Empirical models for Dark Matter Halos. I. Nonparametric Construction of Density Profiles and Comparison with Parametric Models

TL;DR

The paper develops and applies a nonparametric kernel-density estimator to extract smooth ρ(r) and its slope from N-body dark matter halos, enabling unbiased comparisons with several parametric models. Among the models tested, Einasto's r^{1/n} law and the Prugniel–Simien deprojection provide the best overall fits across cluster- and spherical-collapse halos, while the NFW-like (1,3,γ) form underperforms in many cases. The results reveal a clear mass dependence in the density-profile shape parameter n, indicating nonhomology and challenging the notion of a universal halo density profile. These findings support using Einasto-type descriptions for halos and highlight structural differences arising from formation histories, with implications for lensing and dynamical modeling.

Abstract

We use techniques from nonparametric function estimation theory to extract the density profiles, and their derivatives, from a set of N-body dark matter halos. We consider halos generated from LCDM simulations of gravitational clustering, as well as isolated, spherical collapses. The logarithmic density slopes gamma = d(log rho)/d(log r) of the LCDM halos are found to vary as power-laws in radius, reaching values of gamma ~ -1 at the innermost resolved radii (~0.01 r_virial). This behavior is significantly different from that of broken power-law models like the NFW profile, but similar to that of models like de Vaucouleurs'. Accordingly, we compare the N-body density profiles with various parametric models to find which provide the best fit. We consider an NFW-like model with arbitrary inner slope; Dehnen & McLaughlin's anisotropic model; Einasto's model (identical in functional form to Sersic's model but fit to the space density); and the density model of Prugniel & Simien that was designed to match the deprojected form of Sersic's R^{1/n} law. Overall, the best-fitting model to the LCDM halos is Einasto's, although the Prugniel-Simien and Dehnen-McLaughlin models also perform well. With regard to the spherical collapse halos, both the Prugniel-Simien and Einasto models describe the density profiles well, with an rms scatter some four times smaller than that obtained with either the NFW-like model or the 3-parameter Dehnen-McLaughlin model. Finally, we confirm recent claims of a systematic variation in profile shape with halo mass.

Figures (16)

• Figure 1: Nonparametric, bias-variance tradeoff in the estimation of $\rho(r)$ using a single sample of $10^6$ radii generated from a halo having an Einasto $r^{1/n}$ density profile with $n=5$ (see Section \ref{['SecSer']}). From top to bottom, $h_0=(0.1,0.03,0.01,0.003,0.001)\,r_{\rm e}$; all estimates used $\alpha=0.3$ (see equations \ref{['eq:rhohat']}, \ref{['eq:ktilde']}, and \ref{['hrhog']}).
• Figure 2: Five estimates of the logarithmic slope of an Einasto $r^{1/n}$ halo, derived via differentiation of $\hat{\rho}(r)$. The same sample of $10^6$ radii was used as in Figure \ref{['fig:tests1']}. From top to bottom, $h_0=(0.3,0.2,0.1,0.05,0.03)\,r_{\rm e}$; each estimate used $\alpha=0.4$ (see equation \ref{['hrhog']}). Dashed lines show the true slope.
• Figure 3: Nonparametric estimates of the density $\rho(r)$ (left panel) and the slope $d\log\rho/d\log r$ (right panel) for the ten $N$-body halos of Table \ref{['Table1']}. The virial radius $r_{vir}$ is marked with an arrow. Dashed lines in the right hand panels are linear fits of $\log(-d\log\rho/d\log r)$ to $\log r$; regression coefficients are also given.
• Figure 4: Nonparametric estimates of $\rho(r)$ (left panel) and $d\log\rho/d\log r$ (right panel) for the two "collapse" models. Dashed lines in the right hand panels are linear fits of $\log(-d\log\rho/d\log r)$ to $\log r$.
• Figure 5: Residual profiles from application of the 3-parameter (1, 3, $\gamma$) model (equation \ref{['EqNFW']}) to our ten, $N$-body density profiles. The virial radius is marked with an arrow, and the rms residual (equation \ref{['EqChi']}) is inset with the residual profiles.