Cosmogenic Neutrino Spectrum Overview

• Cosmogenic neutrinos are ultra-high-energy particles produced when cosmic rays interact with background photons such as the CMB and EBL.
• The spectrum encodes detailed information on UHECR source distribution, acceleration limits, and propagation physics through photopion production processes.
• Spectral features and distortions can reveal effects beyond standard physics, including Lorentz violation and neutrino self-interactions, guiding experimental strategies.

Cosmogenic neutrinos are ultra-high-energy (UHE) neutrinos produced by the interactions of UHE cosmic rays (UHECRs) with cosmic photon backgrounds primarily via photopion production on the cosmic microwave background (CMB) and extragalactic background light (EBL). The resulting cosmogenic neutrino spectrum encodes information about UHECR source distribution, composition, maximum acceleration energy, and propagation physics. Rigorous modeling incorporates observational constraints from UHECR and VHE $\gamma$-ray spectra, Monte Carlo transport codes, and sensitivity limits of current and future neutrino detectors.

1. Generation Mechanisms and Formalism

The principal production channel is $$p+\gamma \rightarrow \Delta^{+} \rightarrow n+\pi^+$$ (plus analogous interactions for heavy nuclei), followed by $$\pi^\pm \rightarrow \mu^\pm \nu_\mu\rightarrow e^\pm \nu_e \nu_\mu$$. The corresponding comoving proton emissivity in the most widely used models is

$$Q_p(E_p,z) = Q_0\,E_p^{-\gamma}\,\exp[-E_p/E_{\text{max}}]\,f(z)$$

where $Q_0$ sets the normalization (often by requiring the propagated spectrum to match HiRes/Pierre Auger data), injection index $2 \lesssim \gamma \lesssim 2.7$, cutoff energy $E_{\text{max}}\sim 10^{20}-10^{21}$ eV, and redshift evolution $f(z)$ parameterized as $(1+z)^m$ or via astrophysical rate functions (SFR, GRB, AGN). The observed all-flavor differential neutrino intensity is

$$\Phi_\nu(E_\nu) = \frac{c}{4\pi} \int_0^{z_\text{max}} dz\,\frac{dt}{dz}\,\int_{E_{p,\min}}^{E_{p,\max}} dE_p\,Q_p(E_p,z)\,\frac{dN_\nu}{dE_\nu}(E_\nu;E_p,z)$$

with cosmology $dt/dz$ in a flat $\Lambda$CDM universe. Numerical solutions require coupled transport equations for nucleons, pions, $e^\pm$, $\gamma$, $\nu$ or a full Monte Carlo such as CRPropa3 (Vliet et al., 2017).

2. Composition, Source Evolution, and Spectral Features

Composition Effects

• Pure-proton: Maximizes cosmogenic neutrino yield. Nearly all photopion interactions produce charged pions and hence neutrinos. Peak $E^2\Phi_\nu$ at few $\times10^{-9}$ GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$ near $E_\nu\sim5\times10^8$ GeV (Gelmini et al., 2011).
• Mixed/heavy nuclei: Photo-disintegration dominates, decreasing neutrino normalization by up to an order of magnitude and softening the cutoff energy ($E_\nu$ cuts off at lower values) (Vliet et al., 2017, Romero-Wolf et al., 2017).

Source Evolution

• Mild (SFR-type) evolution: Compatible with VHE $\gamma$-ray bounds; $m\lesssim3$ or SFR history leads to allowed flux envelopes (Gelmini et al., 2011).
• Strong evolution (GRB/AGN rates, $m\gtrsim3.5$): Disfavored or excluded by Fermi/LAT VHE $\gamma$-ray background due to excessive electromagnetic cascades (Gelmini et al., 2011).

Energy Range and Spectral Index

The typical spectrum extends $E_\nu\sim10^7$ GeV up to several $\times10^{10}$ GeV. Below the spectral peak, $E^2\Phi_\nu$ rises as $\propto E$, crosses over near $10^8$–$10^9$ GeV, and falls off as $\propto E^{-1}$–$E^{-2}$ above the peak (Gelmini et al., 2011), with variations depending on $E_{\text{max}}$ and $\gamma$. Heavy nuclei compositions further steepen the spectrum.

Illustration from (Gelmini et al., 2011):

Model	$E_{\text{peak}}$ [GeV]	$E^2\Phi_{\text{peak}}$ [GeV/cm${}^2$s sr]	Allowed
Dip (m=2)	$5\times10^8$	$1\times10^{-9}$	Yes
Dip (SFR)	$5\times10^8$	$2\times10^{-9}$	Yes
Ankle (m=3)	$1\times10^9$	$7\times10^{-9}$	Yes
Dip (GRB/AGN)	$5\times10^8$	$5\mbox{–}6\times10^{-9}$	No

3. Model-Independent Lower Bounds

The minimal cosmogenic neutrino flux is set by the observed UHECR spectrum and direct inversion, independent of source class, magnetic field, or arbitrary $E_{\text{max}}$ (Ahlers et al., 2012):

• Pure-proton, no evolution: $E_\nu^2\Phi_\nu(E_\nu)\gtrsim1\times10^{-8}$ GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$ for $E_\nu \sim 10^8$–$10^{10}$ GeV.
• SFR evolution: Limit rises by $\sim\times5$.
• Heavy composition: Minimal yield suppressed by up to an order of magnitude.

This bound is robust against uncertainties in the IR/optical background and energy scale, and cannot be violated without abandoning the proton-dominated UHECR paradigm or standard cosmological source distributions.

4. Spectral Modifications from Beyond-Standard-Model Physics

Lorentz Violation (LV)

Dimension-6, CPT-even LV operators introduce a $p^4/M_{\text{Pl}}^2$ term in the dispersion relation; sizable $\eta_\nu$ triggers neutrino splitting ($\nu\to\nu\nu\nu$) above $E_\text{th} \sim 20$ TeV $\eta_\nu^{-1/4}$ (0911.0521, Gorham et al., 2012). Consequences:

• Sharp cutoff in the spectrum above $E_c\sim6\times10^{18}$ eV $\eta_\nu^{-4/13}$ with $\eta_\nu\sim1$.
• Bump feature below cutoff: Splitting cascades pile up flux just below $E_c/3$ (0911.0521).
• Sensitivity: Non-observation of suppression up to $E_\text{obs}\sim10^{19}$ eV implies $\eta_\nu\lesssim10^{-4}$.

LIV-induced vacuum $e^+e^-$ pair emission produces a "brick-wall" cutoff plus pile-up, with present limits imposed by ANITA/RICE non-observation implying $\delta_\nu\gtrsim10^{-28}-10^{-25}$ (Gorham et al., 2012).

Neutrino Self-Interactions, (Pseudo-)Dirac States, and $\nu$–DM Interactions

Radio array sensitivity projections (e.g., GRAND) show that BSM physics can imprint distinctive dips, oscillations, or monotonic suppressions in the spectrum (Leal et al., 14 Apr 2025):

• Self-interactions (light scalars): Resonant dips near $E_{\rm res}$, plus low-energy pile-up for $g_{\tau\tau} \sim 10^{-2}$–$10^{-1}$, $m_\phi\sim0.1$–$1$ GeV.
• Pseudo-Dirac oscillations: $\Delta m^2\sim10^{-14}$–$10^{-15}$ eV$^2$ generates oscillatory spectral dips.
• Neutrino–DM scattering: Heavy-mediator scenario yields monotonic suppression above $E\sim m_\chi^2/2m_S$, light-mediator below that scale.
• Active–sterile secret interactions (pseudoscalars): For mediator mass $M_\phi\sim250$–$500$ MeV, strong suppression above $E_\text{th}\sim10^9$ GeV, potentially testable at GRAND radio arrays (Fiorillo et al., 2020).

5. Observational and Theoretical Constraints

Gamma-Ray Cascade Bound

Since each neutrino is accompanied by comparable-energy $\pi^0\to\gamma$ decay photons, electromagnetic cascades contribute to the GeV–TeV $\gamma$-ray background. Fermi/LAT isotropic diffuse $\gamma$-ray bounds constrain the cosmogenic neutrino normalization, excluding pure-proton, strong-evolution models with optimistic $E^2\Phi_\nu \gg 10^{-7}$ GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$ (Gelmini et al., 2011, Vliet et al., 2017).

Point and Transient Sources

Modeling of nearby, transient UHECR sources (GRBs, blazars) shows that time-dependent, sub-degree point sources can "pop up" above the diffuse cosmogenic background for short epochs post-burst, especially in regions of low IGMF (Zhang et al., 14 Apr 2025, Das et al., 2021). For sufficiently high $L_{\rm UHECR}/L_\nu$ ($\gtrsim10$), predicted ν-fluxes per source can approach detection thresholds in IceCube-Gen2 or GRAND (Das et al., 2021).

Bayesian Inference and Population Uncertainties

Joint modeling of composition, source evolution, and detector systematics via Bayesian inference yields a broad credible interval for cosmogenic neutrino spectra, with 68% C.I. bands $E_\nu^2\Phi_\nu\sim10^{-9}$–$10^{-8}$ GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$ at $E_\nu\sim10^{17.5}$ eV for mixed composition and weak evolution (Romero-Wolf et al., 2017).

6. Probing Fundamental Physics and Cosmic Backgrounds

• Relic Neutrino Clustering: Resonant $\nu + \bar{\nu} \to \rho^0$ absorption dips in the cosmogenic flux reveal clustering and mass of the cosmic $\nu$ background; position and depth of dip at $E_{\rm res} = m_\rho^2/2m_\nu(1+z)$ directly probe $m_\nu$, relic overdensity $\xi$ (Brdar et al., 2022).
• Constraints on New Physics: Cosmogenic spectra at EeV energies provide some of the most stringent direct bounds on Planck-scale Lorentz violation, secret interactions, and dark-sector couplings (Gorham et al., 2012, Fiorillo et al., 2020, Leal et al., 14 Apr 2025).

7. Detectability and Experimental Prospects

Event-rate estimates for next-generation detectors:

• IceCube/ARA-37: Pure-proton dip-model at maximal allowed normalization yields $\lesssim$0.1–0.2 events/yr (requiring $\sim$10 yr for 3$\sigma$ detection); ankle model achieves sensitivity within 3 yr (Gelmini et al., 2011).
• GRAND/Askaryan arrays: Sensitivity goals $E^2\Phi_\nu\sim10^{-9}$–$10^{-10}$ GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$ at $E\sim1$ EeV can probe proton fraction down to a few percent and distinguish BSM spectral distortions (Vliet et al., 2017, Leal et al., 14 Apr 2025).
• Transient/point sources: Individual bursts can produce detectable excess for $\lesssim$0.1° angular extent if $L_{\rm UHECR}$ is sufficiently high and IGMF sufficiently weak; population stacking may be necessary given the low rates (Zhang et al., 14 Apr 2025, Boxi et al., 22 Nov 2025, Das et al., 2021).

In conclusion, the cosmogenic neutrino spectrum is a sensitive multi-messenger probe of UHECR physics, source evolution, composition, and fundamental neutrino properties. The predicted spectrum and its distortions from known and speculative effects form a critical guide for current and future observational programs.