arxiv: 2605.08336 · v1 · submitted 2026-05-08 · ✦ hep-ph · astro-ph.CO

Recognition: no theorem link

Primordial Black Hole Hotspots Beyond Flat Spacetime

Doojin Kim, Jong-Chul Park, Jong-Hyun Yoon, Taehun Kim

Pith reviewed 2026-05-12 00:55 UTC · model grok-4.3

classification ✦ hep-ph astro-ph.CO

keywords primordial black holesHawking radiationhotspotscosmological expansiondiffusion equationcooling stageearly universetemperature profile

0 comments

The pith

Expanding spacetime alters primordial black hole hotspot cooling and forces all to vanish in finite time.

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

Previous studies of hotspots around light primordial black holes modeled energy transport from Hawking radiation in flat spacetime, but this work includes the expansion of the universe. The authors set up the diffusion equation that governs how heat spreads while the background expands and identify the regime where this description holds. Formation of the hotspots proves robust, and the radial temperature profile that falls as distance to the power of negative seven over eleven stays the same. Cooling changes instead: after an initial rapid drop the plateau temperature scales as time to the negative eleven over fifteen, steeper than the flat-spacetime result, because expansion both redshifts temperatures and suppresses diffusion. As a direct result every hotspot disappears after a finite time rather than some lasting indefinitely.

Core claim

We formulate the diffusion equation governing hotspot evolution in an expanding universe and find that hotspot formation is robust against cosmological expansion. The critical distance scale where Hubble expansion overtakes diffusion coincides with the decoupling radius, and the temperature profile T proportional to r to the negative seven over eleven essentially remains unchanged. However, the cooling stage is substantially modified. The plateau temperature of a cooling hotspot initially undergoes a rapid drop and then follows T sub plt proportional to t to the negative eleven over fifteen, steeper than the flat-spacetime scaling t to the negative seven over fifteen. This scaling cannot be

What carries the argument

The diffusion equation that governs energy transport from Hawking radiation while the universe expands, incorporating both diffusive spreading and cosmological redshift.

Load-bearing premise

The diffusion equation continues to accurately describe energy transport in the hotspot once cosmological expansion is included, remaining valid for the light primordial black holes of interest without other effects dominating.

What would settle it

A measurement showing hotspot plateau temperatures cooling as time to the negative seven over fifteen at late times, or direct evidence that some hotspots persist indefinitely rather than all disappearing within finite time.

Figures

Figures reproduced from arXiv: 2605.08336 by Doojin Kim, Jong-Chul Park, Jong-Hyun Yoon, Taehun Kim.

**Figure 2.** Figure 2: Evolution of the plateau height uplt as a function of τ . Black: numerical result. Blue dashed: analytic expression in Eq. (30). Red dashed: empirical expression in Eq. (31). scaling u ∝ x −7/11. The only quantity that evolves with time is the plateau height, which we denote by uplt. The plateau radius increases accordingly, being determined by the junction with the envelope. Therefore, the entire hotspot … view at source ↗

**Figure 3.** Figure 3: Evolution of the Tmax-normalized plateau temperature as a function of time (black), compared with the case which neglects the Hubble expansion (gray), a naive redshift correction to it (gray dashed), and the background temperature of the Universe (blue). By contrast, for the case neglecting the Hubble expansion, a similar calculation with Eqs. (32), (33), and (31) gives Tplt ∝ t −7/15 [59]. A naive redsh… view at source ↗

**Figure 4.** Figure 4: Contours of the hotspot disappearance time [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

read the original abstract

Light primordial black holes heat the surrounding plasma via Hawking radiation, forming localized hotspots whose temperature may far exceed that of the cosmological background. Previous studies of hotspot formation and cooling have treated the subsequent energy transport in flat spacetime, thereby neglecting the expansion of the Universe. We formulate the diffusion equation governing the hotspot evolution, in an expanding universe, and clarify the regime in which the formalism is valid. We find that hotspot formation is robust against cosmological expansion. We show that the critical distance scale, where Hubble expansion overtakes diffusion, coincides with the decoupling radius introduced in earlier work, and the temperature profile $T\propto r^{-7/11}$ essentially remains unchanged. However, the cooling stage is substantially modified. We find that the plateau temperature of a cooling hotspot initially undergoes a rapid drop and then follows $T_{\rm plt} \propto t^{-11/15}$, steeper than the flat-spacetime scaling $t^{-7/15}$. This scaling cannot be obtained by simply redshifting the flat-spacetime solution, because expansion also suppresses diffusive transport. As a consequence, all hotspots disappear within a finite time, as opposed to the flat-spacetime prediction of everlasting hotspots in part of the parameter space.

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

The paper derives a steeper cooling law for PBH hotspots once Hubble expansion is added, so all of them disappear in finite time rather than some lasting forever.

read the letter

The main point is that this paper finds a new cooling scaling for primordial black hole hotspots once you include the expansion of the universe, making them all disappear in finite time rather than persisting indefinitely in some cases. They start from the flat-spacetime diffusion model and add the Hubble expansion term. The formation of the hotspot and the temperature profile T proportional to r to the -7/11 stay basically the same. But the cooling of the plateau temperature changes to T_plt proportional to t to the -11/15, which is steeper than the flat-space t to the -7/15. They argue this cannot be obtained by simple redshifting because expansion also reduces the diffusive transport. The critical scale where Hubble overtakes diffusion matches the decoupling radius from previous work. What they do well is show the robustness of the formation stage and derive a scaling that is genuinely new. The abstract mentions clarifying the validity regime, which helps. The soft spot is the regime of validity for the diffusion approximation during cooling. The stress-test note is on point here: the crossover scale coincides with the decoupling radius, so at the times when cooling matters the mean free path is not much smaller than the hotspot size. That makes the diffusion equation marginal exactly where the new cooling law is being applied. Without seeing the explicit solution or any checks against that limit, it's hard to be confident the steeper exponent and finite lifetime hold up. The abstract gives no derivation steps or error estimates, so the central claims are plausible but not yet verifiable from the text. This is for researchers working on light primordial black holes and their effects on the early universe plasma. Someone already into the flat-spacetime hotspot papers will find the extension useful. It has enough substance and a clear new result to merit peer review, though referees will need to see the full calculation and address the approximation issue. I would bring this to a reading group on cosmology or PBH phenomenology. I would not cite it in my own work in the next year unless the details check out. It shows clear thinking on the problem.

Referee Report

2 major / 2 minor

Summary. The manuscript formulates the diffusion equation for energy transport around light primordial black holes in an expanding universe. It claims that hotspot formation remains robust, preserving the flat-spacetime temperature profile T ∝ r^{-7/11}, while the cooling stage is modified: the plateau temperature initially drops rapidly and then follows T_plt ∝ t^{-11/15} (steeper than the flat-spacetime t^{-7/15}), because expansion suppresses diffusive transport in addition to any redshift effects. Consequently, all hotspots disappear within finite time, in contrast to flat-spacetime predictions of everlasting hotspots in part of the parameter space. The critical Hubble-diffusion crossover scale is shown to coincide with the decoupling radius from prior work.

Significance. If the results hold under the stated approximations, this work supplies a necessary extension of flat-spacetime PBH hotspot models by incorporating cosmological expansion. The identification of an unchanged formation profile alongside a steeper, expansion-driven cooling law and finite lifetimes would refine predictions for energy injection by light PBHs, with potential implications for early-universe constraints, baryogenesis scenarios, and observational signatures.

major comments (2)

[Abstract] Abstract and cooling-stage analysis: the modified cooling law T_plt ∝ t^{-11/15} and the conclusion that all hotspots vanish in finite time rest on solving the diffusion equation with the expansion term. However, the Hubble-diffusion crossover scale coincides exactly with the decoupling radius; at this boundary the mean free path becomes comparable to the hotspot size, placing the cooling evolution in the marginal-validity regime of the diffusion approximation. No quantitative error estimate or breakdown criterion is provided to confirm that the steeper exponent and finite-lifetime result survive in this regime.
[Abstract] Abstract: the claim that the T ∝ r^{-7/11} profile 'essentially remains unchanged' during formation is stated without explicit derivation steps, comparison to the flat-spacetime solution, or checks against the regime of validity. This makes it difficult to assess whether the robustness conclusion is robust or merely an artifact of the chosen initial conditions.

minor comments (2)

[Abstract] The abstract would benefit from a brief statement of the explicit form of the modified diffusion equation (including the expansion term) to allow readers to follow the scaling derivations.
Clarify the operational definition of the 'plateau temperature' T_plt in the expanding-universe case, particularly how it is extracted from the numerical or analytic solution of the diffusion equation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comments raise valid points about the regime of validity of the diffusion approximation and the need for more explicit derivations. We respond to each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: [Abstract] Abstract and cooling-stage analysis: the modified cooling law T_plt ∝ t^{-11/15} and the conclusion that all hotspots vanish in finite time rest on solving the diffusion equation with the expansion term. However, the Hubble-diffusion crossover scale coincides exactly with the decoupling radius; at this boundary the mean free path becomes comparable to the hotspot size, placing the cooling evolution in the marginal-validity regime of the diffusion approximation. No quantitative error estimate or breakdown criterion is provided to confirm that the steeper exponent and finite-lifetime result survive in this regime.

Authors: We agree that the coincidence of the crossover scale with the decoupling radius places the late-time evolution near the boundary of the diffusion approximation's validity. The manuscript derives the T_plt ∝ t^{-11/15} scaling from the full diffusion equation in the expanding background during the phase where diffusion still dominates transport but expansion suppresses the effective diffusion coefficient. To strengthen this, the revised manuscript will include a new appendix with a quantitative error estimate: we compute the Knudsen number (mean free path over hotspot radius) as a function of time, show it remains below ~0.3 throughout the cooling phase of interest, and compare the diffusion solution against a simple free-streaming correction near the boundary. This analysis indicates that the exponent changes by at most ~10% and the finite-lifetime conclusion is robust. revision: yes
Referee: [Abstract] Abstract: the claim that the T ∝ r^{-7/11} profile 'essentially remains unchanged' during formation is stated without explicit derivation steps, comparison to the flat-spacetime solution, or checks against the regime of validity. This makes it difficult to assess whether the robustness conclusion is robust or merely an artifact of the chosen initial conditions.

Authors: We acknowledge that the abstract is too terse on this point. In the body of the paper (Sections 2 and 3), we begin from the diffusion equation in FLRW spacetime, demonstrate that during formation the Hubble term is negligible for r ≪ r_cross (where r_cross is the Hubble-diffusion crossover), and recover the identical flat-spacetime equation whose steady-state solution is T ∝ r^{-7/11}. We will revise the abstract to note this reduction explicitly and add a short paragraph plus a comparison plot in the revised manuscript showing the numerical solution with and without the expansion term during formation, confirming the profile is unchanged to within 5% inside the valid regime (Knudsen number ≪ 1). revision: yes

Circularity Check

0 steps flagged

No circularity; new cooling law derived independently from modified diffusion equation

full rationale

The paper formulates the diffusion equation including the Hubble expansion term and solves it to obtain both the robust T∝r^{-7/11} formation profile and the new T_plt ∝ t^{-11/15} cooling law. The observation that the Hubble-diffusion crossover coincides with the decoupling radius from earlier work is used only to delineate the validity regime and does not enter the derivation of the exponents or the finite-lifetime conclusion by construction, fitting, or self-definition. No load-bearing self-citation, ansatz smuggling, or renaming of known results occurs in the central steps; the results are obtained directly from the PDE.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the diffusion approximation being sufficient once expansion is added and on the existence of a well-defined regime where this holds for light PBHs.

axioms (2)

domain assumption Energy transport around the PBH is governed by a diffusion equation that can be modified to include cosmological expansion.
Invoked to set up the governing equation whose solution yields the new cooling law.
domain assumption Other transport mechanisms and back-reaction effects remain negligible in the relevant regime.
Required for the diffusion equation to remain the complete description.

pith-pipeline@v0.9.0 · 5521 in / 1323 out tokens · 67125 ms · 2026-05-12T00:55:02.315161+00:00 · methodology

discussion (0)

Reference graph

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