Neutrino Cosmic Horizon Overview

• Neutrino cosmic horizon is a multifaceted concept defining scales from the relic last-scattering surface to effective high-energy attenuation limits.
• It encompasses the particle horizon at neutrino decoupling and the free-streaming scales that influence CMB phase shifts and matter clustering.
• In high-energy neutrino astronomy, redshift-induced and spectral cutoff horizons determine observable anisotropy and source contributions.

“Neutrino cosmic horizon” denotes several closely related scales rather than a single invariant boundary. In the early universe it refers to the neutrino last-scattering surface, the particle horizon at neutrino decoupling, and the neutrino free-streaming horizon that governs the imprint of relic neutrinos on the cosmic microwave background (CMB) and large-scale structure (LSS). In high-energy neutrino astronomy it can instead denote an attenuation horizon defined by optical depth, an observer-induced effective horizon produced by cosmological redshift combined with a source spectral cutoff, or a practical oscillation horizon set by the saturation of the Friedmann–Robertson–Walker (FRW) phase integral. These usages are linked by the same underlying question: from what spacetime region, and under what propagation law, can neutrinos contribute to an observable signal today (Stecker, 2022, Muzio et al., 2023, Lacki, 2011, Wagner et al., 2011).

1. Terminological scope and unifying definitions

In hot big-bang cosmology, the expression is used in “a few closely related ways.” One meaning is the neutrino last-scattering surface of the cosmic neutrino background (CNB), namely the shell from which relic neutrinos last interacted with the primordial plasma at $T_{\nu,\mathrm{dec}}\sim\mathrm{MeV}$ and $t\sim 1\,\mathrm{s}$. A second meaning is the particle horizon at neutrino decoupling, which sets the largest causally connected patch on that surface. A third is the neutrino free-streaming horizon, the comoving scale over which collisionless neutrinos propagate and modify perturbations in the metric and matter distribution (Stecker, 2022).

A distinct astrophysical usage defines the neutrino cosmic horizon as the maximum comoving distance or redshift from which neutrinos of observed energy $E$ can reach Earth without significant attenuation, usually by requiring optical depth $\tau(E)\lesssim 1$. In this sense the horizon is energy dependent and model dependent, because it depends on the interaction cross section, target density, and cosmological line element (Lacki, 2011, Franklin et al., 2024).

A further usage appears in diffuse high-energy neutrino anisotropy studies. There, the “effective neutrino horizon” does not arise from attenuation in intergalactic space. Instead it is induced by cosmological redshifting, $E'=(1+z)E$, together with a high-energy source cutoff $E_{\mathrm{max}}$. If the redshifted source-frame energy required to produce an observed neutrino exceeds $E_{\mathrm{max}}$, distant sources cease to contribute. In that construction the horizon is kinematic and observer side rather than absorptive (Muzio et al., 2023).

Finally, in cosmological oscillation studies the relevant horizon is the maximum effective baseline available for neutrino phase accumulation in an expanding universe. The FRW phase depends on the integral $I(z)$, and because $I(z)$ converges as $z\to\infty$ in standard cosmologies, the oscillation phase has a cosmology-set upper bound. This defines a practical “neutrino cosmic horizon” for oscillations (Wagner et al., 2011).

2. Primordial horizon, decoupling, and free streaming

The CNB consists of relic neutrinos that decoupled when the weak-interaction rate fell below the Hubble rate. A standard estimate gives $t\sim 1\,\mathrm{s}$0 at $t\sim 1\,\mathrm{s}$1. After electron–positron annihilation, the present temperature is

$t\sim 1\,\mathrm{s}$2

with $t\sim 1\,\mathrm{s}$3 per active flavor including neutrino and antineutrino, total $t\sim 1\,\mathrm{s}$4, and $t\sim 1\,\mathrm{s}$5 (Stecker, 2022).

The causal scale at decoupling is small by present standards. In radiation domination the physical particle horizon is $t\sim 1\,\mathrm{s}$6, so at $t\sim 1\,\mathrm{s}$7,

$t\sim 1\,\mathrm{s}$8

Using $t\sim 1\,\mathrm{s}$9, the corresponding comoving horizon today is $E$0, with $E$1. This is the causally connected patch size on the CNB last-scattering surface. By contrast, the comoving distance to that surface is within a percent of the present particle horizon because relic neutrinos have traversed essentially the whole observable universe since decoupling (Stecker, 2022).

The free-streaming distance is

$E$2

and the corresponding instantaneous scale may be written

$E$3

While neutrinos are relativistic, $E$4, so the free-streaming horizon closely tracks the comoving particle horizon. After they become non-relativistic, $E$5. The non-relativistic transition occurs at

$E$6

giving $E$7 for $E$8 and $E$9 for $\tau(E)\lesssim 1$0. A present-day order-of-magnitude free-streaming length is $\tau(E)\lesssim 1$1 for $\tau(E)\lesssim 1$2 (Stecker, 2022).

This horizon is cosmologically observable. Relativistic free-streaming neutrinos carry anisotropic stress, modify $\tau(E)\lesssim 1$3, and imprint a distinct phase shift in photon–baryon acoustic oscillations. Once massive neutrinos become non-relativistic, they suppress matter clustering on scales smaller than their free-streaming scale, with the late-time approximation

$\tau(E)\lesssim 1$4

The free-streaming horizon is therefore both a causal propagation scale and a transfer scale in linear structure formation (Stecker, 2022).

The same horizon can be altered by nonstandard interactions. In a dark neutrino interaction model, a small interacting dark-matter fraction $\tau(E)\lesssim 1$5 scatters with standard left-handed neutrinos through a temperature-independent cross section, and the relevant control parameter is $\tau(E)\lesssim 1$6. When $\tau(E)\lesssim 1$7, neutrinos do not free stream after horizon entry, their anisotropic stress is suppressed, and the standard acoustic phase shift is reduced. Using Planck 2015 CMB and WiggleZ full-shape data, that model yields $\tau(E)\lesssim 1$8 for P15 + W1 + SH0ES, and the Hubble tension is reduced to approximately $\tau(E)\lesssim 1$9 while keeping $E'=(1+z)E$0 (Ghosh et al., 2019).

3. Observer-induced effective horizon in high-energy neutrino astronomy

In extragalactic neutrino anisotropy studies, the effective neutrino horizon arises from redshift kinematics and source spectral cutoffs rather than from absorption. If a neutrino is observed with energy $E'=(1+z)E$1, then the source-frame energy was $E'=(1+z)E$2. For a sharp source cutoff at $E'=(1+z)E$3, the maximal contributing redshift satisfies

$E'=(1+z)E$4

In the threshold-integrated case, “all neutrinos beyond a redshift $E'=(1+z)E$5 will arrive at Earth with energies below this threshold.” With an exponential cutoff $E'=(1+z)E$6, the same constraint becomes a soft horizon through the factor $E'=(1+z)E$7 (Muzio et al., 2023).

The diffuse intensity above threshold is written as

$E'=(1+z)E$8

where $E'=(1+z)E$9 is the source-frame spectral shape per unit neutrino luminosity and $E_{\mathrm{max}}$0 is the luminosity density of sources. The source evolution model is

$E_{\mathrm{max}}$1

The corresponding redshift weighting or window function is

$E_{\mathrm{max}}$2

so that $E_{\mathrm{max}}$3 (Muzio et al., 2023).

To quantify the cutoff-induced horizon together with source evolution, the effective horizon $E_{\mathrm{max}}$4 is defined operationally as the redshift within which $E_{\mathrm{max}}$5 of the above-threshold observed flux originates. For $E_{\mathrm{max}}$6, $E_{\mathrm{max}}$7, and $E_{\mathrm{max}}$8, the results are $E_{\mathrm{max}}$9 ($E_{\mathrm{max}}$0 comoving) for a negatively evolving population with $E_{\mathrm{max}}$1, and $E_{\mathrm{max}}$2 ($E_{\mathrm{max}}$3) for a non-evolving population with $E_{\mathrm{max}}$4. More positive evolution increases the high-$E_{\mathrm{max}}$5 contribution and enlarges $E_{\mathrm{max}}$6 (Muzio et al., 2023).

The physical significance is anisotropy amplification. If sources trace the matter distribution, shrinking the effective horizon enhances the imprint of local LSS because nearby structure is highly anisotropic while the distant universe is nearly isotropic. The anisotropic sky is constructed by dividing the line of sight into redshift shells and weighting them by the local mass distribution from the CosmicFlows-2 quasi-linear density field. The paper then fits a template amplitude $E_{\mathrm{max}}$7 via a Poisson likelihood, with $E_{\mathrm{max}}$8 isotropic and $E_{\mathrm{max}}$9 identical to a maximally anisotropic LSS template. For $I(z)$0, $I(z)$1, $I(z)$2, and $I(z)$3, the sky is dominated by Virgo, Coma, Great Attractor, and Perseus–Pisces structures; for $I(z)$4, it is markedly more isotropic (Muzio et al., 2023).

A common misconception is that this horizon requires intergalactic neutrino opacity. It does not. The underlying paper explicitly attributes the effect to cosmological redshifting plus the high-energy cutoff in the source spectrum, with neutrinos remaining essentially unabsorbed in intergalactic space (Muzio et al., 2023).

4. Attenuation horizons, opacity, and new interactions

The attenuation-based neutrino cosmic horizon is defined by optical depth. For a neutrino observed with energy $I(z)$5 from redshift $I(z)$6,

$I(z)$7

The horizon $I(z)$8 follows from $I(z)$9. In a homogeneous region, $I(z)$0 is the mean free path $I(z)$1 (Franklin et al., 2024).

At TeV energies the Standard Model universe is essentially transparent. Using TeV-scale neutrinos from NGC 1068, TXS 0506+056, and PKS 1424+240, and assuming $I(z)$2, the relic neutrino overdensity is constrained to $I(z)$3 at $I(z)$4 confidence level. For $I(z)$5–$I(z)$6 and $I(z)$7–$I(z)$8, the Standard Model cross section is $I(z)$9–$z\to\infty$0, implying

$z\to\infty$1

so even with $z\to\infty$2, one still finds $z\to\infty$3–$z\to\infty$4 at $z\to\infty$5 (Franklin et al., 2024).

New interactions can shrink this horizon sharply near resonance. For a real scalar mediator $z\to\infty$6 with $z\to\infty$7,

$z\to\infty$8

and current IceCube point-source data probe couplings of approximately $z\to\infty$9 for a boson mass around the MeV scale. The absence of strong attenuation or sharp spectral dips constrains such scenarios and pushes $t\sim 1\,\mathrm{s}$00 above the source–Earth distance except near allowed resonant bands (Franklin et al., 2024).

A closely related framework is the $t\sim 1\,\mathrm{s}$01 model with a light $t\sim 1\,\mathrm{s}$02. There the survival probability is $t\sim 1\,\mathrm{s}$03, and the horizon condition is $t\sim 1\,\mathrm{s}$04. For nonrelativistic CNB targets,

$t\sim 1\,\mathrm{s}$05

The paper computes the optical depth with thermal distributions and identifies regions in $t\sim 1\,\mathrm{s}$06 space where $t\sim 1\,\mathrm{s}$07. For $t\sim 1\,\mathrm{s}$08–$t\sim 1\,\mathrm{s}$09 and $t\sim 1\,\mathrm{s}$10–$t\sim 1\,\mathrm{s}$11, the resonant energies lie in the $t\sim 1\,\mathrm{s}$12–$t\sim 1\,\mathrm{s}$13 window. Representative couplings in the $t\sim 1\,\mathrm{s}$14-favored region are $t\sim 1\,\mathrm{s}$15–$t\sim 1\,\mathrm{s}$16, and the paper describes parameter regions where $t\sim 1\,\mathrm{s}$17, $t\sim 1\,\mathrm{s}$18–$t\sim 1\,\mathrm{s}$19, and high-energy neutrino attenuation can be explained simultaneously (Fardeen et al., 20 Jul 2025).

Standard-model attenuation becomes relevant only at much higher energies through $t\sim 1\,\mathrm{s}$20-resonant annihilation on the CNB,

$t\sim 1\,\mathrm{s}$21

with a characteristic interaction length $t\sim 1\,\mathrm{s}$22–$t\sim 1\,\mathrm{s}$23. Below that scale, late-universe CNB interactions are negligible for ordinary propagation (Pisanti, 2017).

5. Oscillation, superluminal, refractive, and lensing formulations

In FRW spacetime the oscillation phase of an astrophysical neutrino is not proportional to a static baseline but to the integral

$t\sim 1\,\mathrm{s}$24

so that

$t\sim 1\,\mathrm{s}$25

Because $t\sim 1\,\mathrm{s}$26 saturates as $t\sim 1\,\mathrm{s}$27, the available phase also saturates. In Einstein–de Sitter cosmology,

$t\sim 1\,\mathrm{s}$28

and $t\sim 1\,\mathrm{s}$29; already at $t\sim 1\,\mathrm{s}$30, $t\sim 1\,\mathrm{s}$31, or $t\sim 1\,\mathrm{s}$32 of the asymptotic value. This is a practical neutrino cosmic horizon for oscillations: beyond moderate redshift, additional source distance adds little phase (Wagner et al., 2011).

A different propagation-based definition appears in studies of hypothetical superluminal neutrinos. There the neutrino horizon is

$t\sim 1\,\mathrm{s}$33

which reduces to $t\sim 1\,\mathrm{s}$34 if $t\sim 1\,\mathrm{s}$35 is approximately constant over the relevant redshift range. Because the diffuse flux scales linearly with $t\sim 1\,\mathrm{s}$36 under the paper’s assumptions, nondetection of “guaranteed” backgrounds bounds superluminal speeds. The quoted limits are $t\sim 1\,\mathrm{s}$37 at $t\sim 1\,\mathrm{s}$38–$t\sim 1\,\mathrm{s}$39 from star-forming galaxies, $t\sim 1\,\mathrm{s}$40 at $t\sim 1\,\mathrm{s}$41–$t\sim 1\,\mathrm{s}$42 in a pessimistic UHECR model, $t\sim 1\,\mathrm{s}$43 at $t\sim 1\,\mathrm{s}$44 in a typical proton model, and $t\sim 1\,\mathrm{s}$45 at $t\sim 1\,\mathrm{s}$46 in an optimistic proton model. The same formalism implies that extremely subluminal speeds would suppress the diffuse intensity by shrinking the horizon (Lacki, 2011).

By contrast, CNB refraction does not generate an attenuation horizon. The CNB acts as an ultradilute medium producing a diagonal forward-scattering potential $t\sim 1\,\mathrm{s}$47, an energy shift $t\sim 1\,\mathrm{s}$48, and a nondispersive refractive index $t\sim 1\,\mathrm{s}$49. For asymmetries $t\sim 1\,\mathrm{s}$50, the characteristic potential scale is $t\sim 1\,\mathrm{s}$51, the oscillation length is $t\sim 1\,\mathrm{s}$52, and the accumulated phase over a gigaparsec is $t\sim 1\,\mathrm{s}$53 rad. The effect is purely refractive and produces no absorption, so “the CNB does not impose any practical ‘cosmic horizon’ for neutrinos” (Diaz et al., 2015).

Massive relic neutrinos introduce yet another horizon notion through lensing geometry. Because they travel on timelike rather than null geodesics, the comoving distance traveled since decoupling is

$t\sim 1\,\mathrm{s}$54

so the effective neutrino surface of last scattering is closer than the CMB surface for subluminal relics. The lensing is strongly chromatic: both the deflection and the time of lens crossing depend on the neutrino momentum. In the paper’s thin-lens treatment, the deflection angle is enhanced by the factor $t\sim 1\,\mathrm{s}$55, and the momentum dependence implies, in principle, access to the causal volume “not restricted to the light cone” (Lin et al., 2019).

6. Observational programs, assumptions, and open problems

The observational status depends on which horizon definition is under consideration. For CNB anisotropies, numerical line-of-sight and Einstein–Boltzmann hierarchy calculations agree to within about $t\sim 1\,\mathrm{s}$56 over several decades in $t\sim 1\,\mathrm{s}$57 for masses $t\sim 1\,\mathrm{s}$58 and multipoles $t\sim 1\,\mathrm{s}$59. For $t\sim 1\,\mathrm{s}$60, low-$t\sim 1\,\mathrm{s}$61 C$t\sim 1\,\mathrm{s}$62B anisotropies are amplified by more than two orders of magnitude relative to the massless case; for $t\sim 1\,\mathrm{s}$63, the anisotropy level after the non-relativistic transition reaches roughly $t\sim 1\,\mathrm{s}$64 of the C$t\sim 1\,\mathrm{s}$65B monopole temperature, and a polarized tritium PTOLEMY configuration is predicted to see a directional capture-rate modulation at the level of $t\sim 1\,\mathrm{s}$66 across the sky. The dominant correlations are expected with wide-area, low-$t\sim 1\,\mathrm{s}$67 galaxy surveys because the relevant transfer functions are sourced by low-redshift metric perturbations (Tully et al., 2021).

For high-energy extragalactic anisotropy, the limiting factor is event statistics rather than opacity. Using the template-amplitude framework of the LSS-imprinted sky, the rule of thumb is that $t\sim 1\,\mathrm{s}$68 astrophysical events are required to achieve sensitivity to $t\sim 1\,\mathrm{s}$69 at the $t\sim 1\,\mathrm{s}$70–$t\sim 1\,\mathrm{s}$71 level depending on sky coverage and threshold. Ten-year event estimates quoted in the paper include, for $t\sim 1\,\mathrm{s}$72, IceCube $t\sim 1\,\mathrm{s}$73 events and IceCube-Gen2 $t\sim 1\,\mathrm{s}$74; for $t\sim 1\,\mathrm{s}$75, IceCube $t\sim 1\,\mathrm{s}$76 and IceCube-Gen2 $t\sim 1\,\mathrm{s}$77; and for $t\sim 1\,\mathrm{s}$78, IceCube $t\sim 1\,\mathrm{s}$79 and IceCube-Gen2 $t\sim 1\,\mathrm{s}$80. The same analysis concludes that IceCube can measure large anisotropies, $t\sim 1\,\mathrm{s}$81, and constrain down to $t\sim 1\,\mathrm{s}$82 at $t\sim 1\,\mathrm{s}$83, whereas IceCube-Gen2 can constrain to $t\sim 1\,\mathrm{s}$84 and measure $t\sim 1\,\mathrm{s}$85. At $t\sim 1\,\mathrm{s}$86, IceCube-Gen2 and radio arrays become crucial; the paper notes that KM3NeT has already reported a single UHE event of $t\sim 1\,\mathrm{s}$87 (Muzio et al., 2023).

Across all formulations, the literature emphasizes strong model dependence. In the LSS-anisotropy construction, the source density is assumed to trace the local mass density inferred from CosmicFlows-2, with isotropy beyond $t\sim 1\,\mathrm{s}$88; source-type-dependent bias is not explicitly modeled, individual galaxies are not modeled, and small-scale anisotropies are not addressed. Energy resolution blurs the observer-induced horizon, and a dispersion in source $t\sim 1\,\mathrm{s}$89 broadens the effective cutoff. Cosmogenic neutrinos are more isotropic and would dilute anisotropy if they made a large contribution at EeV energies (Muzio et al., 2023).

The resulting picture is therefore plural rather than singular. The neutrino cosmic horizon can be the MeV-era causal boundary of the CNB, the free-streaming scale governing CMB phase shifts and matter-power suppression, a kinematic window function induced by source cutoffs and redshift, an optical-depth boundary in models with strong $t\sim 1\,\mathrm{s}$90-$t\sim 1\,\mathrm{s}$91 scattering, or a phase-saturation limit for cosmological oscillations. What unifies these usages is that each turns neutrino propagation into a probe of spacetime structure, source evolution, or new neutrino-sector interactions, with different observables—CMB acoustic phases, LSS suppression, CNB anisotropies, diffuse spectral dips, and extragalactic sky maps—encoding different versions of the same horizon concept (Stecker, 2022, Muzio et al., 2023, Franklin et al., 2024).