arxiv: 2601.17117 · v2 · submitted 2026-01-23 · 🌌 astro-ph.CO · astro-ph.HE· gr-qc

Recognition: no theorem link

Non-Equilibrium Relativistic Core Collapse of Self-Interacting Dark Matter Halos -- Limits On Seed Black Hole Mass

Hua-Peng Gu , Fangzhou Jiang , Xian Chen , Ran Li

Authors on Pith no claims yet

Pith reviewed 2026-05-16 11:31 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.HEgr-qc

keywords self-interacting dark mattergravothermal collapseseed black holesgeneral relativityMisner-Sharp formalismsupermassive black holesearly universenon-equilibrium dynamics

0 comments

The pith

Non-equilibrium relativistic collapse of self-interacting dark matter halos produces seed black holes of only 3×10^{-8} times the halo mass at horizon formation.

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

The paper models the full dynamical collapse of SIDM halos by dropping the hydrostatic equilibrium assumption and adding general-relativistic gravity through the Misner-Sharp formalism. In the late short-mean-free-path stage, strong outward heat flow inflates the outer envelope, strips mass from the core, and slows the collapse compared with earlier equilibrium models. The simulation tracks the evolution all the way to apparent-horizon formation and finds a seed black-hole mass fraction of roughly 3×10^{-8}. This small ratio implies that pure SIDM collapse cannot supply the massive seeds needed for the supermassive black holes observed at high redshift, so additional processes such as baryonic effects must be included.

Core claim

By evolving SIDM halos without assuming hydrostatic equilibrium and with general-relativistic gravity, the model shows that intense outward heat transport in the short-mean-free-path regime drives rapid expansion of the outer layers, removing mass from the collapsing core and thereby limiting the seed black hole to approximately 3×10^{-8} of the total halo mass when the apparent horizon appears.

What carries the argument

The Misner-Sharp formalism applied to SIDM, which supplies the metric and fluid equations needed to follow non-equilibrium heat flow and mass redistribution until an apparent horizon forms.

If this is right

The final seed mass is set by the amount of mass ejected from the core during the short-mean-free-path phase rather than by the initial halo mass alone.
Pure SIDM collapse without baryons cannot reach the seed masses required to explain z>6 supermassive black holes.
The apparent horizon forms later and at lower central density than predicted by hydrostatic-equilibrium models.
Mass loss from the outer envelope continues after horizon formation and reduces the final black-hole mass still further.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Baryonic cooling or angular-momentum transport inside the halo could counteract the outward heat flux and allow larger seeds to form.
The same non-equilibrium mass-loss mechanism may operate in other self-interacting particle models and could set a general upper limit on seed masses from pure dark-matter collapse.
Future high-resolution simulations that include both SIDM and baryons can test whether the reported mass fraction increases enough to match early black-hole observations.

Load-bearing premise

The treatment of heat transport and mass loss in the short-mean-free-path regime, together with the continued validity of the Misner-Sharp equations, accurately captures the dynamics in the absence of baryons or other additional physics.

What would settle it

A simulation or observation that produces a seed black hole mass fraction larger than a few times 10^{-8} of the halo mass while keeping the same SIDM cross-section and no baryons would contradict the reported outcome.

Figures

Figures reproduced from arXiv: 2601.17117 by Fangzhou Jiang, Hua-Peng Gu, Ran Li, Xian Chen.

**Figure 2.** Figure 2: FIG. 2. Evolution of the specific internal energy (left column) and density profiles (right column) for the fiducial halo. The upper panels [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. The evolution of the bulk velocity profile (upper panel) [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Evolution of the SMFP mass fraction with central density, [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. The evolution of the mass profile at BH formation stage. The [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

read the original abstract

Recent observations of supermassive black holes (SMBHs) at high redshifts pose challenges to standard seeding mechanisms. Among competing models, the collapse of self-interacting dark matter (SIDM) halos provide a plausible explanation for early SMBH formation. While previous studies on modeling the gravothermal collapse of SIDM halos have primarily focused on non-relativistic evolution under the assumption of hydrostatic equilibrium, We advance this framework by relaxing the equilibrium assumption and additionally incorporating general-relativistic effects. To this end, we introduce the Misner-Sharp formalism to the SIDM context for the first time. Our model reproduces the standard hydrostatic models in the early long-mean-free-path (LMFP) regime, but displays interesting distinct behavior in the late short-mean-free-path (SMFP) regime, where intense outward heat flux drives a rapid expansion of the outer envelope, removing mass from the core and significantly decelerating the collapse. Our general relativistic treatment enables us to follow halo evolution to the final stage when the apparent horizon forms. Our simulation yields a seed black hole mass of approximately $3\times10^{-8}$ of the halo mass at horizon formation, suggesting that additional mechanisms such as baryonic effects are critical for seeding black holes that are sufficiently massive to account for SMBHs in the early Universe.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

SIDM halo collapse to horizon in GR gives a seed BH mass fraction of only ~3e-8 due to SMFP envelope expansion and core mass loss, but the number rests on limited numerical evidence.

read the letter

The main thing to know is that this paper runs SIDM halo collapse without assuming hydrostatic equilibrium, adds general relativity via the Misner-Sharp formalism, and finds that the short-mean-free-path phase drives strong outward heat flow. That expands the outer envelope, strips mass from the core, and slows the collapse enough to leave a final seed black hole of only about 3 times 10 to the minus 8 of the halo mass at apparent-horizon formation. They recover the usual long-mean-free-path hydrostatic behavior as a check, which is useful. The new dynamical outcome in the SMFP regime is the clearest addition over earlier equilibrium models. The result is concrete and points to the need for baryonic or hybrid channels to reach the observed high-redshift SMBH masses. The soft spot is that the quoted mass fraction comes from what appears to be a single evolution with no resolution tests, convergence checks, or parameter variations shown. The SMFP heat conductivity closure and the handling of mass flux inside the Misner-Sharp equations are load-bearing, and any overestimate of outward transport would let more mass stay in the core and raise the seed fraction by orders of magnitude. Without those diagnostics it is hard to judge how robust the number is. This is worth sending to referees for people working on SIDM seeding or relativistic gravothermal collapse; the method is new and the outcome is specific enough to be checked. I would accept it for review.

Referee Report

2 major / 2 minor

Summary. The manuscript applies the Misner-Sharp formalism to simulate the non-equilibrium, general-relativistic core collapse of a self-interacting dark matter halo. It reproduces prior hydrostatic long-mean-free-path (LMFP) results but finds qualitatively different short-mean-free-path (SMFP) evolution in which outward heat flux expands the envelope, ejects mass from the core, and slows collapse. The simulation is evolved to apparent-horizon formation, producing a seed black-hole mass fraction of approximately 3×10^{-8} of the halo mass and concluding that baryonic physics is required to reach observationally viable SMBH seeds.

Significance. If the SMFP mass-loss result is robust, the work supplies a concrete, relativistic upper bound on SIDM-only seed masses that is directly relevant to high-redshift SMBH formation. The first use of the Misner-Sharp metric in this context and the explicit reproduction of LMFP hydrostatic equilibria are technical strengths. The small reported fraction, however, rests on a single numerical realization whose sensitivity to the heat-conductivity closure and mass-flux treatment remains unquantified.

major comments (2)

[Results / Simulation Setup] The headline seed-mass fraction of ~3×10^{-8} is obtained from the late-time SMFP regime; the manuscript provides no resolution study, convergence test, or variation of the conductivity closure that would demonstrate this fraction is insensitive to numerical choices or to the precise form of the heat-flux term inside the Misner-Sharp equations.
[Methods / Heat Transport] The treatment of heat transport and mass loss once the mean free path becomes short is load-bearing for the central claim, yet the paper does not compare the adopted closure against known analytic limits or against an independent code in the SMFP regime; an overestimate of outward heat flux would systematically reduce the retained core mass and therefore the final seed fraction.

minor comments (2)

[Figures] Figure captions and axis labels should explicitly state the units and the precise definition of the plotted mass fraction (e.g., enclosed mass within the apparent horizon versus total halo mass).
[Abstract / §3] The abstract states that the model 'reproduces the standard hydrostatic models' in the LMFP regime; a quantitative comparison (e.g., density-profile residuals or collapse timescale) should be shown in the main text rather than asserted.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading and constructive comments, which have helped us identify areas for improvement. We appreciate the acknowledgment of the technical novelty in applying the Misner-Sharp formalism to this problem and the reproduction of prior LMFP results. We address each major comment below and will revise the manuscript to strengthen the numerical robustness of our findings.

read point-by-point responses

Referee: [Results / Simulation Setup] The headline seed-mass fraction of ~3×10^{-8} is obtained from the late-time SMFP regime; the manuscript provides no resolution study, convergence test, or variation of the conductivity closure that would demonstrate this fraction is insensitive to numerical choices or to the precise form of the heat-flux term inside the Misner-Sharp equations.

Authors: We agree that a dedicated resolution and convergence study focused on the SMFP regime is needed to support the robustness of the reported seed-mass fraction. In the revised manuscript we will add simulations at three different radial resolutions (doubling the number of zones each time) and demonstrate that the final seed mass fraction converges to within ~15%. We will also vary the conductivity prefactor by ±25% around the fiducial value and report the resulting range in the retained core mass at apparent-horizon formation. These additions will directly address sensitivity to numerical choices and the heat-flux implementation. revision: yes
Referee: [Methods / Heat Transport] The treatment of heat transport and mass loss once the mean free path becomes short is load-bearing for the central claim, yet the paper does not compare the adopted closure against known analytic limits or against an independent code in the SMFP regime; an overestimate of outward heat flux would systematically reduce the retained core mass and therefore the final seed fraction.

Authors: The heat-conductivity closure we employ is the standard diffusion approximation used in the SIDM literature for the short-mean-free-path limit. In the revision we will add an explicit comparison of our implementation against the expected analytic scaling of heat flux with temperature gradient and density in the optically thick regime, confirming consistency with prior non-relativistic results. A direct comparison against an independent relativistic code is not currently feasible, as no such code exists in the published literature; we will therefore discuss this limitation openly and provide additional internal tests (e.g., recovery of the hydrostatic LMFP equilibria at higher resolution and reproduction of the expected mass-loss behavior when the heat-flux term is artificially suppressed). We maintain that the outward heat flux and associated mass ejection are physical consequences of relaxing the hydrostatic assumption, rather than numerical artifacts. revision: partial

standing simulated objections not resolved

Direct comparison of the SMFP heat-transport implementation against an independent general-relativistic SIDM code (no such code is available in the literature)

Circularity Check

0 steps flagged

No significant circularity: result is direct output of numerical evolution

full rationale

The central result (seed BH mass fraction ~3e-8 at apparent horizon) is obtained from time-dependent numerical integration of the Misner-Sharp equations with SIDM heat transport, not from algebraic reduction, parameter fitting, or self-citation. The paper states it reproduces known LMFP hydrostatic models as a validation check but reports distinct SMFP dynamics as an emergent simulation outcome. No load-bearing step reduces by construction to the inputs; the formalism is introduced as new to the SIDM context without reliance on prior author theorems or ansatzes. This is the expected non-circular outcome for a simulation study whose predictions are falsifiable against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the Misner-Sharp metric for SIDM fluid evolution and on an unspecified heat-transport closure in the SMFP regime; no free parameters are explicitly listed in the abstract.

axioms (2)

domain assumption Misner-Sharp formalism applies to the SIDM fluid throughout collapse
Invoked to incorporate GR effects and track apparent horizon formation
domain assumption Heat flux in the short-mean-free-path regime drives net outward mass transport
Underlies the reported envelope expansion and core deceleration

pith-pipeline@v0.9.0 · 5550 in / 1326 out tokens · 25110 ms · 2026-05-16T11:31:33.885999+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

139 extracted references · 139 canonical work pages · 48 internal anchors

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Non-Equilibrium Relativistic Core Collapse of Self-Interacting Dark Matter Halos -- Limits On Seed Black Hole Mass

which have been interpreted by many studies as actively accreting SMBHs [12–14]. The LRDs appear unexpectedly abundant at redshifts 4<z<9 and are believed to host BHs with masses ranging from 10 7 to 109 M⊙ [15, 16]. However, their stellar masses are surprisingly low relative to the scaling relations observed in local active galactic nuclei (AGNs)[17– 19]...

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or energy conservation [99]. To rigorously determine this mass, a complete time-dependent GR analysis tracing the non- equilibrium evolution towards horizon formation is required. Incorporating GR into the non-equilibrium evolution will introduce profound physical consequences that were previ- ously overlooked. As matter falls deep into the relativistic p...

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Update Velocity (U):Using variables from the previous time stepn−1: Un i =Un−1 i + ∆t " (Γn−1 i )2 ∂ϕ ∂A !n−1 i × 4π(Rn−1 i )2 ¯ρn−1 i −(e ϕ)n−1 i  mn−1 i (Rn−1 i )2 +4πR n−1 i ¯Pn−1 i  # , (A1) where the potential gradient term is given by: ∂ϕ ∂A !n−1 i = −1 2 ¯wn−1 i " Pn−1 i+1/2 −P n−1 i−1/2 ¯ρn−1 i (Ai+1/2−A i−1/2) + σ (eϕ)n−1 i ∆t qn−1 i ...

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Update Radius (R): Rn i =R n−1 i +(e ϕ)n−1 i Un i ∆t.(A3)

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Update Lorentz Factor (Γ):We utilizem n−1 i as an approx- imation form n i to decouple the implicit dependence9: Γn i = s 1+(U n i )2 − 2mn−1 i Rn i .(A4)

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Update Density (ρ): ρn i−1/2 = (Ai −A i−1)(Γn i−1 + Γn i )/2 4π 3 h (Rn i )3 −(R n i−1)3 i .(A5)

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Update Internal Energy (ϵ): ϵn i−1/2 =ϵ n−1 i−1/2 −P n−1 i−1/2  1 ρn i−1/2 − 1 ρn−1 i−1/2  − 4π∆t ¯eϕn i−1/2  F n−1 i − F n−1 i−1 Ai −A i−1  , (A6) whereF i ≡(R i)2qi(eϕ i )2

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Update Pressure (P) and Enthalpy (w): Pn i−1/2 =(γ−1)ρ n i−1/2ϵn i−1/2,(A7) wn i−1/2 =1+ϵ n i−1/2 + Pn i−1/2 ρn i−1/2 .(A8) 9 Strictly,Γ n depends onm n. However, calculatingm n requiresΓ n. For suf- ficiently small time steps, usingm n−1 provides a valid first-order approxi- mation. 12

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Update Metric Potential (ϕ):Integrated inward from the boundary condition (eϕ)n N =1: ln((eϕ)n i )=ln((e ϕ)n i+1) − 1 2(Ai+1 −A i) " ∂ϕ ∂A !n i + ∂ϕ ∂A !n i+1 # . (A9)

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