Cosmogenic Gamma-Ray & Neutrino Fluxes

• Cosmogenic gamma-ray and neutrino fluxes are high-energy secondary messengers produced by UHECR interactions with cosmic photon fields, offering clues about cosmic source composition and evolution.
• Mechanisms such as photopion production, Bethe–Heitler pair production, and electromagnetic cascades are modeled to predict fluxes, with results constrained by Fermi-LAT and IceCube observations.
• Advanced computational tools like CRPropa and SimProp simulate UHECR propagation, enabling multi-messenger diagnostics that critically test and refine cosmic-ray source scenarios.

Cosmogenic gamma-ray and neutrino fluxes are the high-energy secondary messengers produced by the interactions of ultra-high-energy cosmic rays (UHECRs) with cosmic background photon fields during their propagation through extragalactic and circumgalactic space. These fluxes serve as key, nearly unavoidable by-products of the “cosmic-ray beam dump,” carrying critical information about the sources, spectrum, and composition of UHECRs, as well as the properties of cosmic photon backgrounds. They are subject to stringent observational constraints, especially from the extragalactic gamma-ray background and the diffuse flux of astrophysical neutrinos.

1. Physical Mechanisms of Cosmogenic Production

The dominant production channels for cosmogenic gamma rays and neutrinos are triggered as UHECRs—protons or nuclei with energies above a few EeV—propagate through the cosmic microwave background (CMB) and extragalactic background light (EBL):

• Photopion Production (GZK Process): Protons (or nucleons) interact with CMB/EBL photons via $p + \gamma \to \Delta^+ \to p\pi^0, n\pi^+$. The threshold for the process is $E_p \gtrsim 6 \times 10^{19}$ eV for CMB photons. The decay of $\pi^0$ yields prompt gamma rays, while $\pi^\pm$ decay into muons and ultimately neutrinos ($$\pi^+ \to \mu^+ \nu_\mu \to e^+ \nu_e \bar{\nu}_\mu \nu_\mu$$).
• Bethe–Heitler $e^+e^-$ Pair Production: Nucleons (protons, nuclei) scatter on photons: $N + \gamma \rightarrow N + e^+ + e^-$, with much lower energy thresholds. The resulting $e^\pm$ inject energy into electromagnetic cascades that develop on CMB/EBL backgrounds.
• Electromagnetic Cascades: High-energy gamma rays and $e^\pm$ pairs initiate cascades through $\gamma\gamma\to e^+e^-$ and $e^\pm\gamma\to e^\pm\gamma$ processes, redistributing energy until photons pile up in the $\sim$GeV–TeV region, shaping the observed diffuse gamma-ray background (Aloisio et al., 2016, Globus et al., 2017, Batista et al., 2019).
• Photodisintegration of Nuclei: UHECR nuclei (e.g., Fe, N) interact with background photons, leading to spallation and production of lighter fragments and nucleons, which subsequently may undergo photopion processes.

These processes imprint the cosmogenic fluxes with signatures determined by the injection spectrum, composition, redshift evolution of the sources, and properties of the photon fields.

2. Analytical Formalism and Computational Approaches

The calculation of cosmogenic gamma-ray and neutrino fluxes employs transport equations that account for cosmic-ray injection, energy losses, secondary production yields, and propagation effects:

• General Flux Expression:

$$\Phi_i(E) = \frac{c}{4\pi} \int_0^{z_\mathrm{max}} dz\ |\frac{dt}{dz}| \ n_0L_0(1+z)^m \ \frac{dN_i}{dE'}(E' = E(1+z), z) e^{-\tau_i(E', z)}$$

where $i$ denotes $\nu$ (neutrino) or $\gamma$ (gamma ray), $n_0L_0(1+z)^m$ is the emissivity evolution, and $\tau_i$ accounts for optical depth (negligible for neutrinos, crucial for $\gamma$-rays due to pair production on the EBL) (Aloisio et al., 2016).

• Nuclear Transport and Yields:

The comoving number density $n_i(E,z)$ evolves according to:

$$\frac{\partial n_i(E,z)}{\partial t} = \text{(continuous losses)} - \Gamma_{\text{dis}} n_i(E,z) + \sum_{j>i} \Gamma_{j\to i} n_j(E,z) + Q_i(E, z),$$

with $Q_i$ the source term (Globus et al., 2017).

• Monte Carlo Codes: Modern simulations utilize frameworks such as CRPropa 3 and SimProp, which model UHECR propagation, photohadronic interactions, nuclear disintegration, electromagnetic cascades, and redshift evolution. Both codes implement detailed cross-section prescriptions and utilize up-to-date EBL models (e.g., Gilmore 2012, Domínguez 2011) (Batista et al., 2019, Vliet et al., 2017).
• Sensitivity Analysis: The uncertainties in EBL modeling, photodisintegration cross sections, cosmic-ray injection parameters, and propagation assumptions can contribute systematic errors at the 10–50% level in the normalization and spectral shape of predicted fluxes (Batista et al., 2019).

3. Astrophysical Source Scenarios and Model Constraints

UHECR source characteristics—composition, spectral index $\gamma$, maximum energy $E_\mathrm{max}$, and redshift evolution (power law in $1+z$ to $z_\mathrm{max}$ with index $m$)—critically affect cosmogenic flux predictions:

• Pure-Proton “Dip” and “Ankle” Models: Predict robust cosmogenic neutrino and gamma-ray fluxes. However, models with strong source evolution ($m \gtrsim 3$ in $(1+z)^m$), high $E_\mathrm{max}$, and hard injection spectra ($\gamma\sim2$) are ruled out by the Fermi-LAT extragalactic gamma-ray background, while mixed-composition or heavy-dominated scenarios produce much lower secondary yields (Gelmini et al., 2011, Globus et al., 2017, Vliet et al., 2017).
• Mixed-Composition Fits: As motivated by Auger and KASCADE-Grande data, these employ soft proton components ($\gamma_p \sim 2.4$–2.5), moderate evolution ($m\lesssim3$), and a heavy-enriched composition. Such models predict $E^2\Phi_\nu \sim 10^{-10}$–$10^{-9}$ GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$ around EeV energies—below current IceCube/ARIANNA limits but within reach of GRAND and POEMMA (Globus et al., 2017, Globus et al., 2017).
• Blazar Line-of-Sight Scenarios: Individual blazars (e.g., TXS 0506+056) associated with IceCube neutrino events may yield detectable cosmogenic signatures along the line of sight for sufficiently high UHECR luminosity ($L_{\mathrm{UHECR}} \gtrsim 10 L_\nu$), especially if $E_\mathrm{max} \gtrsim 10^{19}$ eV (Das et al., 2021).
• Circumgalactic Gas Interactions: Diffuse gamma-ray flux from cosmic-ray–circumgalactic gas interactions in the Milky Way and other galaxies can be a non-negligible fraction ($\sim$10–30%) of the Fermi-LAT background in the 1–100 GeV band, but the neutrino yield is subdominant—$\lesssim1\%$ of the measured IceCube flux (Kalashev et al., 2016).
• Parameter Dependence Table:

Scenario	$E^2\Phi_\nu$ (all-flavor)	$E^2\Phi_\gamma$ (GeV band)	Allowed by Fermi-LAT?
Mixed comp., $m\lesssim3$	$\sim10^{-10}$–$10^{-9}$	$\lesssim$20% EGB	Yes
Pure-proton, $m \geq 3$	$\gtrsim10^{-8}$	$>$100% EGB	No
Circumgalactic (MW+EG)	$\sim10^{-9}$ (10–100 TeV)	$2$–$5\times10^{-7}$	Yes (if subdominant)

4. Observational Constraints and Multi-Messenger Diagnostics

Current and forthcoming gamma-ray and neutrino observatories provide stringent, largely model-independent upper bounds on cosmogenic fluxes and thus indirectly on UHECR source models:

• Fermi-LAT Diffuse Gamma-Ray Background (EGB): The total cosmogenic gamma-ray flux, when added to contributions from blazars, star-forming galaxies, and misaligned AGN, must not exceed the measured EGB: $E^2\Phi_\gamma \lesssim (5$–$10)\times10^{-7}$ GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$ in the 1–100 GeV band (Globus et al., 2017, Aloisio et al., 2016, Berezinsky et al., 2010).
• IceCube and Auger Neutrino Limits: Current $E^2\Phi_\nu$ upper limits are of order $10^{-8}$ GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$ (single flavor) at EeV energies, ruling out strongly evolving, pure-proton models with high $E_\mathrm{max}$ (Vliet et al., 2017, Aloisio et al., 2016).
• Cascade-Energy Bound: The total energy density deposited into electromagnetic cascades by UHECRs is subject to $\omega_\mathrm{cas} \le 5.8\times10^{-7}$ eV cm$^{-3}$, which provides a robust upper bound on the allowed cosmogenic neutrino flux through the production correlations between $\gamma$ and $\nu$ (Berezinsky et al., 2010, Ahlers et al., 2010).
• Prospects for Detection: Next-generation experiments (GRAND, POEMMA, IceCube-Gen2, CTA) are targeting single-flavor sensitivities $E^2\Phi_\nu \sim 10^{-10}$–$10^{-11}$ GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$. These will critically probe both the standard mixed-composition scenario and any subdominant trans-GZK proton component, which may otherwise be invisible to air-shower measurements (Vliet et al., 2017, Globus et al., 2017).

5. Theoretical Uncertainties and Systematic Effects

The robustness of current predictions and constraints depends on several sources of uncertainty:

• EBL and CMB Modeling: Systematic differences in EBL models ($\sim$10–20% level) and their redshift evolution affect the transparency of the universe to gamma rays and hence the normalization and spectral shape of cosmogenic $\gamma$-ray fluxes (Batista et al., 2019).
• Cosmic-Ray Composition: The presence of heavy nuclei suppresses photopion production and thus both cosmogenic $\nu$ and $\gamma$ yields at the highest energies. Realistic mixed-composition or heavy-dominated scenarios reduce fluxes by 1–2 orders of magnitude relative to pure-proton cases (Globus et al., 2017, Vliet et al., 2017).
• Nuclear Cross-Sections: Updates to photodisintegration and photopion cross-sections (e.g., TALYS, PSB, SOPHIA) affect both the primary UHECR propagation and secondary yields, introducing uncertainties of typically 10–30% (Batista et al., 2019).
• Propagation Approximations: Different implementations in CRPropa and SimProp—such as single-pion or multi-pion treatments, EBL redshift scaling, and cascade development—induce differences in flux predictions by factors up to $\sim$50% in some parameter regimes (Batista et al., 2019).
• Source Evolution and Maximum Redshift: The choice of source evolution index $m$ and cutoff redshift $z_\mathrm{max}$ strongly impacts the normalization and low-energy tail of the secondaries, especially for positively evolving or high–$z_\mathrm{max}$ scenarios (Aloisio et al., 2016).

6. Significance for UHECR Origin and Multi-Messenger Astrophysics

Cosmogenic gamma-ray and neutrino fluxes represent essential, model-discriminating diagnostics for UHECR source scenarios:

• The non-detection of a significant EeV cosmogenic neutrino background, coupled with the measured level of the Fermi EGB, currently excludes strong source evolution or hard (proton-dominated, high–$E_\mathrm{max}$) models, providing evidence for mixed or heavy composition at the highest energies. Any future detection at the $10^{-10}$ GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$ level would set a lower bound on the proton fraction at ultra-high energies and on the source redshift evolution (Globus et al., 2017, Vliet et al., 2017).
• Direct measurements of hard TeV–PeV gamma-ray emission or line-of-sight fluxes from individual nearby UHECR source candidates (e.g., blazars) may offer unambiguous identification of cosmic-ray acceleration sites provided UHECR luminosity is sufficiently high and maximal energies exceed $10^{19}$ eV (Das et al., 2021).
• The combination of multi-messenger observations (gamma, neutrino, UHECR spectra, and composition) provides the only means to break degeneracies among source evolution, composition, and spectrum, thereby constraining the nature and distribution of the highest-energy cosmic accelerators. Continued improvements in modeling, cross-section measurements, and the removal of systematic uncertainties in the backgrounds and source populations are essential for further progress (Globus et al., 2017, Aloisio et al., 2016).

7. Summary Table: Key Cosmogenic Flux Results from Recent Models

Reference	Model Type / Key Assumptions	$E^2\Phi_\nu$ (peak)	$E^2\Phi_\gamma$ (GeV–TeV)	Fermi / IceCube Consistency
(Globus et al., 2017)	Mixed comp., $m\leq3$, soft $\gamma$	$10^{-10}$–$10^{-9}$	$<20\%$ EGB	Fully compatible
(Aloisio et al., 2016)	Pure proton, $m=0,3,5$	$2\times10^{-9}$–$3\times10^{-8}$	$2\times10^{-8}$–$3\times10^{-7}$	$m\gtrsim3$ disfavored
(Berezinsky et al., 2010)	Pure proton, $m=0$–$4$	$\lesssim10^{-9}$	$\lesssim10\%$ EGB	Only extreme $m$ reaches detection
(Vliet et al., 2017)	Auger CTG fit (88% N, 12% Si, low $E_{\max}$)	$\lesssim10^{-11}$	$<3\times10^{-8}$	Well below bounds
(Kalashev et al., 2016)	Circumgalactic gas (MW+EG)	$\sim10^{-9}$ (10–100 TeV)	2–5$\times10^{-7}$ (1–100 GeV)	$\lesssim$30% of Fermi-LAT, $\ll$ IceCube

All fluxes in units of GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$. Peak values refer to total all-flavor neutrinos.

Key references: (Kalashev et al., 2016, Globus et al., 2017, Aloisio et al., 2016, Batista et al., 2019, Vliet et al., 2017, Berezinsky et al., 2010, Ahlers et al., 2010, Das et al., 2021, Gelmini et al., 2011, Globus et al., 2017).