The rest mass of the photon, mγm_\gamma, can be bounded from the frequency-dependent arrival delays of fast radio bursts (FRBs), millisecond radio transients observed at cosmological distances. The obstacle is exact: the delay induced by dispersion in line-of-sight free electrons and the delay induced by a hypothetical photon mass both scale as νl−2−νh−2\nu_{\rm l}^{-2}-\nu_{\rm h}^{-2}, so for any single burst the two effects are perfectly degenerate, and every uncertainty in the Milky Way, intergalactic-medium, and host-galaxy electron columns, or in intrinsic source emission lags, propagates directly into the mass bound. The founding analysis (Wu et al. 2016, on FRB 150418) derived mγ≤5.2×10−47m_\gamma \leq 5.2\times 10^{-47} g under the assumption that the entire observed delay is photon-mass-induced, while explicitly flagging these systematics. A decade of follow-up work, now using Bayesian analyses of tens of well-localized FRBs with spectroscopic redshifts and simulation-based electron-column priors, has tightened the FRB-kinematic limit only to a few ×10−51\times 10^{-51} kg (about 10−1510^{-15} eV/c2c^2); Monte Carlo studies show the bound saturating because intergalactic dispersion uncertainties dominate, the anticorrelation between the intergalactic-medium baryon fraction fIGMf_{\rm IGM} and mγm_\gamma persists in the largest samples, and the one proposed in-plasma separation channel (a ν−4\nu^{-4} subleading term) was refuted from the Proca equations (the field equations of massive electrodynamics). The problem - separating the photon-mass contribution from plasma dispersion and intrinsic delays so that FRB data either yields a tighter bound, a rigorous quantified floor, or an unambiguous attribution of the residual to other physics - remains open.

Background

Fast radio bursts (FRBs) are flashes of radio emission lasting milliseconds, originating in galaxies at cosmological distances; thousands have now been detected and more than one hundred have been localized to identified host galaxies. Because an FRB propagates over cosmological distances, its arrival time depends on the observing frequency in two ways that matter here. First, free electrons along the line of sight disperse the pulse: in a cold plasma the group velocity falls below the speed of light by an amount proportional to the electron density divided by the square of the wave frequency, so the pulse observed at the lower frequency νl\nu_{\rm l} arrives after the one at νh\nu_{\rm h} by a delay Δtplasma=(e2/2πmec) DM (νl−2−νh−2)\Delta t_{\rm plasma}=(e^2/2\pi m_{\rm e}c)\,{\rm DM}\,(\nu_{\rm l}^{-2}-\nu_{\rm h}^{-2}), where DM=∫ne dl{\rm DM}=\int n_{\rm e}\,dl is the dispersion measure, the column density of free electrons in units of pc cm−3^{-3}, contributed by the Milky Way, the intergalactic medium (IGM, the diffuse ionized gas between galaxies), and the FRB host galaxy. Second, if the photon has a nonzero rest mass - Maxwell electromagnetism generalized to a massive gauge field is called Proca theory - the photon group velocity in vacuum becomes frequency dependent, giving, for a source at redshift zz in the standard flat Λ\LambdaCDM cosmology (cold dark matter plus a cosmological constant, with dimensionless Hubble function E(z)E(z)), a delay Δtmγ=[(mγc2/h)2/(2H0)](νl−2−νh−2)H1(z)\Delta t_{m_\gamma}=\left[(m_\gamma c^2/h)^2/(2H_0)\right]\left(\nu_{\rm l}^{-2}-\nu_{\rm h}^{-2}\right)H_1(z) with H1(z)=∫0z(1+z′)−2 dz′/E(z′)H_1(z)=\int_0^z (1+z')^{-2}\,dz'/E(z') and H0H_0 the present-day Hubble rate.

Both delays scale identically with frequency, (νl−2−νh−2)(\nu_{\rm l}^{-2}-\nu_{\rm h}^{-2}), so for a single FRB the plasma and photon-mass contributions are exactly degenerate: attributing the whole residual delay to the photon mass gives an upper limit on mγm_\gamma that is only as good as the independent estimate of the electron column. The founding observation of this problem is Wu et al. (2016, Astrophys. J. Lett. 822, L15), who used the 0.815 s delay of FRB 150418 between 1.2 and 1.5 GHz and its then-claimed redshift z=0.492z=0.492 to derive mγ≤5.2×10−47m_\gamma \leq 5.2\times 10^{-47} g (about 2.9×10−142.9\times 10^{-14} eV/c2c^2), the best bound from light propagation (as opposed to laboratory or magnetostatic methods) at the time, while noting that the host-galaxy electron column is hard to know, that the IGM estimate assumed homogeneous fully ionized hydrogen and helium, that intrinsic source lags and possible Lorentz-invariance-violation effects could contribute to the delay, and that larger, well-localized samples were needed. The companion analysis of Bonetti et al. (2016) showed that accounting for Milky Way, host, and IGM modeling uncertainties weakens the FRB 150418 limit to mγ≲1.8×10−14m_\gamma \lesssim 1.8\times 10^{-14} eV/c2c^2, and the claimed host of FRB 150418 was subsequently shown to be consistent with variability of the active galactic nucleus (AGN, the bright nucleus of the claimed host galaxy) rather than a fading afterglow (Williams & Berger 2016), so later samples use only securely localized FRBs.

The one statistical handle on the degeneracy is that the two contributions grow differently with redshift: the IGM dispersion measure grows roughly linearly with zz (the observed DM-redshift relation), while the photon-mass factor H1(z)H_1(z) grows much more slowly, so a joint fit to many FRBs with measured redshifts can in principle separate them. This route has been implemented with progressively larger samples - nine localized FRBs (2020), 129 FRBs (2021), 23 localized FRBs with galaxy-simulation-based priors for the host and IGM dispersion measures (2023) - and the resulting 68% confidence upper limits have plateaued at (2−5)×10−51(2-5)\times 10^{-51} kg (roughly (1−3)×10−15(1-3)\times 10^{-15} eV/c2c^2). For context, the Particle Data Group lists the solar-wind magnetohydrodynamic limit mγ<10−18m_\gamma < 10^{-18} eV as the tightest model-dependent bound and lists the FRB limits separately, noting the modeling uncertainties; the FRB limits are the tightest obtained purely from the kinematics of light propagation.

Problem Statement

Using the observed dispersion measures, frequency-dependent arrival delays, and (where available) spectroscopic redshifts of fast radio bursts, determine how tightly the photon rest mass mγm_\gamma can be bounded once the degeneracy between plasma dispersion and photon-mass dispersion is confronted directly. A complete answer must do one of the following, with all systematics quantified: (i) derive an upper limit on mγm_\gamma tighter than the current FRB-kinematic plateau of a few ×10−51\times 10^{-51} kg (roughly 10−1510^{-15} eV/c2c^2 at 68% confidence), with the Milky Way, intergalactic-medium, and host-galaxy dispersion-measure contributions and possible intrinsic emission lags treated as uncertainty sources rather than fixed by assumption; (ii) demonstrate rigorously, analytically or by simulation, that the intergalactic and host electron-column uncertainties and the exact ν−2\nu^{-2} frequency degeneracy impose a quantified floor on any FRB-dispersion bound; or (iii) robustly attribute the residual, photon-mass-compatible dispersion to specific alternative physics (for example intrinsic source lags) at stated confidence. Any answer must state explicitly which observable or statistical handle performs, or cannot perform, the separation of the photon-mass term from the plasma term - for instance the different redshift dependences of the two contributions across a sample of localized FRBs, a new observable such as lensing time delays, or an independent electron-column constraint - and whether that separation is or is not absorbed by the degeneracy with the intergalactic baryon fraction, the host dispersion measure, or intrinsic lags. These deliverables are alternative completions of one objective, the robust determination or bounding of the photon mass from FRB dispersion, not independent subproblems.

References

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