Photon Mass Limits from Fast Radio Bursts

The frequency-dependent time delays in fast radio bursts (FRBs) can be used to constrain the photon mass, if the FRB redshifts are known, but the similarity between the frequency dependences of dispersion due to plasma effects and a photon mass complicates the derivation of a limit on $m_\gamma$. The dispersion measure (DM) of FRB 150418 is known to $\sim 0.1$%, and there is a claim to have measured its redshift with an accuracy of $\sim 2$%, but the strength of the constraint on $m_\gamma$ is limited by uncertainties in the modelling of the host galaxy and the Milky Way, as well as possible inhomogeneities in the intergalactic medium (IGM). Allowing for these uncertainties, the recent data on FRB 150418 indicate that $m_\gamma \lesssim 1.8 \times 10^{-14}$eV c$^{-2}$ ($3.2 \times 10^{-50}$kg), if FRB 150418 indeed has a redshift $z = 0.492$ as initially reported. In the future, the different redshift dependences of the plasma and photon mass contributions to DM can be used to improve the sensitivity to $m_\gamma$ if more FRB redshifts are measured. For a fixed fractional uncertainty in the extra-galactic contribution to the DM of an FRB, one with a lower redshift would provide greater sensitivity to $m_\gamma$.

When setting an upper limit on the photon mass, the Particle Data Group (PDG) [1] cites the outcome of modelling the solar system magnetic field: first at 1 AU, m γ < 5.6 × 10 −17 eV c −2 (= 10 −52 kg) [2,3], and later at 40 AU, m γ < 8.4 × 10 −19 eV c −2 (= 1.5 × 10 −54 kg) [2]. However, the laboratory upper limit is four orders of magnitude larger [4]; for reviews see [5,6]. In [6], the authors state the concern that "Quoted photonmass limits have at times been overly optimistic in the strengths of their characterizations. This is perhaps due to the temptation to assert too strongly something one 'knows' to be true". This concern was mainly addressed to the galactic magnetic field model limits [7], but it should be borne in mind also when assessing the solar system limits.
Indeed, the estimates on the deviations from Ampère's law in the solar wind [2,3] are not based simply on in situ measurements. For example: (i) the magnetic field is assumed to be exactly, always and everywhere a Parker spiral; (ii) the accuracy of particle data measurements from, e.g., Pioneer or Voyager, has not been discussed; (iii) there is no error analysis, nor data presentation, instead; (iv) there is extensive use of a reductio ad absurdum approach based on earlier results of other authors, which are often devoted to other issues than establishing a basis for an extremely difficult measurement of a mass that is many orders of magnitude lower than that of an electron or a neutrino.
In order to check these estimates of the solar wind at 1 AU, a more experimental approach has been pursued via a thorough analysis of Cluster data [8], leading to a mass upper limit lying between 1.4 × 10 −49 and 3.4 × 10 −51 kg, according to the estimated potential. The difference between the results of this conservative approach and previous estimates, as well as the need for astrophysical modelling, motivate the development of additional methods for constraining the photon mass.
The time structures of electromagnetic emissions from astrophysical sources at cosmological distances have been used to constrain other aspects of photon/electromagnetic wave propagation, such a possible Lorentz-violating energy/frequency dependence of the velocity of light in vacuo [9][10][11][12][13], and the possibility of dispersion in photon velocities of fixed energy/frequency, as suggested by some models of quantum gravity and space-time foam [14,15]. Similarly, the gravitational waves recently observed by Advanced LIGO from the source GW150914 have been used to constrain aspects of graviton/gravitational wave propagation, including an upper limit on the graviton mass: m g < 1.2 × 10 −22 eV c −2 (= 2.1×10 −58 kg) [16,17] and limits on Lorentz violation [18,19], and the possible observation by Fermi of an associated γ-ray pulse [20] suggests that light and gravitational waves have the same velocities to within 10 −17 [18,21].
The time structures of electromagnetic emissions from astrophysical sources at cosmological distances can also be used to derive an upper limit on the photon mass, m γ .
Since the effect of the photon mass on the velocity of light is enhanced at low frequency ν (energy E): ∆v ∝ −m 2 γ c 4 /h 2 ν 2 (−m 2 γ c 4 /E 2 ), measurements of time structures at low frequency or energy are particularly sensitive to m γ . For this reason, measurements of short time structures in radio emissions from sources at cosmological distances are especially powerful for constraining m γ . This is to be contrasted with probes of Lorentz violation, for instance, where measurements of high-energy photons such as γ rays are at a premium. This is why probes of the photon mass using gamma-ray bursters (GRBs) [22] and active galactic nuclei (AGNs) have not been competitive in constraining m γ . As we mention later, a stronger limit can be obtained by using the apparent coincidence of a radio afterglow with a GRB, but this is also not competitive with the sensitivity offered by fast radio bursts (FRBs).
FRBs are potentially very interesting because their radio signals have well-measured time delays that exhibit the 1/ν 2 dependence expected for both the free electron density along the line of sight and mass effects on photon propagation. Until recently, the drawback was that no FRB had had its redshift measured, though there was considerable evidence that they occurred at cosmological distances. This has now changed with FRB 150418 [23], which has been reported to have occurred in a galaxy with a well-measured redshift z = 0.492 ± 0.008. The identification of its host galaxy has been questioned, and the alternative possibility of a coincidence with an AGN flare has been raised [24], though the likelihood of this is currently an open question [25]. In the following we assume the host galaxy identification made in [23], and also discuss more generally how non-galactic FRBs could be used to constrain photon propagation.
The frequency-dependent time lag of FRB 150418 between the arrivals of pulses with ν 1 = 1.2 GHz and ν 2 = 1.5 GHz is ∆t FRB 12 ≈ 0.8 s, and was used in [23] to extract very accurately the dispersion measure (DM), which is given in the absence of a photon mass by the integrated column density of free electrons along the propagation path of a radio signal, n e dl. The delay of an electromagnetic wave with frequency ν propagating through a plasma with an electron density n e , relative to a signal in a vacuum, makes the following frequency-dependent contribution to the time delay [26,27] where ν p = (n e e 2 /πm e ) 1/2 = 8.98 · 10 3 n 1/2 e Hz (cgs units). As is discussed in [23], plasma effects with DM = 776.2(5) cm −3 pc could be responsible for the entire ∆t FRB 12 that was measured 1 . There are contributions to the DM of this extragalactic object from the free electron density in the host galaxy, estimated to be ∼ 37 cm −3 pc, from the Milky Way and its halo, estimated to be 219 cm −3 pc, and the intergalactic medium (IGM).
Subtracting the other contributions, the IGM contribution to the DM was estimated to be 520 cm −3 pc, with uncertainties ∼ 38 cm −3 pc from the modelling of the Milky Way using NE2001 [28] 2 and ∼ 100 cm −3 pc from inhomogeneities in the IGM. The DM IGM contribution to the dispersion delay (1) for a source at red shift z can be expressed in terms of the density fraction Ω IGM of ionized baryons [26]: where H 0 is the present Hubble expansion rate, G is the Newton constant, m p is the proton mass, and the factor takes proper account of the time stretching in (1) and evolution of the free-electron density due to the cosmological expansion [10,26,27,30]. The relation (2) was used in [23] to estimate the density of ionized baryons in the IGM: Ω FRB IGM = 0.049 ± 0.013, assuming that the helium fraction in the IGM has the cosmological value of 24%. We also assume that the present cosmological constant density fraction Ω Λ = 0.714 and the present matter density fraction Ω m = 0.286, and set the reduced Hubble expansion rate, [31]. This measurement of Ω IGM is quite compatible with the density expected within standard ΛCDM cosmology [31]: Ω ΛCDM IGM = 0.041±0.002. The measurement of ∆t FRB 12 can also be used to constrain the photon mass. For this purpose, we note that the difference in distance covered by two particles emitted by an object at a red shift z with velocity difference ∆u is In case of the cosmological propagation of two massive photons with energies E 2 > E 1 the velocity difference is where the red shifts of the photon energies are taken into account and we use units: = c = k = 1. Thus, difference in arrival times of two photons of different energies 2 For limitations of NE2001, see [29].
from a remote cosmological object due to a non-zero photon mass can be parametrized as follows: where and we include in (6) the contribution ∆t DM to the time delay due to plasma effects and a possible, generally unknown, source time lag b sf in the source frame. Inverting (6) and transforming to experimental units F GHz ( ν 1 1GHz , ν 2 1GHz ) and expressing all time measurements in seconds we arrive at The most conservative bound would be obtained if the entire DM of FRB 150418 were due to m γ = 0, i.e., ∆t lag ∆t FRB 12 , ∆t DM = 0 and b sf = 0 in (8). However, this approach is probably too conservative, and a very reasonable assumption would be to subtract from the DM FRB IGM the IGM contribution corresponding to Ω ΛCDM IGM . In this case, since the 95% CL estimate of the IGM dispersion measure is DM FRB IGM(2σ) 520 ± (2 · 138) cm −3 pc [23], one should assume, according to (2) and (1), that ∆t lag 0.82 s at the 95% CL, ∆t DM ≈ 0.45 s and b sf = 0 in (8). In this case, one would find at the 95% CL. 3 These bounds are much stronger than those obtained from GRBs [22] and AGNs, and are getting within shouting distance of the PDG limit [1][2][3]. We regard this as the most reasonable interpretation of the data on FRB 150418.
The question then arises, how much the FRB limit could be improved in the future?
The DM of FRB 150418 has been measured with an accuracy of 0.1%, but the uncertainties in subtracting the contributions from the host galaxy, the IGM and the Milky Way amount to > 20%. In particular, uncertainties associated with inhomogeneities in the IGM approach 20%, dwarfing uncertainties associated with Ω IGM , which approach 3 Similar bounds were given in [32], which we received while working on this paper. 5%, and in modelling the Milky Way [28,29], which exceed 5%. We doubt that the corresponding uncertainties for other FRBs could soon be reduced to the 0.1% level of the FRB 150418 DM measurement, and consider that a plausible objective may be to constrain the sum of DM IGM and a possible photon-mass effect for any given FRB with an accuracy of 10%. 4 One way to improve the sensitivity to m γ may be to use data from FRBs at different redshifts. As we discuss below, the relative contributions of the IGM and a photon mass vary with the redshift z, and the sensitivity to m γ is greater for FRBs with smaller redshifts. A hypothetical 10% measurements of the non-host and non-Milky Way contributions to the DM of a FRB with z = 0.1 would yield a prospective sensitivity to m γ = 6.0 × 10 −15 eV c −2 (1.1 × 10 −50 kg).
As already commented, the frequency dependences of the IGM and m γ effects, Eqs.
(1) and (8), are similar, but the degeneracy between them is broken by the different z dependences of H e (3) and H γ (7). In particular, we note the m γ effect gains in relative more importance at smaller z because of the difference between the powers of (1 + z ) in the integrands of H e and H γ . In practice, if in the future a statisticaly relevant sample of FRBs at different redshifts is observed one might use the parametrization (6) to recover the intrinsic time lag of every source i from the sample as Assuming identical origins for the FRBs, one could optimize the set of b i sf with respect to a i γ and Ω i IGM (∆t i DM ), separating the non-zero photon mass contribution out from the plasma effect. The optimization can be performed on a basis of some estimator: a simple one could be just a minimization of the RMS of b i sf . 5 As discussed above, we consider that future measurements of the non-host galaxy and non-Milky Way contributions to the DMs of other FRBs at the 10% level may be feasible objectives. Accordingly, we have made a first assessment of their possible future impacts on the photon mass limit. Figure 1 displays an (m γ , Ω IGM ) plane, featuring as a thin horizontal band the ΛCDM expectation that Ω ΛCDM IGM . The other curves have the forms that follows from (8), where A is a numerical pre-factor determined by the factor H γ (z) of an object, the term B represents an observed time lag in terms of intergalactic DM B = (103.1 s) · DM obs IGM 10 5 pc cm −3 (13) and C defines the fraction of an actual contribution of the ionized plasma effect to the observed time lag relative to the prediction of the standard ΛCDM model for a given object The curves in Figure 1 assume an ionization fraction 0.9 but allow Ω IGM to be a free parameter. The curved grey shaded band shows the FRB 150418 constraint discussed above, at the 68% CL, which implies A = 2.96 · 10 −14 eV · s −1/2 , DM obs IGM = DM FRB IGM and ∆t IGM = 0.45 s. The intersection of this band with the Ω IGM = 0 axis corresponds to the (overly?) conservative 95% CL limit (9) and its intersection with the ΛCDM band for Ω IGM corresponds to the 'reasonable' 95% CL bound (10).
We also see that consistency between a 'high' measurement from an FRB with z = 0. For completeness, we mention another way to bound m γ using radio emissions, namely by comparing the the arrival time of radio afterglow and γ-ray emission from a GRB.
The most promising example seems to be GRB 071109 which was observed [33] to exhibit a radio afterglow at 8.46 GHz about 0.03 d after its γ-ray emission. 8 Although the redshift of this GRB was not measured, assuming that its redshift lies within the range z ∈ [0.1, 5], we find an upper limit on the photon mass m γ 2.8 × 10 −11 eV c −2 ( = 5.0 ×10 −47 kg). 9 The weakness of the limit compared to the FRB limit discussed earlier is due to the much larger time delay before the observation of the radio afterglow. Whilst this limit is not competitive with the FRB limit given above or the limit currently quoted by the PDG, this GRB afterglow method has the interest of involving a different type of astrophysical modelling. Moreover, it has potential for future improvement, e.g., if one could use lower-frequency waves and/or observe an afterglow sooner after the parent GRB, and particularly if time structure in the radio emissions analogous to those in the γ-ray emissions could be detected.
We finish our discussion with come comments and speculations. The present lack of redshift measurements for other FRBs is an obstacle for obtaining a more robust upper bound on the photon mass. However, one could also reverse the logic used above for FRB 150418 and, assuming the expected cosmological density of the IGM and the upper limit on the photon mass derived from FRB 150418, estimate the redshifts of other observed FRBs. Their redshift distribution might help pin down their origins.
Another option would be to use gravitational lensing, which would become frequency dependent in the presence of a photon mass [5]. The lensing is independent of the distance from the source, and a photon of mass m γ and energy E from a source of mass M would be gravitationally deflected by an angle θ = 4M G , for a photon of energy E (or frequency ν = E/h), where R is the size of the celestial body and G is the gravitational constant. In [5], the photon-mass deflection ∆θ was set equal to the difference between the value observed for some celestial object, e.g., the Sun, and the standard theoretical case for massless photon, thereby obtaining an upper bound m γ hνc −2 2∆θ/θ 0 , where θ 0 = 4M G R c 2 is the standard massless photon deflection. Limits of the order of m γ 10 −44 kg can be obtained this way. Conversely, using upper bounds of the photon mass obtained from other methods like the FRBs discussed here would remove one uncertainty in the predictions for expected deflection angles, sharpening the use of comparisons with observations to constrain better the properties of lensing objects.