Fermionic Dark Matter spikes: origin and growth of Black Hole seeds

Carlos R. Arg\"uelles; Jorge A. Rueda; Valentina Crespi

REVIEW 3 major objections 5 minor 1 cited by

Fermionic Dark Matter spikes: origin and growth of Black Hole seeds

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Pith's one-line read This paper claims that once dark matter is treated as massive fermions in equilibrium, the spike around a supermassive black hole is generally not a power law, and for fermion masses above roughly 300 keV the black hole depletes the…

desk verdict A serious relativistic treatment of dark-matter spikes for fermionic core-halo halos that shows depletion is real, but the light-fermion curves are not self-consistent because the final DM self-gravity is neglected. read the letter →

arxiv 2412.17919 v2 pith:ZRJB3WHO submitted 2024-12-23 astro-ph.GA

Valentina Crespi , Carlos R. Argüelles , Jorge A. Rueda This is my paper · ORCID

classification astro-ph.GA MSC 83C5785A05 PACS 95.35.+d04.70.-s98.62.Js

keywords darkmatterfermionicspikessupermassiveblackholesRARmodelgeneralrelativityadiabaticgrowthcore-haloprofiles

topics Dark Matter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks what happens to the dark matter overdensity around a supermassive black hole when the dark matter is a self-gravitating gas of massive fermions, rather than a featureless power-law halo. It finds a mass- and degeneracy-dependent outcome: only dilute, Boltzmannian fermions produce the familiar $\rho \sim r^{-3/2}$ spike, while semi-degenerate core-halo fermions develop non-power-law spikes or even a density depletion. The significance is that dark-matter annihilation signals, gravitational-wave dephasing, and stellar-orbit fits all assume a universal power-law spike. The paper also compares two histories of the black-hole seed, a small baryonic seed that grows adiabatically and a heavy seed formed by the baryon-triggered collapse of a fermionic core, and shows that the spike carries information about both the particle mass and the seed's origin.

What carries the argument

The machinery is the adiabatic-invariant method in Schwarzschild spacetime applied to initial states generated by the RAR model, a relativistic self-gravitating Fermi gas with finite temperature, tidal cutoff, and a dense degenerate core surrounded by a dilute halo. In the baryonic-seed scenario, radial action and angular momentum are conserved during slow black-hole growth, so the initial distribution function maps to the final one; in the dark-matter-seed scenario, the collapse of the critical fermion core is treated as an instantaneous change of the metric potential with angular momentum conserved. The load-bearing identity is the relation between initial and final energies defined by equality of the radial actions, which converts the spike calculation into a phase-space integral over bound orbits, with the fermion mass $m$ and degeneracy parameter $\theta_0$ entering through the initial RAR halo.

What would settle it

Measure the density profile within the sphere of influence of a supermassive black hole such as Sgr A*: a clean $r^{-3/2}$ power law extending to the innermost bound orbit would contradict the predicted depletion for fermion masses above roughly 300 keV, while a non-power-law profile with a suppressed central density would support the fermionic core-halo prediction.

Watch

Extended reading notes

Core claim

The central claim is that the standard $r^{-3/2}$ dark-matter spike is a special case, not the generic outcome. Starting from equilibrium fermionic halos of the RAR model and growing a Schwarzschild black hole adiabatically, the authors find that core-halo configurations redistribute into spike profiles that depend on fermion mass and central degeneracy and do not follow a simple power law. For a fixed fermion mass there is a black-hole mass above which the spike density is depleted relative to the initial halo, and for fermion masses above roughly 300 keV no black-hole mass yields any enhancement. In the alternative seed scenario, where the black hole forms by the collapse of a dense fermion core, the same qualitative behavior holds with an additional dependence on the baryon-to-dark-matter fraction $\chi$ inside the collapsing core. Dilute Boltzmannian fermions reproduce the classical $r^{-3/2}$ spike, matching earlier treatments.

Load-bearing premise

The load-bearing premise is that once the black hole is present, the remaining dark matter inside the spike feels only the black hole's gravity, with the self-gravity of the dark matter itself neglected, a regime the authors note is questionable for fermion masses below about 250 keV and black-hole masses below about $4\times10^6$ solar masses.

Editorial extensions

If this is right

Annihilation-flux predictions, gravitational-wave estimates, and stellar-orbit constraints that assume a universal $r^{-3/2}$ spike need revision when the host halo is a fermionic core-halo configuration.
Below the enhancement threshold the spike peak height stays roughly constant while its width shrinks as the black hole grows; above the threshold the density falls below the initial halo value.
For fermion masses above roughly 300 keV, a supermassive black hole suppresses rather than enhances the surrounding dark matter density for every black-hole mass considered.
The two seed scenarios give nearly identical spike shapes for the same final black-hole mass, differing mainly by the flattened peak left when the collapsed core is removed.
Spike observables become probes of the seed's origin and of the halo's degeneracy state, not just of the black-hole mass.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extension: the same mass-dependent logic would predict that fermionic cores around stellar-mass black holes and neutron stars produce smaller or absent spikes, potentially altering dark-matter dynamical-friction and X-ray binary constraints.
Extension: applying the same phase-space machinery to bosonic or solitonic central cores would test whether non-power-law, mass-dependent spike shapes are generic to self-gravitating central cores or specific to fermionic degeneracy pressure.
Extension: if heavy fermions deplete the spike, gamma-ray constraints from the Galactic center would be weaker for that mass range; a detected non-power-law emission morphology could then be turned into a fermion-mass measurement.
Extension: the dependence of the final spike on the baryon fraction $\chi$ suggests that observed spike profiles could encode the accretion history of the black-hole seed, not only the particle mass.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a serious, well-written application of the relativistic adiabatic spike machinery to RAR fermionic core-halo halos, and it shows something genuinely new—the r^-3/2 spike is not universal, and a central BH can deplete the surrounding DM instead of enhancing it. But the paper's general claim overreaches: the authors' own admitted neglect of DM self-gravity in the final state breaks the calculation for the light-fermion cases (mc^2 ≲ 250 keV) that anchor the mass-dependence, so the 100–250 keV curves should be treated as uncomputed, not as predictions.

What is new: applying the Sadeghian et al. GR treatment to finite-temperature, self-gravitating fermion halos. The equations are laid out in enough detail to be checked. In the Boltzmannian limit they recover G&S r^-3/2, which is a good sanity check. The depletion threshold—for a given fermion mass there is a BH mass above which density is suppressed, and for m≳300 keV no enhancement for any BH mass—is a real qualitative result, at least for heavy fermions. Model II, the sudden collapse of a critical DM core, is an interesting new scenario and the authors attempt to model the orbital reshuffling.

Soft spots: the self-gravity neglect is the load-bearing one. They write in Sec. IV that for m≲250 keV and M_BH≲4×10^6 M_sun, the enclosed DM mass is comparable to the BH mass. In that regime the final metric is not Schwarzschild; the effective potential (A6), radial action (A7), and energy relation (A9) all change, so Eq. (3) is not a self-consistent spike. The authors acknowledge this and conclude that m≳300 keV is favored—which is a retreat from the abstract's 'general case.' It means the novel non-power-law, mass-dependent spikes are only established for the heavy-fermion corner. Also, no code or data are released, so the numerical results are not independently reproducible. Model II leans on imported critical masses from prior group papers and a sudden-collapse mapping that is heuristic; that is defensible as prior work, but the mapping itself is less tested than the adiabatic one.

This paper deserves a serious referee. The right outcome is a conditional accept with the authors forced to either restrict the claims to the regime where self-gravity is negligible, or compute the spike with the full self-gravitating final potential. A code release would also help. I'd bring it to a reading group; the central idea trains a useful light on how DM particle physics changes spike predictions.

Referee Report

3 major / 5 minor

Summary. The paper computes relativistic dark-matter (DM) spike profiles around a Schwarzschild supermassive black hole (SMBH) that grows in the center of a fermionic DM halo described by the Ruffini-Argüelles-Rueda (RAR) model. In Model I, a small baryonic BH seed grows adiabatically and the final DM distribution is obtained by conserving radial action and angular momentum between the initial RAR phase-space distribution and final bound orbits around the BH (Sec. II A and Appendix A). In Model II, a dense fermion core grows by accreting baryons to a critical mass and then collapses suddenly to an SMBH, with the remaining particles remapped according to the instantaneous-collapse prescription of Appendix B. The principal claims are that fermionic spikes generally do not follow a simple power law, that the spike structure depends on the fermion mass and degeneracy state, that a sufficiently massive BH or heavy fermion mass can suppress rather than enhance the central DM density, and that the usual r^{-3/2} spike is recovered only in the Boltzmannian regime.

Significance. The paper is a genuine forward calculation, not a fit: it starts from a specified RAR equilibrium distribution, applies radial-action conservation with published numerical methods, and recovers the standard Gondolo-Silk r^{-3/2} spike in the low-degeneracy limit. If the results are correct, they would be relevant for indirect DM searches, gravitational-wave signatures of DM spikes, stellar-orbit constraints near Sgr A*, and the DM-origin SMBH-seed scenario. The numerical implementation is described in enough detail to be reproducible. However, the authors themselves identify the key weakness: in the light-fermion regime that produces the novel non-power-law spikes, the enclosed DM mass is comparable to the BH mass, so the neglect of DM self-gravity in the final state is not justified. That limitation directly affects the main quantitative novelty of the paper.

major comments (3)

[Sec. IV; Sec. II A, Eq. (3); Appendix A, Eqs. (A6)-(A9)] The central claim of mass- and degeneracy-dependent, non-power-law spikes for semi-degenerate fermions is computed in a regime where the authors themselves state that the enclosed DM mass is comparable to the BH mass. For fermions with mc^2 ≲ 250 keV and M_BH ≲ 4×10^6 M_sun, the final gravitational potential is not the pure Schwarzschild potential used in Eq. (3), and the effective potential (A6), radial action (A7), and energy mapping (A9) are all evaluated with the DM self-gravity omitted. The 100 keV and 250 keV curves in Figs. 2 and 3, which exhibit the strongest mass dependence and the claimed depletion/enhancement transition, are therefore not self-consistent solutions. The suggestion in Sec. IV that the approximation favors m ≳ 300 keV does not rescue the general abstract claim, because the novel light-fermion behavior is precisely what is not established. The authors should either recompute the spike with a metric that includes the remaining DM mass self-consistently, or explicitly restrict all quantitative claims to parameters for which M_DM << M_BH is verified for every displayed profile.
[Sec. III B; Appendix B, Eq. (B2)] The sudden-collapse mapping in Model II is not sufficiently validated. Equation (B2) assumes that the particle velocity is unchanged at the same coordinate radius during the instantaneous transition from the critical core to a Schwarzschild BH, but the collapse from R_crit = 9.3 M_crit to the BH horizon occurs on a timescale comparable to orbital periods of particles near that radius. The resulting final density therefore depends on this unverified approximation. Because Model II is presented as an independent scenario for SMBH formation of DM origin, the authors should either demonstrate robustness against a finite-time collapse prescription or explicitly label the Model II spike as schematic pending a dynamical treatment.
[Sec. IV, item 3; Fig. 3] The statement that 'for fermion masses above ~300 keV, there is no enhancement of the density of the DM surrounding the BH for any BH mass, but a suppression' is presented as a general result, but it is derived from a single family of RAR models with fixed core mass (M_c = 3.5×10^6 M_sun) and fixed outer boundary conditions. The threshold mass and the depletion behavior could depend on the chosen halo parameters. The claim should be either demonstrated across the allowed parameter space of the RAR model or restated as a property of the specific configurations studied here.

minor comments (5)

[Title] The title line contains a typo: 'black hol e seeds' should read 'black hole seeds'.
[Sec. II A, Eq. (6)] The variable r in Eq. (6) is used interchangeably with the dimensionless ratio r/M_BH; please state explicitly that the integration limits and energy thresholds are given in units of the black-hole mass.
[Sec. II C] The phrase 'critic mass' appears twice and should be corrected to 'critical mass'.
[Appendix B, Eqs. (B1) and (B3)] The probability in Eq. (B1) is written as P(r_f|E_i,L), while Eq. (B3) uses P(r_f|r_i,v_i); the notation should be made consistent and the conditioning variables clearly defined.
[Sec. II C and Sec. III B] The relation between the initial core mass M_dm = 1.5×10^6 M_sun, the baryon-to-DM fraction χ = 0.5, and the resulting critical mass M_crit = 4×10^6 M_sun is not transparent. Please specify whether M_dm in Eq. (17) is the initial pure-DM core mass or the core mass at the time of collapse, and how M_crit is obtained from χ and M_dm.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the spike profiles are forward-computed from RAR initial halos via conserved action integrals; the self-citations to RAR and core-collapse results are independently grounded and not definitional inputs to the spike prediction.

full rationale

The derivation chain is a genuine forward calculation. Initial halos are generated by solving the RAR equilibrium equations (Sec. II.B, Eqs. 9 and 14), and the final spike is obtained by relating initial and final energies via adiabatic invariant action conservation (Appendix A, Eqs. A8 and A9) and then evaluating Eq. (3). No parameter is fitted to the resulting spike: the non-power-law shapes and the depletion threshold (m ~ 300 keV) are emergent outputs of the integration, and the recovery of the r^{-3/2} Boltzmannian spike is a cross-check against external literature (Gondolo & Silk, Sadeghian et al., Speeney et al.), not an input. The main self-citations (RAR halos, Refs. 38-41; critical core collapse, Refs. 47-48) are load-bearing as inputs, but they are prior, independently falsifiable results based on rotation-curve fits, stellar-stream modeling, and S-star orbital constraints; they are not defined in terms of the spike quantities being predicted. The paper even flags its own most important approximation in Sec. IV: once the BH grows, "the gravitational potential of the remaining DM mass is neglected," and for mc^2 less than or similar to 250 keV and M_BH less than or similar to 4e6 solar masses, the enclosed DM mass is comparable to M_BH. That limits the quantitative reliability of the light-fermion enhanced spikes, but it is a physical correctness caveat, not a circular reduction of the output to an input. No self-definitional, fitted-input, uniqueness-by-self-citation, ansatz-smuggling, or renaming pattern is present.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the RAR equilibrium model, the adiabatic approximation, and the neglect of final dark matter self-gravity. Most of these are domain assumptions imported from prior literature; one is explicitly acknowledged to fail in a portion of the parameter space. No new particles, forces, or conserved quantities are introduced.

free parameters (5)

fermion rest mass mc^2 = 100, 250, 300, and 378 keV (scanned values)
Chosen values, not fitted here; drives core compactness, degeneracy, and critical mass, and the central claim of mass-dependent spike profiles depends on these values.
central degeneracy parameter theta0 = 39 (or 41) for core-halo cases, -33 for the Boltzmannian case
Selects the initial halo morphology in the RAR model; determines whether the resulting spike is a power law or a non-power-law profile.
core mass M_c = 3.5e6 solar masses
Boundary condition adopted to mimic a Milky-Way-like halo from Becerra-Vergara et al.; influences the compactness of the initial core.
baryon-to-DM mass fraction chi = 0.5 (Model II example)
Sets the critical collapse mass M_crit = 4e6 solar masses for the 300 keV fermion example; imported from the self-cited baryon-induced collapse scenario.
outer halo boundary masses = M(29 kpc) = 1.8e11 solar masses, M(58 kpc) = 2.4e11 solar masses
Outer boundary conditions from a Milky-Way-like halo; affect the overall normalization and core-halo matching.

assumptions (6)

domain assumption Self-gravitating fermionic DM halos are described by the truncated Fermi-Dirac distribution of the RAR model in hydrostatic and thermodynamic equilibrium (Eq. 9).
The initial halos are generated from this distribution function and the Einstein equations; if real halos are not in this equilibrium, the spike shapes will differ.
domain assumption The maximum entropy production principle determines the most probable core-halo equilibrium (Eq. 8).
This justifies the dense core plus diluted halo morphology; it comes from self-cited statistical mechanics and is not tested here.
domain assumption The black hole grows adiabatically, so radial action and angular momentum are conserved (Eq. A8).
This is the basis of the Gondolo-Silk method; the paper argues the growth timescale is much longer than orbital periods near the center.
ad hoc to paper In the final spike state, the gravitational potential is that of the Schwarzschild black hole alone, with dark matter self-gravity neglected.
Stated limitation in Section IV; the authors acknowledge it fails where the enclosed dark matter mass is comparable to the black hole mass.
ad hoc to paper In Model II, the core collapse is radial and instantaneous, and particles outside the core conserve angular momentum and their velocity at the moment of collapse (Eq. B2).
Sudden-collapse mapping adopted from Ullio et al.; its validity for this fermionic core collapse is not independently demonstrated.
standard math The bound-orbit integration limits in Eqs. (4)-(7) apply to test particles in the Schwarzschild geometry.
The limits come from the effective potential and are taken from Sadeghian et al.; standard for this calculation.

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Pith. "Pith review of Fermionic Dark Matter spikes: origin and growth of Black Hole seeds." pith.science (2026). https://pith.science/paper/ZRJB3WHO

@misc{pith2026241217919,
  author       = {Pith},
  title        = {Pith review of: Fermionic Dark Matter spikes: origin and growth of Black Hole seeds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZRJB3WHO}},
  note         = {Machine review of arXiv:2412.17919}
}

abstract

We characterize the overdensity (spike) of fermionic dark matter (DM) particles around a supermassive black hole (SMBH) within a general relativistic analysis. The initial DM halo distribution is obtained by solving the equilibrium equations of a self-gravitating system of massive fermions at a finite temperature, according to the Ruffini-Arg\"uelles-Rueda (RAR) model. The final fermionic DM spike is calculated around a Schwarzschild SMBH. We explore two possible interpretations for the origin and evolution of the SMBH seed. One corresponds to the traditional scenario proposed by Gondolo & Silk (1999), where a small BH mass of baryonic origin sits at the halo's center and grows adiabatically. The other one, from DM origin, where the dense and degenerate fermion core predicted by the RAR model grows adiabatically by capturing baryons until its gravitational collapse, providing a heavy SMBH seed, whose specific value depends on the fermion mass. We study different initial fermionic DM profiles that the theory allows. We show that overall dilute (i.e., Boltzmannian) fermionic DM develops the well-known spike with density profile $\rho \sim r^{-3/2}$. Instead, for semi-degenerate fermions with a dense and compact core surrounded by a diluted halo, we find novel spike profiles that depend on the particle mass and nature. In the more general case, fermionic spikes do not develop a simple power-law profile. Furthermore, the SMBH does not always imply an enhancement of the surrounding DM density; it might also deplete it. Thus, the self-consistent inclusion of the DM candidate nature and mass in determining the structure and distribution of DM in galaxies, including the DM spikes around SMBHs, is essential for the specification of DM astrophysical probes such as BH mergers, gravitational waves, or stellar orbits.

Figures

Figures reproduced from arXiv: 2412.17919 by the authors.

**Figure 2.** FIG. 2. Density profiles corresponding to DM halos with fermi [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3. Dependence of the peak heights with the central BH [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: FIG. 5. Evolutionary process of a core-halo DM configuration [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 7.** Figure 7: FIG. 7. Color map of the surface of original energies [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: FIG. 8. Full extent of the gravitational metric potential of [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 10.** Figure 10: FIG. 10. Illustration of the orbital change when the critica [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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This is markedly diﬀerent from the spike density proﬁle from power-law DM density halos, which depends only on the initial power-law index γ

Given a central MBH, the fermionic spike density proﬁle crucially depends on the mass of the DM particles and their physical regime (state of degen- eracy). This is markedly diﬀerent from the spike density proﬁle from power-law DM density halos, which depends only on the initial power-law index γ

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The DM spikes arising from fermionic core-halo conﬁgurations do not develop a simple power-law proﬁle (see, e.g., Figs. 1 and 2)

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keV under the RAR model do not suﬀer from any of those problems, as explained in [ 36, 37]. In this work, we apply the general relativistic approach derived in [ 4] and adopt as the initial DM halo the equi- librium conﬁgurations of self-gravitating fermions, ob- tained with t...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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This is markedly diﬀerent from the spike density proﬁle from power-law DM density halos, which depends only on the initial power-law index γ

Given a central MBH, the fermionic spike density proﬁle crucially depends on the mass of the DM particles and their physical regime (state of degen- eracy). This is markedly diﬀerent from the spike density proﬁle from power-law DM density halos, which depends only on the initial power-law index γ

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1 and 2)

The DM spikes arising from fermionic core-halo conﬁgurations do not develop a simple power-law proﬁle (see, e.g., Figs. 1 and 2)

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Over such a mass, the DM density decreases relative to the initial one (see Fig

For a given fermion mass, there is a speciﬁc mass of the central BH above which there is no density enhancement. Over such a mass, the DM density decreases relative to the initial one (see Fig. 3). For fermion masses above ∼ 300 keV, there is no 9 enhancement of the density of the DM surrounding the BH for any BH mass, but a suppression

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In the RAR case, when the central fermions are in a state of low degeneracy (θ0 ≪ − 1), we recover the resulting DM spike of ρ ∝ r−3/2 typical of cored halos as the NSIS (see Fig

The typically assumed power-law DM spikes occur only in the case of initial DM conﬁgurations in the Boltzmannian regime. In the RAR case, when the central fermions are in a state of low degeneracy (θ0 ≪ − 1), we recover the resulting DM spike of ρ ∝ r−3/2 typical of cored halos as the NSIS (see Fig. 4). If one could constrain the DM distribution at r ≲ 1 ...

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