Prompt Cusps: Early Dark Matter Halo Structures

• Prompt cusps are steep inner dark matter density structures that form at the first nonlinear collapse of primordial density peaks, characterized by a power law ρ ∝ r⁻¹⋅⁵.
• They bridge the initial cutoff-scale peak statistics with later NFW-like halo growth, influencing indirect-detection signals and dwarf-galaxy kinematics.
• Numerical simulations show that prompt cusps evolve via major mergers and environmental interactions, with their survival and annihilation impact tightly linked to early collapse conditions.

Prompt cusps are steep inner dark-matter density structures that form immediately at the first nonlinear collapse of primordial peaks, rather than emerging only after extended hierarchical relaxation. In this literature they are characterized by a prompt-collapse power law, typically written as $\rho \propto r^{-1.5}$, and are treated as relics of the earliest phase of halo assembly. Current work studies them as a bridge between cutoff-scale peak statistics, the internal structure of the first haloes, later NFW-like halo growth, and indirect-detection signals from annihilating dark matter (Delos et al., 2022, Ondaro-Mallea et al., 2023, Wang et al., 15 Dec 2025).

1. Definition and cosmological setting

In the astrophysical usage established by recent dark-matter work, a prompt cusp is the central density cusp that forms immediately at the monolithic collapse of a smooth peak in the early-universe density field (Delos, 2023). The underlying picture is that the first generation of haloes forms near the free-streaming cutoff of the linear power spectrum, where the field is smooth enough that collapse is comparatively monolithic rather than strongly hierarchical (Ondaro-Mallea et al., 2023).

This setting is especially natural in thermal WIMP cosmology and in warm-dark-matter models. In WIMP calculations, free streaming is modeled by multiplying the power spectrum by $\exp(-k^2/k_{\rm fs}^2)$, while in warm dark matter one often writes

$$P_{\rm WDM}(k)=T^2(k)\,P_{\rm CDM}(k),\qquad T(k)=\left[1+(f_{\rm s}k)^{2\nu}\right]^{-5/\nu},$$

with the smoothing scale $f_{\rm s}$ setting the characteristic sizes and masses of prompt cusps (Delos et al., 2022, Delos, 2023). In one widely used WIMP picture, the first nonlinear objects collapse around $z\sim 30$ from isolated peaks and form prompt cusps essentially at the instant of first collapse (Stücker et al., 2023).

Prompt cusps are distinguished from several other inner-halo structures. They are not NFW-like cusps or Einasto profiles, which describe the broader halo after later accretion and mergers; they are not macroscopic constant-density cores; and they are not ultracompact minihalos associated with exceptional primordial enhancements. In this literature, the prompt cusp is instead a fossil of condensation from the smooth early mass distribution (Delos, 2023).

2. Internal structure and scaling relations

The defining prompt-cusp density law is

$\rho(r)=A\,r^{-1.5},$

with the normalization $A$ and outer size fixed by the properties of the progenitor peak (Delos, 2023). A commonly used parameterization is

$$A = 24\, \rho_0\, a_{\rm col}^{-1.5} R^{1.5},\qquad r_{\rm cusp}=0.11\, a_{\rm col} R,$$

where $\rho_0$ is the present mean cosmological dark-matter density, $a_{\rm col}$ is the scale factor at first collapse, and $\exp(-k^2/k_{\rm fs}^2)$0 is the Lagrangian size of the peak (Stücker et al., 2023). In the warm-dark-matter formulation, the same scaling is written as

$\exp(-k^2/k_{\rm fs}^2)$1

with $\exp(-k^2/k_{\rm fs}^2)$2 and $\exp(-k^2/k_{\rm fs}^2)$3 the collapse redshift variable used in that paper (Delos, 2023).

This profile has a distinctive kinematic signature. The enclosed mass and circular velocity are

$\exp(-k^2/k_{\rm fs}^2)$4

so $\exp(-k^2/k_{\rm fs}^2)$5 (Delos, 2023). That scaling underlies the claim that prompt cusps can act as a lower envelope in dwarf-galaxy kinematics in warm-dark-matter models (Delos, 2023).

The $\exp(-k^2/k_{\rm fs}^2)$6 law does not continue to arbitrarily small radius. Liouville’s theorem and the primordial maximum phase-space density imply a core scale $\exp(-k^2/k_{\rm fs}^2)$7. In the WIMP annihilation literature this leads to a finite annihilation coefficient

$\exp(-k^2/k_{\rm fs}^2)$8

or closely related expressions differing only by the core prescription (Stücker et al., 2023, Delos, 2023). For the fiducial $\exp(-k^2/k_{\rm fs}^2)$9 WIMP model used in one cluster of papers, annihilation-dominating cusps typically have $$P_{\rm WDM}(k)=T^2(k)\,P_{\rm CDM}(k),\qquad T(k)=\left[1+(f_{\rm s}k)^{2\nu}\right]^{-5/\nu},$$0, while $$P_{\rm WDM}(k)=T^2(k)\,P_{\rm CDM}(k),\qquad T(k)=\left[1+(f_{\rm s}k)^{2\nu}\right]^{-5/\nu},$$1 (Stücker et al., 2023).

3. Formation, numerical evidence, and the prompt-cusp slope

The strongest numerical evidence for prompt cusps comes from phase-space-sheet simulations designed to avoid artificial fragmentation. These simulations show that a cusp with density $$P_{\rm WDM}(k)=T^2(k)\,P_{\rm CDM}(k),\qquad T(k)=\left[1+(f_{\rm s}k)^{2\nu}\right]^{-5/\nu},$$2 is indeed formed promptly, subsequently accreting a more extended halo and participating in the hierarchical growth of later halo generations (Ondaro-Mallea et al., 2023). They also show why earlier standard $$P_{\rm WDM}(k)=T^2(k)\,P_{\rm CDM}(k),\qquad T(k)=\left[1+(f_{\rm s}k)^{2\nu}\right]^{-5/\nu},$$3-body results were controversial: artificial clumps just before peak collapse can significantly shallow the inner profiles of the cusps (Ondaro-Mallea et al., 2023).

The numerical issue is specific to truncated-power-spectrum cosmologies. Near the cutoff, standard particle methods can generate artificial fragments along filaments, and those fragments can alter the inner structure of the first halo. The phase-space-sheet approach removes those artificial clumps and therefore isolates the monolithic-collapse solution (Ondaro-Mallea et al., 2023). Using controlled $$P_{\rm WDM}(k)=T^2(k)\,P_{\rm CDM}(k),\qquad T(k)=\left[1+(f_{\rm s}k)^{2\nu}\right]^{-5/\nu},$$4-body experiments with added small-scale power, the same paper derives a conservative contamination scale

$$P_{\rm WDM}(k)=T^2(k)\,P_{\rm CDM}(k),\qquad T(k)=\left[1+(f_{\rm s}k)^{2\nu}\right]^{-5/\nu},$$5

and argues that standard $$P_{\rm WDM}(k)=T^2(k)\,P_{\rm CDM}(k),\qquad T(k)=\left[1+(f_{\rm s}k)^{2\nu}\right]^{-5/\nu},$$6-body simulations remain useful provided the potentially contaminated inner region is excised (Ondaro-Mallea et al., 2023).

This numerical literature also refines the expected slope. Analytical work based on self-similar collapse predicts $$P_{\rm WDM}(k)=T^2(k)\,P_{\rm CDM}(k),\qquad T(k)=\left[1+(f_{\rm s}k)^{2\nu}\right]^{-5/\nu},$$7 for a quadratic peak, while the new zoom-in study of eight haloes resimulated with eight different free-streaming wavenumbers finds that haloes initially form prompt cusps and that the stable prompt-cusp phase is closer to $$P_{\rm WDM}(k)=T^2(k)\,P_{\rm CDM}(k),\qquad T(k)=\left[1+(f_{\rm s}k)^{2\nu}\right]^{-5/\nu},$$8 (Wang et al., 15 Dec 2025). The same study reports that the prompt-cusp phase is generally brief and that the transition toward an NFW-like total profile usually occurs before $$P_{\rm WDM}(k)=T^2(k)\,P_{\rm CDM}(k),\qquad T(k)=\left[1+(f_{\rm s}k)^{2\nu}\right]^{-5/\nu},$$9 in its collapse-time normalization (Wang et al., 15 Dec 2025).

4. Evolution inside hierarchical haloes

A central result of the recent cosmological literature is that prompt cusps do not all share a single fate. At the population level, haloes initially form prompt cusps, and their profiles subsequently transition towards the NFW form (Wang et al., 15 Dec 2025). The transition is not interpreted as universal erasure. Instead, the simulations identify three distinct pathways by which prompt cusps evolve: major mergers, accretion of artificial fragments, and interactions with large-scale filaments, each with a distinct impact on the inner density profile (Wang et al., 15 Dec 2025).

Major mergers are the clearest destructive channel. In those cases the original power-law cusp can be flattened or transformed by strong potential fluctuations and mixing (Wang et al., 15 Dec 2025). By contrast, interactions with large-scale filaments are often non-destructive: they raise the total density at larger radii while leaving the inner peak-dominated region close to its original power-law form (Wang et al., 15 Dec 2025). This motivates the paper’s title. 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 (Wang et al., 15 Dec 2025).

Semi-analytic hierarchical modeling reaches a related conclusion from the annihilation side. In the SASHIMI-based treatment, one prompt cusp is assigned to each first-generation microhalo and then propagated through the merger hierarchy, including subhalos, sub-subhalos, and stripped material. The surviving number is written as

$f_{\rm s}$0

with $f_{\rm s}$1 and a separate parameter $f_{\rm s}$2 for cusps associated with stripped substructure (Ando et al., 27 Jan 2026). In that framework, the effect of prompt cusps is substantial but more moderate than in universal-average estimates, chiefly because host-conditioned merger histories and environmental dependence reduce the inferred cusp abundance by about an order of magnitude relative to peak-based universal counts (Ando et al., 27 Jan 2026).

5. Annihilation phenomenology and observational constraints

Prompt cusps are especially important because annihilation scales as $f_{\rm s}$3. In one widely used WIMP model they contain only $f_{\rm s}$4 of the dark matter mass at late times but contribute $f_{\rm s}$5 of the annihilation luminosity in all but the densest regions, where they are tidally disrupted (Stücker et al., 2023). This is the basis for the broader claim that prompt cusps increase the annihilation rate by at least an order of magnitude compared to previous predictions, both in the cosmological average and within galaxy-scale halos (Delos et al., 2022).

The morphology of the signal is also altered. Once prompt cusps dominate, the effective emissivity traces the ambient mass density much more nearly than the square of the smooth-halo density, so the boost is predominantly outside of the centers of galactic halos (Delos et al., 2022). Within the Milky Way, stellar encounters are the dominant disruptors of prompt cusps, and for a $f_{\rm s}$6 WIMP they suppress the mean annihilation luminosity by a factor of two already at $f_{\rm s}$7, making the predicted emission from surviving cusps almost uniform over the sky (Stücker et al., 2023). In that picture, the Galactic Center $f_{\rm s}$8-ray Excess is unaffected by cusps (Stücker et al., 2023).

These annihilation predictions have been turned into observational limits. A 14-year Fermi-LAT analysis of the isotropic gamma-ray background finds that, once prompt cusps are included, annihilation to $f_{\rm s}$9 is excluded at 95% confidence for dark-matter masses below $z\sim 30$0 (Delos et al., 2023). A 15-year Fermi-LAT analysis of nearby galaxy clusters reaches a still stronger fiducial result: for prompt-cusp-dominated cluster models, the canonical thermal relic cross section to $z\sim 30$1 is excluded below about $z\sim 30$2, with Virgo providing the strongest bound (Crnogorčević et al., 24 Jan 2025). A complementary source-population study argues that individual prompt cusps could be detectable only in a restricted part of parameter space: for the canonical annihilation cross section, the dark-matter mass must be of order $z\sim 30$3, and the brightest objects would typically be $z\sim 30$4–$z\sim 30$5 away, under a solar mass, and subtend around $z\sim 30$6 on the sky (Delos, 2023).

6. Warm-dark-matter and SIDM variants, and unresolved issues

Prompt cusps are not restricted to the WIMP microhalo regime. In warm dark matter, free streaming moves the characteristic peak scale into the astrophysical regime, and prompt cusps with masses of order $z\sim 30$7 are predicted for particle masses in the $z\sim 30$8–$z\sim 30$9 range (Delos, 2023). Because the prompt-cusp circular velocity scales as $\rho(r)=A\,r^{-1.5},$0, these objects can alter the observed kinematics of Local Group dwarf galaxies, and the paper presents them as a viable new probe of warm dark matter (Delos, 2023). In the scenarios examined there, $\rho(r)=A\,r^{-1.5},$1 warm dark matter is clearly incompatible with dwarf-galaxy kinematics, $\rho(r)=A\,r^{-1.5},$2 is in serious tension, $\rho(r)=A\,r^{-1.5},$3 can also be threatened, and $\rho(r)=A\,r^{-1.5},$4 remains safe (Delos, 2023).

Self-interacting dark matter introduces a different modification. In isolated $\rho(r)=A\,r^{-1.5},$5 SIDM haloes with embedded prompt cusps, more prominent prompt cusps delay early core formation by a factor of $\rho(r)=A\,r^{-1.5},$6 because of smaller velocity-dispersion gradients in the inner region (Tran et al., 28 Nov 2025). During most of the later core-collapse phase, however, the evolution becomes closely aligned in physical time after rescaling densities, radii, and velocity dispersions, with deviations of order $\rho(r)=A\,r^{-1.5},$7 at late times relative to the reference NFW collapse track (Tran et al., 28 Nov 2025). This indicates that prompt cusps can modify SIDM heat transport without preventing the later approach to a nearly universal collapse track.

Several uncertainties remain central across the literature. Formation times are highly important because earlier collapse gives denser cusps (Delos, 2023). Tidal stripping and baryonic disruption remain environment dependent (Delos, 2023, Crnogorčević et al., 24 Jan 2025). The cusp survival fraction is often treated phenomenologically, with fiducial values such as $\rho(r)=A\,r^{-1.5},$8 or $\rho(r)=A\,r^{-1.5},$9–$A$0 in hierarchical models (Delos et al., 2022, Ando et al., 27 Jan 2026). Numerical discreteness near the cutoff remains a serious issue for ordinary particle methods, even though the phase-space-sheet work concludes that prompt cusps are a generic feature of the collapse of peaks on the free-streaming scale of the initial density field (Ondaro-Mallea et al., 2023). Taken together, these results suggest a consistent but still incomplete picture: prompt cusps are early, steep, and often long-lived, yet their abundance, survival, and visibility remain strongly conditioned by environment and later assembly.