FRB 121102 casts new light on the photon mass

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FRB 121102 casts new light on the photon mass

Luca Bonetti, John Ellis, Nikolaos E. Mavromatos, Alexander S. Sakharov,

Edward K. Sarkisyan-Grinbaum, Alessandro D.A.M. Spallicci

To cite this version:

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

FRB

121102

casts

new

light

on

the

photon

mass

Luca Bonetti

a

,

b

,

c

,

John Ellis

d

,

e

,

Nikolaos

E. Mavromatos

d

,

e

,

Alexander

S. Sakharov

f

,

g

,

h

,

Edward

K. Sarkisyan-Grinbaum

h

,

i

,

∗

,

Alessandro

D.A.M. Spallicci

a

,

b

,

c

a_{ObservatoiredesSciencesdel’UniversenrégionCentre,UMS3116,Universitéd’Orléans,1AruedelaFérollerie,45071Orléans,France} b_{PôledePhysique,CollegiumSciencesetTechniques,Universitéd’Orléans,RuedeChartres,45100Orléans,France}

c_{LaboratoiredePhysiqueetChimiedel’Environnementetdel’Espace,UMR7328CentreNationaledelaRechercheScientifique,LPC2E,CampusCNRS,} 3A Avenue delaRechercheScientifique,45071Orléans,France

d_{TheoreticalParticlePhysicsandCosmologyGroup,PhysicsDepartment,King’sCollegeLondon,Strand,LondonWC2R2LS,UnitedKingdom} e_{TheoreticalPhysicsDepartment,CERN,CH-1211Genève23,Switzerland}

f_{DepartmentofPhysics,NewYorkUniversity,4WashingtonPlace,NewYork,NY10003,UnitedStates} g_{PhysicsDepartment,ManhattanCollege,4513ManhattanCollegeParkway,Riverdale,NY10471,UnitedStates} h_{ExperimentalPhysicsDepartment,CERN,CH-1211Genève23,Switzerland}

i_{DepartmentofPhysics,TheUniversityofTexasatArlington,502YatesStreet,Box19059,Arlington,TX76019,UnitedStates}

a

r

t

i

c

l

e

i

n

f

o

a

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c

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Articlehistory:

Received28January2017

Receivedinrevisedform8March2017 Accepted9March2017

Availableonline14March2017 Editor:M.Trodden

Thephoton mass,mγ,caninprinciplebeconstrainedusingmeasurementsofthedispersionmeasures (DMs)offastradiobursts(FRBs),oncetheFRBredshiftsareknown.TheDMoftherepeatingFRB121102 isknownto<1%,ahostgalaxyhasnowbeenidentifiedwithhighconfidence,anditsredshift, z,has nowbeendeterminedwithhighaccuracy:z=0.19273(8).Takingintoaccounttheplasmacontributions to theDMfromtheIntergalactic medium(IGM)andthe MilkyWay,weusethedata onFRB121102 toderivetheconstraintmγ2.2×10−14eV c−2(3.9×10−50 kg).Sincetheplasmaand photonmass contributionsto DMshave differentredshiftdependences,they couldinprinciple bedistinguishedby measurementsofmoreFRBredshifts,enablingthesensitivitytomγ tobeimproved.

©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

The photon is generallyexpected to be massless, but a num-ber of theorists have challenged this assumption, starting from de Broglieandnowadaysconsideringmodelswithmassivephotons fordarkenergyanddarkmatter.Examplesofmechanismsfor pro-vidingmassincludeStandardModelExtensionswith supersymme-tryandLorentzinvariancebreaking[1]andHiggsmechanisms[2]. Inview of thesepossibilities andthe fundamental importance of the photon mass, it is important to constrain the magnitude of the photon mass as robustly as possible. The mostrobust limits availablearethosefromlaboratoryexperiments[3]–see

[4,5]

for reviews–butthesearemuchweakerthanthosederivedfrom as-trophysical observations. The Particle Data Group (PDG) [6]cites the upper limit mγ

<

8

.

4

×

10−19_{eV c}−2 ₍

₌

₁

_.

₅

_×

₁₀−54_{kg) [7]}

obtainedby modelling themagnetic field of thesolar system[7, 8]. However, this limit relies on assumptions about the form of the magnetic field and does not discuss measurement accuracy anderrors.Anotherlimit(mγ

<

4

×

10−52_{kg)whichhas been}

de-rived from atmosphericradio waves hasbeen reported in [9]. A

*

Correspondingauthor.

E-mailaddress:Edward.Sarkisyan-Grinbaum@cern.ch(E.K. Sarkisyan-Grinbaum).

more conservative approachwas followedin an analysisof Clus-terdata[10],leading toan upperlimit between7

.

9

×

10−14_and

1

.

9

×

10−15eV c−2 (1

.

4

×

10−49 and3

.

4

×

10−51kg). Itisclearly desirable to explore more direct and robust astrophysical con-straintsonapossiblephotonmass.

This was the motivation for a study we made [11] (see also [12])showinghowdatafromfastradiobursts(FRBs)couldbeused to constrain mγ . These have durations in the millisecond range, andtheirsignalsareknowntoarrivewithafrequency-dependent dispersion in time of the 1

/

ν

2 _{form. Thisis the dependence}

ex-pected from plasma effects, but a similar dispersion

∝

m2_γ

/

ν

2

could alsoarise froma photon mass.The dispersionsinducedby plasmaeffectsandmγ bothincreasewithdistance(redshift z),but withdifferentdependencesonz.Wenote inthisconnectionthat the lower frequencies of FRBemissionsgive a distinct advantage overgamma-rayburstersandothersourcesofhigh-energy

γ

rays forconstrainingmγ ,sincemasseffectsaresuppressedfor higher-energy photons.1 _{Moreover,using FRBemissionsto constrainmγ}

1 _{Incontrast,sourcesofhigh-energyphotonsarebettersuitedforprobingmodels}

ofLorentzviolation[13].

http://dx.doi.org/10.1016/j.physletb.2017.03.014

ismuch more directand involvesfewer uncertainties than using thepropertiesofastrophysicalmagneticfields.2

Thatsaid, although the large dispersion measures (DMs) and other arguments led to the general belief that FRBs occur at cosmological distances, until recently no FRB redshift had been measured. The first claim to measure a redshift was made for FRB 150418 [20], andthis was the example we used in [11] to show howFRBmeasurements could in principlebe used to con-strain mγ . However, the identification of the host galaxy of FRB 150418hassubsequentlybeenchallenged[21],andisnow gener-allynotaccepted[22].

Ourinterest in the possibility of using FRBs to constrain mγ

hasrecentlybeenrevived,however,bytheobservationofrepeated emissionsfromFRB121102[22].Thesehavepermittedprecise lo-calisation of its host galaxy, which has made possible a precise determinationofitsredshift,z

=

0

.

19273

(

8

)

[23].Thisredshift de-terminationmakesitpossible,inturn,tousedataonFRB121102 toprovidearobustconstraintonmγ ,aswediscussinthispaper.

The dispersion measure (DM) is related to the frequency-dependenttimelagofanFRBby



tDM

=

415



_ν

1 GHz



₋2 _DM

105_{pc cm}−3s

.

(1)

Intheabsenceofa photonmass,the time-lagofanFRBisgiven by integratingthe column densityne offree electrons along the lineofflightofitsradiosignal



tDM

=



_dl c

ν

2 p 2

ν

2

,

(2)

where

ν

p

= (

nee2

/

π

me

)

1/2

=

8

.

98

·

103n1e/2Hz. Several sources contributetothisintegratedcolumndensityoffreeelectrons, no-tably the MilkyWay galaxy, theintergalactic medium (IGM)and thehostgalaxy.ThecontributiontotheDM(1)ofan FRBat red-shiftz fromtheIGM isgivenbythecosmologicaldensityfraction



IGMofionised baryons[24,25]:

DMIGM

=

3c H0



IGM

8

π

Gmp

He(z

) ,

(3)

where H0

=

67

.

8

(

9

)

km

/

s

/

Mpc [6] isthe presentHubble

expan-sionrate,3 G is theNewtonconstant,mp istheprotonmass,and theredshift-dependentfactor

He(z

)

≡

z



0

(

1

+

z

)

dz







+ (

1

+

z

)

3

m

,

(4)

where





=

0

.

692

(

12

)

and



m

=

0

.

308

(

12

)

[6].For comparison, thedifferenceintime lagsbetweenphotonsofenergies E1,2 due

toanon-zerophotonmasshastheform:



tmγ

=

m2_γ 2H0

·



1 E2₁

−

1 E2₂



·

Hγ

(

z

) ,

(5)

2 _{Foranearlyconsiderationofpossibleastrophysicalphotonpropagationdelays,}

see[14].Forpioneeringstudiesusingastrophysicalsources,see[15](flarestars) and[16](Crabnebula),andforananalogoussubsequentstudywithgreater sensi-tivitytothephotonmass,see[17](GRB980703,mγ<4.2×10−47_kg).Themost

recentsuchstudiesarethosein[18](GRB050416A,mγ<1.1×10−47_kg)and[19]

(radiopulsarsintheMagellanicclouds,mγ<2×10−48_{kg).Our limitonmγ is}

significantlystronger.

3 _{Wediscusslater theimpactofassumingabroaderrangeH}

0=70(4)km/s/

Mpc[26].

whereweusenaturalunitsh

=

c

=

1,and[27,28] Hγ

(

z

)

≡

z



0 dz

(

1

+

z

)

2



_



+ (

1

+

z

)

3

m

.

(6)

As alreadycommented,thetime lagsdueto theIGM anda pos-siblephoton masshavedifferentdependences (4)and

(6)

on the redshift.Theuncertaintiesinthecosmologicalparametersandthe measurement of the redshift measurement of FRB 121102 are taken into account in ouranalysis, withthe uncertainties inthe formerbeingmuchmoreimportant,aswewill seelater.

The top (green) band in

Fig. 1

showsthe total DM

=

558

.

1

±

3

.

3 pc cm−3 measured for FRB 121102 [22]. The most conserva-tiveapproachtoconstrainingmγ wouldbetosettozerotheother contributions,andassignthistotalDMtoapossiblephotonmass. However, this is surely over conservative, and a reasonable ap-proachistosubtractfromthetotalDMtheexpectedcontribution fromthe MilkyWay [22],DMMW,which is thesumof

contribu-tions from the disk[29]: DMNE2001



188 pc cm−3 and the halo:

DMhalo



30 pc cm−3 [23], to which we assign an overall

uncer-tainty of20%, namely44 pc cm−3 _{[23], leavingthemiddle (blue)}

band in

Fig. 1

that iscentred at340 pc cm−3_{.From thiswe may}

also subtract the contribution from the IGM, which is estimated withinthe



CDMmodeltobe



200 pc cm−3 _{[23–25].Tothisan}

uncertaintyof85 pc cm−3 _{associatedwithinhomogeneitiesinthe}

IGM [23,30]isassigned, whichismuchlargerthanthe 1.2% vari-ationassociatedwithuncertaintiesinthecosmologicalparameters (shown as the narrow magenta band). The bottom (pink) band in

Fig. 1

,centredat140 pc cm−3, showsthe effectofsubtracting thesecontributionsfromthemeasuredDMforFRB121102.

Aftersubtractingthesecontributions,we areleft witha resid-ual DM

=

140 pc cm−3 witha total uncertainty of

±

96 pc cm−3, shown asthe outer pink band, where the error is calculated by combining in quadrature the uncertainties in the experimental measurementofthetotalDM,theuncertaintyinDMMW,andthe

uncertainties in DMIGM associated with the cosmological

param-eters H0,



 and



m andpossible inhomogeneities.One cannot exclude thepossibility thatall the residualDMof FRB121102 is duetothehostgalaxy,whichisestimatedtolie withintherange 55



DMHost



225 pc cm−3 [23].However, in theabsence of

de-tailed information about the host galaxy, when constraining the photon mass we allow conservatively for the possibility that all theresidualDMisduetomγ

=

0.

The curved band in Fig. 1 shows the possible contribution to the DM of FRB 121102 of a photon mass, as a function of

mγ : DM

=

105_m2

γ Hγ

/(

415 A2h0

)

, where Hγ isgiven in (6), A

=

1

.

05

·

10−14_{ev s}−1/2 _{and h}

0

≡

H0

/

100 km

/

s

/

Mpc. The width of

thisbandisduetotheuncertaintiesinthecosmological parame-tersH0,



and



m[6],andtheuncertaintyinthedetermination of the redshift of FRB 121102. Assuming that the photon mass contributiontothe totalDMofFRB121102 lieswithin therange allowed for the residual DM, aftersubtraction of the MilkyWay andIGMcontributionsandtakingtheiruncertaintiesintoaccount, we findmγ



2

.

2

×

10−14eV c−2 (3

.

9

×

10−50kg).4 This limit is similar to, though slightly weaker than, that obtained from sim-ilar considerations ofFRB 150418 [11,12], whoseredshift is now contested,asdiscussedearlier[21].IfFRB150418wasindeedata cosmologicaldistance,usingits DMvaluedeterminedin [20]and thesamevaluesofHe

(

z

)

andHγ

(

z

)

asinthepresentanalysis,we

4 _{Thislimitwouldincreasetomγ 2.3×10}−14_{eV c}−2_(4.1×10−50_kg)ifthe

morerelaxedrange H0=70(4)km/s/Mpc[26]wereusedfor H0.Ontheother

hand,itwoulddecreasetomγ1.8×10−14eV c−2(3.2×10−50kg)ifthe

Fig. 1. Contributionstothedispersionmeasure(DM)budgetforFRB121102.Thetop(green)bandrepresentstheexperimentalmeasurementofthetotal DM=558.1± 3.3 pc cm−3_{.Themiddle(blue)bandshowstheextragalacticcontribution,asobtainedbysubtractingfromthe centralvalue ofthe totalDMthegalacticcontribution}

DMMW≡DMNE2001+DMhalo=218 pc cm−3[23],towhichisassignedanestimateduncertaintyof20%.Thebottom(pink)bandisobtainedbysubtractingalsotheestimated

contributionfromtheintergalacticmedium(IGM):DMIGM200 pc cm−3[23],whichhasanuncertaintyof85 pc cm−3associatedwithinhomogeneitiesintheIGM,anda

smalleruncertaintyassociatedwiththecosmologicalparameters(indicatedbythenarrowmagentaband).Theouterpinkbandshowsthetotaluncertaintyintheresidual DMaftersubtractionofDMMWandDMIGM,witherrorsaddedinquadrature.Thecurved(black)bandshowsthepossiblecontributionofanon-zerophotonmass,mγ,also

includingtheuncertaintiesincosmologicalparametersandtheredshiftofFRB121102.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderis referredtothewebversionofthisarticle.)

findthattheinferredlimitsonmγ wouldcoincide ifFRB150418 hada redshift z

=

0

.

38, instead of the value z

=

0

.

492 reported in[20]andchallengedin[21].

Howcould thisconstraint be improved inthe future? Clearly it is desirable to reduce the uncertainties in the modelling of the Milky Way and IGM contributions. Also, the limit could be strengthenedbyaredshiftmeasurementforanFRBathigherz,if theuncertaintyintheIGMcontributioncanbecontrolled.Finally, asremarked in [11], comparing the DMs forFRBs with different redshiftscouldenabletheIGMandmγ contributionstobe disen-tangled,inviewoftheirdifferentdependences

(4)

and

(6)

on z.

Ahitherto unexploredwindow atvery low frequencies inthe MHz–KHzregioncouldbeopenedbyaspacemissionconsistingof a swarm ofnanosatellites [31].One possibleconfiguration would beorbitingtheMoon,whereitwouldbesufficientlyawayfromthe ionospheretoavoidterrestrialinterference,andwouldhavestable conditionsfor calibration during observations. Such low frequen-cieswouldofferasensitiveprobeofanydelaysduetoanon-zero photonmass.

Acknowledgements

The research of J.E. and N.E.M. was supported partly by the STFCGrantST/L000326/1.TheworkofA.S.S.wassupported partly bytheUSNationalScienceFoundationunderGrantsPHY-1505463 andPHY-1402964.

References

[1] L. Bonetti,L.R. dos SantosFilho, J.A.Helayël-Neto,A.D.A.M.Spallicci, Effec-tivephotonmassbysuperandLorentzsymmetrybreaking,Phys.Lett.B764 (2017) 203,http://dx.doi.org/10.1016/j.physletb.2016.11.023,arXiv:1607.08786 [hep-ph].

[2] E. Adelberger, G.Dvali, A.Gruzinov,Photon-mass bound destroyed by vor-tices,Phys.Rev.Lett.98(2007)010402,http://dx.doi.org/10.1103/PhysRevLett. 98.010402,arXiv:hep-ph/0306245.

[3] 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.

[4] 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,ActaPhys.Pol.B37(2006)565,http:// www.actaphys.uj.edu.pl/fulltext?series=Reg&Vol=37&page=565, arXiv:hep-ph/ 0602036;

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

[5] 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].

[6] C.Patrignani,etal.,ParticleDataGroup,Reviewofparticlephysics,Chin.Phys. C40(2016)100001,http://dx.doi.org/10.1088/1674-1137/40/10/100001. [7] D.D.Ryutov,Theroleoffinitephotonmassinmagnetohydrodynamicsofspace

plasmas,PlasmaPhys.Control.Fusion39(1997)A73,http://dx.doi.org/10.1088/ 0741-3335/39/5A/008.

[8] D.D.Ryutov,Usingplasmaphysicstoweighthephoton,PlasmaPhys.Control. Fusion49(2007)B429,http://dx.doi.org/10.1088/0741-3335/49/12B/S40. [9] M.Füllekrug,Probingthespeedoflightwithradiowavesat extremelylow

radiofrequencies,Phys.Rev.Lett.93(2004)043901,http://dx.doi.org/10.1103/ PhysRevLett.93.043901.

[10] A. Retinò,A.D.A.M.Spallicci, A.Vaivads,Solar wind testofthe deBroglie– Proca’smassive photon with Cluster multi-spacecraft data, Astropart. Phys. 82(2016)49,http://dx.doi.org/10.1016/j.astropartphys.2016.05.006,arXiv:1302. 6168[hep-ph].

[11] L. Bonetti,J. Ellis, N.E.Mavromatos,A.S.Sakharov, E.K.Sarkisyan-Grinbaum, A.D.A.M.Spallicci,Photonmasslimitsfromfastradiobursts,Phys.Lett.B757 (2016)548,http://dx.doi.org/10.1016/j.physletb.2016.04.035,arXiv:1602.09135 [astro-ph.HE].

[12] X.-F. Wu,S.-B.Zhang,H. Gao,J.-J.Wei,Y.-C. Zou,W.-H.Lei,B.Zhang, Z.-G. Dai, P. Mészáros, Constraints on the photon mass with fast radio bursts, Astrophys.J.822(2016) L15,http://dx.doi.org/10.3847/2041-8205/822/1/L15, arXiv:1602.07835[astro-ph.HE].

[13] G. Amelino-Camelia, J.R. Ellis, N.E. Mavromatos,D.V. Nanopoulos,S. Sarkar, Testsofquantumgravityfromobservationsofgamma-raybursts,Nature393 (1998)763,http://dx.doi.org/10.1038/31647,arXiv:astro-ph/9712103; SeealsoJ.R.Ellis,K.Farakos,N.E.Mavromatos,V.A.Mitsou,D.V.Nanopoulos, Astrophysicalprobesoftheconstancyofthevelocityoflight,Astrophys.J.535 (2000)139,http://dx.doi.org/10.1086/308825,arXiv:astro-ph/9907340. [14]L.deBroglie,LaMécaniqueOndulatoireduPhoton.UneNouvelleThéoriedela

Lumière,Hermann,Paris,1940.

[15] B.Lovell,F.L.Whipple,L.H.Solomon,Relativevelocityoflightandradiowave inspace,Nature202(1964)157,http://dx.doi.org/10.1038/202377a0. [16] B. Warner,R.E.Nather,Wavelengthindependenceofthevelocityoflightin

[17] B.E.Schaefer,Severelimitsonvariationsofthespeedoflightwithfrequency, Phys.Rev.Lett.82(1999)4964,http://dx.doi.org/10.1103/PhysRevLett.82.4964, arXiv:astro-ph/9810479.

[18] B.Zhang,Y.-T.Chai,Y.-C. Zou,X.-F. Wu,Constrainingthe massofthe pho-tonwithgamma-raybursts,J.HighEnergyAstrophys.11–12(2016)20,http:// dx.doi.org/10.1016/j.jheap.2016.07.001,arXiv:1607.03225[astro-ph.HE]. [19] J.-J. Wei, E.-K. Zhang, S.-B. Zhang, X.-F. Wu, New limits on the photon

mass with radio pulsars inthe Magellanic clouds, Res. Astron. Astrophys. 17(2017) 13,http://dx.doi.org/10.1088/1674-4527/17/2/13, arXiv:1608.07675 [astro-ph.HE].

[20] E.F. Keane, et al., A fast radioburst host galaxy, Nature 530(2016) 453,

http://dx.doi.org/10.1038/nature17140,arXiv:1602.07477[astro-ph.HE]. [21] P.K.G.Williams,E.Berger,NopreciselocalizationforFRB150418:claimed

ra-diotransientisAGNvariability,Astrophys.J.821(2016)L22,http://dx.doi.org/ 10.3847/2041-8205/821/2/L22,arXiv:1602.08434[astro-ph.CO];

H.K.Vedantham, V.Ravi,K. Mooley, D.Frail, G.Hallinan, S.R. Kulkarni,On associating fast radio bursts with afterglows, Astrophys. J. 824(2016) L9,

http://dx.doi.org/10.3847/2041-8205/824/1/L9,arXiv:1603.04421[astro-ph.HE]. [22] S.Chatterjee,etal.,Thedirectlocalizationofafastradioburstanditshost, Nature541(2017)58,http://dx.doi.org/10.1038/nature20797,arXiv:1701.01098 [astro-ph.HE].

[23] S.P.Tendulkar,etal., Thehostgalaxy andredshift ofthe repeatingfast ra-dioburstFRB121102,Astrophys.J.834(2017)L7,http://dx.doi.org/10.3847/ 2041-8213/834/2/L7,arXiv:1701.01100[astro-ph.HE].

[24] K.Ioka,Cosmicdispersionmeasurefromgamma-rayburstafterglows:probing thereionizationhistoryandtheburstenvironment,Astrophys.J.598(2003) L79,http://dx.doi.org/10.1086/380598,arXiv:astro-ph/0309200.

[25] S.Inoue, Probingthe cosmic reionization historyand localenvironmentof gamma-ray burststhrough radio dispersion,Mon. Not. R. Astron. Soc. 348 (2004)999,http://dx.doi.org/10.1111/j.1365-2966.2004.07359.x,arXiv:astro-ph/ 0309364.

[26] N.Jackson,TheHubbleconstant,LivingRev.Relativ.18(2015)2,http://dx.doi. org/10.1007/lrr-2015-2.

[27] J.R.Ellis,N.E.Mavromatos,D.V.Nanopoulos,A.S.Sakharov,E.K.G.Sarkisyan, Ro-bustlimitson Lorentzviolationfromgamma-ray bursts,Astropart.Phys. 25 (2006) 402,http://dx.doi.org/10.1016/j.astropartphys.2006.04.001, arXiv:astro-ph/0510172;

J.R. Ellis, N.E.Mavromatos,D.V.Nanopoulos,A.S.Sakharov,E.K.G. Sarkisyan, Erratumto“RobustlimitsonLorentzviolationfromgamma-raybursts” [As-tropart.Phys.25(2006)402],Astropart.Phys.29(2008)158,http://dx.doi.org/ 10.1016/j.astropartphys.2007.12.003,arXiv:0712.2781[astro-ph].

[28] U. Jacob, T. Piran, Lorentz-violation-induced arrival delays of cosmologi-cal particles,J. Cosmol.Astropart. Phys. 0801 (2008)031, http://dx.doi.org/ 10.1088/1475-7516/2008/01/031,arXiv:0712.2170[astro-ph].

[29]J.M.Cordes,T.J.W.Lazio,NE2001.1.Anewmodelforthegalacticdistribution offreeelectronsanditsfluctuations,arXiv:astro-ph/0207156.

[30] M. McQuinn, Locating the “missing” baryons with extragalactic dispersion measure estimates, Astrophys. J. 780 (2014) L33, http://dx.doi.org/10.1088/ 2041-8205/780/2/L33,arXiv:1309.4451[astro-ph.CO].