Dark matter spikes with strongly self-interacting particles

Alejandro Ibarra; Boris Betancourt Kamenetskaia; Motoko Fujiwara; Takashi Toma

REVIEW 3 major objections 4 minor 1 cited by

Dark matter spikes with strongly self-interacting particles

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Number-changing dark matter self-interactions that convert three or more particles into fewer ones can deplete the dense spikes around supermassive black holes, capping the central density far below the standard power-law profile and…

desk verdict Competent spike/SIMP framework with correct rate equations, but the abstract's n≥3 depletion claim contradicts the paper's own benchmarks and needs reconciling. read the letter →

arxiv 2506.12642 v1 pith:O6LY4WR7 submitted 2025-06-14 hep-ph

Boris Betancourt Kamenetskaia , Motoko Fujiwara , Alejandro Ibarra , Takashi Toma This is my paper

classification hep-ph

keywords darkmatterspikeself-interactingnumber-changingprocesses3to2annihilationsemi-annihilationSIMPJ-factorsupermassiveblackhole

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 whether dark matter self-interactions that change particle number can be tested in the ultra-dense dark matter spikes expected around supermassive black holes. It argues that processes converting $n$ initial dark matter particles into $m$ final ones ($n>m$), which are generic in self-interacting dark matter models, can proceed at observable rates there. For $n\geq 3$, cross-sections favored by thermal freeze-out production of the dark matter significantly deplete the spike, while the $2\to 1$ semi-annihilation process generally preserves its structure. These density changes affect the $J$-factors that control predicted signals, so the spike's particle physics must be included in phenomenological forecasts.

What carries the argument

The central objects are the plateau density $\rho_{\mathrm{pl}}$ from the dissolution equation (the density below which a number-changing process cannot deplete the spike within the halo age) and the three radius scales $R_c$ (isothermal core), $R_{n\to m}$ (self-heating core), and $R_{\mathrm{diss}}$ (dissolution radius) that partition the spike profile. The argument operates by evolving the local dark matter number density $n_\chi(r,t)$ with $\dot{n}_\chi = -\langle \sigma_{2\to 0}v\rangle n_\chi^2 - (n/n!)\langle \sigma_{n\to m}v^{n-1}\rangle n_\chi^n$, assuming all $n$ initial particles are lost because the $m$ final-state particles are relativistic and escape. The radius where the resulting density saturates at $\rho_{\mathrm{pl}}$ gives the depletion boundary; comparing $R_{\mathrm{diss}}$, $R_c$, and $R_{n\to m}$ determines which effect dominates the observed profile. Cross-sections are parametrized as $\sigma_{n\to m}v^{n-1} \equiv \alpha_{n\to m}^n / m_\chi^{3n-4}$, following the freeze-out literature, so the benchmark values correspond to couplings that also set the relic abundance.

What would settle it

Compute the optical depth for a boosted final-state particle from an $n\to m$ reaction to undergo a $2\to 2$ scattering before it leaves the spike; if a non-negligible fraction is recaptured for the benchmark cross-sections, the dissolution equation overcounts particle loss and the predicted depletion and $J$-factors would need to be revised.

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Extended reading notes

Core claim

The paper's central claim is that the fate of a dark matter spike is governed by a competition among four effects: isothermal core formation from $2\to 2$ self-scattering, self-heating by boosted final-state particles from $n\to m$ reactions, dissolution of the central density by number-changing processes, and (when present) $2\to 0$ annihilation. For representative spike parameters and $n\geq 3$ processes such as $3\to 2$, the dissolution plateau density $\rho_{\mathrm{pl}} = m_\chi \big( (N-2)!\,/\,\langle \sigma_{N\to M} v^{N-1}\rangle\, t_{\mathrm{age}} \big)^{1/(N-1)}$ bounds the spike density at the level needed for freeze-out, substantially flattening the inner profile. For the $2\to 1$ semi-annihilation the plateau is not restrictive, so the spike shape is instead set by core formation and self-heating. The paper concludes that these effects significantly modify the $J$-factors for photon, neutrino, and boosted dark matter signals relative to naive NFW-based expectations.

Load-bearing premise

The load-bearing assumption is that in every $n\to m$ process all $n$ initial dark matter particles are lost from the local density because the $m$ final-state particles are relativistic and escape the spike, with no account of the fraction that is recaptured by strong $2\to 2$ self-scattering before escaping.

Editorial extensions

If this is right

For $n\geq 3$ processes with freeze-out-favored cross-sections, the central spike density is capped near the plateau density, so annihilation fluxes and boosted dark matter fluxes from the inner spike are markedly lower than collisionless-spike predictions.
For $2\to 1$ semi-annihilation, the spike structure is preserved in general, so semi-annihilating dark matter can still produce strong boosted-dark-matter signals from galactic centers.
The $J_2$ factor is enhanced relative to the NFW expectation only when $\sigma_{2\to 2}/m_\chi \lesssim 10^{-4}\,\mathrm{cm^2\,g^{-1}}$ and $\langle\sigma_{2\to 1}v\rangle \lesssim 10^{-22}\,\mathrm{cm^3\,s^{-1}}$; otherwise the spike yields less signal than the naive profile.
When $\sigma_{2\to 2}/m_\chi \gtrsim 10^{-4}\,\mathrm{cm^2\,g^{-1}}$, self-heating and core formation lower $J_3$ below the NFW expectation even for perturbative $3\to 2$ couplings.
Phenomenological studies that use dark matter spikes should include these density modifications rather than assuming the bare power-law spike.

Reading between the lines

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

If recapture of boosted final-state particles proves efficient, the effective particle loss per $n\to m$ event drops from $n$ to $n-m$, weakening the dissolution depletion; the paper's plateau-density bounds would then be upper limits rather than typical densities.
The same dissolution logic should apply to other high-density dark matter environments, such as the centers of some dwarf galaxies or halos around smaller black holes, where the plateau density could be tested without relying on the Milky Way spike's uncertain stellar-heating history.
Combining the plateau-density cap with stellar-heating constraints suggests that the observable spike signal may be dominated by the outer spike region, making line-like semi-annihilation signatures more promising than searches for $n\geq 3$ process signals.
A numerical simulation tracking boosted particles as they propagate through the spike, rather than the single-scattering efficiency $\xi(r)$, would settle whether self-heating and dissolution act in the same direction or partially compensate.

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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

Quick take: the paper builds a clean framework for n→m number-changing processes in DM spikes — dissolution plateaus, self-heating, isothermal cores — and the rate equations and plateau densities (Eqs. 3.10–3.12) are correct for the stated model. The genuinely new piece is applying 3→2 and 4→2 processes to spikes and computing the generalized J-factors; the 2→1 semi-annihilation result (the spike structure survives) is a clean, useful finding. The calculations are self-contained and the citation pattern is honest, with prior work by Chu-Garcia-Cely, Kamada-Kim, Shapiro-Paschalidis, and Hochberg et al. properly credited.

The first soft spot is the mismatch between the abstract and the benchmarks. The abstract says n≥3 processes significantly deplete the spike at freeze-out-favored cross-sections. The E–H benchmarks at mχ=100 MeV show the opposite, and Sec. 3.3 goes further, claiming the subdominance holds for all n≥3 with perturbative couplings. That is not the same statement as the abstract, and the 'verified for general n≥3' claim is not backed by an actual argument in the text. The depletion regime actually lives at the canonical reference point, mχ ~ 10 MeV with α3→2 ~ 1, where ρ_pl ≈ 1.9×10^13 GeV/cm³ cuts below the spike and would suppress J_3 substantially. The authors should add that regime to the benchmark coverage or narrow the abstract; as written, the paper argues against its own headline.

Second, the all-escape assumption in Eq. (3.10). Counting all n initial particles as lost because the m final particles are relativistic is too strong when σ2→2 is large: the boosted particles can scatter, deposit energy, and be recaptured before escaping, making the net loss closer to n−m. The stress-test version of this overstates the correction — capture plausibly needs many scatterings, not one, and the plateau density shifts by only √3 or 2^{1/3} — but the tension is real. Self-heating assumes these same final particles deposit energy in the spike, while dissolution assumes they leave, and the paper never reconciles the two. A paragraph of honest discussion would fix it.

Who this is for: people computing indirect-detection signals or boosted-DM fluxes from galactic centers, and SIMP phenomenologists. The J-factor formulas are a useful reference even where the depletion claim is contested. Send it to review — it deserves a serious referee, but expect a required revision that aligns the abstract with Sec. 3.3 and the benchmark coverage.

Referee Report

3 major / 4 minor

Summary. This paper studies how number-changing dark-matter self-interactions (n→m processes) affect the density profile of dark-matter spikes around supermassive black holes. It combines three effects: isothermal core formation from 2→2 scattering, self-heating from boosted final-state particles, and density dissolution from 2→0 and n→m processes. The authors derive analytic plateau densities, present benchmark profiles for 2→1 and 3→2 processes at mχ=100 MeV, and compute generalized J-factors. The central claim is that for n≥3 processes with cross-sections favored by thermal freeze-out the spike is significantly depleted, while semi-annihilation 2→1 largely preserves the spike.

Significance. The framework is timely and useful: if the depletion is real, it caps the central spike density and suppresses J-factors for annihilation and boosted-DM searches. The rate equations and plateau-density formulas are straightforward and correctly derived for the stated model, and the J-factor comparison with the NFW baseline is a clear phenomenological output. No parameters are fitted to the target result; the benchmark cross-sections are taken from prior SIMP studies. However, the manuscript's central claim is not currently supported by its own benchmarks and rests on an unquantified all-escape assumption for n→m final states.

major comments (3)

[Sec. 3.3, Eq. (3.10)] Equation (3.10) removes n particles per n→m event, justified by the sentence following it that final-state particles are relativistic and escape the orbit. However, in the strongly self-interacting regime adopted in benchmarks E-H (σ2→2/mχ up to 5×10^-2 cm^2/g), the optical depth ξ(r) defined in Eq. (3.8) is of order unity throughout the region where dissolution is relevant, so most final-state particles scatter before escaping. If a scattered particle is recaptured, the net number loss is n−m rather than n; for 3→2 this raises the plateau density in Eq. (3.14) by sqrt(3), and for 4→2 it raises Eq. (3.15) by 2^(1/3). The statement in Sec. 3.2 that 'the capture rate is much smaller than 1' concerns the boosted flux observed at Earth, not the density profile, so it does not resolve the inconsistency. Please quantify the recapture probability self-consistently and either modify Eq. (3.10) or justify the all-escape limit.
[Abstract vs. Sec. 3.3 and Fig. 2] The abstract claims 'for n≥3, the spike is significantly depleted for n→m cross-sections favored by DM production via thermal freeze-out.' However, Fig. 2 and the text following it state that for the benchmark 3→2 rates (E-H), neither self-heating nor dissolution significantly alters the density profile, and that for general n≥3 the cross-sections required to make these effects dominant are no longer perturbative. This is an internal contradiction between the headline claim and the paper's own benchmarks. Please revise the abstract and conclusions to match the benchmark results, or demonstrate explicitly that the benchmarks are not representative of the freeze-out-favored parameter region.
[Sec. 3.2, self-heating discussion] The manuscript states that 'We have checked the capture rate is much smaller than 1 in our interesting parameter region,' but no calculation, equation, or figure is provided for this check. Because this statement is used to separate the effect on the boosted flux from the effect on the density profile, and because the density-profile effect is the load-bearing part of the n≥3 depletion claim, the check should be written out explicitly or replaced by a proper treatment of recapture in the dissolution equation.

minor comments (4)

[Table 1] The header for the 2→1 column should read [cm^3 s^-1] rather than [cm^3 s]; the current notation is dimensionally inconsistent.
[Figs. 1 and 2 captions] The labels R2→1 and R3→2 appearing in the figures are not defined in the captions; please define them as the self-heating core radii for the respective processes.
[Sec. 3.3, final paragraph] The sentence 'We have verified that this behavior remains for general n→m processes with n≥3' is an unsupported assertion in the text; if it is to be retained, please provide the underlying calculation or a supplementary figure showing a representative 4→2 case.
[Sec. 3.2, Eq. (3.9)] The heat time-scale estimate uses 'the typical radius' without a precise specification; please state exactly how that radius is chosen from the isothermal/self-heating profile.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the spike-depletion derivation is a self-contained application of external spike, self-heating, and SIMP freeze-out inputs, not a reduction to its own outputs.

full rationale

The paper's derivation chain is not circular in the structural sense captured by the seven patterns. The spike profile (Eq. 3.1) is the standard Gondolo-Silk result [53]; the isothermal-core radius (Eq. 3.5) and profile index (Eq. 3.6) come from Kaplinghat-Tulin-Yu [61] and Shapiro-Paschalidis [63]; the self-heating efficiency xi(r) and heat time (Eqs. 3.8-3.9) follow Chu-Garcia-Cely [44] and Kamada-Kim [45], all external to this author set. The dissolution plateau (Eqs. 3.10-3.12) is an internally derived ODE solution whose parameters are the externally fixed freeze-out cross-sections from Ref. [27]; no parameter is fitted to the target J-factors or to the depletion claim. The only self-citations (Refs. [36-38], plus topical model references [31,32,47,48]) motivate boosted-DM signatures and specific particle models; they do not carry the density-depletion argument, and no uniqueness theorem is imported from the authors' prior work. Several flagged weaknesses are genuine but they are robustness or correctness concerns, not circularity. First, Sec. 3.3 assumes all n initial particles are lost per event because the m final-state particles are 'boosted to relativistic velocities' and 'able to escape from their orbit at r', which sits in tension with the same paper's capture efficiency xi(r) <= 1 (Eq. 3.8) in the regime of large sigma2->2/mchi; efficient recapture would soften depletion by factors such as sqrt(3) for 3->2 and 2^(1/3) for 4->2. Second, Sec. 3.2 asserts without demonstration that 'the capture rate is much smaller than 1 in our interesting parameter region'. Third, Sec. 3.3 asserts 'we have confirmed that this is a general statement for n->m processes with n>=3' with no proof shown. Fourth, the abstract's claim of significant depletion for n>=3 sits in tension with the body's statement that in benchmark scenarios E-H 'neither the self-heating nor the dissolution induced by the 3->2 interactions significantly alter the DM density profile'. These issues should be weighed as physical-modeling risk and internal-consistency risk, not as the derivation reducing to its own inputs.

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

No new particles or forces are introduced. The central claim rests on standard spike modeling plus the prior parametrization of n to m cross-sections and self-heating from the SIMP and SIDM literature. The main hand-chosen inputs are the DM mass, the effective couplings, the halo age, and the Milky Way fiducial parameters.

free parameters (6)

DM mass mχ = 100 MeV in benchmarks; 10 MeV (3 to 2) and 100 keV (4 to 2) in reference plateau formulas
Chosen by hand for illustration; the reference values correspond to SIMP freeze-out predictions from Ref. [27].
2 to 2 self-scattering coupling α2→2 = 2.14e-5 to 4.78e-1 across benchmarks (σ2→2/mχ from 1e-10 to 5e-2 cm^2/g)
Controls isothermal core formation and self-heating efficiency; scanned over many orders of magnitude.
2 to 1 semi-annihilation coupling α2→1 = 2.93e-6
Set by ⟨σv⟩_2→1 = 1e-26 cm^3/s at mχ = 100 MeV.
3 to 2 coupling α3→2 = 4.81e-2
Set by ⟨σv^2⟩_3→2 = 1e-57 cm^6/s at mχ = 100 MeV.
Halo age t_age = 10 Gyr
Assumed value used in the plateau density formulas (Eqs. 3.13 to 3.15).
Milky Way fiducial halo parameters = v0=140 km/s, ρs=0.184 GeV/cm^3, rs=24.42 kpc, MBH=4.15e6 M_sun
Used for the spike profile and J-factor ratios; taken from Refs. [54,57,64] without uncertainty propagation.

assumptions (5)

domain assumption Spike profile follows the Gondolo and Silk power law (Eq. 3.1) with γ_sp = 7/3 and inner cutoff at 4 R_sch.
Adopted from Ref. [53]; neglects stellar heating and the relativistic inner edge at 2 R_sch. The paper explicitly states this choice makes the boosted DM flux conservative.
domain assumption All n initial DM particles are removed from the local density in an n to m process because the final-state particles are relativistic and escape (Sec. 3.3).
Load-bearing for dissolution; if a significant fraction is recaptured by 2 to 2 scattering before escape, depletion is overestimated.
domain assumption Self-heating efficiency is given by ξ(r) = r ρχ σ2→2 / mχ, with 100% energy transfer when ξ=1 (Sec. 3.2, Eqs. 3.8 to 3.9).
Taken from Refs. [44,45]; the 100% assumption overestimates energy injection and is acknowledged as conservative for flux.
standard math The rate equation (3.10) can be reduced to a single dominant process, giving the plateau density (3.12).
Exact for one process; the paper solves the full equation numerically when multiple processes coexist.
domain assumption The cross-section parametrization (2.1) to (2.2), with α_n→m, α_2→2, α_2→0 independent, and the freeze-out reference values from Ref. [27], are valid.
The effective couplings are treated as free parameters spanning the perturbative range; the freeze-out normalization is imported from the SIMP literature.

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Cite this review

Pith. "Pith review of Dark matter spikes with strongly self-interacting particles." pith.science (2026). https://pith.science/paper/O6LY4WR7

@misc{pith2026250612642,
  author       = {Pith},
  title        = {Pith review of: Dark matter spikes with strongly self-interacting particles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O6LY4WR7}},
  note         = {Machine review of arXiv:2506.12642}
}

abstract

An unavoidable prediction of scenarios with Dark Matter (DM) self-interactions is the existence of number changing processes that convert $n$ initial DM particles into $m$ final ones ($n\to m$ processes), possibly accompanied by Standard Model particles. We argue that the $n\rightarrow m$ processes could be probed in DM spikes at the center of galaxies, where the high density may allow sizable rates. We systematically study the implications of the $n \to m$ processes in DM spikes, including other possible interactions involving DM, such as annihilation and self-scattering. We find that for $n\geq3$, the spike is significantly depleted for $n\to m$ cross-sections favored by DM production via thermal freeze-out. On the other hand, the semi-annihilation of two DM particles into one DM particle and one Standard Model particle preserves in general the structure of the spike. Such density modifications significantly affect phenomenological studies of both astrophysics and particle DM processes around DM spikes.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Dark matter energy exchange in stars orbiting supermassive black holes
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Orbit-averaged elastic DM scattering in S4714 reaches stellar luminosity at σ_χp ∼ 10^{-36} cm² (MeV–GeV) and σ_χe ∼ 5×10^{-38} cm² (sub-MeV) for a spiked profile.

Reference graph

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