The cusp-halo relation

TL;DR

This work establishes a framework to predict the central cusp of collisionless dark matter halos by linking the prompt cusp formed at the initial density peak to the later halo via a universal cusp-halo relation. It introduces the cusp-NFW density profile to describe halos with a central $\rho \propto r^{-1.5}$ cusp and demonstrates how the cusp properties scale with halo mass through a cosmology-dependent growth factor $\chi(\sigma_0) = e^{-\kappa/\sigma_0}$, enabling practical modeling across varied power spectra. The authors quantify the intrinsic scatter and show mild cosmology dependence, with implications for warm dark matter where a cutoff in the initial power spectrum enhances central cusps in small halos. A Python package is provided to implement the cusp-halo relation, compute concentrations, and generate cusp-NFW profiles, facilitating incorporation into halo models and observational analyses of dark matter. The results highlight that prompt cusps can significantly influence the inner structure of halos and therefore impact constraints on dark matter properties from small-scale observations.

Abstract

Simulations have established that each halo of collisionless dark matter is expected to contain a $ρ= A r^{-1.5}$ density cusp at its center. This prompt cusp is a relic of the halo's earliest moments and has a mass comparable to the cutoff scale in the spectrum of initial density perturbations. In this work, we provide a framework to predict for each halo the coefficient $A$ of its central cusp. We also present a "cusp-NFW" functional form that accurately describes the density profile of a halo with a prompt cusp at its center. Accurate characterization of each halo's central cusp is of particular importance in the study of warm dark matter models, for which the spectral cutoff is on an astrophysically relevant mass scale. To facilitate easy incorporation of prompt cusps into any halo modeling approach, we provide a code package that implements the cusp-halo relation and the cusp-NFW density profile.

Figures (24)

• Figure 1: Radial structure of a dark matter halo. The black curve shows the spherically averaged density profile of the simulated halo W1 from 2023MNRAS.518.3509D in arbitrary units. The dashed blue line marks the prompt cusp predicted from the initial conditions, while the dashed orange curve is a fitted Einasto profile.
• Figure 2: Outline of how we approach the cusp-halo relation. Starting with a halo of a given mass at some late time (black), we can trace its growth history backward in time (orange) until we meet the mass of a newly formed prompt cusp (blue), which depends on the epoch. The halo may then be assigned a central cusp corresponding to the intersection point. We can also obtain from this process an estimate of the halo's density profile (parameterized by the concentration), since that is set by the mass accretion history.
• Figure 3: Linear-theory matter power spectra for the simulations used in this work, shown in dimensionless form at the time when $\sigma_0=1$. The vertical markers indicate the smallest wavenumber represented in the simulation volume, $k=2\pi/L_\mathrm{box}$. Units are given in table \ref{['tab:units']}.
• Figure 4: Halo mass functions in the three simulations (different panels), expressed as the differential mass fraction $\mathrm{d} f/\mathrm{d}\ln M$ in halos of mass $M$. We show a range of times, which are parameterized by $\sigma_0$, and we use the $M_{200}$ mass definition; see table \ref{['tab:units']} for units. Dashed curves show the mass function of all FOF groups, which include spurious halos formed by artificial fragmentation. The solid curves are restricted to groups to which we can assign central prompt cusps; this procedure should exclude numerical artifacts. Mass functions are binned in intervals of $\Delta\ln M\simeq 0.4$, and we only consider masses $M>32m_\mathrm{p}$, where $m_\mathrm{p}$ is the simulation particle mass.
• Figure 5: Distribution of density peaks in the initial conditions of the three simulations (different colors) in terms of the resulting cusp mass $m$ and cusp coefficient $A$. The dotted curves show the overall distribution of peaks, while the solid curves show the subset of peaks that are interpreted as forming cusps in the simulations, such that we are able to assign them as halo centers. The main panel shows the joint distribution of $m$ and $A$ (as contours enclosing 68 percent and 95 percent of the distribution), while the left and bottom panels show the marginalized distributions of $m$ and $A$ separately (with measure $\ln A$ or $\ln m$). The diagonal dashed line indicates $A=0.8m^{1.9}$ (equation \ref{['A-m']}), which matches the solid-line distributions reasonably well (see also figure \ref{['fig:cusps_A-m']}). The cross marks the characteristic properties $(\tilde{m},\tilde{A})|_{\sigma_0=1}$ of cusps forming at the time when $\sigma_0=1$, according to equations (\ref{['mAchar']}); it conveniently lies near the center of the cusp distribution. See table \ref{['tab:units']} for units.