The Weak Equivalence Principle (WEP) can be tested for neutrinos by comparing the arrival times of neutrinos and photons from the same astrophysical transient: if the two species coupled differently to gravity, they would accumulate different gravitational (Shapiro) propagation delays, and the observed arrival-time offset would bound the difference Δγ=γγ−γν\Delta\gamma=\gamma_\gamma-\gamma_\nu between their post-Newtonian γ\gamma parameters. Using the two well-identified neutrino–blazar associations IC-170922A/TXS 0506+056 and IC-211208A/PKS 0735+178 and the gravitational potential of the Laniakea supercluster, Wang et al. (2026) placed upper limits of order 10−810^{-8}–10−610^{-6} on Δγ\Delta\gamma. The open problem, stated in that work and unresolved in the literature, is that these bounds can only be conservative upper limits, not systematics-controlled constraints, because (i) the intrinsic emission-time offset between neutrinos and gamma rays at the blazar source is not independently constrained by any observation, forcing every measured time difference to be treated as an upper limit on the propagation delay, and (ii) source distances inferred from redshifts are uncertain — most severely for PKS 0735+178, whose redshift is bracketed only between roughly 0.42 and 0.65 — and enter the Shapiro delay directly. The problem is to determine whether, and by what route — physical modeling of neutrino and gamma-ray emission in blazar jets, statistical treatment of a population of associations, improved timing, or better distance determinations — the WEP limit for neutrinos versus photons can be tightened beyond the current conservative values and extended to future facilities such as KM3NeT and CTAO.

Background

Einstein's Weak Equivalence Principle (WEP) — the universality of free fall — states that any freely falling test body follows a trajectory independent of its internal structure and composition. It is a pillar of general relativity and all metric theories of gravity. In the parametrized post-Newtonian (PPN) framework, which expands metric theories of gravity in powers of the gravitational potential and encodes deviations from general relativity in a set of dimensionless parameters, the parameter γ\gamma measures the amount of spacetime curvature produced per unit rest mass (general relativity predicts γ=1\gamma=1 for all species). A species-dependent γ\gamma would mean different species fall differently, violating the WEP.

For relativistic messengers the classic test uses the Shapiro delay: the extra propagation time a signal accumulates when traversing a gravitational potential, a firm prediction of general relativity. If photons and neutrinos had different γ\gamma parameters, their Shapiro delays from the same source would differ by a fraction Δγ=γγ−γν\Delta\gamma=\gamma_\gamma-\gamma_\nu, so an observed arrival-time offset Δtobs\Delta t_{\rm obs} between a neutrino and a photon from the same event implies ∣Δγ∣≲Δtobs/ΔtS|\Delta\gamma|\lesssim\Delta t_{\rm obs}/\Delta t_{\rm S}, where ΔtS\Delta t_{\rm S} is the (general-relativistic) Shapiro delay along the line of sight. This method was first applied to the MeV neutrinos and photons from supernova SN1987A, and was sharpened when IceCube — the cubic-kilometer neutrino telescope at the South Pole — began detecting TeV–PeV astrophysical neutrinos in spatial and temporal coincidence with flaring blazars (blazars are active galactic nuclei whose relativistic jet points nearly at us). Wang, Liu & Wang (2016) used the tentative association of a PeV IceCube neutrino with a giant flare of the blazar PKS B1424-418 to constrain Δγ\Delta\gamma to an accuracy of at least 10−510^{-5}; Wei et al. (2019) applied the same logic to the neutrino IceCube-170922A and the flaring blazar TXS 0506+056, computing the Shapiro delay in the potential of the Laniakea supercluster (the galaxy supercluster, roughly 160 Mpc across, that contains the Milky Way) and obtaining 10−610^{-6}–10−710^{-7}; Boran, Desai & Kahya (2019) analyzed the same event with a different gravitational-potential model and quoted a far weaker bound, illustrating how strongly the result depends on the assumed potential. Wang et al. (2026) extended the analysis to the neutrino IceCube-211208A and the blazar PKS 0735+178, reaching Δγ≤6.13×10−8\Delta\gamma\leq6.13\times10^{-8} for an assumed 4-day neutrino–photon offset — the tightest neutrino-versus-photon WEP limit to date.

All of these results share a structural limitation, emphasized by Wang et al. (2026) and already present in Wei et al. (2019): the observed arrival-time difference is a single number that conflates propagation effects with an intrinsic emission-time offset at the source. Neutrinos in blazars are expected to be produced in hadronic processes inside the jet, while gamma rays may come from different regions or zones of the same outflow, and nothing in current observations pins down the moment of neutrino emission relative to the gamma-ray flare. With no independent handle on this intrinsic lag, the measured Δtobs\Delta t_{\rm obs} can only be treated as an upper limit on the propagation delay, so the resulting Δγ\Delta\gamma bound is necessarily conservative. A second systematic enters through the distance: the Shapiro delay scales with the distance traveled, and blazar redshifts — especially for BL Lac objects, whose optical spectra are nearly featureless — are hard to secure. For PKS 0735+178 the literature provides only a lower limit z≥0.424z\geq0.424, an imaging-based estimate z=0.45±0.06z=0.45\pm0.06, and a group-of-galaxies-based estimate z∼0.65z\sim0.65. A third, methodological complication has been raised by Minazzoli, Johnson-McDaniel & Sakellariadou (2019), who argue that the standard computation of Shapiro-delay-based equivalence-principle tests is not well posed on cosmological scales: with the usual convention that the Newtonian potential vanishes at infinity, the delay diverges as more and more remote sources of the gravitational field are included, and no conservative lower bound on the delay follows from considering only a subset of the sources (such as Laniakea alone) without further assumptions. Most recently, Britzen et al. (2026) presented evidence that PKS 0735+178 is gravitationally lensed by an intervening system, which, if confirmed, would add further differential-timing systematics for this source. Meanwhile, equivalence-principle tests between photons of different frequencies — for example with fast radio bursts, where plasma dispersion provides an independent handle on the frequency-dependent delay — reach parts of 10−2110^{-21} in Δγ\Delta\gamma; the neutrino–photon sector lacks any comparable handle, which is precisely why it remains stuck at conservative 10−810^{-8}-level limits.

Problem Statement

Determine whether the conservative upper limits on the WEP-violation parameter Δγ=γγ−γν\Delta\gamma=\gamma_\gamma-\gamma_\nu for neutrinos versus photons — currently at the 6.13×10−86.13\times10^{-8} to 3.11×10−63.11\times10^{-6} level from the IceCube neutrino–blazar associations TXS 0506+056 and PKS 0735+178 — can be tightened into systematics-controlled constraints, and to what precision. As stated by Wang et al. (2026), this requires overcoming the two unconstrained effects that presently force the bounds to remain conservative: (i) the intrinsic emission-time offset between neutrinos and gamma rays at the blazar source, for which no observational guidance exists, so that observed arrival-time differences can only be interpreted as upper limits on the propagation delay; and (ii) the uncertainty in source distances derived from redshift measurements (for PKS 0735+178, bracketed only between roughly z=0.424z=0.424 and z∼0.65z\sim0.65), which enters the computed Shapiro delay directly. A complete answer would establish, with explicit physical reasoning or statistical treatment, either an improved bound on Δγ\Delta\gamma in which the intrinsic-offset and distance contributions are bounded, modeled, or marginalized rather than assumed maximal, or a quantitative demonstration that current associations cannot support such a tightening; because such a bound presupposes that the computed Shapiro delay is itself a controlled lower bound, a complete answer must also state explicitly how it treats the further complications raised in the Background — the cosmological well-posedness of the Shapiro-delay convention (Minazzoli et al. 2019) and the possible gravitational lensing of PKS 0735+178 (Britzen et al. 2026) — whether as additional systematics to be controlled or as premises assumed under the convention of Wang et al. (2026). It should also assess whether the same precision can be extended to future high-energy neutrino and gamma-ray facilities such as KM3NeT and CTAO.

References

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