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REVIEW 4 major objections 7 minor 14 references

Origin likelihood functions for extreme-energy cosmic rays

T0 review · 4 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Extreme-energy cosmic-ray origins can be located analytically: the photodisintegration cascade is a matrix-exponential distribution, yielding arbitrarily precise likelihoods over source distance and rigidity, with some observed charge…

desk verdict Central Eq. (2) is degenerate as printed, but the underlying matrix-exponential method is promising enough to deserve a referee once the author fixes it. read the letter →

arxiv 2501.06677 v1 pith:UINNREOP submitted 2025-01-12 astro-ph.HE

Leonel Morejon This is my paper · ORCID
classification astro-ph.HE
keywords ultra-high-energycosmicraysextreme-energyphotodisintegrationmatrix-exponentialdistributionsoriginlikelihoodAmaterasueventcosmic-raycompositionstochasticinteractions
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

Ultra-high-energy cosmic rays above $10^{20}$ eV are so strongly absorbed by the cosmic microwave background that any detected event must originate within a few tens of megaparsecs, but stochastic photodisintegration of the nuclear species makes pinpointing the source difficult. This paper claims that the full probability distribution for the distance and rigidity at the origin can be computed analytically as a matrix-exponential expression built from photodisintegration transition rates, requiring no Monte Carlo sampling. Applied to the Amaterasu event, the method yields arbitrarily precise normalized likelihood surfaces, resolving features that event-by-event Monte Carlo codes cannot constrain. A striking consequence is that for some observed charge groups, such as $2 < Z \le 7$, the origin likelihood is nearly independent of whether the injected nucleus was silicon, calcium, or iron, suggesting that source localization may be possible without knowing the initial composition.

What carries the argument

The load-bearing object is the matrix-exponential distribution $f(L) = \alpha_0 \exp(\Lambda L) \Lambda e$ for the distance to complete a photodisintegration cascade. $\Lambda(\gamma)$ is an upper-triangular matrix whose off-diagonal entries are the rates $\lambda_{S_i \to S_j}$ for one nucleus to disintegrate into another and whose diagonal entries are the total removal rates; the boost $\gamma$ (energy per nucleon) is treated as constant because photodisintegration conserves it to good approximation, and the rigidity at the source follows from the observed energy and charge. For light products such as protons that are produced at multiple cascade steps, the same distribution is transformed by a 'rewards' construction that counts the yield along the chain. The likelihood surfaces in Figures 3–5 are these densities, normalized and restricted by a Gaussian on the observed energy, evaluated over source distance and rigidity.

What would settle it

Run a Monte Carlo propagation that allows fragments to lose energy between disintegrations (relaxing exact boost conservation) and compare the resulting two-dimensional origin-likelihood surface for an Amaterasu-like event; if the peak distances shift by more than a few megaparsecs relative to the matrix-exponential prediction, the method's resolution claim fails.

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Extended reading notes

Core claim

The central discovery is that the origin likelihood of an extreme-energy cosmic ray is a matrix-exponential distribution. Writing the photodisintegration cascade as a continuous-time Markov chain in which each species has an interaction rate $\lambda(\gamma)$ depending on the Lorentz boost $\gamma = E/A$, the probability density for the distance $L$ needed to go from an initial species $S_0$ to a set of final species is $f(L) = \alpha_0 \exp(\Lambda L) \Lambda e$, where $\Lambda$ is the matrix of transition rates and $\alpha_0$ is the initial-state vector. Because photodisintegration conserves the boost almost exactly, the same $\Lambda$ applies along the whole path, and the observed particle energy fixes the rigidity at the source through $R = E_{\rm obs}/(A_{\rm obs}\eta)$. The paper evaluates these functions for three injected species ($^{28}$Si, $^{40}$Ca, $^{56}$Fe) and compares the resulting likelihood maps with the Monte Carlo–based analysis of the Amaterasu event, showing higher resolution and the ability to assign relative probabilities in regions where Monte Carlo has few or no events.

Load-bearing premise

The calculation assumes that, as a cosmic-ray nucleus breaks apart, every fragment continues with the same energy per nucleon as the original, so that one fixed set of interaction rates governs the whole journey from source to Earth.

Editorial extensions

If this is right

  • Origin likelihoods for extreme-energy cosmic rays can be computed with arbitrary numerical precision, eliminating the sparse-event phase-space artifacts of Monte Carlo methods.
  • For observed charge groups such as $2 < Z \le 7$, the origin likelihood is nearly independent of the injected composition, so source localization may not require knowing the initial nucleus.
  • Combined with an event-by-event composition measurement at the highest energies, the method can narrow the origin to within a few megaparsecs, potentially identifying a specific host galaxy.
  • The method reproduces the qualitative features of the Monte Carlo–based Amaterasu analysis while providing continuous, normalized likelihood values across the full distance–rigidity plane.

Reading between the lines

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

  • The composition-independence claim for $2 < Z \le 7$ is stated for single injected species; a natural testable extension is to mix the injected species with weights from realistic source spectral fits and check whether the likelihood maps still coincide.
  • The single-boost Markov chain approximation ignores energy loss of protons via photopion production after they are produced mid-cascade; the author notes this is expected to be small, but including a distance-dependent boost for light fragments would show how much the likelihood peaks shift.
  • The matrix-exponential machinery may allow analytic marginalization over unknown source parameters such as the maximum rigidity $R_{\rm max}$ or the injection spectrum, offering a direct route from event likelihoods to Bayesian source inference.
  • If the method's resolution survives the inclusion of energy losses and galactic magnetic fields, the same formalism could be applied to smaller statistical ensembles of sub-$10^{20}$ eV events for which the extragalactic magnetic field cannot be neglected.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 7 minor

Summary. The paper proposes an analytic method for computing origin likelihood functions of extreme-energy cosmic rays, replacing Monte Carlo propagation with matrix-exponential distributions for photodisintegration cascades. Using the Telescope Array Amaterasu event as a case study, it presents two-dimensional likelihood surfaces over source distance and rigidity for a range of final nuclear species, and argues that for intermediate charge groups (2 < Z ≤ 7) the origin distribution is nearly independent of the injected composition. The method is compared qualitatively with the CRPropa-based analysis of Unger and Farrar (2024b), and the analytic approach is claimed to offer higher resolution and much lower computational cost.

Significance. If the framework is correct, it provides a fast, deterministic alternative to Monte Carlo for a problem where event statistics are extremely sparse, and it yields likelihood surfaces with arbitrary numerical precision. The observation that some final charge groups give composition-independent origin distributions is practically valuable for single-event source localization. The paper also benefits from building on explicit interaction-rate inputs and from a clear application to a concrete event. However, the central formula as printed is internally inconsistent because the exit vector is undefined, and the main physical approximation—exact rigidity boost conservation across the cascade—is asserted without quantitative validation. These issues must be resolved before the claimed advantages can be accepted.

major comments (4)
  1. [Section 2, Eq. (2)-(3)] The vector e in f(L) = α0 exp(ΛL) Λe is never defined. With the standard phase-type convention e = (1,...,1)^T, the expression vanishes identically because each row of the displayed matrix Λ sums to zero (including rows for the final species, whose total rate is zero). The correct density requires either restricting Λ to the transient states and using the exit vector t = −Λ1, or keeping absorbing states and multiplying by an indicator vector selecting the final species. As written, Eq. (2) cannot be evaluated and is the central result of the paper; this must be corrected and the notation made explicit.
  2. [Section 2, boost-conservation approximation] The paper treats the entire photodisintegration cascade as a Markov chain at a single fixed boost γ = ηR, justified by 'the boost conservation characteristic in photodisintegration interactions.' However, η = Z/A changes between fragments (e.g., 0.464 for 56Fe and 0.5 for 4He), and the interaction rates in Figure 1 vary strongly with γ, especially in the GDR-dominated range. The claim that η changes only slightly is not quantified, and no validation is offered against a propagation model that tracks each fragment's own boost after each disintegration. Since all likelihood surfaces in Figures 3-5 depend on this single-boost assumption, please provide a quantitative error estimate or a comparison with a per-species boost cascade.
  3. [Section 2, Gaussian energy constraint] The text states that the observed nuclei energies are distributed as a Gaussian with mean E_low = 1.64 × 10^20 eV, but the Amaterasu event is reported at 244 EeV = 2.44 × 10^20 eV. This discrepancy directly affects the rigidity constraints applied in Figures 3-5: a wrong mean shifts all reported likelihood surfaces. If E_low is not a typographical error, its role must be explained; if it is a typo, it must be corrected to 2.44 × 10^20 eV (or the appropriate value from Unger and Farrar 2024b).
  4. [Section 2, proton energy losses] For final-state protons and deuterons, the paper explicitly neglects photopion energy losses ('this effect is expected to be reduced and was not included in the results shown in here'), yet the likelihood functions in Figures 3-5 include p and d panels. Because the observed-energy Gaussian is applied at Earth, energy losses during propagation change which source rigidities and distances can contribute to the observed energy window. Please either include the losses within the rewards-transform framework or demonstrate quantitatively that their effect on the displayed likelihoods is negligible.
minor comments (7)
  1. [Abstract] The phrase 'makes difficult the task' is ungrammatical; use 'makes the task difficult' or 'complicates the task.'
  2. [Section 2, after Eq. (3)] The notation λ_{S_n→S_f} = Σ_{j=1}^{k} λ_{S_n→S_f_j} uses S_f for both the set of final species and an individual final species; please disambiguate, for example by writing S_f^set and S_f^j.
  3. [Section 1] The phrase 'a maximal rigidity of Rmax ∈ [3, 5] EV (?)' contains a placeholder '?' and no citation; this must be completed.
  4. [Section 3] 'corner stone' should be spelled 'cornerstone.'
  5. [Figures 3-5] The label 'starting species (blue)' in the text is ambiguous; specify in each caption that blue denotes the likelihood for the injected nucleus itself.
  6. [Section 1 / Section 3] The statement that the origin is 'independent of the original composition' is stronger than the evidence presented, since only three starting species (28Si, 40Ca, 56Fe) are tested and the independence is shown only for some final charge groups; please soften the wording accordingly.
  7. [Overall] Since Eq. (2) depends on the rate matrix Λ, and the rates are taken from Morejon (2023) without a self-contained definition, adding a short appendix or explicit references to the equations in that work would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the origin likelihood is a forward phase-type computation from interaction rates and the measured event energy, with no fitted parameter renamed as a prediction.

full rationale

The paper's central likelihood function is a forward application of matrix-exponential (phase-type) distributions to photodisintegration rates: Eq. (2) is the standard phase-type density, with rates computed from Eq. (1). No free parameter is fitted to the Amaterasu event; the only data input is the reported energy and its Gaussian uncertainty, used to restrict rigidity ranges. The self-citations to Morejon (2020, 2023) motivate the Markov-chain description but are not load-bearing in a circular sense: the underlying theory is also cited to Bladt and Nielsen (2017), and the claim of composition independence is an output of the forward calculation compared across injected species, not an input. The printed Eq. (2) does have an algebraic consistency issue (as written, Lambda e = 0 for the displayed row-sum-zero Lambda if e denotes the ones vector), but that is an error in presentation, not evidence that a derived quantity reduces to a fitted input or a self-citation. Hence no circularity under the specified rubric.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper contributes the likelihood construction and application, but its quantitative content rests on the author's prior interaction models (Morejon 2020, 2023) and on the measured energy Gaussian from Unger and Farrar (2024b). No new entities are introduced. The freedom in the calculation sits mainly in the unstated rate tables and in the energy normalization.

free parameters (2)
  • E_low (gaussian mean observed energy) = 1.64e20 eV (inconsistent with 244 EeV quoted for Amaterasu)
    This Gaussian constraint on the energy of the observed nucleus determines the rigidity ranges in Figures 3-5; it is adopted from Unger and Farrar (2024b), not derived here.
  • sigma_E (gaussian width) = 0.19e20 eV
    Taken as the measurement uncertainty from Unger and Farrar (2024b); it controls how broad the rigidity bands are for each final species.
assumptions (5)
  • domain assumption Photodisintegration preserves Lorentz boost gamma = E/A (up to small changes in eta = Z/A), so all cascade products have nearly the same rigidity across the whole propagation path.
    Used in Section 2 to reduce the cascade to a Markov chain on species at a single boost; Eq. 2 and Eq. 3 would not be valid without it.
  • domain assumption Extragalactic propagation is ballistic for ExECR nuclei of interest: magnetic deflections are negligible and energy losses are dominated by photodisintegration over distances up to about 100 Mpc.
    Invoked in Sections 1 and 2 to interpret the propagation length L as the source distance; neglects any effect of extragalactic magnetic fields.
  • domain assumption The transition rates lambda_{Si->Sj} and total interaction lengths in Figure 1 and Eq. 3 are correct and complete, as computed in Morejon (2020, 2023).
    No derivation, cross-section table, or code is given in this paper; the likelihood surfaces inherit all their quantitative content from this prior work.
  • ad hoc to paper For protons and neutrons, the likelihood is obtained by a rewards transform convolved with a flat homogeneous emission profile over distance.
    This treatment is stated in Section 2 without derivation, and it omits photopion energy losses after production, which the paper itself notes.
  • domain assumption Only three injected species (28Si, 40Ca, 56Fe) are considered, despite the earlier argument that Amaterasu could have had A in [106, 176].
    The composition-independence claim is demonstrated only for these three species; heavier primaries are not tested.

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Pith. "Pith review of Origin likelihood functions for extreme-energy cosmic rays." pith.science (2026). https://pith.science/paper/UINNREOP

@misc{pith2026250106677,
  author       = {Pith},
  title        = {Pith review of: Origin likelihood functions for extreme-energy cosmic rays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UINNREOP}},
  note         = {Machine review of arXiv:2501.06677}
}
abstract

Unlike neutrinos and photons arriving from extra-galactic sources, ultra-high energy cosmic rays (UHECRs) do not trace back to their origins due to propagation effects such as magnetic deflections and energy losses. For ankle energies, UHECRs can propagate for hundreds of megaparsecs with negligible energy losses but the directional information is lost after a few megaparsecs. On the other hand, at the highest energies the directions are kept for larger distances due to the increased rigidity but the interaction rates with the cosmic microwave background strongly suppress the cosmic rays within a few to tens of megaparsecs. Therefore, UHECRs with energies $E > 10^{20}$ eV (extreme-energy cosmic rays (ExECRs)) such as the Amaterasu event recently reported by Telescope Array, are of particular interest to identify the sources within our galactic neighborhood. However, photonuclear interactions are stochastic in nature and produce changes in the nuclear species emitted, which makes it difficult the task of estimating the likelihood distribution of its origin. This work discusses a novel procedure to estimate the likelihood of the origin for extreme-energy cosmic rays based on probability distributions for UHECR stochastic interactions. The method is applied to the Amaterasu event and compared to recently published works which employ Monte Carlo codes (e.g. CRPropa) in their analysis. The advantages of the method presented here are demonstrated by the increased resolution and the ease of computation unlike other approaches employed so far. The results presented indicate that the localization of the origin of extreme energy cosmic rays could be possible in some cases without knowledge of the original composition.

Figures

Figures reproduced from arXiv: 2501.06677 by the authors.

Figure 1
Figure 1. Energy loss lengths versus Lorentz boost for different nuclear species. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Left: Probability distributions with distance for the disintegration of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Origin likelihood distribution for the injection of iron. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Origin likelihood functions for the injection of silicon. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Origin likelihood functions for the injection of calcium. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

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Reference graph

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