High Energy Neutrinos from Astrophysical Sources: An Upper Bound

Eli Waxman; John Bahcall

REVIEW

High Energy Neutrinos from Astrophysical Sources: An Upper Bound

Eli Waxman, John Bahcall

High Energy Physics - Phenomenology · Submitted 8 Jul 1998 · 38 references

Pith's verdict: A clean, correct energy-conservation bound that became a standard reference; the abstract's '>10^14 eV, model-independent' framing is looser than the derivation, and the paper itself says so in the Discussion.

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arXiv IDhep-ph/9807282 (version 2)

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CitationEli Waxman, John Bahcall. "High Energy Neutrinos from Astrophysical Sources: An Upper Bound." arXiv:hep-ph/9807282 (1998). https://pith.science/paper/DWMCDJ6R

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Editor's note

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The bound is real and it's the reason this paper gets cited. Take the observed UHECR generation rate, multiply by the pion-production fraction, the ~0.25 share into muon neutrinos, and the Hubble time, and you get E^2 Φ < 2×10^-8 GeV/cm^2 s sr for optically thin sources. That's a two-line argument and it holds. The application to AGN jets — the bound sits two orders of magnitude below the Mannheim and Halzen-Zas predictions, so the gamma-ray-background-from-pions story doesn't work — is the part that made people pay attention, and it's correct. The magnetic-field and evolution sections are short but they close the obvious escape routes, and the ξ_Z factor of ~3 is stated honestly as a factor-of-a-few uncertainty rather than dressed up as a precise number. The GRB section also does a genuine independent check: the flux estimated from the BATSE gamma-ray fluence agrees with the cosmic-ray-based estimate, which is not circular even though the normalization is tied to the same rate. The paper says that tie out loud, which is the right instinct. The stress-test concern about the 10^14 eV claim lands, but only partly. The bound is rigorous where the extragalactic proton flux is actually measured, above ~3×10^18 eV, i.e. neutrinos above ~5×10^17 eV. Extending the horizontal line in Fig. 1 down to 10^4 GeV assumes the same sources produce a proton spectrum down to ~10^15 eV with the same E^-2 shape. The paper says this explicitly in the Discussion: sources below 3×10^18 eV could produce a flux of <10^17 eV neutrinos above the bound. So the authors know. What they don't do is walk the abstract back — 'model-independent' and '>10^14 eV' in the same sentence is a bit more than the derivation delivers. For AGN jets it doesn't matter, since their emission peaks at ~10^18 eV. For GRBs it matters less than it looks, because the GRB normalization is tied to the same cosmic-ray rate. The E^-2 injection and proton-dominated composition are stated, not stress-tested. If the trans-GZK flux is heavy nuclei, the normalization shifts. That's a real caveat but it's a factor-of-a-few, not an order of magnitude. Who should read it: anyone building or proposing a km^2 detector, and anyone working on AGN jet or GRB neutrino models. It's short and the argument is worth knowing cold. Recommendation: send it to referees. The bound is correct, the caveats are stated, and the abstract overreach is a revision-level fix, not a reason to reject.

Referee report

Significance

If the derivation holds, this is an important and durable result: it converts an observed cosmic-ray flux into a constraint on neutrino emission using only energy conservation and the charged-to-neutral pion ratio, with no model-dependent input about the source. The bound is stated as a falsifiable number, it excludes a specific class of published AGN-jet models, and it yields a concrete detection-rate prediction for km^2 telescopes. The paper also does real work in closing off escape routes: the neutron-escape argument of §III.A is a clean and convincing demonstration that magnetic confinement cannot evade the bound, and the §III.B-III.C analysis of intergalactic and large-scale-structure fields is careful. The independent cross-check of the GRB normalization against the observed gamma-ray fluence in §V.A is a genuine consistency test rather than a restatement of the cosmic-ray-based estimate, and the explicit correction of ref. [31] in footnote 1 is a checkable, useful contribution.

Major objections (1)

[Abstract, sentence s0; §IIA Eqs. (1)-(3); Fig. 1; §VI] The abstract's headline claim of a 'model-independent upper bound' on '> 10^14 eV' neutrinos is broader than what Eqs. (1)-(3) actually establish. The normalization ε̇[10^19, 10^21] is inferred from the observed cosmic-ray flux above ~3 × 10^18 eV, so the bound constrains neutrino production by sources that also produce the observed ultra-high-energy protons with an E^-2 spectrum extending down to the relevant proton energy. The authors themselves state the caveat in §VI: extra-galactic sources producing cosmic rays below 3 × 10^18 eV at a rate higher than Eq. (1) 'may produce a flux of < 10^17 eV neutrinos which is higher than the upper limit implied by Eq. (2).' That acknowledged caveat is exactly what limits the abstract's '> 10^14 eV' and 'model-independent' wording, and Fig. 1 draws the bound as a horizontal line down to 10^4 GeV without marking where the extrapolation begins to rely on the unobserved low-energy extension of the source spectrum. The fix is local but should be made: qualify the abstract to say the bound applies to sources that also produce the observed ultra-high-energy cosmic rays, and mark or truncate the low-energy end of the 'CR Limit' curves in Fig. 1. The exclusion of the AGN-jet models, which peaks near 10^18 eV, is unaffected by this point.

Minor objections (8)

[Abstract; §IIA Eq. (2); §IIC; §VI] The abstract and §VI quote E^2 Φ_ν < 2 × 10^-8 GeV cm^-2 s^-1 sr^-1, but Eq. (2) gives I_max = 1.5 × 10^-8 ξ_Z and §IIC finds ξ_Z ≈ 3 for rapid QSO-like evolution. With ε = 1, Eq. (3) then allows (1/2) × 1.5 × 10^-8 × 3 ≈ 2.3 × 10^-8, slightly above the quoted 2 × 10^-8. The bound should be quoted together with its ξ_Z assumption, or rounded upward to cover the evolutionary case.
[§IIA, Eqs. (1)-(2)] The derivation assumes that the trans-GZK flux is dominated by protons with injection spectrum ∝ E^-2. If the composition were substantially heavy, the proton generation rate feeding photo-meson production would be lower and the bound correspondingly weaker. Since the direction is conservative this does not threaten the AGN exclusion, but a sentence stating how I_max scales with the assumed proton fraction (linearly) would make the assumption explicit.
[Fig. 1] The figure caption should state the assumed ξ_Z values for the two 'CR Limit' curves (0.6 for no evolution, 3 for rapid evolution) directly, rather than leaving the reader to recover them from §IIC.
[§V.A] The fluence-based estimate uses F_γ ≈ 6 × 10^-6 erg cm^-2 and a rate of ≈ 10^3 bursts per year over 4π sr. Please give the source or averaging procedure for these two numbers, since the cross-check with Eq. (6) depends on them.
[§IV] The inference τ_pγ ≤ 10^-3 at E_p ~ 10^19 eV rests on the observed transparency of blazar spectra to ~1 TeV photons. It would help to state explicitly that this is an inference from the TeV measurements of refs. [22,24] and to indicate how the conclusion would change if the TeV spectra were attenuated by an unknown factor.
[Eq. (7)] The factor Γ_300^2 Δt_ms in Eq. (7) is dimensionally implicit (units of c and of milliseconds are absorbed). A short note on the units, as is done for Eq. (A2), would prevent misreading.
[Eq. (A2) and footnote 1] The typeset form of Eq. (A2) should make unambiguous that Γ_300^4 and Δt_ms appear in the denominator; the footnote's numerical check (f_π = 1.6 × 10^-4 for L_γ = 10^50 erg/s, Δt = 10 s, Γ = 100) only reproduces with that reading, so an explicit fraction will help readers verify it.
[Title page] The date line reads '(November 26, 2024)', which appears to be a LaTeX date artifact and should be replaced with the submission date.

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The paper

The paper claims

Cosmic-ray observations cap the high-energy neutrino flux that any optically thin source can produce, and the cap rules out a popular AGN-jet explanation of the gamma-ray background.

The reading

The paper asks whether the observed flux of ultra-high-energy cosmic rays can be used to bound the neutrinos made in the same sources. If protons are accelerated to $10^{19}$–$10^{21}$ eV and lose some energy to pions before escaping, the pion decays must produce a neutrino background tied to the measured proton production rate. The authors turn that logic into a ceiling, $E_�nu^2�Phi_{�nu_�mu} �lesssim 2�times10^{-8}�,�mathrm{GeV�,cm^{-2}�,s^{-1}�,sr^{-1}}$, for sources that are optically thin to proton-photon or proton-proton interactions. The ceiling matters because it is low enough to rule out a popular explanation of the cosmic gamma-ray background as pion production in AGN jets, while leaving the GRB neutrino prediction intact.

What carries the argument

The load-bearing mechanism is an energy-conversion argument that treats the observed ultra-high-energy cosmic-ray flux as a calorimeter for pion production. The observed cosmic-ray flux above $10^{19}$ eV is converted into a proton energy generation rate per comoving volume, $E_{CR}^2 d�dot N_{CR}/dE_{CR}�approx10^{44}�,�mathrm{erg�,Mpc^{-3}�,yr^{-1}}$. If a fraction $�epsilon$ of each proton's energy goes into pions, the resulting combined muon-neutrino intensity is $I_{�max}�approx0.25�,�xi_Z t_H (c/4�pi) E_{CR}^2 d�dot N_{CR}/dE_{CR}$, where the $0.25$ accounts for neutral pions taking roughly half the pion energy and muon neutrinos taking roughly half the charged-pion energy. The $1/2$ in $E_�nu^2�Phi_{�nu_�mu}=�frac12�epsilon I_{�max}$ splits the combined muon-neutrino intensity between $�nu_�mu$ and $�bar�nu_�mu$. Redshift evolution enters through $�xi_Z$, which is about 0.6 for no evolution and about 3 for QSO-like evolution; even the larger value leaves the AGN-jet models two orders of magnitude above the ceiling.

Extended reading notes

Core claim

The central claim is that cosmic-ray data alone fix an upper bound on high-energy neutrinos from any source whose size is not much larger than the proton interaction mean free path. The observed extra-galactic cosmic-ray spectrum above about $3�times10^{18}$ eV implies a proton energy generation rate of roughly $5�times10^{44}�,�mathrm{erg�,Mpc^{-3}�,yr^{-1}}$ over $10^{19}$–$10^{21}$ eV, or $E_{CR}^2 d�dot N_{CR}/dE_{CR}�approx10^{44}�,�mathrm{erg�,Mpc^{-3}�,yr^{-1}}$ for an $E^{-2}$ injection spectrum. If a fraction $�epsilon<1$ of that proton energy is lost to photo-meson pions, the resulting muon-neutrino intensity is $E_�nu^2�Phi_{�nu_�mu}=�frac12�epsilon I_{�max}$ with $I_{�max}�approx1.5�times10^{-8}�xi_Z�,�mathrm{GeV�,cm^{-2}�,s^{-1}�,sr^{-1}}$. Allowing for redshift evolution raises $�xi_Z$ to about 3, giving the stated ceiling of about $2�times10^{-8}$. The paper then compares this ceiling with models: AGN-jet predictions exceed it by about two orders of magnitude, while the GRB prediction is of order $10^{-9}$ and therefore survives.

Load-bearing premise

The bound assumes that cosmic rays above about $3�times10^{18}$ eV are protons from a cosmologically distributed population with an $E^{-2}$ injection spectrum, so that the measured local energy generation rate can be used as the normalization for all optically thin neutrino sources.

Editorial extensions

If this is right

The cosmic gamma-ray background cannot be dominated by photo-pion emission from AGN jets, so the high-energy blazar emission is more likely inverse-Compton scattering by electrons.
GRB fireballs should produce a neutrino intensity of order $10^{-9}�,�mathrm{GeV�,cm^{-2}�,s^{-1}�,sr^{-1}}$, yielding of order 20 muon events per year in a $1�,�mathrm{km^2}$ detector and favoring searches coincident with bursts.
AGN-jet neutrino models that exceed the ceiling are excluded unless the sources are optically thick, in which case they become 'hidden-core' neutrino-only factories that cannot explain the observed cosmic-ray flux.
Magnetic confinement, intergalactic fields, and redshift evolution do not raise the ceiling enough to rescue the AGN-jet models.
The GRB neutrino spectrum should cut off above about $10^{16}$ eV because muons and pions lose energy to synchrotron radiation before decaying.

Reading between the lines

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

The bound is best read as a calorimeter statement: for an optically thin source, neutrino astronomy measures the fraction of ultra-high-energy cosmic-ray energy converted to pions, so a detection at the ceiling would be a measurement of source efficiency rather than a surprise.
If the cosmic-ray composition is mixed or heavy at $10^{19}$ eV, the numerical ceiling weakens; a composition measurement is therefore an indirect neutrino-flux measurement.
Stacking searches on GRB catalogs should be more sensitive than a diffuse search because the GRB prediction supplies a known temporal and spatial template.
The 'hidden-core' loophole defines a neutrino-only source class that cosmic-ray data cannot bound; detecting such neutrinos would require an independent argument for the proton accelerator, not a cosmic-ray flux measurement.

Original abstract

We show that cosmic-ray observations set a model-independent upper bound to the flux of high-energy, > 10^14 eV, neutrinos produced by photo-meson (or p-p) interactions in sources of size not much larger than the proton photo-meson (or pp) mean-free-path. The bound applies, in particular, to neutrino production by either AGN jets or GRBs. This upper limit is two orders of magnitude below the flux predicted in some popular AGN jet models, but is consistent with our predictions from GRB models. We discuss the implications of these results for future km^2 high-energy neutrino detectors.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,636 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.57 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.90 · compares with prior results 0.92 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We show that cosmic-ray observations set a model-independent upper bound to the flux of high-energy, > 10^14 eV, neutrinos produced by photo-meson (or p-p) interactions in sources of size not much larger than the proton photo-meson (or pp) mean-free-path. The bound applies, in particular, to neutrino production by either AGN jets or GRBs. This upper limit is two orders of magnitude below the flux predicted in some popular AGN jet models, but is consistent with our predictions from GRB models. We discuss the implications of these results for future km^2 high-energy neutrino detectors.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show that cosmic-ray observations set a model-independent upper bound to the flux of high-energy, > 10^14 eV, neutrinos produced by photo-meson (or p-p) interactions in sources of size not much larger than the proton photo-meson (or pp) mean-free-path.	result 0.91	0.90	0.92	asserted 1.00
s1	The bound applies, in particular, to neutrino production by either AGN jets or GRBs.	interpretation 0.68	0.47	0.09	asserted 0.99
s2	This upper limit is two orders of magnitude below the flux predicted in some popular AGN jet models, but is consistent with our predictions from GRB models.	result 0.51	0.74	0.27	asserted 0.86
s3	We discuss the implications of these results for future km^2 high-energy neutrino detectors.	interpretation 0.75	0.40	0.03	unclear 0.92

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.54.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Comparison of muon neutrino intensities (

Figures reproduced from arXiv:hep-ph/9807282 by the authors.

References

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Searching for Leptoquarks at IceCube and the LHC
hep-ph 2017-09 conditional novelty 5.0 of 10

No scalar leptoquark mass or coupling in the 400 to 1500 GeV range significantly improves the fit to the 53 IceCube high-energy starting events once the full energy range and backgrounds are fitted simultaneously; onl...
Constraining High-Energy Cosmic Neutrino Sources: Implications and Prospects
astro-ph.HE 2016-07 accept novelty 5.0 of 10

Non-detection of multiple high-energy muon tracks from any sky direction implies a lower bound on the local density of steady neutrino sources, which rules out rare bright blazars as the dominant source class.
Heavy right-handed neutrino dark matter and PeV neutrinos at IceCube
hep-ph 2016-06 conditional novelty 5.0 of 10

A softly broken Z2 in an SU(2)' extension makes a multi-PeV right-handed neutrino a long-lived decaying dark matter candidate whose three-body decay into taus and neutrinos can reproduce the PeV IceCube event rate and cutoff.
Choked Jets and Low-Luminosity Gamma-Ray Bursts as Hidden Neutrino Sources
astro-ph.HE 2015-12 conditional novelty 5.0 of 10

Choked GRB jets in low-luminosity GRBs can plausibly supply a significant share of the diffuse high-energy neutrino flux, with precursor neutrinos arriving 100 to 1000 seconds before the gamma-ray breakout.
Decaying Leptophilic Dark Matter at IceCube
hep-ph 2015-07 conditional novelty 5.0 of 10

A leptophilic PeV-scale dark matter decay can reproduce the highest-energy IceCube neutrinos if the astrophysical flux cuts off near 100 TeV, though the data do not require it.
IceCube neutrinos, decaying dark matter, and the Hubble constant
hep-ph 2015-06 conditional novelty 5.0 of 10

A 76 PeV relic with a 6×10^15 s lifetime and a 3×10^-9 neutrino branching fraction is claimed to reproduce the IceCube PeV flux while a 5% decaying dark-matter fraction relieves the Hubble-constant tension.
Testing the Dark Matter Scenario for PeV Neutrinos Observed in IceCube
hep-ph 2015-03 conditional novelty 5.0 of 10

Very heavy decaying dark matter stays marginally consistent with the Fermi diffuse gamma-ray background, and the scenario can be settled by sub-PeV gamma-ray data or by IceCube-Gen2 muon searches toward Virgo.
Mind the gap on Icecube: Cosmic neutrino spectrum and muon anomalous magnetic moment in the gauged L_{\mu} - L_{\tau} model
hep-ph 2014-09 conditional novelty 5.0 of 10

A gauged L_mu - L_tau interaction with a ~1.9 MeV mediator and coupling ~5e-4 can, for inverted neutrino hierarchy, reproduce the 400 TeV to 1 PeV IceCube gap while sitting in the muon g-2 favored band.
Pinning down the cosmic ray source mechanism with new IceCube data
astro-ph.HE 2013-06 conditional novelty 5.0 of 10

A single power-law neutrino spectrum with index about 2.3 fits the 28-event IceCube excess at roughly 1.5 sigma, and if the sources are Galactic the implied pion production efficiency is compatible with proton-proton ...
Demystifying the PeV Cascades in IceCube: Less (Energy) is More (Events)
astro-ph.HE 2013-06 conditional novelty 5.0 of 10

Two PeV cascade events in IceCube are inconsistent with atmospheric conventional and cosmogenic neutrino fluxes, and dedicated sub-PeV cascade searches would identify the true source fastest.
Neutrino Decays over Cosmological Distances and the Implications for Neutrino Telescopes
astro-ph.CO 2012-08 conditional novelty 5.0 of 10

Correctly accounting for cosmic expansion, neutrino decay over cosmological distances makes complete decay an energy threshold rather than a distance threshold, so high-energy astrophysical neutrinos cannot probe life...
How astrophysical neutrino sources could be used for early measurements of neutrino mass hierarchy and leptonic CP phase
hep-ph 2006-04 accept novelty 5.0 of 10

Astrophysical neutrino flavor ratios could enhance early measurements of the neutrino mass hierarchy and CP phase when combined with reactor and beam experiments.
Long-Lived Staus at Neutrino Telescopes
hep-ph 2006-04 conditional novelty 5.0 of 10

Long-lived stau pairs from cosmic neutrino interactions could be detected as parallel tracks at IceCube at rates up to about 50 per year under optimistic but still allowed SUSY and neutrino flux assumptions.
Searching for MeV-Scale Gauge Bosons with IceCube
hep-ph 2015-07 conditional novelty 4.0 of 10

A light Z' coupling to neutrinos would carve an absorption dip in the IceCube spectrum, and existing data already rule out the lightest such particles in the inverted-hierarchy scenario.
IceCube-Gen2: A Vision for the Future of Neutrino Astronomy in Antarctica
astro-ph.HE 2014-12 conditional novelty 4.0 of 10

A design study argues that a 10 km^3 expansion of IceCube, using roughly 240 m string spacing and modernized optical modules, would deliver about fivefold better point-source sensitivity and larger samples of all neut...
A Possible Two-component Flux for the High Energy Neutrino Events at IceCube
hep-ph 2014-11 conditional novelty 4.0 of 10

A two-component astrophysical neutrino flux, electron-flavored below 1 PeV and pion-decay-flavored above, is fitted to 3-year IceCube data and claimed to explain the 400 TeV to 1 PeV energy gap and the track-to-shower...
Cosmic Neutrino Pevatrons: A Brand New Pathway to Astronomy, Astrophysics, and Particle Physics
astro-ph.HE 2013-12 conditional novelty 3.0 of 10

The 28 IceCube neutrino events are consistent with either Galactic or extragalactic sources, and a single power law of index 2.3 fits the spectrum without requiring a cutoff above 2 PeV.
Astrophysics Uniquely Enabled by Observations of High-Energy Cosmic Neutrinos
astro-ph.HE 2019-03 conditional novelty 2.0 of 10

Neutrino astronomy, given roughly ten times better point-source sensitivity, is argued to be the decisive route to identifying the sources of the highest-energy cosmic rays.
Opening a New Window onto the Universe with IceCube
astro-ph.HE 2018-05 conditional novelty 2.0 of 10

IceCube's diffuse cosmic neutrino flux, whose energy density matches that of gamma rays and ultra-high-energy cosmic rays, points to hadronic processes dominating the nonthermal universe.
Neutrino masses and mixings and...
hep-ph 2006-06 accept novelty 2.0 of 10

The review consolidates the evidence that solar and atmospheric neutrino deficits are explained by three-neutrino oscillations with large mixing angles, and surveys remaining anomalies and theoretical models.

How later papers used it

134 later papers cite it

Evidence: substantial. 131 of the citing papers share no author with it, and 25 of those carry a confident label from Pith's citation classifier.

Pith read 29 of the 175 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

6 later papers by other authors are listed below: 3 used its method or tool, 2 used its data or values and 1 compared its own results with it. 1 more paper by the same authors is listed: 1 used its data or values.

Uses its method or tool 3 of 4 papers with this label

Estimated the cumulative background intensity using this paper's method.

Particle Reacceleration by Turbulence and Radio Constraints on Multi-Messenger High-Energy Emission from the Coma Cluster (2021) · 0.98 · paraphrased: the licence does not allow quoting
Connected the neutrino flux to the cosmic ray flux using this paper's relation.

Can tidal disruption events produce the IceCube neutrinos? (2016) · 0.89 · paraphrased: the licence does not allow quoting
Following the form of Eq. (5) in [ref] , the evolution factor $ξ_z$ is given by [equation] where $dt/dz = -1/((1+z)H(z))$ is the cosmic time-redshift relation and $H(z) = H_0E(z)$ is the Hubble parameter at redshift $z$ with $E(z) = √(Ω_m(1+z)^3 + Ω_Λ)$.

Little Red Dots as Hidden Neutrino Sources (2026) · Population-luminosity diagram for LRDs · 0.90

Uses its data or values 3 of 8 papers with this label · 1 by the same authors

For the fiducial parameter values, the diffusive neutrino intensity at $∼0.1$ PeV from GRBs can be estimate by $I_ν,GRB≈(c/4π)f_z ṅ_GRB t_H (E_ν^ob)^2N_ν^ob ∼3×10^-11 GeV cm^-2 s^-1 sr^-1$, with the local GRB event rate density $ṅ_GRB ∼1 Gpc^-3 yr^-1$ [ref] , $f_z∼3$ accounts for the redshit distribution of GRB event rate density [ref] , and $t_H∼10$Gyr the universe age.

Overall Ultra-high Energy Neutrino Emission from GRBs during Jet Expansion (2025) · Conclusion and discussion · 0.98
Used this paper's cosmic-ray energy generation rate per unit volume of about 5x10^44 erg Mpc^-3 yr^-1.

Ultrahigh-energy neutrino flux as a probe of large extra-dimensions (2007) · 1.00 · paraphrased: the licence does not allow quoting

By the same authors

We note that $ξ_z≈3$ is obtained for redshift evolution following that of the star formation rate or AGN luminosity density, $∝(1 + z)^3$ up to $z = 2$ and constant at higher $z$. $ξ_z ≈0.6$ is obtained for no evolution [ref] .

High-Energy Neutrinos from Supernova Shock Breakouts in Circumstellar Media: Light Curves, Spectra, and Contribution to the Extragalactic Neutrino Background (2026) (same authors) · Contribution to the Neutrino Background · 0.80

Compares with it 1 of 14 papers with this label

Found its own luminosity slightly below the value estimated in this paper's Waxman-Bahcall calculation.

Obscured $pp$-channel neutrino sources (2020) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

94 arXiv papers cite this in 119 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 86 sentences without a confident label

Uses its method or tool 1 paper

Following the form of Eq. (5) in [ref] , the evolution factor $ξ_z$ is given by [equation] where $dt/dz = -1/((1+z)H(z))$ is the cosmic time-redshift relation and $H(z) = H_0E(z)$ is the Hubble parameter at redshift $z$ with $E(z) = √(Ω_m(1+z)^3 + Ω_Λ)$.

Little Red Dots as Hidden Neutrino Sources · Population-luminosity diagram for LRDs · 0.90

Uses its data or values 5 papers

In Fig. [ref], we show the sensitivity of detectors that are expected to collect 12 and 100 neutrino events[The current Auger exposure is $Ξ∼10^16$ cm$^2$ s sr [ref] , and an optimistic estimate for the flux of cosmogenic neutrinos is close to the Waxman-Bahcall bound [ref] , $E^2 ϕ_ν(E) ∼10^-8$ GeV cm$^-2$ s$^-1$ sr$^-1$ [ref] .

Neutrino Oscillations, Lorentz/CPT Violation, and Dark Energy · Proposed Measurement · 0.96
The CR energy generation rate per unit volume in our local universe in the energy range $10^19-21$ eV is $P_CR ≈5×10^44$ erg Mpc$^-3$ yr$^-1$ [ref] .

Ultrahigh-energy neutrino flux as a probe of large extra-dimensions · Ultrahigh-energy neutrino flux · 1.00
For the cosmic neutrino flux we take the optimistic Waxman-Bahcall upper bound $E_ν^2Φ_ν<2×10^-8 GeV cm^-2 s^-1 sr^-1$ [ref] .

Sensitivity of the JEM-EUSO telescope to gravity effects in neutrino-induced air showers · Detection of quasi-horizontal air showers with JEM-EUSO · 1.00

Compares with it 12 papers

For a Centaurus neutrino flux that goes as $E^-2$, this evaluates to [equation] approximately one-tenth of the Waxman-Bahcall flux [ref] .

Neutrinos from Auger Sources · Gamma-ray Observations · 0.99
These have put upper limits on the existence of UHE particles but these are appreciably higher than the predictions by Waxman and Bahcall [ref] .

Prospects for detection of the lunar Cerenkov emission by the UHE Cosmic Rays and Neutrinos using the GMRT and the Ooty Radio Telescope · Introduction · 0.95
This is consistent with upper bounds from IceCube [ref] , and is smaller than the upper range of the Waxman-Bahcall bound, $E^2 dΦ/dE ≃(0.3 - 1.5) ×10^-8$ GeV cm$^-2$ s$^-1$ sr$^-1$ [ref] .

Demystifying the PeV Cascades in IceCube: Less (Energy) is More (Events) · Astrophysical neutrinos: plausible · 0.96

Cites it as background 13 papers

The Waxman-Bahcall (WB) bound on UHE $ν$ flux [ref] is based on CRs that interact at their sources and lose all their energy equally to charged and neutral pions.

Ultrahigh-energy neutrino flux as a probe of large extra-dimensions · Ultrahigh-energy neutrino flux · 0.88
The GRBs are expected to be the sources of high-energy cosmic rays [ref] , and the accelerated protons will interact with ambient photons, producing TeV neutrinos and gamma rays via photohadronic processes and subsequent decays [ref] .

Invisible Neutrino Decays as Origin of TeV Gamma Rays from GRB221009A · TeV Neutrinos and Photons from GRBs · 0.82
The charged pions decay to high-energy neutrinos, $π^± →μ ν_μ →e ν_e ν_μν_μ$ (e.g. [ref] ), and the neutral pions to gamma rays, $π^0 →γγ$ (e.g. [ref] ).

Investigating starburst-driven neutrino emission from galaxies in the Great Observatories All-Sky LIRG Survey · Motivating GOALS galaxies as candidate neutrino sources · 0.81

Discussion

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