The Inner Structure of LambdaCDM Halos III: Universality and Asymptotic Slopes

TL;DR

The paper tackles the inner structure of $\Lambda$CDM halos across five decades in mass using 19 high-resolution simulations to test universality and constrain central slopes of the density profile.It shows that density profiles deviate from simple power laws and lack a well-defined core, yet collapse to a nearly universal shape when scaled by $r_{-2}$ and $\rho_{-2}$, with inner slopes shallower than isothermal and no evidence for a single asymptotic $\beta_0$.To improve extrapolation and fit quality, the authors introduce a flexible profile with $\beta_{\alpha}(r)=2\left(r/r_{-2}\right)^{\alpha}$ and $\alpha\approx0.17$, which better matches the radial variation of $\beta(r)$ and provides better per-halo fits than NFW or M99.The work also discusses scaling relations for halo structure via $r_{-2}$ and $\rho_{-2}$, emphasizes the need for direct comparison between observations and simulations, and provides practical guidance on applying these results to interpret inner halo physics.

Abstract

We investigate the mass profile of LambdaCDM halos using a suite of numerical simulations spanning five decades in halo mass, from dwarf galaxies to rich galaxy clusters. Our analysis confirms the proposal of Navarro, Frenk & White (NFW) that the shape of LambdaCDM halo mass profiles differs strongly from a power law and depends little on mass. The logarithmic slope of the spherically-averaged density profile, as measured by beta=-dln(rho)/dln(r), decreases monotonically towards the center and becomes shallower than isothermal (beta<2) inside a characteristic radius, r_{-2}. Although the fitting formula proposed by NFW provides a reasonably good approximation to the density and circular velocity profiles of individual halos, systematic deviations from the best NFW fits are also noticeable. Inside r_{-2}, the profile of simulated halos gets shallower with radius more gradually than predicted and, as a result, NFW fits tend to underestimate the dark matter density in these regions. This discrepancy has been interpreted as indicating a steeply divergent cusp, but our results suggest a different interpretation. We use the density and enclosed mass at our innermost resolved radii to place strong constraints on beta_{0}: density cusps as steep as r^{-1.5} are inconsistent with most of our simulations, although beta_{0}=1 is still consistent with our data. Our density profiles show no sign of converging to a well-defined asymptotic inner power law. We propose a simple formula that reproduces the radial dependence of the slope better than the NFW profile, and so may minimize errors when extrapolating our results inward to radii not yet reliably probed by numerical simulations.

Figures (5)

• Figure 1: Spherically-averaged density profiles of all our simulated halos. Densities are computed in radial bins of equal logarithmic width and are shown from the innermost converged radius ($r_{\rm conv}$) out to about the virial radius of each halo ($r_{200}$). Our simulations target halos in three distinct mass groups: "dwarf", "galaxy", and "cluster" halos. These groups span more than five decades in mass. Thick solid lines in the top panels illustrate the expected halo profile for each mass range according to the fitting formula proposed by NFW (top-left) or M99 (top-right). Bottom panels indicate the deviation from the best fit achieved for each individual halo (simulation minus fit) with the NFW profile (eq. \ref{['eq:nfw']}) or with its modified form, as proposed by M99 (eq. \ref{['eq:mooreetal']}).
• Figure 2: Spherically-averaged circular velocity ($V_c(r)=\sqrt{GM(r)/r}$) profiles of all our simulated halos. As in Figure \ref{['figs:rhoprof']}, circular velocities are computed in radial bins of equal logarithmic width and are shown from the innermost converged radius ($r_{\rm conv}$) out to about the virial radius ($r_{200}$) of each halo. Our simulations target halos in three distinct mass groups: "dwarf", "galaxy", and "cluster" halos, spanning more than a factor of $\sim 50$ in velocity. Thick solid lines in the top panels illustrate the expected profile for each mass range according to the fitting formula proposed by NFW (top-left) or M99 (top-right). Bottom panels indicate the deviation from the best fit achieved for each individual halo (simulation minus fit) with the NFW profile (eq. \ref{['eq:nfw']}) or with its modified form, as proposed by M99 (eq. \ref{['eq:mooreetal']}).
• Figure 3: Logarithmic slope of the density profile of all halos in our sample, plotted versus radius. Thick solid and dotted curves illustrate the radial dependence of the slope expected from the NFW profile (eq. \ref{['eq:nfw']}) and the modification proposed by M99 (eq. \ref{['eq:mooreetal']}), respectively. Note that although both fitting formulae have well-defined asymptotic inner slopes ($-1$ and $-1.5$, respectively) there is no sign of convergence to a well-defined value of the central slope in the simulated halos. At the innermost converged radius, the simulated halo profiles are shallower than $-1.5$, in disagreement with the Moore et al profile. Also, inside the radius at which the slope equals $-2$, $r_{-2}$, the profiles appear to get shallower more gradually than in the NFW formula. A power-law radial dependence of the slope seems to fit the results of our simulations better; the dot-dashed lines indicate the predictions of the $\rho_{\alpha}$ profile introduced in eqs. \ref{['eq:newbeta']} and \ref{['eq:newprof']} for $\alpha=0.17$. Best fits to individual halos yield $\alpha$ in the range $0.1$-$0.2$ (see Table \ref{['tab:fitpar']}).
• Figure 4: Maximum asymptotic inner slope compatible with the mean density interior to radius $r$, $\bar{\rho}(r)$, and with the local density at that radius, $\rho(r)$. This provides a robust limit to the central slope, $\beta_0<\beta_{\rm max}(r)=-3(1-\rho(r)/{\bar{\rho}(r)})$, under the plausible assumption that $\beta$ is monotonic with radius. Note that there is not enough mass within the innermost converged radius in our simulations to support density cusps as steep as $r^{-1.5}$. The asymptotic slope of the NFW profile, $\beta_0=1$, is still compatible with the simulated halos, although there is no convincing evidence for convergence to a well defined power-law behavior in any of our simulated halos. The thick dot-dashed curves illustrate the expected radial dependence of $\beta_{\rm max}$ for the $\rho_{\alpha}$ profile introduced in § \ref{['ssec:newprof']}, for $\alpha=0.17$.
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