Neutrino Fluence from Gamma-Ray Bursts: Off-Axis View of Structured Jets

Lea Halser; Markus Ahlers

REVIEW 3 major objections 5 minor 69 references

Neutrino Fluence from Gamma-Ray Bursts: Off-Axis View of Structured Jets

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For structured gamma-ray burst jets, neutrinos observed far off the jet axis can arrive at fluence comparable to an on-axis view, and far above the naive uniform-jet off-axis rescaling.

desk verdict Clean Doppler-integral formalism and a useful scaling relation for off-axis structured jets, with an honest but model-dependent GRB 170817A application. read the letter →

arxiv 1908.06953 v2 pith:XASZL7EQ submitted 2019-08-19 astro-ph.HE hep-exhep-ph

Markus Ahlers , Lea Halser This is my paper · ORCID

classification astro-ph.HEhep-exhep-ph

keywords gamma-rayburstshigh-energyneutrinosstructuredjetsoff-axisemissioninternalshocksproton-photoninteractionsGRB170817Aneutrinofluence

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 tries to establish that neutrino fluence from gamma-ray bursts cannot be treated with the usual on-axis uniform-jet formula when the jet is structured and the observer is off-axis. It derives an exact angular integral for the fluence of any particle species emitted by thin relativistic shells and packages it into a revised scaling relation using a jet factor and an average Doppler factor. Its central result is that in structured jets the neutrino emission has its own angular dependence, set by the proton-photon opacity, which is much broader and stronger at intermediate angles than the gamma-ray emission. Applied to GRB 170817A with an afterglow-inferred structured jet, the predicted off-axis neutrino fluence at about 15 degrees is similar to the on-axis prediction in the TeV range and orders of magnitude larger than the expected fluence from an off-axis uniform jet.

What carries the argument

The machinery is the Doppler-boosted fluence integral for thin relativistic shells, $F = (1+z)/(4\pi d_L^2)\int d\Omega_*\, D^3(\Omega_*)/\Gamma(\theta_*)\, dE_*/d\Omega_*$, from which the paper defines the jet scaling factor $N_{\rm jet}$ and average Doppler factor $D_{\rm jet}$ and derives the revised off-axis scaling $F_{\rm off}(\epsilon)\simeq (N_{\rm jet}(\theta_v)/N_{\rm jet}(0))\, \eta^{-2} F_{\rm on}(\epsilon/\eta)$ with $\eta = D_{\rm jet}(\theta_v)/D_{\rm jet}(0)$. For neutrinos the load-bearing new ingredient is the proton-photon opacity $\tau_{p\gamma}(\theta_*) \propto \Gamma^{-5}(\theta_*)\, dE_\gamma^*/d\Omega_*$, which gives the neutrino angular distribution a different, broader profile than the gamma-ray distribution. That opacity scaling, rather than geometry alone, produces the claimed off-axis enhancement.

What would settle it

Recompute the off-axis neutrino fluence using Eq. (20) with a prompt-emission-constrained jet profile for GRB 170817A, of the kind fixed by the gamma-ray time structure and Doppler factor rather than by the afterglow fit; if the TeV fluence at $\theta_v\simeq 15^\circ$ falls to within a factor of a few of the naive uniform-jet $D_{\rm off}^3/D_{\rm on}^3$ rescaling, the claimed enhancement rests on the afterglow-profile assumption and would not survive alternative prompt-jet structures.

Watch

Extended reading notes

Core claim

The central claim is that for a structured jet, the neutrino fluence is not obtained by rescaling an on-axis calculation with a single Doppler factor; one must integrate the Doppler-boosted emissivity over the jet and, for neutrinos, include an angular-dependent proton-photon opacity. In the low-opacity regime, the neutrino energy per solid angle scales as $(dE_{\rm IC}/d\Omega_*)^2/\Gamma^5$, which enhances emission from jet angles around 10 to 20 degrees relative to the gamma-ray brightness. For the structured jet model inferred from the afterglow of GRB 170817A ($s_1=5.5$, $s_2=3.5$, $\Delta\theta\simeq 3.4^\circ$, $\hat{\Gamma}\simeq 250$, $\theta_v\simeq 15^\circ$), the paper predicts that the off-axis muon-neutrino fluence is comparable to the on-axis prediction in the TeV energy range and orders of magnitude above the off-axis uniform-jet expectation.

Load-bearing premise

The load-bearing premise is that the afterglow-derived jet structure and Lorentz-factor profile also describe the prompt-emission outflow at the internal-shock radius, and that the internal photon target spectrum is captured by one of the two adopted peak models; if the prompt jet is narrower, slower, or has a different photon spectrum, the claimed off-axis neutrino enhancement can weaken or disappear.

Editorial extensions

If this is right

The reference signal for neutrino searches from compact-binary mergers must be the structured-jet fluence, not the on-axis uniform-jet prediction rescaled by a single Doppler factor; using the latter can underestimate the expected off-axis signal by orders of magnitude.
Equation (23) gives a practical route to convert existing on-axis neutrino calculations into off-axis predictions for any jet structure and viewing angle whenever the internal spectrum varies only mildly across the shell.
For GRB 170817A-like events, the TeV off-axis neutrino fluence can be comparable to the on-axis fluence, so a large viewing angle does not by itself suppress the neutrino signal as much as it suppresses the gamma-ray signal.
The predicted fluence remains below the current 90% confidence upper limits, so the immediate consequence is not a detection but a corrected expected signal for stacking and future searches.

Reading between the lines

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

Because the angular profile of neutrino energy follows $\tau_{p\gamma}\propto \Gamma^{-5}$ in the low-opacity regime, the same broadening should apply to other products of proton-photon interactions, such as ultra-high-energy cosmic rays that escape before energy losses.
The paper's comparison assumes the afterglow-derived jet also describes the prompt outflow; if future prompt-emission fits for GRB 170817A favor a narrower core, the enhancement at 15 degrees would shrink, and comparing prompt and afterglow structures becomes a direct test of jet dissipation physics.
Using the same formalism with alternative short-GRB jet profiles, for example a sharper core or different power-law indices, would bracket how generic the off-axis enhancement is across structured-jet models.

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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

Markus,

The thing to know: the lasting value is the formalism and the scaling relation, not the GRB 170817A neutrino number. The derivation from emissivity to fluence for arbitrary viewing angles and jet structures (Eqs. 1-22) is standard in outline—building on Salafia, Granot, and others—but it is done cleanly and it fixes a real gap. The improved scaling relation Eq. (23) shows when the naive D^3 rescaling used earlier is wrong: it only works for top-hat jets at large viewing angle. The paper's Eq. (29) is the more interesting result: in low-opacity regions the neutrino energy scales as (1/Gamma^5)(dE_IC/dOmega)^2, making off-axis structured jets far more emissive in neutrinos than uniform jets. That qualitative point is solid.

The authors are honest that Eq. (23) requires a mildly varying spectrum across the shell, and for the GRB 170817A application they use the exact expression (20), so that concern does not bite. They also normalize to the observed gamma-ray fluence, so there is no circularity in the neutrino prediction.

The soft spots are where the stress-test lands. The quantitative claim that the 15-degree off-axis fluence is comparable to on-axis in the TeV range is driven by emission from the jet tail at theta* ~ 10-20 degrees, where the p-gamma opacity is low. The calculation adopts the Ghirlanda et al. (2019) afterglow-fit jet profile and the Appendix-A efficiency model (eta_infinity=0.2) without propagating parameter uncertainties or testing alternative afterglow fits. Nothing guarantees that the angular profile of kinetic energy and Lorentz factor at the internal-shock radius is the same as the late-time afterglow structure. If the true prompt jet is wider, slower, or has a different low-Gamma efficiency, the enhancement can shrink. That is a genuine uncertainty, not a fatal flaw—the authors flag the model dependence, and the predicted fluence sits orders of magnitude below current limits anyway, so the result is a fair illustration rather than a testable prediction.

Minor points: the two ad hoc photon-peak models (a) and (b) are exactly that—the paper admits model (a) implies an on-axis peak at 20 MeV, in tension with observations. The off-axis result depends weakly on this choice, so it is a minor issue. The modified SOPHIA code is not released, so the cascade calculation is not independently reproducible; that is a small but real loss. The citation pattern is solid, with Salafia, Biehl, and Ghirlanda as the right anchors.

Who this is for: GRB neutrino modelers and the multimessenger community doing kilonova and off-axis jet follow-up. It deserves a serious referee. My recommendation: send it out, and ask for a robustness section that varies the jet profile and efficiency parameters, plus code or numerical tables behind Fig. 4.

Referee Report

3 major / 5 minor

Summary. This paper derives a general relation between the internal emissivity of a relativistic, axisymmetric structured jet and the observed photon or neutrino fluence at an arbitrary viewing angle, using the standard D^3 Doppler transformation and a thin-shell geometry (Eqs. 1-20). It introduces a jet scaling factor Njet and an average Doppler factor Djet, leading to a revised off-axis scaling relation for particle fluences (Eq. 23) that reduces to the naive (D_off/D_on)^3 scaling only for top-hat jets at large viewing angle. The formalism is then applied to neutrino production from p-gamma interactions in internal shocks, with a numerical illustration for GRB 170817A using the structured jet profile inferred from afterglow fits by Ghirlanda et al. (2019). The central numerical result is that the predicted off-axis neutrino fluence at theta_v = 15 degrees is comparable to the on-axis prediction in the TeV range and orders of magnitude above the expectation from an off-axis observation of a uniform jet, because low-opacity regions at jet angles of about 10-20 degrees contribute strongly.

Significance. If the derivation is accepted, the paper provides a clean and useful generalization of the standard on-axis uniform-jet neutrino fluence calculation, and it clarifies why structured jets can produce neutrino angular distributions that are much broader than the gamma-ray angular distribution. The exact expression (20), the reduction of the approximate scaling relation (23) to known limits, and the explicit identification of the assumptions behind the approximation are genuine strengths. The application to GRB 170817A is a concrete and in principle falsifiable model prediction, although the predicted fluence is currently orders of magnitude below the available ANTARES, Auger, and IceCube upper limits shown in Fig. 4. The main weakness is not the Lorentz-transformation formalism, which is standard and internally consistent, but the dependence of the numerical result on unquantified input assumptions about the prompt-phase jet structure and the internal photon target spectrum.

major comments (3)

[§5.2, Fig. 4; Eqs. (16), (17), (29), (A6)] The headline claim that the off-axis neutrino fluence at theta_v = 15 degrees is comparable to the on-axis prediction in the TeV range and orders of magnitude above the uniform-jet expectation is driven by low-opacity emission from jet angles of about 10-20 degrees (Fig. 3). In this regime Eq. (29) gives dE_nu/dOmega proportional to Gamma^{-5} (dE_IC/dOmega)^2, so the result is extremely sensitive to the Lorentz-factor and energy profiles adopted from the afterglow fit of Ghirlanda et al. (2019) and to the internal-shock efficiency model of Eq. (A6) with eta_infinity = 0.2. The afterglow fit constrains the external forward shock, not the prompt internal-shock dissipation radius, and the manuscript neither propagates the fit uncertainties nor tests alternative structured-jet profiles such as those of Lazzati et al. (2018), Troja et al. (2018), Margutti et al. (2018), and Lamb et al. (2019). A quantitative variation of s1, s2, Delta-theta, and Gamma-hat within the allowed ranges is needed to determine whether the claimed enhancement is a robust prediction or an artifact of the chosen prompt-jet model.
[§5.1, Eqs. (32)-(33), Fig. 4] The neutrino fluence calculation requires the internal photon target spectrum, which is fixed by one of two ad hoc peak-energy models. The paper itself states that model (a) implies an on-axis photon peak of about 20 MeV, in tension with the Fermi-GBM GRB peak-energy distribution, and model (b) is introduced phenomenologically; Fig. 4 shows that the two models change the on-axis neutrino fluence by up to two orders of magnitude at EeV energies. The conclusion that the off-axis and on-axis fluences are 'similar in the TeV range' should therefore be accompanied by a demonstration that this comparison is insensitive to the target-spectrum model, or the claim should be explicitly hedged as applying only under one of the adopted spectral assumptions.
[§4, Eq. (23)] The approximate scaling relation (23) is derived under the assumption that the relative emission spectrum n'(theta*, epsilon')/u'(theta*) is nearly angle-independent, and the paper correctly notes in Section 4 that this condition can fail in structured jets with strong local variations of magnetic fields and photon densities. Since the paper recommends the exact expression (20) for such cases, the numerical neutrino results should state explicitly whether they are obtained from Eq. (20) or from the approximate Eq. (23), particularly because the approximate relation is one of the paper's stated main results and is presented without this caveat in the conclusions.

minor comments (5)

[§5.2] The sentence 'For low-opacity (tau_pgamma >> 1) regions' should read 'tau_pgamma << 1'; the opposite inequality is used in the same paragraph and in Fig. 3.
[§5.2] There is a typo in 'GRB 170717A'; it should be 'GRB 170817A'.
[§5.2] The reference to 'the thick green line in Fig. 4' appears to be a mis-reference: the angular distributions of neutrino emissivity are shown in Fig. 3, whereas Fig. 4 shows fluence spectra.
[§6] The conclusion states that the average Doppler factor is 'defined by Eq. (23)', but Djet is defined in Eq. (22); this should be corrected.
[Throughout] The name 'Thompson scattering' appears in a few places; it should be 'Thomson scattering'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the scaling formalism is derived from first principles, jet parameters come from an external afterglow fit, and neutrino fluence is normalized to observed gamma rays rather than fitted to neutrino data.

full rationale

The paper's central derivation is self-contained. Equations (3) through (23) follow from standard Lorentz transformations of specific emissivity, the Doppler factor, and the definition of an average Doppler boost; they are not set up to reproduce the off-axis neutrino enhancement by construction. The structured jet parameters (s1=5.5, s2=3.5, Gamma-hat=250, E-hat=2.5e52 erg, theta_v=15 deg) are taken from the external afterglow analysis of Ghirlanda et al. (2019), not from the present authors' prior work and not from the neutrino result. The neutrino calculation normalizes the internal photon density using the observed Fermi-GBM gamma-ray fluence (Eq. 26) and uses the standard p-gamma opacity expression (Eq. 27); the claimed off-axis enhancement emerges from the angular integral over the structured jet, and is not equivalent to any fitted parameter. The two photon-peak models (Eqs. 32 and 33) are explicitly presented as assumptions, and the paper itself notes the tension of model (a) with the on-axis peak photon energy distribution and the phenomenological character of model (b); these are model uncertainties, not circular inputs. The self-citations (Ahlers et al. 2011; Bustamante & Ahlers 2019) are used only for IceCube constraints and standard neutrino flavor mixing, respectively, and are not load-bearing for the claimed off-axis enhancement. No circular step can be identified from the paper's equations or citations.

Assumptions & free parameters 11 free parameters · 6 assumptions · 0 invented entities

The central scaling derivation is self-contained, but the application to GRB 170817A stands on many fitted and assumed parameters. The list above counts those that are load-bearing for the quantitative predictions.

free parameters (11)

s1 (structured jet energy profile index) = 5.5
Best-fit from GRB 170817A afterglow modeling (Ghirlanda et al. 2019), used in Eq. (16) for dE*/dOmega*.
s2 (structured jet Lorentz factor profile index) = 3.5
Best-fit from afterglow modeling, used in Eq. (17) for Gamma(theta*).
Delta-theta (jet half-opening angle) = 3.4 degrees
Best-fit from afterglow modeling; sets the angular width of the core energy and Lorentz factor.
Gamma-hat (core Lorentz factor) = 250
Best-fit from afterglow modeling; controls Doppler factors and opacity scalings.
theta_v (observer viewing angle) = 15 degrees
Inferred from gravitational wave and afterglow observations; determines the off-axis normalization Njet.
E-hat (core kinetic energy) = 2.5e52 erg
Best-fit from afterglow modeling; normalization of dE*/dOmega* in Eq. (16).
epsilon-gamma (gamma-ray energy fraction) = 0.41 +/- 0.09
Chosen so the internal energy model reproduces the observed Fermi-GBM fluence of GRB 170817A.
eta-infinity (asymptotic internal shock efficiency) = 0.2 (assumed, x about 0.75)
Assumed in Appendix A Eq. (A6) to convert kinetic to internal energy; affects the absolute neutrino normalization.
xi_p (baryonic loading) = 1 (assumed)
Non-thermal baryonic loading epsilon_p/epsilon_gamma; the neutrino fluence scales linearly with this factor.
xi_B (magnetic energy ratio) = 0.1 (assumed)
Ratio of magnetic to gamma-ray energy density; sets synchrotron losses and the neutrino cutoff.
Peak photon energy model (a) or (b) = Fixed comoving 75 keV, or fixed engine frame equivalent to 178 keV on-axis
Two ad hoc prescriptions for the internal photon target spectrum, chosen to match the observed spectrum at 15 degrees; affects the on-axis neutrino peak strongly.

assumptions (6)

domain assumption Prompt gamma-ray emission is produced by internal shocks in a relativistic outflow.
Central model assumption stated in Section 1; all neutrino predictions inherit it.
domain assumption The jet is axisymmetric, radial, with isotropic rest-frame emissivity and no counter-jet.
Used in Section 2 to derive Eqs. (3) to (5); the counter-jet is omitted as trivial.
domain assumption Thin-shell geometry with dissipation radius r_dis = 2 Gamma^2 c Delta-t_eng and shell width c Delta-t_eng.
Used in Eq. (6) and throughout opacity estimates; if the dissipation radius differs, the opacity and neutrino spectra change.
domain assumption SOPHIA Monte Carlo with synchrotron-loss modifications correctly models p-gamma cascades.
Neutrino spectra in Section 5.2 and Appendix B rely on this tooling; no independent verification is provided.
standard math Standard Lorentz transformations of emissivity and the relation d_L = (1+z)^2 d_A are valid.
Eqs. (2), (5), and (10) use standard Doppler transformation and cosmological distance relations.
domain assumption The afterglow-inferred structured jet model applies to the prompt emission phase.
The paper transfers afterglow-derived jet structure to the internal-shock emission; this is a key unvalidated step for the quantitative GRB 170817A predictions.

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Cite this review

Pith. "Pith review of Neutrino Fluence from Gamma-Ray Bursts: Off-Axis View of Structured Jets." pith.science (2026). https://pith.science/paper/XASZL7EQ

@misc{pith2026190806953,
  author       = {Pith},
  title        = {Pith review of: Neutrino Fluence from Gamma-Ray Bursts: Off-Axis View of Structured Jets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XASZL7EQ}},
  note         = {Machine review of arXiv:1908.06953}
}

read the original abstract

We investigate the expected high-energy neutrino fluence from internal shocks produced in the relativistic outflow of gamma-ray bursts. Previous model predictions have primarily focussed on on-axis observations of uniform jets. Here we present a generalization to account for arbitrary viewing angles and jet structures. Based on this formalism, we provide an improved scaling relation that expresses off-axis neutrino fluences in terms of on-axis model predictions. We also find that the neutrino fluence from structured jets can exhibit a strong angular dependence relative to that of gamma-rays and can be far more extended. We examine this behavior in detail for the recent short gamma-ray burst GRB 170817A observed in coincidence with the gravitational wave event GW170817.

Figures

Figures reproduced from arXiv: 1908.06953 by the authors.

**Figure 1.** Sketch of the GRB coordinate frame. The red arrow indicates the orientation of the jet-axis. The blue arrow points into the line-of sight of the observer. The grey cone shows a top-hat jet with half-opening angle . ✓ unit vector . The relative viewing angle between the observer and n jet core is denoted as . The Doppler factor can then be expressed v✓ as ⌦ D( ⇤ =) ⇥ ✓( ⇤ 1 )( ⌦( ⇤ n )· ) obs ⇤ 1 , (3) where… view at source ↗

**Figure 2.** The scaling factor Njet (left) defined in Eq. (19) and effective Doppler factor Djet (right) defined in Eq. (22) for a top-hat jet (top) and a structured jet (bottom). The dotted lines in the plots indicate the expected scaling of Njet and Djet/bΓ for a top-hat jet observed at a large viewing angle θv . where we introduce the jet scaling factor Njet(θv) ≡ ∫ dΩ ∗ D3 (Ω∗ ) Γ(θ ∗ ) 1 Eb dE ∗ dΩ∗ . (19) The top left pan… view at source ↗

**Figure 3.** Relative angular distribution of the energy associated with the bulk flow (solid black line), neutrinos at low and high opacity (thin & thick green line), and γ-rays corrected for Thomson scattering in the shell (dotted blue line). loss of the initial protons and secondary charged particles before their decay. The mechanism was initially introduced by Waxman & Bahcall (1997) for the case of an on-axis jet with wide … view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Predicted fluence of muon neutrinos (νµ + ν¯µ) associated with the prompt emission in the best-fit structured jet model of Ghirlanda et al. (2019). We show the predictions based on a fixed photon peak in the shell frame (“fixed 0 peak”, solid lines) using Eq. (32) …

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