Contents lists available atScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Limit on graviton mass from galaxy cluster Abell 1689
Shantanu Desai
DepartmentofPhysics,IndianInstituteofTechnology,Hyderabad,Telangana- 502285,India
a r t i c l e i n f o a b s t ra c t
Articlehistory:
Received1August2017
Receivedinrevisedform18January2018 Accepted18January2018
Availableonline3February2018 Editor: H.Peiris
To date, theonly limitongravitonmass usinggalaxy clusters was obtainedbyGoldhaber and Nieto in 1974, using the fact that the orbits of galaxy clusters are bound and closed, and extend up to 580 kpc.FrompositingthatonlyaNewtonianpotential givesrisetosuchstableboundorbits,alimit onthegravitonmassmg<1.1×10−29 eV wasobtained[1].Recently,ithasbeenshownthatonecan obtainclosedboundorbitsforYukawa potential[2],thusinvalidatingthemainansatzusedinRef.[1]
toobtainthegravitonmassbound.Inordertoobtainarevisedestimateusinggalaxyclusters,weuse dynamical mass modelsof theAbell1689 (A1689)galaxy clustertocheck theircompatibility witha Yukawagravitationalpotential.Weusethemassmodelsforthegas,darkmatter,andgalaxiesforA1689 fromRefs.[3,4],whousedthisclustertotestvariousalternategravitytheories,whichdispensewiththe need fordark matter.We quantify thedeviations intheaccelerationprofile usingthesemass models assumingaYukawapotentialandthatobtainedassumingaNewtonianpotential bycalculatingtheχ2
residualsbetweenthetwoprofiles.Ourestimatedboundonthegravitonmass(mg)istherebygivenby, mg<1.37×10−29 eV orintermsofthe gravitonComptonwavelength of,λg>9.1×1019 km at 90%
confidencelevel.
©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
1. Introduction
Acentury afterits inception,Generalrelativity(GR)passesall observationaltestsatsolarsystemandbinary pulsarlengthscales withflyingcolors[5–7].Therecentdirectdetectionofgravitational waveshas confirmed the validity of generalrelativity in the dy- namicalstrong-fieldregime[8].Despitethis,awholeslewofmod- ifiedtheories ofgravityhavebeenexplored, eversincethe equa- tionsofGRwerefirstwrittenmorethanacenturyago.Mostofthe recentresurgenceinstudyingandproposingtheplethoraofmod- ifiedtheories of gravity has beendriven by the need to address problemsinCosmologysuchasDarkmatter,DarkEnergy,Inflation, andBaryogenesis[9–18],whichcannotbeexplainedusingGRand theStandardmodelofparticlephysics.Independentofcosmologi- calproblems,anumberofalternatives havealsobeenextensively proposedtoresolveconceptualproblemsinclassicalGRatthein- terfaceoffundamental Physics,such asBig Bangsingularity [19], arrowoftime [20,21], orthe quantizationofgravity [22–24].An updated summary of almost all the modified theories of gravity canbe found in monographs such as[25] and alsoin recentre- views[26–28]andreferencestherein.
E-mailaddress:[email protected].
OnesuchmodificationtoGR,namelymassivegravityinwhich agravitonisendowedwithnon-zeromassdatesbacktomorethan 70years.The firstevertheoryofmassivegravityintheperturba- tivelimitwasproposedbyPauliandFierz[29].However,thisthe- orydoesnotreproducetheGRresult,inthelimitwhenthegravi- ton massgoestozero,usually referredto intheliterature asthe vDVZ discontinuity[30,31].However,Vainshtein showedthat this discontinuity isdue tohow the gravitationaldegrees offreedom are treated during the linearization procedure and can be fixed in a non-linear version of massive gravity [32]. Bouleware and Deser then showedthat thisnon-linearversion has aghost [33].
Therefore, the field of massive gravity theories lay dormant be- causeof theseconceptual problems. However inthe last decade, theBouleware–Deserghostproblemhasbeensolved,leadingtoa resurgence of interest in these massive gravity theories [34–37].
These massive gravity models can address multiple problems in cosmology such as dark energy [38], dark matter [39,40], infla- tion [41]andalso infundamental physicsrelatedto quantization ofgravity[42].
Onegenericfeatureofmassivegravitymodelsisthatthegrav- itationalpotential hasa Yukawabehaviorinthelinearweakfield limit,typicallyparameterizedas[1,43–45]:
V
=
G Mr exp
( −
r/λ
g),
(1)https://doi.org/10.1016/j.physletb.2018.01.052
0370-2693/©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.
whereλg istheComptonwavelengthofthegravitonandisgiven byλg≡mhgc,wheremg isthegravitonmass.
Basically there are three model-independent methods, which have been used to obtain graviton mass bounds [45]. The first methodinvolveslookingforaweakeningofthegravitationalforce due to a Yukawa-like potential. The second type of constraint comesfromlookingforfifthforceinteractions,whichariseinmas- sivegravitymodels.Thethird typeoflimitcomesfromtheprop- agation ofgravitational waves,either due to modified dispersion relationsorfromdifferenceinarrivaltimesbetweengravitational wavesandotherastrophysicalmessengers(photons,neutrinos).In thegravitationalwaveliterature,thelimitsfromthefirsttwotypes ofmeasurements are referred to as“static” bounds, whereas the limitsfromgravitationalwaveobservationsarereferredtoas“dy- namical” bounds. In addition to these three traditional methods, onecan alsoobtainboundson gravitonmassby studyingits im- plications for cosmology, such as large scale structure and late time evolution. But these are strongly model-dependent. In ad- dition, one can also get constraints on Yukawa gravity fromthe weak-fieldlimitofcertain modifiedgravity theoriessuchas f(R) gravityorMoffat’sScalar-Vector-Tensor gravity[46,47].Acompre- hensivesummaryofalltheobservational/experimentalboundson themassofthegravitonaswellasfutureprospectscanbefound inRef.[45]andatabularsummarycanbefoundinTable 1ofthe samepaper.We nowbriefly recapthelimitsfromthesedifferent typesofmethods.
Twoyears ago there was a watershedevent inthe historyof physics, due to the direct detection of gravitational waves from the two LIGO detectors in Hanford and Livingston [48]. These observations enabled us to obtain the most stringent dynamical bounds on the graviton mass, by looking for dispersion in the observed signal as it propagated towards the detectors. The first detectionfromGW150914[48] providedalimitofλg>1013 km, ormg<10−22 eV [49], based on looking for a modified disper- sion relationfora non-zero graviton mass. Subsequently, a more stringent bound of mg<7.7×10−23 eV has been obtained us- ingGW170104[50].WenotehoweverthatDeser[51]haspointed outthat nostrongfield generationofradiationinmassivegravity models can reproduce the observed ringdown patterns observed inLIGO. Fora gravitationalwave sourcein ourgalaxy,one could alsoconstrainthismassusingthelineofsightShapirodelay from thesourceofthegravitationalwave[52].Mostrecently,thedirect detectionof photonsin coincidence withthe gravitationalwaves froma binary neutron star mergerhave enabled usto constrain the mass of the graviton to mg10−22 eV [53]. In the future, eLISA could obtain boundsofmg<10−26 eV [43], andfromthe detectionofinflationarygravitationalwavesfromstageIVCMBex- periments,onecouldgetaboundofmg<3×10−29eV[54].
Manymassivegravitymodelsgiverisetoafifthforce.However, these results are theory dependent andin particular depend on howthe non-linearVainshtein mechanismoperatesinthesethe- ories[45]. However, one commonfeature inthese models isthe existenceofaGalileon-likescalar.Asofnow,theboundsongravi- tonmassinthiscategoryhavebeenobtainedfromthedecoupling limit of DGP [55] and dRGT [56] theories. Data from lunar laser rangingexperimentsgivea massboundofmg<10−30 eV within thecontextofthedecouplinglimitofdRGTtheory[45].Fromthe corresponding decouplinglimit ofDGP, future surveysongalaxy- galaxylensingcouldsetabounduptomg<10−33eV[57].
We finally recap the limits on graviton mass by looking for Yukawa-typefalloffofthegravitationalforce.Thefirstsuchbound was obtained by Hare [58], by assuming that the gravitational forcefromthecenterofthegalaxy isdiminishedby factorofless than 1e.Fromthisargument,amassboundofmg<6.7×10−28eV
wasobtained[58].AsimilarreasoningwasthenextendedbyGold- haberandNietotoextragalacticobservationsofgalaxyclusters[1].
The current best limit (from all the three types of methods) onthemassofagraviton comesfromthemeasurementsofweak lensingcosmic shear[59],obtainedby comparingthe varianceof themodified shearconvergencepowerspectruminmassivegrav- itymodelsto theobserveddata[60].1 Byimposingthecondition thattheobserveddeviationsfromtheCDMpowerspectrumare lessthan1
σ
,alimitofmg<6×10−32eV wasobtained[60].One assumptionhowevermadeinobtainingthislimitisthatthegravi- ton masshasno effectonthe cosmologicalexpansion,growthof structureandalsotheCDMtransferfunction.Furthermore,thereis alsoadegeneracyinthemodifiedpowerspectrumbetweenanon- zero graviton massand other cosmological parameters. To evade thisdegeneracy,the otherparameters weredetermined usingthe valuesofthe powerspectrumatsmaller valuesoftheradius,for which the effect of a non-zero graviton mass is assumed to be negligible.Thesefittedparameterswerethenusedforthelimiton gravitonmassusingthemeasurementsforlargervaluesofthera- dius[60].TheconstraintongravitonmassinRef.[1]usinggalaxyclusters wasobtainedbyassumingthattheorbitsofgalaxiesinclustersare boundaswellasclosedandusingthefactthatthemaximumsep- arationbetweengalaxiesfromtheHolmberggalaxyclustercatalog is about 580 kpc [61]. The limit on graviton mass was obtained by positing e−1≤exp(−
μ
gr), (whereμ
g is thereciprocal ofthe reduced Compton wavelength) and assuming r=580 kpc. This condition implies that there are at most O(1) departures from Newtoniangravityattheedgeofthegalaxycluster.Theestimated limitongraviton massthusobtainedwasμ
g<5.6×10−25cm−1 ormg<1.1×10−29 eV orλg>1020 km.Wenote that thisis a very rough estimate.Thislimit doesnotuseanydynamical mass informationforthegalaxycluster oranyansatzforthepotentials of the different cluster components (gas, galaxies, dark matter).Also,noconfidenceintervalwasgivenforthisupperlimit.
Furthermore,very recentlyit hasbeen shownthat Newtonian gravity is not theonly central force that gives rise to bound or- bits and one can also get bound orbits for potentials which do not satisfy Bertrand’s theorem [2,62]. In particular, Mukherjeeet al.[2]haveshownthatonecangetsingle-particleboundorbitsin aYukawapotential forcertainvalues.Thus,themainedificeupon which thisbound ofmg<10−29 eV hasbeen obtained[1]is no longervalid.
Galaxy clusters are the most massive gravitationally bound objects in the universe and provide an excellent laboratory for studying a diverse range of topics fromgalaxy evolution to cos- mology (for reviews, see [63–66]). In the past two decades a large number of galaxy cluster surveys in the optical [67–70], microwave [71–73], and X-ray [74–79] have come online. These surveys have enabled the discovery of a large number ofgalaxy clusters up to very high redshifts, allowing us to probe a wide rangeofquestionsinastrophysicsandcosmology.
Multipleobservablesfromthesesurveyssuch asgalaxy cluster counts,gasmassfraction,anddynamicsofgalaxieswithinclusters havebeenwidelyusedtodesigntestsandconstrainalargeclassof modifiedtheoriesofgravity,whichdispensewithdarkenergy[66, 80–88].Galaxyclustershavealsobeenusedtoconstrainmodified gravitytheories,whichdispensewithdarkmatter,e.g.,variousin- carnationsofMOND-liketheories,Verlinde’sentropicgravity,Mof- fat’s MOG theory,nonlocal gravity [89–97]. However, despite the wealth ofexquisite multi-wavelengthgalaxy cluster observations,
1 Wenotethatthepaper[60]incorrectlystatesthatthelensingsignalinRef.[59]
isfromaclusterofstarsataroundanaverageredshiftofz=1.2.
no improvement to the initial estimate on graviton mass using galaxyclustershasbeenobtainedafterGoldhaberandNieto’s1974 paper.2 The only related result is by De Martino and Laurentis, whoobtainedaconstraintonavariantoftheYukawagravitational potentialconsideredhere(fromthepost-Newtonianlimitof f(R) gravity),usingthethermalSZEprofileoftheComaclusterfromthe 2013Planckobservations[46,47].Inprinciplehowever,theanaly- sisinthisworkcouldbeextendedtoobtainalimitonthegraviton mass.
Therefore,to thebest of ourknowledge, we are not awareof anydirect constraintongraviton massusinggalaxy clustersfrom completed stage II or ongoing stage III dark energyexperiments, oranyforecastontheestimatedsensitivitytogravitonmassfrom upcoming stage IV experiments such as LSST [98], Euclid [99], WFIRST[100],etc.Thisisdespitethefactthatoneofthekeysci- encedriverfortheseupcomingsurveysistotest modifiedgravity theories[101].
Therefore,torectifythesituationandtoseehowsensitivecur- rent galaxy cluster data is to graviton mass compared to very rough estimatesfromfourdecades ago,we do afirst end-to-end study to obtain a limit on graviton mass using the Abell 1689 galaxycluster, forwhichexquisite multi-wavelengthdataisavail- able,allowing thereconstructionofdetailedmassmodelsforthis clusterinliterature.
Thismanuscriptisorganizedasfollows.Wediscussthedynam- icalmodeling and massestimates in Section 2. Ouranalysis and resultscan befound inSection 3. Weexamine therobustness of ourlimit to differentmass models in Section 4. We concludein Section5.
2. DynamicalmodelingofA1689
WeusethegalaxyclusterAbell1689(hereafterA1689)forour analysis. A1689 is one of the largest and most massive galaxy cluster located at a redshift of 0.18. In the past decade, it has beensubjected to intensive dynamical modeling within the con- textoftheCDM cosmologicalparadigm,usingmulti-wavelength observations from weak and strong lensing, SZE and X-Ray ob- servations[102–105] (andreferencestherein). Theseobservations haveenabledustoobtain estimatesseparately forthedarkmat- ter,gas,andgalaxycomponentsforthiscluster.Mostrecently,this clusterwasextensivelystudiedtoseeifitsavailable dataiscom- patiblewithMOND-liketheories,whichprovideasolutionforthe dark matter problem from a modification of Newtonian gravity andwithouttheneedfordarkmatter[3,4,106].Weusethesame modelingfromthesepaperstotesttowhatextentthedataisvi- able with a Yukawa potential in order to constrain the graviton mass.The firststep inthisprocedure involvesestimatingthe to- talmassofthedifferentcomponentsofthegalaxy cluster,viz.its darkmattercontent,galaxies,andgasintheintra-clustermedium.
WefollowthesameprocedureasinRef.[4].
The totaldark matter mass can be obtainedby assuming the densityobeystheNavarro–Frenk–Whiteprofile[107]andisgiven by[4]:
Mdm
=
4πρ
sr3slog
rs+
rrs
−
r rs+
r,
(2)wherers and
ρ
s representthedarkmatter haloscale radius and scaledensityrespectively.Theyareusuallyobtainedfromtherela-2 Infactthispaperhasnotobtainedanycitationsfromanyothergalaxycluster orcosmologypaper,accordingtotheADSdatabase.Theonlycitationstothispaper arefrom thegravitationalwaveliteratureor reviewpapers,whichconstrainthe massofthegraviton.
tionbetweentheNFWconcentrationparameter(c200)andtheto- talmassataradius200timesthecriticaluniversedensity(M200).
Tocalculatethetotal darkmattermass,weusetheNFW concen- trationparametersforthisclustermeasuredbyUmetsuetal.[104], viz. c200=10.1±0.82, M200=(1.32±0.09)×1015Mh−1, ob- tained using a combinationof weak and stronglensing observa- tions.Massesobtainedfromweakorstronglensingdonotdepend onthedynamicalstateoftheclusterandhencedonotrelyonas- sumptionsofhydrostaticequilibrium.Wehowevernotethatspher- icalsymmetryhasbeenassumedinthedynamicalmassmodeling, whereasthereisobservationalevidencethatthisclusterhastriax- ialsymmetry[104].Althoughthisclusterhasbeenmodeledusing ellipsoidalhalo[104],forthiswork sphericalsymmetry hasbeen assumedthroughout.
The centralgalaxy (oftencalledBCG,which isan acronymfor BrightestClusterGalaxy)massdistributionismodeledbypositing adensitydistributionoftheform[3,102]:
ρ
gal(
r) =
Mcg(
Rco+
Rcg)
2
π
2(
r2+
R2co)(
r2+
R2cg) ,
(3) where Mcg and Rcg representthe BCG massand coreradius re- spectively; Rco represents thecore size.The values forMcg, Rcg, and Rco that we use forour analysiscan be found inRefs. [3,4, 106],whichwe useforouranalysis. Thegasmassisobtainedus- ingacoredSersicprofileandgivenby[3,108]:ρ
gas=
1.
167mpne0exp⎧ ⎨
⎩
kg−
kg 1+
r2 R2g 1/(2ng)⎫
⎬
⎭ ,
(4) wheremp istheprotonmass,ne0 isthecentral electrondensity;Rg representsthe radial extent of the gas; while kg and ng are dimensionlessparameterswhichcontroltheshapeofthegaspro- file.ThevaluesforalltheseparameterscanbefoundinRefs.[3,4].
ThetotalbaryonicmassMbar upto agivenradius R canbefound byintegratingthegalaxyandgasdensityprofilesfromEq.(3)and Eq. (4): Mbar=R
0 4
π
[ρ
gal+ρ
gas]r2dr.We note that these mass estimateshavebeenmadebypositingsphericalsymmetry.Once,wehavecalculatedthemassofthedifferentcomponents, the total acceleration assuming only Newtonian Gravity (ane wt) fromthecenterofthegalaxyclusterisgivenby
ane wt
=
G(
Mdm+
Mbar)/
R2.
(5)3. Results
InordertotesttheviabilityofYukawagravity,thegravitational acceleration can be obtained from the derivative of the Yukawa potential(Eq.(1))andisgivenby[43]:
ayuk
=
G(
Mdm+
Mbar)
R exp
−
Rλ
g1
λ
g+
1R
.
(6)Inthe limitthat mg→0, Eq.(6)willasymptoteto Eq.(5). In ordertotestforthevalidityofthismodified accelerationlaw,we assumethatthetotalmassisthesameasthatinNewtoniangrav- ityandweonly lookfordeviationscomparedto ordinarygravity as a function of distance from the cluster center. This is similar to theapproaches usedto constrain modified theories ofgravity, whichdispensewiththedarkmatterparadigm[3,4,106].
Forthisanalysis,theNFWdensityprofile,theBCGdensitypro- file, and gas density estimates from literature, used to calculate thetotalmasshavebeenobtainedafterpositingaNewtonian po- tential.Strictlyspeaking,thetotalestimatedmasswouldbelarger within the context of Yukawa gravity, because of the weakness
of gravity in a Yukawa potential compared to the corresponding Newtonianone.However,aself-consistentconstraintonthegravi- ton mass is out of the scope of the current paper and at this time wouldrequiresignificant additionalwork fromthe commu- nity, since one wouldneed to determine three unknown density profiles (the darkmatter, gas, galaxy)in additionto the graviton massfromtheobservationaldata.Forthedarkmattercomponent, one would need to do a suiteof N-body simulations in Yukawa gravityasafunctionofgraviton massandthenobtainaparamet- ricestimateasafunctionofgravitonmass.Similarly,thebaryonic masswouldneedtobe determinedassuming hydrostaticequilib- riuminsuch amodifiedpotential. Therefore,becauseofthelarge number of additional free parameters, doing the whole problem self-consistentlyin ordertoobtain morerobust boundsongravi- tonmasswouldposeformidablechallenges.
However, ifthe graviton mass isvery small, the deviations in the totalmass estimatesshould be small compared to thoseob- tainedusingNewtonian gravity andthe errorsinour estimateof graviton mass should be negligible. Furthermore, since we shall only be interested indeviations in theacceleration profile(com- paredtoaNewtonianpotential)forthesamemass,thetotalmass wouldonlybeanormalizationconstantandwouldnotmakeadif- ferencetothefinallimit.Therefore,similartowhatisusuallydone inconstrainingalternategravitytheories,whichdispensewiththe darkmatterparadigm (seee.g.[4]andreferencestherein),weas- sume that the total density profile is the same asin Newtonian gravityandthen lookfordeviationsintheaccelerationprofile as afunction ofdistancefrom thecenterofthe clusterto constrain departures from a standard Newtonian acceleration profile. Due to the above assumption, the limit is of course conservative. In Sect.4,weshallexaminehowthelimitongravitonmasschanges whenvaryingthemassmodelsfortheclusterusedhere.
We also point out that if we posit that dark matter is made upofmassivegravitons(see e.g. [39]),then onlythedarkmatter potentialwouldbemodifiedwhiletheothertermsinthepotential wouldbeunchangedandourlimitswouldbe different. Here,we assume that all the distinct mass components (gas, galaxy, dark matter)uniformly obey the Yukawapotential anddarkmatter is somehypotheticalelementaryparticle,withthesamegravitational lawsasthebaryoniccomponents.
To quantify the deviations between Newtonian and Yukawa gravity as a function of distance from the center of the cluster, weconstructa
χ
2functionalgivenby,χ
2=
N i=1 ane wt−
ayukσ
a 2,
(7)whereane wt andayuk aregivenbyEqs.(5)and(6);
σ
a istheun- certainty in theestimated acceleration.Toget the 90% c.l. upper limiton themassofthe graviton,we findthe thresholdvalueof mgforwhichχ
2>2.71[109],whereχ
2=χ
2−χ
min2 .Wenote thatχ
min2 =0 correspondstoazerogravitonmass.Therefore,χ
2isidenticallyequalto
χ
2fromEq.(7).Since,themassofthegravi- toncannot benegative, mg=0 isaphysicalboundary. Therefore, insuchcasesχ
2 valuesforagivenconfidenceintervalcould in principlegetmodifiedcomparedtothevaluesinRef.[109].Toob- tain the modifiedχ
2, we use the procedure recommended by theParticleDataGroup[110,111],whichhaspreviouslybeenused for neutrino oscillation analysis [112]. The effect of the physical boundaryis determinedby thedifference betweentheminimum value ofχ
2 andthe value ofχ
2 atthe boundaryofthe physical region.Iftheminimumvalueofχ
2 occursatthephysicalbound- ary (which is true in our case), thenχ
2 intervals for a given confidence interval are the same as without a physical bound- ary[111].Therefore,toobtainthe90%confidencelevelupperlimitFig. 1.χ2asafunctionofgravitonmass.Thehorizontallineatχ2=2.71 gives the90%c.l.upperlimitongravitonmassof1.37×10−29eV oralowerlimiton theComptonwavelengthofλg>9.1×1019km.Wenotethatsinceχmin2 =0 for mg=0,thisismathematicallyequivalenttotheχ2functionaldefinedinEq.(7).
Fig. 2. Fractional absolute deviation between acceleration computed assuming Yukawagravity(foragravitonmassofmg=1.37×10−29eV,correspondingtothe 90%clupperlimit)asafunctionofdistancefromthecenterofthecentralgalaxy ofthecluster(usuallyreferredtoasBCG).Thefractionaldeviationisabout10%at 1Mpc.
we choose
χ
2=2.71, whichis the same asthe value without a physicalboundary. Alternately,themodifiedχ
2 thresholdcan alsobeobtainedusingtheFeldman–Cousinsmethod[113],which requires extensive Monte-Carlo simulations. However, they have been showntonot differtoomuch comparedtothe methoduse here[111].We calculated the
χ
2 for24 points betweenroughly 1and 3000 kpc, for which errors in acceleration have been estimated fromexisting observations [3],for whichsphericalsymmetry has been assumed. The first12 points were located atradii between 3and271kpc,forwhichtheerrorsinaccelerationhavebeenes- timated fromthelineofsightmassdensity[108],obtainedusing strong lensing observations [102]. The remaining 12 data points were distributed between125 kpc and3 Mpc and the errors in acceleration were estimated from the weak lensing shear pro- files [104]. We note that the errors in acceleration data do not includeanyerrorsindeterminationoftheradii.χ
2 was thenes- timatedfromEq.(7)forthese24datapointsby calculatingane wt and ayuk at these radii and using the errors in acceleration es- timated in Ref. [3].This plot is shown as a function ofgraviton mass in Fig. 1. The 90% c.l. upperlimit on the mass of a gravi- tonobtainedfromχ
2=2.71,isgivenbymg<1.37×10−29eV, corresponding to a Compton wavelength of λg>9.1×1019 km.Forthisvalue ofmass,we alsoshow the fractionaldeviationbe- tweentheordinaryNewtonianaccelerationandthatassumingthe YukawapotentialinFig. 2.Wecanseethatforthisgravitonmass,
Table 1
Sensitivityofthegravitonmasslimittodifferentmodelsofdarkmatterpotential (differentNFWfits),gasmass,andBCGmass.Thefirsttwocolumnsindicatec200 andM200valuesusedfortheNFWprofiletocalculatethetotaldarkmattermass fromEq.(2).Thenexttwocolumnsindicatethecorrespondingequation(orprofile), fromwhichthegasandBCGmasswasestimated.Thefinalcolumnindicatesthe upperlimitongravitonmass(expressedasamultipleof10−29eV).
M200 (1015Mh−1)
c200 Gas mass BCG mass mg(10−29eV)
1.32 10.1[104] Eq.(4) Eq.(3) <1.37
1.81 5.7[114] Eq.(4) Eq.(3) <1.18
0.83 12.2[116] Eq.(4) Eq.(3) <1.43
1.32 10.1[104] Eq.(4) Hernquist <1.37
thedifferencesare lessthan1% up to 200 kpcandabout10%at about1 Mpc.
4. Effectofsystematicerrors
Wenowexaminethesensitivityofourlimitongraviton mass todifferentmassmodelsforthethreecomponentsofA1689,com- paredtotheresultsintheprevioussection.Atabularsummaryof alltheseupperlimitsonvaryingthemassmodelscanbefoundin Table 1.
Westartwiththedarkmatterpart.Alarge numberofgroups have obtained different NFW parameters for this cluster using weak and strong lensing data. (See Table 9 of Ref. [104].) We firstexaminehow ourresultchangeswithdifferentNFW param- eters from the literature by considering two values of the con- centrationparameter, whichspan thefull rangeof theestimated values and for which spherical symmetry is assumed. The low- est value for c200 forthis cluster corresponds to c200=5.71 for M200=1.81×1015Mh−1 [114]. The corresponding upper limit on graviton mass is at 1.18×10−29 eV. At the other extreme, when we choose c200=12.2 for M200=0.83×1015M, we get mg<1.43×10−29eV.
Eventhough,mostofthemassinthisgalaxyclusterismadeup ofdarkmatter, wedon’t expectanymajorchanges withdifferent BCG orgas mass profiles. Nevertheless, to check this, instead of Eq.(3), weused theHernquistprofile [115] (similarto Ref.[94]) fortheBCGmass withthesameparameters asinRef.[94].With thisnewgalaxy mass profile,the newgraviton mass limitis the sameasbefore.
Therefore,thechangeinthelimitonthegravitonmassbyvary- ing our ansatz for the mass models is less than 15% and does not change the ballpark estimate on the limit on graviton mass ofmg10−29eV.
5. Conclusions
In1974,alimit ongravitonmassofmg<1.1×10−29eV was obtained from galaxy clusters, using the fact that the orbits of galaxyclustersareboundupto580kpc[1]andsuchclosedbound orbitscanonlyexistwithinNewtoniangravity.However,recentlyit hasbeenshownthatonecangetclosedboundorbitsforaYukawa potential[2].Therefore,themainpremiseusedtoobtainthemass boundlimitfromgalaxyclustersinRef.[1]cannolongerbejusti- fiedandthisresultshouldnolongerbequotedintheliterature.
Subsequently, even though a huge amount of work has been doneintestingaplethoraofmodifiedgravitytheorieswithgalaxy clustersusing optical, X-ray, and SZE data, we are not aware of anyotherworkonestimatingaboundongravitonmassfromclus- ters,despiteawealthofnewpreciseobservationaldatainthepast decade,courtesyawholeslewofmulti-wavelengthsurveys.
We obtain a limit on graviton mass fromA1689using an in- dependentmethodcomparedtoRef.[1].Weuserecentdynamical
massmodelsofthedifferentcomponentsofgalaxyclusterA1689, obtainedusingX-ray,weakandstronglensingdata[3,4,106]toob- tainalimitonthegravitonmass.Forthispurpose,weassumethat thepotential dueto thegas,galaxy,anddarkmatter allfollowa Yukawabehavior,duetonon-zerogravitonmass.Wethenlookfor deviationsfromtheestimatedaccelerationdata(assumingvalidity ofNewtonian gravity)andaYukawapotential, andfindthecriti- cal gravitonmassforwhichthe
χ
2 difference betweenthetwo potentialscrosses2.71.Thisgivesusa90%c.l.upperboundonthe gravitonmassofmg<1.37×10−29eV orontheComptonwave- length λg>9.1×1019 km.Wealsocheckedhowthelimitvaries with differentmass models for the darkmatter andBCG poten- tial.Wefindthatthemaximumvariationinthelimitongraviton massisabout15%andthusdoesnotchangetheballparkestimate ofourlimit,whichisO(10−29)eV.We should pointout that thefact that ourupper limit isap- proximately ofthe same orderof magnitudeasthat obtainedby GoldhaberandNieto[1]isonlyacoincidence.Themaximumsize theyassumedforthegalaxyclusterorbitsisabout580kpc,asthis was thesizeofthelargestknownclustersin1974.Usingthises- timateforthesize,theyobtainedanupperlimitofO(10−29)eV, whichisofthesameorderofmagnitudeasours.Inprinciple,one could trivially apply thesame method [1]to some ofthe galaxy superclusters currently known. For example,the recently discov- eredSaraswatisupercluster[117](whosesizeisatleast200Mpc) wouldyieldamorestringentupperlimit onthegravitonmassof about 3×10−32 eV. However, asmentioned earlierthe underly- ingassumptionsbehindthisargumentusedtoobtainthelimitare incorrect.
We howevernote that toobtain ourlimit, mass estimatesfor the different components (dark matter, gas, galaxy) have been obtainedassuming Newtonian gravity, since otherwise thewhole problemofsimultaneouslydeterminingthemassofthethreeun- known components in addition to the graviton mass becomes currently intractable, giventhe large numberof free parameters.
However, inthiscasesince thelimit hasbeen obtainedby using deviationsfromNewtonianaccelerationprofileasafunctionofthe distancefromthegalaxyclustercenterandthetotalmassofeach componentwouldmainlyactasanormalizationconstantandnot makeabigdifferencetoourfinallimit.
GiventhelargenumberofupcomingStage IVdarkenergyex- periments such as LSST [98], Euclid [99], WFIRST [100] etc., it wouldbeinterestingtoestimatetheexpectedimprovementinthe limit on graviton mass compared to the result obtained in this work. We now carry out an order of magnitudeestimate of the same.
Someofourerrorsinaccelerationdata(atradiilessthanabout 150kpc)comefromstronglensingmeasurements[102],whichuse HubbleSpaceTelescope data.Therefore, we donot expectsignif- icant improvements in the strong lensing based error estimates.
The acceleration errors at higher radii are obtained from weak lensing measurements using Suprime-Cam data [104]. For Euclid and other stage IV experiments, the multiplicative bias from the shearmustbelessthan0.1%[99,118].Ifweevaluatetheaccelera- tionerrorsfromthesepredictedshearerrorsfordistancesgreater than150kpcandcombineitwithcurrenterrorsfromstronglens- ing for smaller radii, we expect a 90% confidence upper limit of mg<2.75×10−30eV. Thisisstillnot assensitiveasthecurrent bestlimitongravitonmass[60].However, wecautionthatthisis onlyaballparkestimateofexpectedimprovementsensitivity.More detailedforecastingstudiesneedtobedonebytherelevantwork- inggroupsfromthevariousstageIVdarkenergyexperiments.One key missingingredientneededforthat purposeisthe generation ofN-bodysimulationsinYukawagravityandthecalculationofthe correspondinghalomassfunctions.
Acknowledgements
We are thankful to the anonymous referee for detailed criti- cal feedback on the manuscript. We would like to thank Sabine Hossenfelder for explaining the recent resurgence of interest in massivegravity theories.We aregratefulto TheoNieuwenhuizen forprovidingusthedatain[3]andforcommentsonthedraft.We alsoacknowledge Ivan De Martino, Fred Goldhaber,Alistair Hod- son,andMarkMessierforusefulcorrespondenceanddiscussions.
References
[1]A.S.Goldhaber,M.M.Nieto,Phys.Rev. D9(1974)1119.
[2]R. Mukherjee, S. Sounda, Indian J. Phys. 92 (2) (2017) 197–203, arXiv:
1705.02444.
[3]T.M.Nieuwenhuizen,Fortschr.Phys.65(2017)1600050,arXiv:1610.01543.
[4]A.O.Hodson,H.Zhao,ArXive-prints,arXiv:1703.10219,2017.
[5]C.M.Will,LivingRev.Relativ.17(2014)4,arXiv:1403.7377.
[6]S.G.Turyshev,Annu.Rev.Nucl.Part.Sci.58(2008)207,arXiv:0806.1731.
[7]I.H.Stairs,LivingRev.Relativ.6(2003)5,arXiv:astro-ph/0307536.
[8]N. Yunes, K. Yagi, F. Pretorius, Phys. Rev. D 94 (2016) 084002, arXiv:
1603.08955.
[9]T.Clifton,P.G.Ferreira,A.Padilla,C.Skordis,Phys.Rep.513(2012)1,arXiv:
1106.2476.
[10]K.Koyama,Rep.Prog.Phys.79(2016)046902,arXiv:1504.04623.
[11]A.Joyce,B.Jain,J.Khoury,M.Trodden,Phys.Rep.568(2015)1,arXiv:1407.
0059.
[12]R.Woodard,in:L.Papantonopoulos(Ed.),TheInvisibleUniverse:DarkMatter andDarkEnergy,in:LectureNotesinPhysics,vol. 720,Springer-Verlag,Berlin, 2007,p. 403,arXiv:astro-ph/0601672.
[13]A.Joyce,L.Lombriser,F.Schmidt,Annu.Rev.Nucl.Part.Sci.66(2016)95, arXiv:1601.06133.
[14]J.Martin,C.Ringeval,V.Vennin,Phys.DarkUniverse5(2014)75,arXiv:1303.
3787.
[15]S.Alexander,N.Yunes,Phys.Rep.480(2009)1,arXiv:0907.2562.
[16]N.J.Poplawski,Astron.Rev.8(2013)108,arXiv:1106.4859.
[17]K.Bamba,S.Capozziello,S.Nojiri,S.D.Odintsov,Astrophys.SpaceSci.342 (2012)155,arXiv:1205.3421.
[18]K.Bamba,S.D.Odintsov,Symmetry7(2015)220,arXiv:1503.00442.
[19]S.W.Hawking,R.Penrose,Proc.R.Soc.Lond.Ser.A314(1970)529.
[20]G.F.R.Ellis,Stud.Hist.Philos.Mod.Phys.44(2013)242,arXiv:1302.7291.
[21]T.Padmanabhan,Gen.Relativ.Gravit.42(2010)2743,arXiv:1001.3380.
[22]S.Carlip,Rep.Prog.Phys.64(2001)885,arXiv:gr-qc/0108040.
[23]A.Ashtekar,NewJ.Phys.7(2005)198,arXiv:gr-qc/0410054.
[24]R.P.Woodard,Rep.Prog.Phys.72(2009)126002,arXiv:0907.4238.
[25]C.M.Will,TheoryandExperimentinGravitationalPhysics,1993.
[26]C.M.Will,ArXive-prints,arXiv:1409.7871,2014.
[27]T.Baker,E.Bellini,P.G.Ferreira,M.Lagos,J.Noller,I.Sawicki,ArXive-prints, arXiv:1710.06394,2017.
[28]I.Debono,G.F.Smoot,Universe2(2016)23,arXiv:1609.09781.
[29]M.Fierz,W.Pauli,Proc.R.Soc.Lond.Ser.A173(1939)211.
[30]H.vanDam,M.Veltman,Nucl.Phys.B22(1970)397.
[31]V.I.Zakharov,Sov.J.Exp.Theor.Phys.Lett.12(1970)312.
[32]A.I.Vainshtein,Phys.Lett.B39(1972)393.
[33]D.G.Boulware,S.Deser,Phys.Rev. D6(1972)3368.
[34]C.deRham,G.Gabadadze,A.J. Tolley,Phys.Rev.Lett.106(2011)231101, arXiv:1011.1232.
[35]S.F.Hassan,R.A.Rosen,Phys.Rev.Lett.108(2012)041101,arXiv:1106.3344.
[36]K.Hinterbichler,Rev.Mod.Phys.84(2012)671,arXiv:1105.3735.
[37]C.deRham,LivingRev.Relativ.17(2014)7,arXiv:1401.4173.
[38]Y. Akrami, T.S. Koivisto, M. Sandstad, J. High Energy Phys. 3 (2013) 99, arXiv:1209.0457.
[39]M.Pshirkov, A.Tuntsov, K.A. Postnov, Phys. Rev.Lett.101(2008) 261101, arXiv:0805.1519.
[40]K.Aoki,S.Mukohyama,Phys.Rev. D94(2016)024001,arXiv:1604.06704.
[41]Y. Sakakihara, T. Tanaka, J. Cosmol. Astropart. Phys. 9(2016) 033,arXiv:
1605.05790.
[42]K.A.Milton,Can.J.Phys.92(2014)964,arXiv:1312.4298.
[43]C.M.Will,Phys.Rev. D57(1998)2061,arXiv:gr-qc/9709011.
[44]A.S.Goldhaber,M.M.Nieto,Rev.Mod.Phys.82(2010)939,arXiv:0809.1003.
[45]C.deRham,J.T.Deskins, A.J.Tolley,S.-Y.Zhou,Rev.Mod. Phys.89(2017) 025004,arXiv:1606.08462.
[46]I.DeMartino,Phys.Rev. D93(2016)124043,arXiv:1605.08223.
[47]I.DeMartino,M.DeLaurentis,Phys.Lett.B770(2017)440,arXiv:1705.02366.
[48]B.P.Abbott,etal.,Virgo,LIGOScientific,Phys.Rev.Lett.116(2016)221101, arXiv:1602.03841.
[49]B.P.Abbott,etal.,Virgo,LIGOScientific,Phys.Rev.Lett.116(2016)061102, arXiv:1602.03837.
[50]B.P.Abbott,etal.,VIRGO,LIGOScientific,Phys.Rev.Lett.118(2017)221101, arXiv:1706.01812.
[51]S.Deser,Gen.Relativ.Gravit.48(2016)103,arXiv:1604.04015.
[52]E.O.Kahya,S.Desai,Phys.Lett.B756(2016)265,arXiv:1602.04779.
[53]T.Baker,D.Psaltis,C.Skordis,Astrophys.J.802(2015)63,arXiv:1412.3455.
[54]K.N.Abazajian,etal.,CMB-S4,arXiv:1610.02743,2016.
[55]G.Dvali,G.Gabadadze,M.Porrati,Phys.Lett.B485(2000)208,arXiv:hep- th/0005016.
[56]C.deRham,G.Gabadadze,Phys.Rev. D82(2010)044020,arXiv:1007.0443.
[57]Y.Park,M.Wyman,Phys.Rev. D91(2015)064012,arXiv:1408.4773.
[58]M.G.Hare,Can.J.Phys.51(1973)431.
[59]L.VanWaerbeke,Y.Mellier,M.Radovich,E.Bertin,M.Dantel-Fort,H.J.Mc- Cracken,O.LeFèvre,S.Foucaud,J.-C.Cuillandre,T.Erben,etal.,Astron.As- trophys.374(2001)757,arXiv:astro-ph/0101511.
[60]S.R. Choudhury,G.C. Joshi, S.Mahajan, B.H.J.McKellar,Astropart.Phys. 21 (2004)559,arXiv:hep-ph/0204161.
[61]E.Holmberg,Ark.Astron.5(1969)305.
[62]D.M.Christodoulou,D.Kazanas,ArXive-prints,arXiv:1705.09356,2017.
[63]G.M.Voit,Rev.Mod.Phys.77(2005)207,arXiv:astro-ph/0410173.
[64]A.V. Kravtsov, S. Borgani, Annu. Rev. Astron. Astrophys. 50 (2012) 353, arXiv:1205.5556.
[65]S.W.Allen,A.E.Evrard,A.B.Mantz,Annu.Rev.Astron.Astrophys.49(2011) 409,arXiv:1103.4829.
[66]F.Schmidt,A.Vikhlinin,W.Hu,Phys.Rev. D80(2009)083505,arXiv:0908.
2457.
[67]E.S.Rykoff,etal.,DES,Astrophys.J.Suppl.224(2016)1,arXiv:1601.00621.
[68]E.S. Rykoff, E. Rozo, M.T. Busha, C.E. Cunha, A. Finoguenov, A. Evrard, J.
Hao,B.P.Koester,A.Leauthaud,B.Nord,etal.,Astrophys.J.785(2014)104, arXiv:1303.3562.
[69]M.Radovich,E.Puddu,F.Bellagamba,M.Roncarelli,L.Moscardini,S.Bardelli, A.Grado,F.Getman,M.Maturi,Z.Huang,etal.,Astron.Astrophys.598(2017) A107,arXiv:1701.02954.
[70]M.Radovich,E.Puddu,F.Bellagamba,M.Roncarelli,L.Moscardini,S.Bardelli, A.Grado,F.Getman,M.Maturi,Z.Huang,etal.,Astron.Astrophys.598(2017) A107,arXiv:1701.02954.
[71]L.E.Bleem,etal.,SPT,Astrophys.J.Suppl.216(2015)27,arXiv:1409.0850.
[72]M. Hasselfield, et al., J. Cosmol. Astropart. Phys. 1307 (2013) 008, arXiv:1301.0816.
[73]P.A.R.Ade,etal.,Planck,Astron.Astrophys.581(2015)A14,arXiv:1502.00543.
[74]H. Ebeling, A.C.Edge, H. Bohringer,S.W. Allen,C.S. Crawford,A.C.Fabian, W.Voges,J.P.Huchra,Mon.Not.R.Astron.Soc.301(1998)881,arXiv:astro- ph/9812394.
[75]H. Ebeling,A.C.Edge,J.P.Henry,Astrophys.J.553(2001)668,arXiv:astro- ph/0009101.
[76]H.Böhringer,P.Schuecker,L.Guzzo,C.A.Collins,W.Voges,S.Schindler,D.M.
Neumann,R.G.Cruddace,S.DeGrandi,G.Chincarini,etal.,Astron.Astrophys.
369(2001)826,arXiv:astro-ph/0012266.
[77]A.K.Romer,P.T.P.Viana,A.R.Liddle,R.G.Mann,Astrophys.J.547(2001)594.
[78]R.A.Burenin,A.Vikhlinin,A.Hornstrup,H.Ebeling,H.Quintana,A.Mesch- eryakov,Astrophys.J.Suppl.Ser.172(2007)561,arXiv:astro-ph/0610739.
[79]M.Pierre,etal.,Astron.Astrophys.592(2016)A1,arXiv:1512.04317.
[80]D.Rapetti,S.W.Allen,A.Mantz,H. Ebeling,Mon.Not.R.Astron.Soc. 406 (2010)1796,arXiv:0911.1787.
[81]S.Peirone,M.Raveri,M.Viel,S.Borgani,S.Ansoldi,Phys.Rev.D95(2017) 023521,arXiv:1607.07863.
[82]S.Bocquet,etal.,SPT,Astrophys.J.799(2015)214,arXiv:1407.2942.
[83]L. Pizzuti, et al., J. Cosmol.Astropart.Phys. 1707 (2017) 023,arXiv:1705.
05179.
[84]L.Lombriser,K. Koyama,G.-B.Zhao,B.Li,Phys.Rev. D85(2012) 124054, arXiv:1203.5125.
[85]A.Terukina,L.Lombriser,K.Yamamoto,D.Bacon,K.Koyama,R.C.Nichol,J.
Cosmol.Astropart.Phys.4(2014)013,arXiv:1312.5083.
[86]M. Cataneo,D.Rapetti,F.Schmidt, A.B.Mantz,S.W.Allen, D.E.Applegate, P.L.Kelly,A.von derLinden,R.G.Morris,Phys. Rev. D92(2015) 044009, arXiv:1412.0133.
[87]H. Wilcox,D. Bacon, R.C. Nichol,P.J. Rooney, A. Terukina, A.K. Romer, K.
Koyama,G.-B.Zhao,R.Hood,R.G.Mann,etal.,Mon.Not.R.Astron.Soc.452 (2015)1171,arXiv:1504.03937.
[88]B. Li, J.-h. He, L. Gao, Mon. Not. R. Astron. Soc. 456 (2016) 146, arXiv:1508.07366.
[89]E.Pointecouteau,J.Silk,Mon.Not.R.Astron.Soc.364(2005)654,arXiv:astro- ph/0505017.
[90]S.Rahvar,B.Mashhoon,Phys.Rev. D89(2014)104011,arXiv:1401.4819.
[91]J.R. Brownstein, J.W. Moffat, Mon. Not. R. Astron. Soc. 382 (2007) 29, arXiv:astro-ph/0702146.
[92]S.Ettori,V.Ghirardini, D.Eckert,F.Dubath,E.Pointecouteau,Mon.Not.R.
Astron.Soc.470(2017)L29,arXiv:1612.07288.
