Photon mass limits from fast radio bursts

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Photon mass limits from fast radio bursts

Luca Bonetti, John Ellis, Nikolaos E. Mavromatos, Alexander S. Sakharov, Edward K. Sarkisyan-Grinbaum, Alessandro D.A.M. Spallicci

To cite this version:

Luca Bonetti, John Ellis, Nikolaos E. Mavromatos, Alexander S. Sakharov, Edward K. Sarkisyan-

Grinbaum, et al.. Photon mass limits from fast radio bursts. Physics Letters B, Elsevier, 2016, 757,

pp.548-552. �10.1016/j.physletb.2016.04.035�. �insu-01316496�

Contents lists available atScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

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Photon mass limits from fast radio bursts

Luca Bonetti

^a,b

, John Ellis

^c,d

, Nikolaos E. Mavromatos

^c,d

, Alexander S. Sakharov

^e,f,g

, Edward K. Sarkisyan-Grinbaum

^g,h

, Alessandro D.A.M. Spallicci

^a,b

aObservatoiredesSciencesdel’UniversenrégionCentre,UMS3116,Universitéd’Orléans,1AruedelaFérollerie,45071Orléans,France

bLaboratoiredePhysiqueetChimiedel’Environnementetdel’Espace,UMR7328,CentreNationaledelaRechercheScientifique,LPC2E,CampusCNRS,3AAvenue delaRechercheScientifique,45071Orléans,France

cTheoreticalParticlePhysicsandCosmologyGroup,PhysicsDepartment,King’sCollegeLondon,Strand,LondonWC2R2LS,UnitedKingdom dTheoreticalPhysicsDepartment,CERN,CH-1211Genève23,Switzerland

eDepartmentofPhysics,NewYorkUniversity,4WashingtonPlace,NewYork,NY10003,UnitedStates fPhysicsDepartment,ManhattanCollege,4513ManhattanCollegeParkway,Riverdale,NY10471,UnitedStates gExperimentalPhysicsDepartment,CERN,CH-1211Genève23,Switzerland

hDepartmentofPhysics,TheUniversityofTexasatArlington,502YatesStreet,Box19059,Arlington,TX76019,UnitedStates

a r t i c l e i n f o a b s t ra c t

Articlehistory:

Received5March2016

Receivedinrevisedform30March2016 Accepted17April2016

Availableonlinexxxx Editor:G.F.Giudice

Wededicatethispapertothememoryof LevOkun,anexpertonphotonmass

The frequency-dependenttimedelaysinfastradiobursts(FRBs) canbeused toconstrain thephoton mass,iftheFRBredshiftsareknown,butthesimilaritybetweenthefrequencydependencesofdispersion duetoplasmaeffects and aphoton masscomplicatesthederivationofalimitonmγ.Thedispersion measure (DM)ofFRB150418 isknownto∼⁰.1%, andthereis aclaim tohavemeasureditsredshift withanaccuracy of∼^{2%,butthestrengthoftheconstraintonm}γ is limitedbyuncertainties inthe modellingofthehostgalaxyandtheMilkyWay,aswellaspossibleinhomogeneitiesintheintergalactic medium (IGM). Allowing for theseuncertainties,the recent data on FRB150418 indicatethat m_γ 1.8×^{10−14eV c−2(3}.2×^{10−50kg),ifFRB150418indeedhasaredshiftz}=⁰.492 asinitiallyreported.

Inthe future,thedifferentredshiftdependencesoftheplasmaandphoton masscontributionstoDM canbeusedtoimprovethesensitivitytom_γ ifmoreFRBredshiftsaremeasured.Forafixedfractional uncertainty inthe extra-galactic contribution tothe DMof anFRB, one withalower redshiftwould providegreatersensitivitytomγ.

©^{2016PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

When setting an upper limit on the photon mass, the Parti- cleDataGroup (PDG)[1]cites theoutcome ofmodelling theso- larsystemmagneticfield:firstat1 AU,mγ<5.6×^{10−17eV c−2} (=^{10−52kg)[2,3], and later at 40 AU, m}γ<8.4×^{10−19eV c−2} (=¹.5×^{10−54kg)[2].However,thelaboratoryupperlimitisfour} orders of magnitude larger [4]; for reviewssee [5,6]. In [6], the authors state the concern that “Quotedphoton-masslimitshaveat timesbeenoverlyoptimisticinthestrengthsoftheircharacterizations.

Thisisperhapsduetothetemptationtoasserttoostronglysomething one‘knows’tobetrue”. Thisconcernwas mainlyaddressedto the galacticmagneticfield modellimits[7],butitshouldbebornein mindalsowhenassessingthesolarsystemlimits.

Indeed,theestimateson thedeviations fromAmpère’s lawin thesolarwind[2,3]arenotbasedsimplyoninsitumeasurements.

Forexample: (i) the magnetic field is assumedto be exactly, al-

E-mailaddress:alexandre.sakharov@cern.ch(A.S. Sakharov).

waysandeverywherea Parkerspiral;(ii)the accuracyofparticle 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 useof a reductioadabsurdum ap- proachbased onearlier resultsofother authors, whichare often devotedtoother issuesthanestablishingabasisforanextremely difficultmeasurementofamassthatismanyordersofmagnitude lowerthanthatofanelectronoraneutrino.

In order to check theseestimates of the solar wind at 1 AU, a more experimental approach has beenpursued via a thorough analysis of Cluster data [8], leading to a mass upper limit lying between 1.4×^{10−49 and 3}.4×^{10−51kg, according to the esti-} mated potential. The difference between the results of this con- servativeapproachandpreviousestimates,aswellastheneedfor astrophysical modelling, motivates the development of additional methodsforconstrainingthephotonmass.

The time structures of electromagnetic emissions from astro- physicalsourcesatcosmologicaldistanceshavebeenusedtocon- strain other aspects ofphoton/electromagnetic wave propagation, http://dx.doi.org/10.1016/j.physletb.2016.04.035

0370-2693/©^{2016PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

JID:PLB AID:31900 /SCO Doctopic: Theory [m5Gv1.3; v1.175; Prn:20/04/2016; 12:02] P.2 (1-5)

2 L. Bonetti et al. / Physics Letters B•••⁽••••⁾•••^–•••

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sucha possibleLorentz-violatingenergy/frequencydependenceof thevelocityoflightinvacuo[9–13],andthepossibilityofdisper- sion 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 Ad- vancedLIGOfromthe sourceGW150914havebeen usedto con- strainaspectsofgraviton/gravitationalwavepropagation,including an upper limit on the graviton mass: mg<1.2×^{10−22eV c−2} (=².1×^{10−58kg) [16,17] and limits on Lorentz violation [18,} 19],andthepossibleobservationby Fermiofanassociated

γ

-ray pulse [20] suggests that light and gravitational waves have the samevelocitiestowithin10⁻¹⁷[18,21].

The time structures of electromagnetic emissions from astro- physicalsourcesatcosmologicaldistancescanalsobeusedtode- riveanupperlimitonthephotonmass,mγ.Sincetheeffectofthe photonmassonthevelocityoflightisenhancedatlowfrequency

ν

(energy E): v∝ −^m2_γ^c4/h²

ν

² (−^m2_γ^c4/E²), measurements of timestructuresatlow frequencyorenergyare particularlysensi- tivetomγ.Forthisreason,measurementsofshorttimestructures inradio emissionsfromsources atcosmologicaldistancesare es- peciallypowerfulforconstrainingmγ.Thisistobecontrastedwith probes ofLorentz violation, forinstance,where measurements of high-energyphotonssuchas

γ

raysareatapremium.Thisiswhy probesofthephotonmassusinggamma-raybursters (GRBs)[22]

and active galactic nuclei (AGNs) have not been competitive in constrainingmγ.As wementionlater,a strongerlimitcanbe ob- tainedbyusingtheapparentcoincidenceofaradioafterglowwith aGRB,butthisisalsonotcompetitivewiththesensitivityoffered byfastradiobursts(FRBs).

FRBsarepotentiallyveryinterestingbecausetheirradiosignals havewell-measuredtimedelaysthatexhibitthe1/

ν

² dependence expectedforboththefreeelectrondensityalongthelineofsight andmasseffectsonphotonpropagation. Untilrecently,thedraw- backwasthatnoFRBhadhaditsredshiftmeasured,thoughthere wasconsiderableevidencethattheyoccurredatcosmologicaldis- tances. This has now changed with FRB150418 [23], which has beenreportedtohaveoccurredina galaxywithawell-measured redshiftz=⁰.492±⁰.008.Theidentificationofitshostgalaxyhas been questioned, andthe alternative possibility of a coincidence withan AGNflarehas beenraised [24], though thelikelihood of thisiscurrentlyanopenquestion[25].Inthefollowingweassume thehostgalaxyidentificationmadein[23],andalsodiscussmore generallyhownon-galacticFRBscouldbeusedtoconstrainphoton propagation.

Thefrequency-dependenttimelagofFRB150418 betweenthe arrivalsofpulseswith

ν

1=¹.2 GHz and

ν

2=¹.5 GHz ist₁₂^FRB≈ 0.8 s,andwas usedin [23]to extractvery accurately thedisper- sion measure (DM), which is given in the absence of a photon massbytheintegratedcolumndensityoffreeelectronsalongthe propagation path of a radio signal,

nedl. The delay of an elec- tromagneticwavewithfrequency

ν

propagatingthroughaplasma withanelectrondensityn_e,relativetoasignalinavacuum,makes the following frequency-dependent contribution to the time de- lay[26,27]

t_DM

=

_dl

c

ν

_p²

2

ν

²

⁼

⁴¹⁵

ν

1 GHz

₋2 DM

10⁵pc cm⁻³ s

,

(1) where

ν

p =(nee²/

π

me)^1/2=⁸.98·¹⁰³ⁿ¹e^/2Hz. As is discussed in[23],plasma effects withDM=⁷⁷⁶.2(5)cm⁻³pc could bere- sponsiblefortheentiret^FRB₁₂ thatwasmeasured.¹ Therearecon-

1 In[23]adifferentmethodhasbeenusedtoobtaintheDMvalue.However,for thisletteritisenoughtocomparethearrivaltimesofthesetwofrequencies,which reproducesquiteaccuratelytheresultof[23].

tributions to the DM of this extragalactic object from the free electrondensityinthehostgalaxy,estimatedtobe∼^{37 cm−3pc,} from the Milky Way and its halo, estimated to be 219 cm⁻³pc, and the intergalactic medium (IGM). Subtracting the other con- tributions, the IGM contribution to the DMwas estimated to be ^{520 cm−3pc, with} uncertainties ∼^{38 cm−3pc from the mod-} elling of the Milky Way using NE2001 [28]² and∼^{100 cm−3pc} frominhomogeneitiesintheIGM. TheDMIGM contributiontothe dispersiondelay(1)forasourceatredshiftzcanbeexpressedin termsofthedensityfractionIGMofionizedbaryons[26]:

DMIGM

=

^{3c H0}

_IGM 8

π

Gmp

He

(

z

) ,

(2)

where H0 isthepresentHubbleexpansionrate, G isthe Newton constant,mp istheprotonmass,andthefactor

H_e

(

z

) ≡

z

0

(

1

+

z

)

dz

+ (

1

+

^z

)

³

m

,

(3)

takes proper account of the time stretching in (1) and evolu- tion of the free-electron densitydue to the cosmological expan- sion [26,27,10,30].The relation (2)was used in [23] to estimate thedensityofionizedbaryonsintheIGM:^FRB_IGM=⁰.049±⁰.013, assuming that the heliumfraction inthe IGMhas thecosmolog- ical value of 24%. We also assume that the presentcosmological constantdensityfraction=⁰.714 andthepresentmatterden- sity fraction m=⁰.286, andset the reduced Hubble expansion rate, h0≡^H0/(100 km s⁻¹Mpc⁻¹=⁰.69 [31]. This measurement ofIGMisquitecompatiblewiththedensityexpectedwithinstan- dardCDMcosmology[31]:_IGM^CDM=⁰.041±⁰.002.

The measurement of t₁₂^FRB can also be used to constrain the photonmass.Forthispurpose,wenotethatthedifferenceindis- tancecoveredbytwoparticlesemittedbyan objectataredshift zwithvelocitydifferenceu is

L

=

H⁻₀¹

z

0

udz

+ (

1

+

^z

)

³

m

.

(4)

In case of the cosmologicalpropagation of two massive photons withenergies E2>E1thevelocitydifferenceis

umγ

=

^m

2γ 2

(

1

+

z

)

²

1 E²₁

−

¹

E²₂

,

(5)

wheretheredshiftsofthephotonenergiesaretakenintoaccount andwe useunits:^h¯ =^c=^k=^{1.Thus, differenceinarrival times} of two photonsof differentenergies froma remote cosmological objectduetoanon-zerophotonmasscanbeparametrizedasfol- lows:

t_lag

=

^m

2γ

2H₀

·

^F

(

E₁

,

E₂

) ·

^Hγ

(

z

) +

t_DM

+

^bsf

(

1

+

^z

) ,

(6) where F(E1,E2)≡ _E¹2

1−_E¹2 2

,

H_γ

(

z

) ≡

z

0

dz

(

1

+

^z

)

²

+ (

1

+

^z

)

³

m

,

(7)

and we include in (6) the contribution tDM to the time delay due to plasma effects anda possible, generally unknown, source

2 ForlimitationsofNE2001,see[29].

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time lag b_sf in the source frame. Inverting (6)and transforming to experimental units FGHz(_{1 GHz}^ν1 ,_{1 GHz}^ν2 ) and expressingall time measurementsinsecondswearriveat

m_γ

= (

1

.

05

·

^{10−14eV s−1/2}

)

×

h₀

F_GHzH_γ

(

t_lag

−

t_DM

−

^bsf

(

1

+

^z

)) .

(8) Themostconservativebound

m_γ

<

2

.

6

×

^{10−14eV c−2}

(

4

.

6

×

^10−50kg

)

(9) wouldbe obtained ifthe entireDM ofFRB 150418 were due to mγ =^{0, i.e.,} t_lagt₁₂^FRB, t_DM=^{0 and b}sf=^{0 in (8). How-} ever,this approach is probablytoo conservative, anda very rea- sonable assumption would be to subtract from the DM^FRB_IGM the IGMcontributioncorresponding to_IGM^CDM.Inthiscase, sincethe 95% CL estimate of the IGM dispersion measure is DM^FRB_IGM(2σ) 520±(2·138)cm⁻³pc[23],oneshould assume,accordingto(2) and (1), that t_lag ⁰.82 s at the 95% CL, t_DM≈⁰.45 s and b_sf=^{0 in(8).Inthiscase,onewouldfind}

m_γ

<

1

.

8

×

^{10−14eV c−2}

(

3

.

2

×

^10−50kg

)

(10) atthe 95% CL.³ Thesebounds are much stronger than thoseob- tainedfromGRBs[22] andAGNs,andaregettingwithin shouting distanceofthePDGlimit[2,1,3].We regardthisasthemostrea- sonableinterpretationofthedataonFRB150418.

Thequestionthenarises,howmuchtheFRBlimitcouldbeim- provedinthefuture?

The DMof FRB150418 hasbeen measured with an accuracy of0.1%,buttheuncertaintiesinsubtractingthecontributionsfrom the host galaxy, the IGM andthe MilkyWay amount to >20%.

Inparticular,uncertaintiesassociatedwithinhomogeneitiesinthe IGM approach 20%, dwarfing uncertainties associated withIGM, which approach 5%, and in modelling the Milky Way [28,29], whichexceed 5%.We doubt that the corresponding uncertainties forotherFRBscouldsoonbe reducedtothe0.1%leveloftheFRB 150418DMmeasurement,andconsiderthataplausibleobjective may be to constrain the sum of DM_IGM and a possible photon- masseffectforanygivenFRBwithan accuracyof10%.⁴ Oneway to improvethe sensitivity to mγ maybe to use datafrom FRBs atdifferentredshifts. As we discussbelow, therelative contribu- tionsoftheIGMandaphotonmassvarywiththeredshiftz,and the sensitivity to mγ is greater for FRBs with smaller redshifts.

Ahypothetical10%measurements ofthenon-hostandnon-Milky WaycontributionstotheDMofaFRBwithz=⁰.1 wouldyielda prospectivesensitivitytomγ=⁶.0×^{10−15eV c−2(1}.1×^10−50kg).

Asalreadycommented,thefrequencydependencesoftheIGM andmγ effects, Eqs. (1) and(8), are similar, butthe degeneracy betweenthemisbrokenbythedifferentzdependencesof He (3) andHγ (7).Inparticular, wenote themγ effectgainsinrelative moreimportance atsmaller z because ofthe difference between thepowersof(1+^z)intheintegrandsofHeandHγ.Inpractice, ifin thefutureastatistically relevantsample ofFRBsatdifferent redshiftsisobservedonemightusetheparametrization(6)tore- covertheintrinsictimelagofeverysourcefromthesampleas

bⁱ_sf

=

¹

(

1

+

^zi

) (

aⁱ_γ

·

^F

(

E1

,

E2

) ·

^Hγ

(

z

) +

tⁱ_DM

−

tⁱ_lag

) .

(11)

3 Similarboundsweregivenin[32],whichwereceivedwhileworkingonthis paper.

4 Inthisrespectweareconsiderablylessoptimisticthantheauthorsof[32].

Fig. 1.The (mγ,IGM)plane,showingasathinhorizontal redbandthe CDM expectationthatIGM=⁰.041±⁰.002,acurvedgreyshadedbandrepresentingthe FRB150418constraintasdiscussedinthetext,andotherbandsrepresentingthe impactsofhypotheticalfuture10%measurementsoftheextragalacticDMforFRBs withredshiftsz=0.1 (greenandmauve)andz=1.0 (blue).(Forinterpretationof thereferencestocolorinthisfigure,thereaderisreferredtothewebversionof thisarticle.)

AssumingidenticaloriginsfortheFRBs,onecouldoptimizetheset ofbⁱ_sfwithrespecttoaⁱγ andⁱ_IGM(tⁱ_DM),separatingthenon-zero photon mass contribution out from the plasma effect. The opti- mizationcanbeperformedonabasisofsomeestimator:asimple onecouldbejustaminimizationoftheRMSofbⁱ_sf.⁵

As discussed above,we consider that future measurements of thenon-hostgalaxyandnon-MilkyWaycontributionstotheDMs ofotherFRBsatthe10%levelmaybe feasibleobjectives.Accord- ingly, we have made a first assessment of their possible future impacts on the photon masslimit. Fig. 1 displays an (mγ,IGM) plane, featuringasa thinhorizontalbandthe CDMexpectation that_IGM^CDM.Theothercurveshavetheforms

m_γ

=

^A

√

B

−

^{C (12)}

that follows from (8), where A is a numerical pre-factor deter- mined by the factor Hγ(z) of an object, the term B represents anobservedtimelagintermsofintergalacticDM

B

= (

103

.

1 s

) ·

^DM

obs IGM

10⁵pc cm⁻³ (13)

andC definesthefractionofanactualcontributionoftheionized plasmaeffecttotheobservedtimelagrelativetothepredictionof thestandardCDM modelforagivenobject

C

=

t_IGM

·

IGM

_IGM^CDM

.

(14)

The curves in Fig. 1 assume an ionization fraction 0.9 but al- low IGM to be a free parameter. The curved grey shaded band shows the FRB 150418 constraint discussed above, at the 68%

CL,whichimplies A=².96·^{10−14eV s−1/2,DMobs}IGM=^DMFRBIGM and t_IGM=⁰.45 s. The intersectionof thisband with the IGM=⁰ axiscorrespondstothe(overly?)conservative95%CLlimit(9)and itsintersectionwiththeCDMbandforIGM correspondstothe

‘reasonable’95%CLbound(10).

TheFigurealsodisplaysotherbands,showingthepotentialim- pacts ofhypothetical 10% measurements of the extragalactic DM

5 Avariantofsuchalgorithmhasbeenusedin[34]forneutrinomassestimations fromasupernovasignal.

JID:PLB AID:31900 /SCO Doctopic: Theory [m5Gv1.3; v1.175; Prn:20/04/2016; 12:02] P.4 (1-5)

4 L. Bonetti et al. / Physics Letters B•••⁽••••⁾•••^–•••

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for FRBs with redshift z=⁰.1 (green and mauve) and z=¹.0 (blue).⁶ The hypothetical z=⁰.1 green band has the same cen- tral value as expected for _IGM^CDM and a massless photon, for which case A=¹.97·^{10−14eV s−1/2, DMobs}IGM=^{83 pc cm−3 and} t_IGM=⁰.086 s havebeen used in (13) and (14).⁷ The z=¹.0 blue band has been calculated with A=⁴.60·^{10−14 eV s−1/2,} DM^obs_IGM=^{903 pc cm−3and}tIGM=⁰.94 s appliedin(13)and(14).

Thehypotheticalz=⁰.1 mauvebandhasthesameupperlimiton IGM astheFRB150418 measurementanddiffersfromthegreen oneinhavingDM^obs_IGM=^{103 pc cm−3 usedin(13)and(14).Asex-} pected, we see that a 10% measurement of an FRB with z=⁰.1 yielding theexpectedcentral value (greenband) wouldimposea morestringentconstraintonmγ,namely

m_γ

<

6

.

0

×

^{10−15eV c−2}

(

1

.

1

×

^10−50kg

) .

(15) ifone(veryconservatively)allowsanyIGM≥^0,strengtheningto

<3×^{10−15eV c−2 for}_IGM^CDM. Alternatively,we see thatconsis- tency of the green band with the FRB 150418 constraint would requiremγ<2.5×^{10−15eV c−2,withoutanyassumptionon}IGM. We also see that consistency between a ‘high’ measurement froman FRB with z=⁰.1 (mauve band)and an ‘expected’ mea- surementfromanFRBwithz=¹.0 (blue band)wouldbeconsis- tentwith_IGM^CDM onlyifonerequiresanon-zeromγ∈ [².5,4.0]× 10⁻¹⁵eV c². These are just examples of possible future develop- ments in the interpretation of possible DM measurements from futureFRBswithmeasuredredshifts, andspecificallyhow theef- fectsoftheIGM anda photonmass could inprinciplebe distin- guished.Significantimprovementsontheseestimatedsensitivities wouldrequiremorecarefulestimatesofpossiblereductionsinthe uncertainties in DMIGM, in particular, and would benefit from a combinedanalysisofalargernumberofFRBs.

Forcompleteness, we mention anotherway to boundmγ us- ingradioemissions,namelybycomparingthe arrivaltimeofradio afterglow and

γ

-ray emission from a GRB. The most promising exampleseemstobeGRB 071109whichwasobserved[33]toex- hibit a radio afterglow at 8.46 GHz about0.03 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} findanupperlimit onthephotonmassmγ².8×^{10−11eV c−2} (=⁵.0×^{10−47kg).9 The weakness ofthe limit comparedto the} FRBlimit discussed earlier is dueto the much larger time delay beforethe observationofthe radioafterglow. Whilst thislimitis not competitive withthe FRBlimit givenabove or thelimit cur- rently quoted by the PDG, this GRB afterglow method has the interest of involving a different type of astrophysical modelling.

Moreover,ithaspotentialforfutureimprovement,e.g.,ifonecould uselower-frequencywavesand/orobserveanafterglowsooneraf- tertheparentGRB,andparticularlyiftime structureintheradio emissionsanalogoustothoseinthe

γ

-rayemissionscouldbede- tected.

We finish our discussion with come comments and specula- tions.Thepresentlackofredshiftmeasurementsforother FRBsis anobstacleforobtainingamorerobustupperbound onthepho- ton mass. However, one could also reverse the logic used above forFRB150418 and, assumingthe expectedcosmologicaldensity oftheIGMandtheupperlimitonthephotonmassderived from

6 ThelowluminositiesofFRBswouldrenderthemdifficulttodetectatlargerz.

7 Forallhypotheticalsourcesa10%uncertainty inDM^obs_IGMisapplied.

8 OtherGRBshavelesssensitivity,becausetherewerelargerdelaysbeforetheir afterglowsweredetected.

9 Hereweassumesimultaneousemissionoftheradiowavesandγ rays,which maynotbethecase.Iftheradiowaveswereemittedbeforetheγrays(foreglow), anydelayduetothephotonmasswouldbemaskedbytheearliertimeofemission.

FRB 150418,estimate the redshifts ofother observed FRBs.Their redshift distribution might help pin down their origins. Another optionwouldbetousegravitationallensing,whichwouldbecome frequency dependent in the presence of a photon mass [5]. The lensingisindependentofthedistancefromthesource,andapho- ton of massmγ and energy E froma source of mass M would be gravitationallydeflectedby an angleθ= ^4M_Rc2^G

1+^m_2E^2γc2⁴

γ

,for aphoton ofenergy E (orfrequency

ν

=^E/h), whereR isthesize of the celestial body and G is the gravitational constant. In [5], thephoton-massdeflectionθ wassetequaltothedifferencebe- tween the value observedforsome celestial object,e.g., theSun, andthestandardtheoreticalcaseformasslessphoton,therebyob- taining anupper boundmγ^h

ν

c⁻²√

2θ/θ0,where θ0=^4M_Rc2^G

is the standard massless photon deflection. Limits of the order of mγ ^{10−44 kg can be obtained this way.} Conversely, using upper bounds ofthe photon mass obtainedfrom other methods liketheFRBsdiscussedherewouldremoveoneuncertaintyinthe predictions for expecteddeflection angles, sharpeningthe use of comparisons with observationsto constrain better the properties oflensingobjects.

Acknowledgements

The research of JE and NMwas supported partlyby the Lon- don Centre for Terauniverse Studies (LCTS), using funding from theEuropeanResearchCouncilviatheAdvancedInvestigatorGrant 26732,andpartlybytheSTFCGrantST/L000326/1.JEthanksDale FrailforusefulcommunicationsandtheUniversidaddeAntioquia, Medellín, for its hospitality while this work was initiated, using GrantFP44842-035-2015fromCOLCIENCIAS (Colombia).Thework ofASwassupportedpartlybytheUSNationalScienceFoundation underGrantsNo. PHY-1205376andNo. PHY-1402964.

References

[1] K.A. Olive, et al., Particle Data Group Collaboration, Review of particle physics,Chin.Phys.C38(2014)090001,http://dx.doi.org/10.1088/1674-1137/

38/9/090001.

[2] D.D.Ryutov,Theroleoffinitephotonmassinmagnetohydrodynamicsofspace plasmas,PlasmaPhys.Control.Fusion39(1997)A73,http://dx.doi.org/10.1088/

0741-3335/39/5A/008.

[3] D.D.Ryutov,Usingplasmaphysicstoweighthephoton,PlasmaPhys.Control.

Fusion49(2007)B429,http://dx.doi.org/10.1088/0741-3335/49/12B/S40.

[4] E.R.Williams,J.E.Faller,H.A.Hill,NewexperimentaltestofCoulomb’slaw:a laboratoryupperlimitonthephotonrestmass,Phys.Rev.Lett.26(1971)721, http://dx.doi.org/10.1103/PhysRevLett.26.721.

[5] D.D. Lowenthal, Limits on the photon mass, Phys. Rev. D 8(1973) 2349, http://dx.doi.org/10.1103/PhysRevD.8.2349;

L.C.Tu,J.Luo,G.T.Gillies,Themassofthephoton,Rep.Prog.Phys.68(2005) 77,http://dx.doi.org/10.1088/0034-4885/68/1/R02;

L.B.Okun, Photon: history,mass, charge, Acta Phys. Pol.B 37(2006) 565, arXiv:hep-ph/0602036;

G.Spavieri,J.Quintero,G.T.Gillies,M.Rodriguez,Asurveyofexistingandpro- posedclassicalandquantumapproachestothephotonmass,Eur.Phys.J.D61 (2011)531,http://dx.doi.org/10.1140/epjd/e2011-10508-7.

[6] A.S.Goldhaber,M.M.Nieto,Photonandgravitonmasslimits,Rev.Mod.Phys.

82(2010)939,http://dx.doi.org/10.1103/RevModPhys.82.939,arXiv:0809.1003 [hep-ph].

[7] Y.Yamaguchi,Acompositetheoryofelementaryparticles,Prog.Theor.Phys.

Suppl.11(1959)1,http://dx.doi.org/10.1143/PTPS.11.1;

G.V.Chibisov,Astrophysicalupperlimitsonthephotonrestmass,Sov.Phys.

Usp. 19 (1976) 624, http://dx.doi.org/10.1070/PU1976v019n07ABEH005277, Usp.Fiz.Nauk119(1976)551;

E.Adelberger,G. Dvali,A. Gruzinov,Photon-mass bounddestroyed byvor- tices,Phys.Rev.Lett.98(2007)010402,http://dx.doi.org/10.1103/PhysRevLett.

98.010402,arXiv:hep-ph/0306245.

[8] A.Retinò,A.D.A.M.Spallicci, A.Vaivads,Solar wind testofthe deBroglie–

Proca’smassivephotonwithClustermulti-spacecraftdata,versionfromFebru- ary2016,arXiv:1302.6168v3[hep-ph].