A Bayesian Framework to Constrain the Photon Mass with a Catalog of Fast Radio Bursts

A Bayesian Framework to Constrain the Photon Mass with a Catalog of Fast Radio Bursts

Lijing Shao Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam D-14476, Germany    Bing Zhang Department of Physics and Astronomy, University of Nevada, Las Vegas, NV 89154, USA Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China Department of Astronomy, School of Physics, Peking University, Beijing 100871, China

A hypothetical photon mass, , gives an energy-dependent light speed in a Lorentz-invariant theory. Such a modification causes an additional time delay between photons of different energies when they travel through a fixed distance. Fast radio bursts (FRBs), with their short time duration and cosmological propagation distance, are excellent astrophysical objects to constrain . Here for the first time we develop a Bayesian framework to study this problem with a catalog of FRBs. Those FRBs with and without redshift measurement are both useful in this framework, and can be combined in a Bayesian way. A catalog of 21 FRBs (including 20 FRBs without redshift measurement, and one, FRB 121102, with a measured redshift ) give a combined limit , or equivalently (, or equivalently ) at 68% (95%) confidence level, which represents the best limit that comes purely from kinematics. The framework proposed here will be valuable when FRBs are observed daily in the future. Increment in the number of FRBs, and refinement in the knowledge about the electron distributions in the Milky Way, the host galaxies of FRBs, and the intergalactic median, will further tighten the constraint.

14.70.Bh, 41.20.Jb, 52.25.Os, 95.85.Bh
\stackMaththanks: lijing.shao@aei.mpg.dethanks: zhang@physics.unlv.edu

I Introduction

The special relativity postulates the “constancy of light speed” as a fundamental principle in physics Einstein (1905). It is extended into the general relativity and quantum field theories. In quantum mechanics, the particle-wave duality translates the constancy of light speed into the “masslessness of photons” Weinberg (2005). Nevertheless, there exist theories with massive photons. The Maxwell-de Broglie-Proca theory is a famous example where photons obtain mass in the cost of the gauge invariance Proca (1936); de Broglie (1922). Another example where “effectively massive photons” arise due to a possible oscillation between the canonical U(1) photons and hypothetical U(1) “photons” Georgi et al. (1983). Despite the celebrated success of the postulate, these scenarios are interesting and worthy to investigate, whereas the ultimate word on the photon mass roots in empirical facts.

Great efforts were put to various kinds of experiment to push the empirical boundary on the “masslessness of photons” Goldhaber and Nieto (2010); Spavieri et al. (2011). These tests start from early experiment in testing the Coulomb law Williams et al. (1971), to Schumann resonance in cavity Kroll (1971), gravitational deflection of electromagnetic waves Lowenthal (1973), mechanical stability of magnetized gas in galaxies Chibisov (1976), Jupiter’s magnetic field Davis et al. (1975), toroid Cavendish balance Lakes (1998); Luo et al. (2003), magneto-hydrodynamics of the solar wind Ryutov (1997, 2007), black hole bombs Pani et al. (2012), and spindown of a white-dwarf pulsar Yang and Zhang (2017). These tests involve the anomalous dynamics introduced by the mass term of photons, while there exist cleaner tests which only involve the anomalous kinematics introduced by the mass Lovell et al. (1964); Wu et al. (2016); Bonetti et al. (2016, 2017), thus independent of the underlying massive photon theory. The latter kind of tests use the propagation of photons through a cosmological distance, and examine the time delay between photons with different energies. In this paper we will study the empirical mass constraint on the photon mass from the propagation of electromagnetic waves of fast radio bursts (FRBs) Lorimer et al. (2007); Thornton et al. (2013); Champion et al. (2016).

If photon has a mass , the Lorentz-invariant dispersion relation reads,


where is the limiting velocity for high energy photons. The group velocity for a photon with energy is,


where the last derivation holds when . It is evidently clear from Eq. (2) that when the energy of a photon is smaller, the relative modification is larger.111This is opposite to the test of Lorentz invariance violation with light propagation Amelino-Camelia et al. (1998); Mattingly (2005); Shao et al. (2010), where high energy photons are preferred to put constraints. Because we here study the test that involves the accumulative time delay from light propagation, it is easy to understand that (i) finer time structure of the event, (ii) longer propagation distance, and (iii) lower energy for photons, define the figure of merit of the test. With this setting, FRBs provide a superb celestial laboratory for testing the photon mass, because —

  1. they are very short and even not temporally resolved by the receivers in general, and

  2. they are believed to be cosmological objects with non-negligible redshifts, and

  3. they are observed at radio frequency, which leverages the smallness of in Eq. (2).

The first work using FRBs to constrain the photon mass, performed by Wu et al. (2016) and Bonetti et al. (2016), used FRB 150418 Keane et al. (2016). However, the measurement of the redshift for this FRB was challenged with a flare in a radio-variable active galactic nucleus Williams and Berger (2016); Vedantham et al. (2016), and now the measurement is generally believed to be unreliable Chatterjee et al. (2017). A reliable measurement of the redshift was made for FRB 121102 Chatterjee et al. (2017); Tendulkar et al. (2017), and Bonetti et al. (2017) followed up the measurement to constrain the photon mass to be , or equivalently, . The method proposed in these papers needs a measurement of the redshift for FRBs, however, up to now, only one FRB is fortunate enough to identify the host galaxy and gets a redshift measurement. Since the localization of an FRB is facilitated if the source is repeating and since none of the other FRBs are observed to repeat so far, the sample of redshift-measured FRBs may not grow significantly in the near future Palaniswamy and Zhang (2017). Therefore, we here extend these work to FRBs where the redshift is not available. We construct a Bayesian formula to derive a combined constraint from a catalog of FRBs, where uninformative prior is made to the redshift. Figure 1 shows the sky distribution of FRBs that are used in this work (see also Table 3Petroff et al. (2016).

The paper is organized as follows. In the next section, the theoretical framework for data analysis is laid which includes a hypothesis for the -behaved time delay, and a Bayesian framework to constrain . In Section III we examine our uninformative treatment of the redshift with FRB 121102, and present the constraint that comes from a combination of a catalog of 21 FRBs where only one of them has a redshift measurement. Section IV summarizes the work and discusses future prospects in constraining the photon mass with FRBs.

FRB Telescope Sky position
010125 Burke-Spolaor and Bannister (2014) Parkes 0.77
010621 Keane et al. (2011) Parkes 0.27
010724 Lorimer et al. (2007) Parkes 0.38
090625 Champion et al. (2016) Parkes 0.98
110220 Thornton et al. (2013) Parkes 1.02
110523 Masui et al. (2015) GBT 0.66
110626 Thornton et al. (2013) Parkes 0.76
110703 Thornton et al. (2013) Parkes 1.20
120127 Thornton et al. (2013) Parkes 0.60
121002 Champion et al. (2016) Parkes 1.77
121102 Spitler et al. (2014) Arecibo, GBT 0.43
130626 Champion et al. (2016) Parkes 1.00
130628 Champion et al. (2016) Parkes 0.48
130729 Champion et al. (2016) Parkes 0.93
131104 Ravi et al. (2015) Parkes 0.80
140514 Petroff et al. (2015) Parkes 0.60
150418 Keane et al. (2016) Parkes 0.67
150807 Ravi et al. (2016) Parkes 0.23
160317 Caleb et al. (2017) UTMOST 0.95
160410 Caleb et al. (2017) UTMOST 0.26
160608 Caleb et al. (2017) UTMOST 0.51
Table 1: A catalog of FRBs333http://www.astronomy.swin.edu.au/pulsar/frbcat/ Petroff et al. (2016) that are used to constrain the photon mass. Sky position is given in right ascension, , and declination, , at vernal equinox of J2000.0 epoch. Dispersion measure is in unit of , where is from the fitting of the behavior in the frequency-dependent time delay, and is based on the NE2001 Galactic electron density model Cordes and Lazio (2002). is the upper limit on the true redshift, obtained by assuming that the excess dispersion measure, , entirely comes from the IGM; since here we consistently use the parameter given in the main text and the full expression of , their values are close to, but larger than, that given in the catalog Petroff et al. (2016).
Figure 1: Distribution of FRBs that are used in constraining the photon mass Petroff et al. (2016) in (upper) celestial coordinate, and (lower) galactic coordinate.

Ii Theoretical Framework

We review a hypothesis on the behavior observed in the time delay of FRBs in Section II.1, and then construct a Bayesian framework in Section II.2 to analyze the observed FRB data.

ii.1 A hypothesis on the time delay

Figure 2: The dependence of functions and on the redshift. Their function values evaluated at the maximum redshift, , for 21 FRBs in Table 3, are shown with circles.

Here we present the hypothesis of time delay for FRBs when photon has a mass,  Wu et al. (2016); Bonetti et al. (2016, 2017). The hypothesis, , will be used in the Bayesian inference in Section II.2.

From observations, all FRBs show an indisputable -dependent time delay,  Petroff et al. (2016); Katz (2016). In our hypothesis, we attribute the delay to two causes, (i) the propagation of electromagnetic wave through ionized median, and (ii) the mass term of photon. Some remarks come as follows —

  • The interaction between the propagating electromagnetic wave and the ionized median introduces a time delay, , for a photon with energy, , relative to a photon with an infinite energy Lorimer and Kramer (2012),


    where the plasma frequency with the number density of electrons, the charge of an electron, the mass of an electron, and the permittivity of free space. The dispersion measure is defined as the integral of the electron number density along the path, . In a cosmological setting, where is the redshift and is the electron number density in the rest frame Deng and Zhang (2014).

    In Eq. (3), different sources contribute to the dispersion measure,  Petroff et al. (2016); Wu et al. (2016); Bonetti et al. (2016, 2017), notably from the Milky Way, , from the intergalactic median (IGM), , and from the host galaxy, . Therefore, the total dispersion measure reads,


    where we have included contributions from the host galaxy and the near-source plasma (e.g. supernova remnant, pulsar wind nebula, HII region Yang et al. (2017)) collectively in . We will present in Section II.2 how different pieces in the above equation are treated in a Bayesian framework.

  • With Eq. (2), it is straightforward to show that, in the CDM universe, a photon with energy, , propagates slower relative to that with an infinite energy by Jacob and Piran (2008); Wu et al. (2016); Bonetti et al. (2016, 2017),


    where is the Hubble constant Ade et al. (2016), and is a dimensionless function of the source redshifit (see Figure 2),


    where the matter energy density , and the vacuum energy density  Ade et al. (2016). In deriving Eq. (5), we have assumed a flat universe with the curvature energy density , and the expansion of universe has been properly taken into account Jacob and Piran (2008); Wu et al. (2016).

In our hypothesis , the total time delay is,


The two terms in the above equation both depend on the frequency in the same way, therefore, it conforms to the observational fact that the total time delay . The observational dispersion measure, , is obtained from the fit of the behavior from the total time delay. In our hypothesis, it equals to,


where is given in Eq. (4), and we have denoted the “effective dispersion measure” caused by the non-vanishing photon mass as,


ii.2 A Bayesian framework

In Bayesian analysis, given data and a hypothesis (here Eq. 8), the posterior distribution of can be obtained by,


where is all other relevant prior background knowledge. In the above equation, given and , is the prior on , is the likelihood for the data, and is the model evidence.

We choose a uniform prior on in the range . The lower end is chosen because it corresponds to the ultimate upper limit that in principle we can probe in one observation due to the uncertainty principle of quantum mechanics and the finite age of our universe Barrow and Burman (1984), while the upper end is chosen because beyond which the approximation in Eq. (2) breaks down. Such a wide prior across 27 orders of magnitude in reflects our conservativeness.

To display the likelihood that we adopt in our calculation, we first investigate different contributions in Eq. (4) —

  • : albeit with uncertainties, there are electron distribution models for the Milky Way that incorporate different astrophysical observational results Cordes and Lazio (2002); Yao et al. (2017). We here use the NE2001 model. For different FRBs, is calculated from their line of sight. We assign a uncertainty to the value given by the NE2001 model to account for possible model inaccuracy.

    In principle, there is an additional contribution from the Galactic halo, which is not captured by the NE2001 model because pulsars in general do not probe this regime Cordes and Lazio (2002). The halo contribution is not easy to model, but in our case it could already have been included effectively in  Yang et al. (2017), which is obtained from the excess dispersion measure of FRBs, (see below). The large uncertainty that we assign to could also account for (at least part of) this unknown contribution.

  • : the dispersion measure due to the intergalactic medium (IGM) is given by Deng and Zhang (2014),


    where is the mass of a proton, is the baryonic matter energy density Ade et al. (2016), is its fraction to the IGM Fukugita et al. (1998), and


    where , , and the ionized fractions of IGM for hydrogen at  Fan et al. (2006) and for helium at  McQuinn et al. (2009). Therefore for , we have,


    where the effective energy density of ionized baryons  Ioka (2003); Inoue (2004); Deng and Zhang (2014). To be conservative, we associate a uncertainty to  Bonetti et al. (2016); Yang et al. (2017), in the hope that such a large uncertainty could account for, at least partially, the inhomogeneity of IGM along different line of sight. In Eq. (13), the dimensionless redshift function, , reads,


    In Figure 2 we depict and with cosmological parameters from the CDM model Ade et al. (2016). Worthy to mention that, Bonetti et al. (2016, 2017) pointed out that the different behavior of these two redshift functions might be able to break parameter degeneracy in testing the photon mass at the point when a handful measurements of redshift for FRBs become available. For now we leave this point to future work.

    To predict the contribution of , we need a redshift measurement, which is only available for FRB 121102 up to now Chatterjee et al. (2017).444We will not consider the redshift measurement for FRB 150418 Keane et al. (2016), which is challenged with a flare in an active galactic nucleus Williams and Berger (2016); Vedantham et al. (2016); Chatterjee et al. (2017). Nevertheless, the inclusion of this measurement into our framework is straightforward if the redshift measurement is proven genuine. Due to our lack of knowledge, we assume a prior for such that the prior for the FRB’s spatial distribution is uniform in the comoving spherical volume,


    where is the comoving distance to a source at redshift ,


    and is the maximum possible redshift value of redshift (see Table 3) by setting and in Eq. (8Petroff et al. (2016). Such a prior for will be denoted as uniform for where is the comoving spherical volume within redshift . In Section IV we will in addition present the results with a prior of that traces the star formation rate Yüksel et al. (2008), and confirm the robustness of results. For FRB 121102, the measured redshift will be used as the prior for (see Section III.1).

  • : From a relation, Yang et al. (2017) derived a statistical result, in the rest frame of FRB, under the assumption that the isotropic-equivalent luminosity of FRBs has a characteristic value. Simulations show that a Gaussian dispersion in still keeps the result valid Yang et al. (2017). The conclusion in Yang et al. (2017) is supported by the large scattering time of FRBs and the inferred from FRB 121102 Chatterjee et al. (2017); Yang et al. (2017). We will use this result in our estimation of , after multiplying it by a factor, , which takes the cosmological evolution into account. 555Strictly speaking, the obtained in Yang et al. (2017) uses the assumption . A global re-analysis that closely follows the MCMC simulations in that work but allowing a nonzero would be ideal, however, this goes beyond the scope of current work. Instead we perform the following simulation to assess the influence to our result from using the in Yang et al. (2017). Notice that, when is underestimated, the constraint on is more conservative. Therefore, we perform the most conservative simulation that artificially sets . We observe that, even under such an assumption, our results only change by a factor less than three. Consequently, the results in the paper are robust to possible changes in the value we adopt.

Finally the logarithm of likelihood is constructed as,


where indexes FRBs, is the dispersion measure obtained with the above listed prescriptions in Markov-chain Monte Carlo (MCMC) runs, includes all uncertainties added in quadratic (including uncertainties in , , , and ), and denotes the second term in Eq. (8). In writing Eq. (17), an assumption is made that the observations of different FRBs are independent.

Iii Results

Figure 3: Distribution of MCMC samples from FRB 121102 (left) in the - plane: without using the redshift measurement, and (right) in the - plane using the redshift measurement as the prior on  Chatterjee et al. (2017). The pink bands show the redshift measurement and its 1- uncertainty obtained by Chatterjee et al. (2017); the band of is too narrow to be visible in the left panel. The CL constraints on are depicted as dashed lines.
Figure 4: The cumulative posterior probability distributions on from FRB 121102 without using the redshift measurement (in blue) and using the redshift measurement (in green). The excluded values for at and CLs are shown with shadowed areas for the case where the redshift is used.
Figure 5: The MCMC samples in the - plane for a catalog of FRBs in Table 3 except FRB 121102. The excluded regions at CL are shadowed for each FRB.
Figure 6: The cumulative posterior probability distributions for 20 FRBs (the catalog of FRBs in Table 3 except FRB 121102) are shown in grey. The same quantity is shown in green for FRB 121102 (the same green curve in Figure 4). The combination of these 21 FRBs is given in red. The excluded values for at and CLs are shown with shadowed areas for the combination.

As said, we use MCMC techniques to explore the posterior in Eq. (10). Ideally, one would use the log-likelihood in Eq. (17) to simultaneously analyze all FRBs in one go, whereas here the computational cost would be very high due to the large dimensionality of the parameter space. The dimensionality equals to the number of FRBs (their redshifts) plus one (the photon mass ). We adopt a sub-optimal strategy where the posteriors of , from different individual FRBs, are combined after independent MCMC run is performed on each single FRB Lyons and Chapon (2017). This is not a down-graded choice because we know that the redshifts of different FRBs are unlikely to correlate with each other. Such an approach is also the strategy adopted in constraining the strong equivalence principle in Ref. Stairs et al. (2005), the local Lorentz invariance of gravity in Refs. Shao et al. (2013); Shao (2014a), and the parameterized tests of general relativity with the Advanced LIGO events in Ref. Abbott et al. (2016a).

We use the PYTHON implementation of an affine-invariant MCMC ensemble sampler Goodman and Weare (2010); Foreman-Mackey et al. (2013), emcee,666http://dan.iel.fm/emcee to explore the posterior distributions. This algorithm generally has better performance over the traditional MCMC sampling methods (e.g., the Metropolis-Hasting algorithm), as measured by the smaller autocorrelation time and fewer hand-tuning parameters Foreman-Mackey et al. (2013). We set up MCMC runs to investigate the pair for each FRB. As mentioned, the priors are uniform in and in . Each MCMC run samples the posterior distribution according to Eq. (10), with the log-likelihood given by Eq. (17), with 20 chains. For each FRB, samples are accumulated. The first half of the samples are discarded as the burn-in phase Brooks et al. (2011). We check the convergence of different chains with the Gelman-Rubin statistic Gelman and Rubin (1992),


where the estimate of the marginal posterior variance for each parameter (indices denote the -th posterior sample in the -th chain) is,


with the between-chain variance, , and the within-chain variance, ,


where and are respectively the number of chains (in our case ) and the number of samples per chain (in our case after discarding the burn-in samples). Our convergence test shows for all cases, indicating very good convergence in MCMC runs. The posteriors and constraints on are presented in the following subsections.

iii.1 Limit from FRB 121102

Because the redshift of FRB 121102 was measured to great precision in Refs. Chatterjee et al. (2017); Tendulkar et al. (2017), , we would like to compare the constraints on the photon mass with and without this measurement. By including the redshift measurement, we mean using a Gaussian prior for , centered around its measured value with a spread of the uncertainty.

In Figure 3 we show the samples returned by the MCMC sampler (after discarding the burn-in samples) in both cases for FRB 121102. We immediately see that if the samples are marginalized over the photon mass, priors on are more or less recovered in both cases. This means that the Bayesian framework proposed here does not add more information to the redshift, as it should not.

In Figure 4 we show the accumulative posterior probability on , marginalized over the redshift , for both cases. As we can see, the result from the use of redshift measurement is very close to the one that does not use the redshift. We read out, at a 95% confidence level,


when using an uninformative uniform prior on , and


when is used. The latter agrees well with the result presented by Bonetti et al. (2017) for this FRB with a less sophisticated method. The marginalized 1 D probability distribution on with the uninformative prior has a long tail which reflects our ignorance in the redshift. The ultimate closeness of the results in Eqs. (2223) is a bit coincident, but it also shows the reasonableness of the use of the uninformative prior.

iii.2 Limits from individual FRBs

Except for FRB 121102 discussed above, the other 20 FRBs in the catalog (see Table 3) unfortunately have no redshift measurement Petroff et al. (2016). Therefore, we can only rely on the uninformative priors. The distribution of MCMC samples are shown in Figure 5 for these FRBs. As in the FRB 121102 case with the uninformative prior, the distributions have long tails towards large . Especially for the FRBs with large (e.g., FRB 110703 and FRB 121002), a large is needed to account for part of the dispersion measure in when the redshift is very small, as expected. From their panels in Figure 5, we see that some regions with small and small have no support from MCMC runs. Because of the conservatively large uncertainties that we use in and in , these FRBs individually only constrain at at 95% confidence level, as shown by the shadowed regions in the figure. The uncertainties in these two terms dominate the test, hence, in terms of order of magnitude, all FRBs here have comparable constraints.

iii.3 Combined limit from a catalog of FRBs

We now have 21 individual constraints on . Assuming that these FRBs are independent, we can combine their posterior distributions, in the spirit of Eq. (10). Similar combination of posteriors was done in Refs. Stairs et al. (2005); Shao et al. (2013); Shao (2014a); Abbott et al. (2016a) under different subjects. Here, since for FRB 121102 a reliable redshift is available Chatterjee et al. (2017); Tendulkar et al. (2017), we use the result that takes this measurement into account. In Figure 6, we plot the marginalized accumulative posterior distributions for 20 FRBs (see Figure 5) in gray, that generally give at 95% confidence level, and the accumulative posterior distribution for FRB 121102 in green (same as the green curve in Figure 4), that gives at 95% confidence level (see Eq. 23). We also give the accumulative posterior distribution that combines the 21 FRBs with a red curve in the figure. In the Bayesian sense, it is unlikely that multiple FRBs reside in the long tails of their distributions. This result demonstrates the collective power of these “deceptively boring” FRBs that have no redshift measurement. It has strong implication for future study using FRBs to constrain the photon mass. The final combination of 21 FRBs (the red curve in Figure 6) give a tight constraint on ,


at 68% and 95% confidence levels respectively. These limits improve over previous results that only used a single FRB Wu et al. (2016); Bonetti et al. (2016, 2017) by a factor of .

Iv Discussions

Recently, the first direct observation of gravitational waves from a binary black hole merger at a cosmological distance by the Advanced LIGO puts a constraint on the graviton mass, , at 90% confidence level Abbott et al. (2016b). Because most of the power of the gravitational-wave event is at , even significantly lower than the radio waves we here use to constrain the photon mass, a tighter constraint on the mass is expected (cf. discussion below Eq. 2). Nevertheless, the constraints on the photon mass (see Eqs. 2425) pertain to a different sector of species. As far as we are aware, this is the tightest limit on the photon mass that is obtained solely depending on the propagation kinematics, therefore completely avoiding assumptions about the underlying dynamical theory for the massive photon.

The Bayesian framework proposed here will be even more valuable in future, when more and more FRB observations become available (for example, with the ALERT project777http://www.alert.eu.org/). The improvement could come from —

  1. new discovery of more FRBs;

  2. more coincident measurements of FRB redshift;

  3. a better understanding of the various astrophysical contributions to the observed dispersion measure, including those from the Milky Way, the IGM, and the host galaxies of FRBs.

If future FRBs are observed with the same quality as current ones, we expect a rough improvement on the photon mass where is the number of FRBs. The improvement from the measurement of the redshift is very hard to predict. It depends on the redshift value that is measured. For example, in the case of FRB 121102, the measured redshift resides in the lower end of its possible values up to . Taking the uniform prior in volume into account, one would expect a chance of approximately to observe a redshift as low as for FRB 121102.888For sources with low redshift, the comoving distance , thus the comoving volume . Even in such a case, the inclusion of the measured redshift does not provide a worse constraint compared with the case where we are uninformative about the redshift. Were the measured redshift larger, the constraint would be better. For now, the constraint on the photon mass is limited by our assumptions about the uncertainties from the Milky Way, the IGM, and the host galaxies of FRBs (cf. Section II.2). Better determination of these contributions leads to tighter limits for individual FRBs, and when combined through Eq. (10), a better combined constraint results. We expect all three points listed above will make progresses in observations soon.

The reason that in our analysis those FRBs without the redshift measurement still contribute to the constraint is the use of an uninformative prior for the redshift and the inclusion of it in a Bayesian way. In addition to the uniform-in-volume prior used in Section III, we here use another physical prior that traces the star formation rate for a robustness test. We use the fit for the star formation rate given by Yüksel et al. (2008),


where , , , , , , with the breaking points and . We obtain a combined limit () at 68% (95%) confidence level, showing that reasonable changes in the prior of redshift do not lead to large difference. The slight improvement here results from the fact that the star formation rate in Eq. (26) favours larger when where most FRBs in Table 3 reside.

Lastly, worthy to mention that, because FRBs are distributed nearly isotropically in the sky (see Figure 1), they will also be useful to constrain the anomalous anisotropic inertial mass tensor of photons Laemmerzahl (1998); Kostelecky and Mewes (2002, 2009) in a similar way that pulsars are used to set constraints on a Lorentz-violating tensor Shao (2014b). We leave this point for future work.


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