphasize details in the comparison. Radii have been scaled to $r_{-2}$, the radius where the logarithmic slope has the "isothermal" value, $-2$. Thick lines show the profiles from $r_{\rm conv}^{(7)}$ outward; thin lines extend inward to $r_{\rm conv}^{(1)}$. For comparison, we also show the NFW and M99 profiles, which are fixed in these scaled units. This scaling makes clear that the inner profiles curve inward more gradually than NFW, and are substantially shallower than predicted by M99. The bottom panels show residuals from the best fits (i.e., with the radial scaling free) to the profiles using various fitting formulae (Sec. \ref{['ssec:fitform']}). Note that the Einasto formula fits all profiles well, especially in the inner regions. The shape parameter, $\alpha$, varies significantly from halo to halo, indicating that the profiles are not strictly self-similar: no simple physical rescaling can match one halo onto another. The NFW formula is also able to reproduce the inner profiles quite well, although the slight mismatch in profile shapes leads to deviations that increase inward and are maximal at the innermost resolved point. The steeply-cusped Moore profile gives the poorest fits. Right: Same as left, but for the circular velocity profiles, scaled to match the peak of each profile. This cumulative measure removes the bumps and wiggles induced by substructures and confirms the lack of self-similarity apparent in the left panel.
• Figure 4: Minimum-$Q$ values as a function of the Einasto parameter $\alpha$ for best fits to all level-2 halo profiles in the radial range $0.01<r/r_{-2}<5$. Colors identify different halos, and line types the number of bins chosen for the profile. The minimum-$Q$ values obtained for NFW and M99 best fits are also shown, and are plotted at arbitrary values of $\alpha$ for clarity. Note that Einasto fits are consistently better than NFW which are consistently better than M99, and that a significant improvement in $Q$ is obtained when letting $\alpha$ vary in the Einasto formula. $Q$ is approximately independent of the number of bins used in the profile, and is minimized for different values of $\alpha$ for each individual halo. See text for further details.
• Figure 5: Logarithmic slope of the density profile as a function of radius for our Aq-A convergence series. As in other plots, thick lines show results for $r>r_{\rm conv}^{(7)}$, thin lines extend the profiles down to the less strict convergence radius $r_{\rm conv}^{(1)}$. Comparison shows that excellent numerical convergence for the slope is achieved down to a radius intermediate between these two convergence radii. Applied to the highest-resolution Aq-A-1 simulation, this implies that the slope is shallower than the asymptotic value of the NFW profile ($r^{-1}$) in the inner regions. We see no sign of convergence to an asymptotic inner power-law. Instead, the profiles get shallower toward the centre as predicted by the Einasto formula (a straight line in this plot). The "critical solution" of Taylor2001 (which has a $r^{-0.75}$ asymptotic inner cusp) does better than NFW but not as well as Einasto in reproducing the inner profile of the halo.