Buried but not destroyed: the evolution from prompt cusps to NFW haloes

TL;DR

Prompt cusps form rapidly from the collapse of primordial density peaks, but their inner density profiles are not final; they evolve toward NFW-like centers under cosmological mass assembly. The authors develop a peak-based method to identify artificial fragmentation and to separate peak-origin material from later-accreted matter, enabling a clean study of cusp evolution. Across 64 zoom-in simulations varying the small-scale power, they find three evolutionary pathways—major mergers, disturbances from filaments, and mergers with artificial fragments—that differentially erode or bury the prompt cusp, yet all routes tend toward an NFW-like inner profile. The results reinforce the view of NFW as a dynamical attractor and reveal how the imprint of early collapse persists in peak material even as the total halo becomes NFW-like, with implications for the universality of halo structure and dark matter annihilation signals.

Abstract

The internal structure of dark matter haloes encodes their assembly history and offers critical insight into the nature of dark matter and structure formation. Analytical studies and high-resolution simulations have recently predicted the formation of 'prompt cusps' - steep power-law density profiles that emerge rapidly from the monolithic collapse of primordial peaks. Yet, by $z=0$ most haloes are well described by Navarro-Frenk-White (NFW) density profiles, raising the question of how these early cusps are transformed in a cosmological setting. We address this problem using 64 zoom-in $N$-body simulations of eight haloes, each resimulated with eight different free-streaming wavenumbers to control the abundance of small-scale structure while keeping the large-scale environment fixed. To mitigate numerical discreteness effects, we classify artificial fragments and genuine subhaloes with a physically motivated procedure based on matching subhaloes to their progenitor peaks. At the population level, we demonstrate that haloes initially form prompt cusps, and their profiles subsequently transition towards the NFW form. Our study reveals three distinct pathways by which prompt cusps evolve: major mergers, accretion of artificial fragments, and interactions with large-scale filaments, each having a distinct impact on the inner density profile. In particular, we show that the original power-law cusp remains visible in the profile of particles associated with the primordial peak even when the total halo profile is already NFW-like. This work highlights the imprint of early collapse on present-day halo structure and provides new insights into the origin of the universality of the NFW profile.

Figures (16)

• Figure 1: Dimensionless power spectrum $\Delta^2(k)$ for halo zoom-in simulations with varying small‐scale suppression scales $k_{\rm fs}$ (solid curves). Dotted lines mark the peak wavenumber, $k_{\rm max}$, for each power spectrum, while dashed lines indicate the characteristic scale, $k_{\rm c}$, of eight haloes, ranked from least to most massive. Both axes are shown in logarithmic units.
• Figure 2: Detection performance of artificial fragments that appear as subhaloes across different free-streaming scales, with each column corresponding to a set of simulations with a distinct free-streaming wavenumber $k_{\mathrm{fs}}$, as indicated in the title. Top row: scatter plots of the maximum progenitor mass of the subhaloes, $M_{\rm sub}$, normalized by the artificial mass threshold $M_{\rm lim}$Wang2007, versus the Lagrangian axis ratio $c/a$ for subhaloes matched (red, +) or unmatched (blue, $\times$) to density peaks; marker colours indicate the host halo rank. The fraction of spurious subhaloes is annotated in the bottom-right corner of each panel. Second to fourth rows: density maps showing Youden's $J$ statistic, the true positive rate (TPR), and the false positive rate (FPR) as functions of joint threshold cuts on $\log_{10}(M_{\mathrm{sub}}/M_{\mathrm{lim}})$ and $c/a$. Grey dashed and dotted lines in the top row, white markers in the second row and red markers in the two bottom rows mark the optimal thresholds for Youden's $J$. And as the optimal threshold spans a finite range in the parameter space, it is indicated in the top panels as shaded regions. Colour bars to the right of each row provide the corresponding metric scales.
• Figure 3: Schematic illustration of the three mechanisms breaking the self-similarity of prompt cusp formation (Sec. \ref{['subsec:self-similar']}). The left panel shows the physical evolution of the prompt cusp (purple) while the right panel shows the initial condition corresponding to this configuration. The upper right shows the spatial distribution of the main primordial peak (purple) and environmental perturbations (orange, blue and grey) on a projected coordinate plane, while the lower right displays the initial linear density field ($\delta$) as a function of Lagrangian distance from the centre ($q$) for these structures. The environmental perturbations manifest as: (1) Subhalo infall (orange), arising from overlapping neighboring peaks in the initial conditions; (2) Smooth accretion (blue), arising from diffuse background mass inflow; and (3) Artificial fragments (grey), arising from numerical noise.
• Figure 4: Distribution of the fitted peak shape index $\gamma$. The histogram aggregates all haloes and all free–streaming realisations included in the analysis, with the summary statistics are annotated in the panel. The mean, standard deviation and median are 1.77, 0.31 and 1.91, respectively.
• Figure 5: Diagnostic of the peak-selection and profile-fitting procedure described in Sec. \ref{['subsec:self-similar']}. Each panel corresponds to a different peak selected from one $\gamma$-bin in Figure \ref{['fig:Gamma']}. From left to right the Halo ID = $\{150, 150, 260, 108, 108\}$ and $k_{\rm sf}$ are $=\{30, 35, 30, 24, 30\}\,h\,\mathrm{Mpc}^{-1}$ we find $\gamma=\{1.052,\,1.598,\,1.686,\,1.986,\,2.003\}$. Points show the linear overdensity $\delta$ of all particles in the peak neighbourhood as a function of their elliptical radius $r_{\rm ell}$: non-peak particles are plotted in grey, and peak particles - defined by the envelope criterion in Sec. \ref{['subsec:self-similar']} - are plotted in salmon. The teal and dark purple curves denote the minimum and maximum envelope fits, respectively. The dark red dashed curve shows the best-fit free-index $\gamma$ profile while the blue dashed curve shows the comparison with fixed $\gamma=2$. The fitted $\gamma$ for each peak is reported in the legend of the corresponding panel. A dashed grey rectangle on each main panel marks the exact extent of the central region used for the fits, with its zoom-in view in the sub-panel. A grey dashed line in the main panel shows the maximum $\delta$ at the $r_{\rm ell} = 0$. Above each main panel, a subpanel shows the fractions of peak particles as functions of $r_{\rm ell}$, providing a quantitative view of the central dominance of peak material and the progressive increase of non-peak material with increasing radius. The horizontal dashed lines mark the fraction of 1 and 0.9.