Convergence and scatter of cluster density profiles

TL;DR

Problem: determine the inner density structure and potential universality of cluster-scale dark matter halos within ΛCDM. Approach: six high-resolution N-body cluster simulations with cross-code validation, thorough convergence testing, and density-profile fitting using NFW, M99, and generalized three-parameter forms. Findings: density profiles from different codes agree; at small radii the mean inner slope is $\gamma = 1.16 \pm 0.14$ (and $\gamma \approx 1.26 \pm 0.16$ at $0.01\,r_{vir}$), with three-parameter fits providing superior descriptions over two-parameter models; substantial scatter is modest ($\sim$0.15 in slope). Significance: tightens empirical constraints on halo cusps, informs interpretations of observations and annihilation signals, and sets the stage for even higher-resolution simulations to settle the true central behavior.

Abstract

We present new results from a series of LCDM simulations of cluster mass halos resolved with high force and mass resolution. These results are compared with recently published simulations from groups using various codes including PKDGRAV, ART, TPM, GRAPE and GADGET. Careful resolution tests show that with 25 million particles within the high resolution region we can resolve to about 0.3% of the virial radius and that convergence in radius is proportional to the mean interparticle separation. The density profiles of 26 high resolution clusters obtained with the different codes and from different initial conditions agree very well. The average logarithmic slope at one percent of the virial radius is $γ= 1.26$ with a scatter of $\pm 0.17$. Over the entire resolved regions the density profiles are well fitted by a smooth function that asymptotes to a central cusp $ρ\propto r^{-γ}$, where we find $γ=1.16\pm 0.14$ from the mean of the fits to our six highest resolution clusters.

Figures (6)

• Figure 1: Numerical convergence tests for the cluster profiles: Panel (a): Density profiles of cluster $D$ resolved with $N_{\rm vir} = 205k, 1.8M, 6M$ and $14M$ particles. Panel (b): Logarithmic slope for the profiles from (a). Panel (c): Density profiles of cluster $F$ simulated with different numerical parameters: $F9ft$ used 4096 fixed timesteps and constant $\epsilon$ in physical coordinates as in Fukushige2003. $F9cm$ and $F9$ used adaptive timesteps $0.2 \sqrt{\epsilon(z)/a}$ with comoving softening in $F9$ and mixed comoving/physical softening in $F9$ ($\epsilon_{\rm max} = 10 \epsilon_0$). Panel (d): Logarithmic slope for the profiles from (c).
• Figure 2: Ratios of the mass enclosed in low resolution runs to mass enclosed in the high resolution run $D12$. By comparing runs with equal softening (smaller than one third of the convergence scale) like $D3h$ and $D6h$ one finds that the resolved radii scale like $r \propto N^{-1/3}$. A larger softening (see run $D6$) can increase the converged scales and change this scaling.
• Figure 3: The triangles show the timestep criterium $\eta\sqrt{\epsilon(z)/a}$ as a function of radius for run $D9$ at $z=0$. The dashed line is for run $D9lt$, which has $\eta = 0.3$, and the long dashed line for run $D12$. The open squares give $15 (\Delta t/t_0)^{5/6} t_{\rm circ} (r_{vir})$ form Power2003, the circles are the circular orbit timescale $2\pi r/v_{\rm circ}(r)$. Lines without symbols show $t_{\rm dyn}/15 = 1/(\sqrt{G\rho(<r)} 15)$. The two horizontal lines are the timesteps and $15 (\Delta t/t_0)^{5/6} t_{\rm circ} (r_{vir})$ for run $F9ft$.
• Figure 4: Density profiles of the six clusters in our sample, clusters $B$ to $F$ are shifted downwards for clarity. Clusters are ordered by mass form top to bottom. Profiles of cluster $A$ and $C$ are shown at redshifts $-0.14$ and $-0.17$, i.e. when they have reached a 'relaxed' state with one well defined centre. Best fit NFW and M99 profiles and residual are shown, obtained by minimising the squares of the relative density differences.
• Figure 5: Same as Figure \ref{['proHRfits.eps']}, but with fitting functions that have one additional free parameter. The dashed dotted lines show the profile (\ref{['Npro']}) proposed by Navarro2003. The dashed lines show a general $\alpha\beta\gamma$-profile (\ref{['Gpro']}). We fitted the inner slope $\gamma$ to the data and used fixed values for the outer slope $\beta = 3$ and turning parameter $\alpha = 1$. $\gamma=1$ corresponds to the NFW profile. The fit parameters and rms of the residuals are given in Table \ref{['fitParams']}.