Probing Double-Peaked Gamma-Ray Spectra from Primordial Black Holes with Next-Generation Gamma-Ray Experiments

Anthony G. Williams; C. J. Ouseph; Giorgio Busoni; John Gargalionis; Sin Kyu Kang

REVIEW 3 major objections 5 minor 81 references

Probing Double-Peaked Gamma-Ray Spectra from Primordial Black Holes with Next-Generation Gamma-Ray Experiments

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Double-peaked gamma rays could betray black-hole dark matter.

desk verdict Useful first f1-f2 map for double-peaked PBH gamma rays, but the blue 'distinguishable from single-peak' contours rest on an undefined single-peaked True model, so the central claim needs a fix before it can be used. read the letter →

arxiv 2507.16244 v1 pith:JGXAIJUF submitted 2025-07-22 hep-ph astro-ph.COastro-ph.GAastro-ph.HE

C. J. Ouseph , Giorgio Busoni , John Gargalionis , Sin Kyu Kang , Anthony G. Williams This is my paper

classification hep-phastro-ph.COastro-ph.GAastro-ph.HE

keywords primordialblackholesHawkingradiationdouble-peakedgamma-rayspectrumdarkmattere-ASTROGAMlikelihoodanalysisasteroid-massPBHsmulti-modalmassdistribution

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

This paper claims that a future gamma-ray telescope with the projected sensitivity of e-ASTROGAM can tell apart two ways primordial black holes (PBHs) might shine: a single population evaporating into one spectral peak, and two populations of different masses evaporating into a spectrum with two peaks. The relevant mass window is the asteroid range $10^{15}$ g to $10^{17}$ g, where Hawking radiation peaks between MeV and GeV. Using a likelihood-based Poisson test statistic on simulated observations of a 5-degree region around the Galactic Center, the authors map regions in the abundance plane $f_1$ vs $f_2$ where a double-peaked spectrum is distinguishable from both the astrophysical background and a single-peaked PBH spectrum at TS $\geq 9$. If these regions are real, an instrument like e-ASTROGAM could identify multi-modal PBH mass distributions predicted by inflationary models with multiple features, rather than merely setting upper limits on PBH abundance.

What carries the argument

The statistical engine is the Poisson likelihood ratio $\mathrm{TS}=-2\ln(L/L_{\mathrm{true}})=\Sigma^2$, assumed to follow a $\chi^2$ distribution, with $\mathrm{TS}=9$ (about $3\sigma$) taken as the detection threshold. The signal model is built from the Hawking emission rate $\partial N_i/(\partial E_i\partial t)$ with graybody factors computed by BlackHawk v2.0, including primary photons, neutral-pion decay, and final-state radiation; the flux normalization comes from the decay J-factor $\bar{J}_D$ of an NFW profile integrated over a $|R|<5^\circ$ cone around the Galactic Center. The toy model that motivates the two populations is a double-peaked curvature power spectrum, Eq. (1), whose two lognormal peaks produce PBHs at two mass scales. The comparison of a double-peaked Test model against either a background-only True model or a single-peaked True model across the e-ASTROGAM energy bins carries the argument.

What would settle it

Take the same $(M_1,M_2)$ benchmarks, replace the assumed background model with the actual measured diffuse gamma-ray background (including its normalization and shape uncertainties), and recompute the $\mathrm{TS}=9$ contours; if the blue identification regions in Figs. 4--7 shrink significantly or vanish under this variation, the claimed discrimination power does not survive contact with data. Alternatively, an e-ASTROGAM-like observation of $10^8$ s toward the Galactic Center that finds no double-peaked excess in a region predicted to be blue would directly falsify that benchmark.

Watch

Extended reading notes

Core claim

The central claim is that the spectral shape of Hawking radiation is a diagnostic of the PBH mass distribution, not just of the total PBH abundance. For two PBH populations with masses $(M_1,M_2)$ and abundance fractions $f_1,f_2$, the gamma-ray flux is the sum of the two monochromatic-population spectra computed with the BlackHawk package, and the peaks sit at energies inversely proportional to each mass. The paper shows, for several mass combinations in the $10^{15}$--$10^{17}$ g window, that a likelihood ratio test with threshold $\mathrm{TS}=9$ carves out regions in the $(f_1,f_2)$ plane where the double-peaked signal can be distinguished from both the background-only hypothesis and the single-peaked PBH hypothesis. The lighter PBH dominates emission because evaporation is faster, so peak heights are controlled by $f/M$; when $M_1 \approx M_2$ the two peaks merge and detector energy resolution becomes the limiting factor. The conclusion is that next-generation MeV observatories can discover single-peaked PBH signals and, in a more restricted region of parameter space, identify the double-peaked structure that would reveal a multi-modal mass distribution.

Load-bearing premise

The whole reach map rests on assuming the modeled e-ASTROGAM background is exactly the real gamma-ray sky toward the Galactic Center and that the test statistic $\mathrm{TS}$ is $\chi^2$-distributed with a fixed threshold $\mathrm{TS}=9$, with no systematic uncertainty folded in for the background, the dark-matter profile, or the detector response.

Editorial extensions

If this is right

A positive detection of the double-peaked signature would be direct evidence that PBHs formed from a multi-modal curvature power spectrum, consistent with inflationary models containing multiple inflection points or features.
A null observation after $10^8$ s toward the Galactic Center would exclude the corresponding $(f_1,f_2)$ values shown in the blue regions of Figs. 4--7, tightening limits on PBH dark matter in the asteroid-mass window.
The single-peaked discovery curve (Fig. 3) provides a projected sensitivity benchmark for future MeV telescopes independent of the two-peak scenario, since it marks the minimum abundance $f$ detectable above background for each mass.
Because the same double-peaked curvature power spectrum also generates a double-peaked stochastic gravitational-wave background, an electromagnetic gamma-ray excess and a gravitational-wave excess at related frequencies would reinforce each other as evidence.
Detector energy resolution sets the smallest mass ratio $M_2/M_1$ for which the two peaks can be resolved; improving resolution extends the distinguishable region in the $(f_1,f_2)$ plane.

Reading between the lines

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

Beyond what the paper states: the TS=9 contours assume the adopted background model is exactly true; folding in systematic uncertainties on the diffuse gamma-ray background, the NFW $J$-factor, and the detector response would likely shrink or shift the blue regions, so the quoted abundance reach should be read as idealized rather than guaranteed.
Beyond what the paper states: the same likelihood framework can be applied to extended (non-monochromatic) PBH mass distributions, where the two-peak analysis would become a template for reconstructing the shape of the mass function rather than just its peaks.
Beyond what the paper states: the method suggests a concrete multi-messenger test, correlating a candidate double-peaked gamma-ray excess with the predicted double-peaked induced gravitational-wave background in the nHz-to-Hz band probed by pulsar timing arrays and LISA.
Beyond what the paper states: if the memory burden effect is real, PBHs with masses as low as $10^9$ g could survive today, shifting the peaks to higher energies; a similar shape analysis could distinguish such scenarios from the standard evaporation picture.

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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

Short version: this is the first forecast I know of that maps where an e-ASTROGAM-like instrument could distinguish a two-population PBH gamma-ray signal from a single-population one. The gray 'detectable above background' contours in Figs. 4-7 look fine. The blue 'distinguishable from single-peak' contours, however, rest on a single-peaked True model that is never defined, so the paper's central discrimination claim is not yet well-posed.

What is actually new is the f1 vs f2 scan for fixed mass pairs. Previous work, including the authors' own single-peak forecast, stopped at one population. The flux calculation, BlackHawk v2 usage, NFW J-factor, and Poisson likelihood are all standard and correctly assembled. The gray regions, where a double-peaked spectrum is compared only against background, are a straightforward extension and I have no serious objection to them.

The main soft spot is in Section III B. The text says the Test model (double-peaked with background) is compared to a True model (single-peaked with background), but it never specifies the mass and abundance of that single-peaked True model. If it is fixed to (M1, f1), the TS mostly measures whether the second component is visible, which is not equivalent to discriminating between single- and double-peaked scenarios. If, instead, the single-peaked model is optimized over M and f, a single Hawking spectrum at an intermediate mass could soak up part of the double-peaked signal, shrinking or erasing the blue regions. The figures do not tell us which convention was used, and the contours as plotted should not be taken at face value until this is resolved.

Two smaller issues. The background is treated as the exact true model and the TS is assumed to follow a chi-square with threshold 9, with no Monte Carlo calibration; for a sensitivity forecast that is common, but it should be stated as a limitation. Also, the toy double-peaked power spectrum used to motivate the scenario predicts specific correlations between f1 and f2, and the paper does not check whether any of its blue regions are actually realizable by that model. The scan over f1 and f2 is more general than the motivating cosmology, which is fine, but the connection is left dangling.

This is a conditional accept-type paper: the science case is worth a serious referee, but the undefined comparator is a load-bearing gap. I would not cite the blue discrimination regions as they stand; the gray detectability maps are usable. Recommendation: send it to review, but ask the authors to define the single-peaked True model explicitly, recompute the contours, and provide code/data if possible.

Referee Report

3 major / 5 minor

Summary. The paper studies gamma-ray signatures from two monochromatic primordial black hole (PBH) populations in the asteroid-mass window (10^15-10^17 g), using BlackHawk v2.0 spectra, an NFW Galactic profile, and the projected e-ASTROGAM sensitivity. It builds a Poisson likelihood test statistic to first map the single-peak discovery threshold in the f-M plane (Fig. 3), and then maps f1-f2 regions where a double-peaked spectrum is distinguishable from the astrophysical background alone (gray regions) and from a single-peaked PBH spectrum (blue regions), for several mass pairs (Figs. 4-7). The paper also connects the two-mass population to a toy double-peaked curvature power spectrum and to the associated induced gravitational-wave background.

Significance. If the quantitative contours are correct, the paper provides a useful forecast for e-ASTROGAM and similar MeV missions, framing the two-mass PBH signature as a spectral-shape discriminant rather than a simple total-flux search. The technical backbone is standard and mostly sound: BlackHawk v2.0 spectra, a standard J-factor calculation, a Poisson likelihood, and forward-model parameter scans rather than fits. The main value lies in the f1-f2 identification regions, but those regions depend critically on the definition of the single-peaked comparator and on the statistical calibration of the test statistic; neither is specified precisely enough for the contours to be taken at face value. The paper also does not propagate systematic uncertainties from the background model, J-factor, or detector response, so the quantitative reach should be read as a perfect-background forecast.

major comments (3)

[Sec. III B (paragraph defining the blue regions in Figs. 4-7)] The 'True model (single-peaked with background)' used for the identification contours is never defined. The text gives no equation or rule specifying the mass and abundance of the single-peaked spectrum being compared, nor whether it is fixed to (M1,f1), fixed to (M2,f2), allowed to vary f at one of the two masses, or optimized over both M and f. This ambiguity is load-bearing: if the comparator is fixed to (M1,f1), the likelihood ratio largely reduces to the visibility of the second component, whereas an optimized comparator could partially absorb the double-peaked signal at an intermediate mass. Please define the comparator explicitly and, if it is optimized, state the number of free parameters and how the TS distribution is calibrated for that scan.
[Sec. III, Eqs. (16)-(17)] The test statistic is asserted to equal Sigma^2 and to follow a chi-square distribution with threshold TS = 9, but the paper does not specify whether the observed counts n_i are taken as the Asimov expectation of the True model or as pseudo-experiment draws, nor does it state the number of degrees of freedom or whether a trials factor is applied when scanning f1 and f2. Because the signal parameters sit at a physical boundary (f_i >= 0) and the BSM-versus-single-peaked comparison may not be nested in the standard sense, the null distribution of TS need not be the assumed chi-square. Please state the evaluation ensemble and either justify the TS = 9 threshold analytically or calibrate it with Monte Carlo pseudo-experiments.
[Sec. III B and Figs. 3-7] The analysis treats the e-ASTROGAM background model of Refs. [67,68,77] as the exact True model and propagates no systematic uncertainties from the background normalization or shape, the NFW J-factor, or the detector response. The TS >= 9 contours, including the single-peak discovery curve in Fig. 3, are therefore forecasts under a perfect-background assumption; if the true diffuse background differs from the adopted model, the contours shift or disappear. Please add a quantitative or at least explicit qualitative discussion of these systematics, or moderate the abstract's claim to 'under the adopted background model'.

minor comments (5)

[Fig. 2 caption] The caption refers to 'the test statistic analysis ... as discussed in Sec. IV,' but Sec. IV is the conclusion; the relevant discussion is in Sec. III B. The phrase 'chosen from the allowed parameter space' is also potentially circular in presentation and should be reworded as, for example, 'representative values satisfying TS >= 9.'
[Eq. (1)] The notation ln^2(k/k_f) is ambiguous; it should be written as [ln(k/k_f)]^2 to avoid confusion with ln(ln(...)).
[Eq. (17)] The symbol Sigma is used before it is defined, and the relation TS = Sigma^2 is stated without explaining the one-sided significance convention. Please define Sigma and clarify the sense in which the relation holds.
[Sec. III B, Fig. 5 (right panel)] The statement that 'once observational constraints are imposed, even these higher-TS regions become inaccessible' depends on the gray hatched regions, but the text does not describe how constraints on the two individual masses are combined for a two-population model. Please specify the combination rule.
[Sec. I (introduction)] The memory-burden effect is introduced as motivation for lighter PBHs, but the numerical analysis uses standard Hawking evaporation without memory burden. Clarify that the memory-burden scenario is not included in the computed spectra and contours, or remove the discussion if it is outside the scope.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the TS contours are forward-model scans over (f1,f2) against an external background model and projected detector response, not fits that are then relabeled as predictions.

full rationale

The paper's central claim is a sensitivity projection, not a fitted result. Equations (16) and (17) define a Poisson likelihood and a test statistic, and the 'True' model in Sec. III A is the e-ASTROGAM astrophysical background from external references [67,68,77]. The PBH Test spectra are forward-modeled through BlackHawk and Eq. (13); f1 and f2 are scanned over a grid, and the TS>=9 contours in Figs. 4-7 are derived from those scans. No parameter is fitted to the quantity that is later called a prediction, so the fitted-input-called-prediction pattern does not apply. The selection of the Fig. 2 f1,f2 values from the already-allowed TS regions is a presentational choice, not an input to the analysis. The only overlapping-author citations ([10] and [35]) are contextual references for known mechanisms and prior single-peak studies; they are not used as load-bearing justification for the discrimination claim, and the memory-burden [38-40] and inflationary multi-peak [41,42] references are external to the authors. The single-peaked True model in Sec. III B is not fully specified (its mass and abundance are not given explicitly), which is a reproducibility and correctness ambiguity rather than a circular reduction: the TS compares two distinct forward spectra, and circularity would require the comparator to be constructed from the test model's fitted parameters, which is not the case here. Overall, the derivation is self-contained against external data and code, and no Eq. X reduces to Eq. Y by construction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The gamma-ray forecast does not introduce new physics entities. It rests on standard PBH evaporation via BlackHawk, an assumed NFW halo, an assumed background model, and an uncalibrated TS criterion. The f1, f2, M1, and M2 parameters are scanned; the toy power-spectrum parameters in Eq (2) are hand-picked and not propagated into the f1-f2 analysis.

free parameters (4)

PBH abundance fractions f1, f2 = scanned (order 1e-11 to 1e-1 depending on mass)
These are the model parameters that set the peak heights; the TS contours in Figures 4-7 are curves in this plane, so the central claim depends on scanning them.
PBH masses M1, M2 = chosen from 1e15 to 1e17 g grid
The peak positions are set by the masses; four combinations are considered.
TS threshold TS = 9 (Sigma = 3) = 9
Chosen by hand as the significance threshold for detection and discrimination; not derived from a Monte Carlo calibration.
Toy power-spectrum parameters (P0, kf, kp1, kp2, A1, A2, sigma0) = values in Eq (2)
Chosen by hand to give a double-peaked curvature power spectrum and CMB normalization. They motivate the two-mass scenario but are not propagated into the f1-f2 scans, so they are decorative for the gamma-ray forecast.

assumptions (5)

domain assumption NFW dark matter profile with rs = 11 kpc, rho_s = 0.838 GeV/cm3, r200 = 193 kpc describes the Galactic DM distribution.
Used in Eqs (14)-(15) to compute the J-factor 1.597e26 MeV cm^-2 sr^-1; profile uncertainties are not propagated.
domain assumption BlackHawk v2.0 graybody factors and secondary spectra are correct.
The photon spectra in Eq (8) are taken from BlackHawk v2.0; no independent check or run parameters are given.
domain assumption The e-ASTROGAM projected sensitivity and background model from Refs. [67,68,77] are accurate.
The likelihood analysis uses these as the True model and detector response; inaccurate background or response would shift all TS contours.
ad hoc to paper The Poisson likelihood ratio TS = Sigma^2 follows a chi-squared distribution with threshold TS = 9.
The paper states this in one line after Eq (17) with no Monte Carlo validation or look-elsewhere treatment.
domain assumption Each PBH population has a monochromatic mass distribution.
Footnote 1 says the flux is computed per population as monochromatic; extended mass distributions could alter peak shape and detectability.

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Cite this review

Pith. "Pith review of Probing Double-Peaked Gamma-Ray Spectra from Primordial Black Holes with Next-Generation Gamma-Ray Experiments." pith.science (2026). https://pith.science/paper/JGXAIJUF

@misc{pith2026250716244,
  author       = {Pith},
  title        = {Pith review of: Probing Double-Peaked Gamma-Ray Spectra from Primordial Black Holes with Next-Generation Gamma-Ray Experiments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JGXAIJUF}},
  note         = {Machine review of arXiv:2507.16244}
}

abstract

Primordial black holes (PBHs), hypothesized to form in the early universe from gravitational collapse of density fluctuations, represent a well-motivated dark matter (DM) candidate. Their potential detection through gamma-ray signatures arising from Hawking radiation would provide definitive evidence for their existence and constrain their contribution to the DM abundance. Unlike conventional DM candidates, PBHs emit a unique, thermal-like spectrum of particles as they evaporate, including photons, neutrinos, and possible beyond-the-Standard Model particles. Future high-sensitivity gamma-ray observatories, such as e-ASTROGAM and other next-generation telescopes, will play a pivotal role in this search. With improved energy resolution and sensitivity, these missions can disentangle PBH-originating photons from astrophysical backgrounds, probe subtle spectral features such as multi-peak structures, and test exotic evaporation models. Such observations could either confirm PBHs as a viable DM component or place stringent limits on their abundance across critical mass windows. In this work, we explore the distinguishing features of a double-peaked gamma-ray spectrum produced by PBHs, focusing on the asteroid-mass window ($10^{15}$ g to $10^{17}$ g), where Hawking radiation peaks in the MeV to GeV range. Using a likelihood-based analysis, we demonstrate how future missions could discriminate between single- and double-peaked PBH scenarios, the latter arising in cosmological models predicting multi-modal PBH mass distributions. Our results highlight the diagnostic power of spectral shape analysis in identifying PBH populations and constrain the parameter space for which a double-peaked signal could be detectable above background.

Figures

Figures reproduced from arXiv: 2507.16244 by the authors.

**Figure 2.** FIG. 2: Double-peaked gamma-ray spectra from Hawking radiation of PBHs for different [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** FIG. 3: Bounds on PBH detectability in the [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4: Constraints on PBH detectability in the [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: FIG. 5: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: FIG. 6: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: FIG. 7: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

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