The Train Wreck Cluster Abell 520 and the Bullet Cluster 1E0657-558 in a Generalized Theory of ...

19 Pages·2017·1.12 MB·English

Article
The Train Wreck Cluster Abell 520 and the Bullet
Cluster 1E0657-558 in a Generalized Theory
of Gravitation
NormanIsrael1,*andJohnMoffat2,3
1 DepartmentofPhysics,SchoolofNaturalandAppliedSciences,UniversityofTechnology,Jamaica,
237OldHopeRoad,Kingston6,Jamaica
2 PerimeterInstituteforTheoreticalPhysics,Waterloo,ONN2L2Y5,Canada;[email protected]
3 DepartmentofPhysicsandAstronomy,UniversityofWaterloo,Waterloo,ONN2L3G1,Canada
* Correspondence:[email protected]
Received:19January2018;Accepted:20March2018;Published:26March2018
Abstract: Amajorhurdleformodifiedgravitytheoriesistoexplainthedynamicsofgalaxyclusters.
AcaseismadeforageneralizedgravitationaltheorycalledScalar-Tensor-Vector-Gravity(STVG)
orMOG(ModifiedGravity)toexplainmergingclusterdynamics. Thepaperpresentstheresults
ofare-analysisoftheBulletCluster,aswellasananalysisoftheTrainWreckClusterintheweak
gravitationalfieldlimitwithoutdarkmatter. TheKing-βmodelisusedtofittheX-raydataofboth
clusters,andtheκ-mapsarecomputedusingtheparametersofthisfit. Theamountofgalaxiesin
theclustersisestimatedbysubtractingthepredictedκ-mapfromtheκ-mapdata. Theestimatefor
theBulletClusteristhat14.1%oftheclusteriscomposedofgalaxies. FortheTrainWreckCluster,
iftheJeeetal. dataareused,25.7%oftheclusteriscomposedofgalaxies. Thebaryonmatterinthe
galaxiesandtheenhancedstrengthofgravitationinMOGshiftthelensingpeaks,makingthemoffset
fromthegas. Theworkdemonstratesthatthisgeneralizedgravitationaltheorycanexplainmerging
clusterdynamicswithoutdarkmatter.
Keywords: scalartensorvectorgravity;BulletCluster;TrainWreckCluster
1. Introduction
In the past few decades, observations of galactic rotation speeds [1,2], cluster mergers via
gravitational lensing [3] and the cosmic microwave background [4] have revealed gravitational
anomaliesthatareinterpretedbymanyastronomersasevidencethat26%ofthemassintheuniverse
isunseen. Manyexpectthattheexistenceofthismissingmass(darkmatter)willbeconfirmedby
thediscoveryofnewparticles, whichmaybeinteractingornon-interacting[5]. Therearevarious
candidatesforthis. However,atthistime,experimentshaveturnedupempty[6–8]. Theinterpretation
of the anomalies as a problem of missing mass is based on the assumption that general relativity
isapplicabletolarge-scalestructuressuchasgalaxiesandclusters. Whilegeneralrelativityisvery
wellcheckedatthesolarsystemscale,thereisnoreasontobelievethatitshouldsurvivelarge-scale
structure tests unscathed. With this in mind, with no dark matter particle candidate yet detected
andtheeffectsonlyobservedtobegravitational,theauthorsseeitjustifiedtoinvestigateapossible
modification of general relativity that does not involve the addition of any extra mass. In either
approach,atleastoneextradegreeoffreedomisadded. Thevariousapproachescanbebrokendown
toadoptthefollowingstrategies:
1. Addextramassintheregionswheregravityisstrongest. Themasscouldbe:
(a) Non-interacting(thisisthecaseforcolddarkmatter),
Galaxies2018,6,41;doi:10.3390/galaxies6020041 www.mdpi.com/journal/galaxies
Galaxies2018,6,41 2of19
(b) Interacting(thiscreatesamuchbroaderclasscalledthedarksector),
2. Addextrafieldsthatimplementvariousmechanisms.
Somehavechosentocalltheextradegreesoffreedomdarkmatter,regardlessofwhetherthey
taketheformof(1)or(2). However,relativisticmodificationsofgravityfallintothesecondcategory
andimplementverydifferentstrategiesfromthosethatfallintothefirstcategory. Asaresult,inthis
paper,theauthorshaveonlydecidedtocallCategory(1),darkmatter(i.e.,anytheorythatimplements
thestrategyof(1)is,totheauthors,atheorythatcontainsdarkmatter,whereasthosethatdonotfall
intoCategory(2)andaresaidtopossessnodarkmatter). Afewideashavebeendevelopedtotakethe
secondapproach[9,10]. Someoftheseideasfallunderwhatarecalledmimeticgravitytheories[11–19].
Amajorobstacleformanymodifiedgravitytheoriesisexplainingclusterdynamics. Thisbegsthe
question,doesthereexistamodificationofgeneralrelativitythatcanaddressclusterdynamicswithout
theadditionofextramass(darkmatter)? Inthispaper,wewilldemonstratethattheModifiedGravity
(MOG)theorycalledScalar-Tensor-Vector-Gravity(STVG)[20]hasthepotentialtoexplainmerging
galaxyclusterdynamics,thusansweringthequestionintheaffirmative. Thetheoryisconsistentwith
theLIGOGW170817results[21]thatgravitationalwavesandelectromagneticwavestravelatthesame
speed[22]. Recallthattherearethreekeyareaswheretheanomaliesshowup:
• Galacticrotationcurves
• Thedynamicsofgalaxyclusters
• Thecosmicmicrowavebackground
Anymodificationofgeneralrelativitythatisintendedtosolvethe‘missingmass’anomalywithout
darkmattermustaddressthesethreekeyareaswheretheanomaliesshowup. Anearliermodified
gravitationaltheoryhasalreadybeenshowntoresolveanomaliesofgalaxyrotationspeed[23–25],
andSTVGhassucceededinthisrealm,aswell.
TwokeyaspectsofSTVGarethefollowing:
1. GravityisstrongerthanisdepictedinGR.
2. Arepulsivegravitationalforcemediatedbyaspin-onevectorfield(φ)screensgravity,makingit
appearweakatsmalldistancescalessuchasthesolarsystemandbinarypulsars.
In addition to its success in explaining galaxy rotation curves, STVG has shown success in
cosmology. InSTVGcosmology,thereisanerawhereφbecomesdominant. Thiseraisthenfollowed
bythelaterdominanceofmatterandthelaterenhancementof G,whichinturnisfollowedbyan
acceleratederadrivenbythecosmologicalconstantΛ. Thisreproducestheacousticpowerspectrum,
aswellasthematterpowerspectrum. STVGcosmologyisexplainedfurtherin[26–29].
Thispaperisfocusedonclusterdynamics. ThestudyusestheBulletClusterandTrainWreck
Clusterastestcasesforthis. InSTVG,(1)isnoticeableatgalacticandgalaxyclusterdistancescales,
while(2)guaranteesagreementwiththesolarsystemandbinarypulsarsystemsandisbuiltintothe
Newtonianaccelerationlaw. Recallthatdarkmattermodelsassumegeneralrelativityisfineasitis
andaddsextramasstoproducetheextragravityneededtoexplainthegalaxyrotationcurvesand
galaxyclusterdynamics. AsthegravityinSTVGissourcedonlybyordinarymatter,itisconsidered
analternativetodarkmattermodels. Thekeymodificationtothegeneralrelativitytheorythatleads
toSTVGisaspin-onemassiveparticlecalleda“phion”. Thephionfieldinteractswithordinarymatter
withgravitationalstrength,andthefitsofthetheorytorotationcurvesandclusterdynamicsyielda
massforthephionofm =2.6×10−28eV.
φ
In2006,apaperbyCloweetal.[30]claimedthattheyhaddiscovereddirectevidencefordark
matterthroughaclustermergerknownastheBulletCluster(BC).TheBulletClusterisamergeroftwo
clusterswherethegasesinbothclustersinteractasthemergeroccurs. Thegasslowsdownandheats
upwhileemittingX-rays. Thegalaxiesdonotinteractmuch,movingtothesidesofthegas. Asaresult,
oneendsupwithasystemthathasthegasinthemiddleandthegalaxiesofftothesides. Figure1[30]
showsthelensingmap(blue)superimposedonthegas(red). Noticetheoffsetofthelensingmap
Galaxies2018,6,41 3of19
from the X-ray gas. It is this offset that resulted in the authors claiming a direct detection of dark
matter,asoneexpectsthelensingmaptobeonthegasandnotoffsetfromit. Thecommonexplanation
forthisinthecontextofdarkmatteristhatthereisnon-interacting(cold)darkmatterpresentinthe
sub-cluster(pinkshockwavefrontontherightofFigure1)andthemaincluster(pinkregiononthe
leftofFigure1). Whenthemergeroccurred,thedarkmatterfromthemainandsub-clusterspassed
through each other moving off to the sides (blue regions of Figure 1) and centred on the galaxies.
Whiletheresultsarenotbeingquestionedbytheauthors,thedarkmatterexplanationis. Theauthors
willshowinthispaperthatthereisanexplanationinthecontextofSTVGwithoutpostulatingthat
extraexoticmassissourcingthegravity. Rather,theapparentextragravityisduetothesignificant
decreaseintheshieldingeffectoftherepulsivevectorfieldrelativetotheincreaseofthestrengthof
gravityforthesizesofclusters. Inaddition, theoffsetsofthelensingpeaksareduetothebaryon
matterinthegalaxiesshiftingthemawayfromthegas.
Figure1. κ-map(blue)superimposedonthegas(pink)fortheBulletCluster. Theblueregionsare
theregionswiththelargestamountofgravityandarewheredarkmatterissaidtobeconcentrated
whenviewedintheEinsteinianparadigm; theredregionisthevisiblemassobtainedfromX-ray
observations.ThisimageisaChandraX-rayObservatoryimage.
Inalaterdevelopment,Mahdavietal.[31]analysedtheclustermergerknownasAbell520(Train
WreckCluster), claimingalensingdistribution, whichchallengesthenotionthatcolddarkmatter
isallthatisneededtoexplainclusters. WegiveinFigure2someplotsfroma2014studydoneby
Jeeetal. [32]. Figure2bshowsthecontoursfortheκ-mapsuperimposedoverthephotonmap(red).
Figure2ashowsthelensingmapfromJeeetal. Thereisatotalofsixlensingpeaks, oneofwhich
(P3(cid:48))coincideswiththegas. Thisisnotexpectedinamergingsystemifonethinksofthesystemas
dominatedbycolddarkmatter,ascolddarkmatter(beingnon-interacting)isexpectedtoproduce
a lensing peak offset from the gas in all cases. In the present study, we will explain all six peaks.
ThepresenceofapeakcoincidingwiththegaswasreaffirmedbyastudyperformedbyJeeetal.[33].
Cloweetal.[3]disputedtheexistenceofthispeak,butamorerecentdetailedstudybyJeeetal.[32]
comparing the observational data used by Clowe et al. with theirs reaffirmed the existence of the
significantlensingpeakatthegaslocation. SuchacentralpeakisnaturallyincorporatedinSTVG.
Thepeaksthatareoffsetfromthegasareagainduetothegalaxiescontainingbaryonsshiftingthem
awayfromthegas.
Galaxies2018,6,41 4of19
Finally,apaperbyHarveyetal.[34],whichplacedfurtherconstraintsonadarkmatterinteracting
cross-section,includedinitsstudytheAbell520cluster. Thispaperdidnotcontainacentraldark
peak. Thisseemstoleavetheissueregardingtheexistenceofthispeakunresolved. Thetaskofthis
paperisnottoresolvethecontroversyofwhetherornotthecentralpeakexists,buttorathermakea
preliminaryattemptatdemonstratingthatitispossibleforScalarTensorVectorGravity(STVG)to
explainthissystemifthepeakispresent.
(a)κmapdata (b)κmapcontoursongas
Figure2.Abell520(TrainWreckCluster)κ-map.Onepeak(P3(cid:48))coincideswiththegas(red).
WestartwiththeweakfieldapproximationoftheSTVGfieldequationsandusetheKing-βmodel
tofittheX-raydata. Wethenusetheparametersofthefittocomputetheκ-mapforbothsystems.
BrownsteinandMoffathavealreadydoneananalysis[35]ofthebulletclusterusingapreviousversion
ofamodifiedgravitationaltheory[23],showingthatthetheorycanexplaintheBulletClusterdata
withoutexoticdarkmatter. However,thisstudyappearstohavelackedaclearexplanationofthe
offsetofthelensingpeaksfromthegas. WeperformedareanalysisoftheCloweetal. datausing
themorerecentSTVGtheory[20,36]andattemptedtosuggestaclearexplanationoftheoffsetofthe
lensingpeaks. TheresultsareconsistentwiththeresultsoftheBrownstein–Moffatstudy, andthe
suggestedoffsetofthepeaksisthatthebaryonsinthegalaxiesshiftedthepeaksfromthegasasthey
movedtothesides. WebelievethatthisstudythereforeshowsthatSTVGprovidesaviablealternative
tothecolddarkmatterexplanation.
InadditiontoreaffirmingtheBulletClusterresults,theauthorsinvestigatedtheclustermerger
Abell520inthecontextofSTVG.TheAbell520studyuseslensingdatafromtheJeeetal. studies[32].
We show that the lensing peak centred on the gas that has defied a cold dark matter explanation
iseasilyincorporatedinSTVG.AswiththeBulletCluster,thepeaksthatareoffsetfromthegasare
duetothegalaxiesthatmovedtothesides.
2. Scalar-Tensor-Vector-Gravity
STVG (usually termed Modified Gravity (MOG)) is a modification of general relativity with
amassivevectorfieldφ ,correspondingtoarepulsivegravitationalforcewithchargeQ .Theaddition
µ g
ofthisvectorfieldleadstoamodificationofthelawofgravitationbeyondsolarsystemscales.Acentral
featureofthetheoryisthedynamicalgravitationalcouplingG. Thiscouplingvariesbothspatially
andtemporally. Inaddition,thetheoryintroducesthescalarfieldµ,whichisthemassofthevector
field. Thetheoryhastwo(2)degreesoffreedom: αandµ.
Galaxies2018,6,41 5of19
2.1. STVGFieldEquations
TheMOGactionisgivenby:
S = S +S +S +S , (1)
G φ S M
whereS isthematteractionand:
M
S = 1 (cid:90) d4x(cid:112)−g(cid:20)1(R+2Λ)(cid:21), (2)
G
16π G
S = (cid:90) d4x(cid:112)−g(cid:20)−1BµνB + 1µ2φµφ (cid:21), (3)
φ µν µ
4 2
S = (cid:90) d4x(cid:112)−g(cid:20) 1 (cid:18)1gµν∂ G∂ G−V (cid:19)+ 1 (cid:18)1gµν∂ µ∂ µ−V (cid:19)(cid:21), (4)
S G3 2 µ ν G µ2G 2 µ ν µ
where B = ∂ φ −∂ φ andV andV arepotentials. Notethatwechooseunitssuchthatc = 1,
µν µ ν ν µ G µ
usingthemetricsignature[+,−,−,−]. TheRiccitensoris:
R = ∂ Γλ −∂ Γλ +Γλ Γσ −Γσ Γλ . (5)
µν λ µν ν µλ µν λσ µλ νσ
The matter stress-energy tensor is obtained by varying the matter action S with respect to
M
themetric:
Tµν = −2(−g)−1/2δS /δg . (6)
M M µν
VaryingS +S withrespecttothemetricyields:
φ S
Tµν = −2(−g)−1/2[δS /δg +δS /δg ]. (7)
MOG φ µν S µν
Combiningthesegivesthetotalstress-energytensor:
Tµν = Tµν+Tµν . (8)
M MOG
TheMOGfieldequationsaregivenby:
G −Λg +Q =8πGT , (9)
µν µν µν µν
(cid:18) (cid:19)
√1 ∂ (cid:112)−gBµν +µ2φν = −Jν, (10)
−g µ
∂ B +∂ B +∂ B =0. (11)
σ µν µ νσ ν σµ
Here,G = R − 1g RistheEinsteintensorand:
µν µν 2 µν
2 1
Q = (∂αG∂ Gg −∂ G∂ G)− ((cid:50)Gg −∇ ∂ G) (12)
µν G2 α µν µ ν G µν µ ν
isatermresultingfromthethepresenceofsecondderivativesof g in RinS . Wealsohavethe
µν G
fieldequations:
(cid:50)G = K, (cid:50)µ = L, (13)
where(cid:50)= ∇µ∇ ,K = K(G,µ,φ )andL = L(G,µ,φ ).
µ µ µ
Galaxies2018,6,41 6of19
Combining the Bianchi identities, ∇ Gµν = 0, with the field Equation (9) yields the
ν
conservationlaw:
1 1
∇ Tµν+ ∂ GTµν− ∇ Qµν =0. (14)
ν ν ν
G 8πG
It is a key premise of MOG that all baryonic matter possesses, in proportion to its mass M,
positive gravitational charge: Q = κ M. This charge serves as the source of the vector field
√ g √g
φµ. Moreover, κ = G−G = αG , where G is Newton’s gravitational constant and
g N N N
α = (G−G )/G ≥0. Variation of S with respect to the vector field φµ then yields the MOG
N N M
four-current J = −(−g)−1/2δS /δφµ.
µ M
Forthecaseofaperfectfluid:
Tµν = (ρ+p)uµuν−pgµν, (15)
M
where ρ and p are the matter density and pressure, respectively, and uµ is the four-velocity of an
elementofthefluid. Weobtainfrom(15)anduµu =1thefour-current:
µ
J =κ T uν =κ ρu . (16)
µ g Mµν g µ
Itisshownin[37]that,withtheassumption∇ Jµ =0,(14)reducesto:
µ
∇ Tµν = BµJν. (17)
ν M ν
2.2. STVGWeakFieldApproximation
Intheweakfieldapproximation,weperturbg abouttheMinkowskimetricη :
µν µν
g = η +(cid:101)h . (18)
µν µν µν
Theconditiong gµρ = δρ demandsthatgµρ = ηµρ−(cid:101)hµρ.
µν ν
Thefieldequationforthevectorfieldφµ isgivenby:
∂ Bµν+µ2φµ = −Jµ. (19)
ν
Assuming that the current Jµ is conserved, ∂ Jµ = 0, then in the weak field approximation,
µ
wecanapplythecondition∂ φµ = 0. Inthestaticlimit,oneobtainsforthesource-freespherically
µ
symmetricequation:
∇2φ0−µ2φ0 =0, (20)
whichhasthepointparticlesolution:
exp(−µr)
φ (r) = −Q , (21)
0 g
r
√
wherethechargeQ = αG M =κ M. Thissolutioncanbewrittenforadistributionofmatteras:
g N g
(cid:90) e−µ|(cid:126)x−(cid:126)x(cid:48)|
φ = − J ((cid:126)x(cid:48))d3x(cid:48), (22)
0 |(cid:126)x−(cid:126)x(cid:48)| 0
where:
(cid:90)
Q = J d3x. (23)
g 0
Galaxies2018,6,41 7of19
Weassumethatintheslowmotionandweakfieldapproximationdr/ds ∼ dr/dtand2GM/r (cid:28)1,
thenfortheradialaccelerationofatestparticle,weget:
d2r GM qgQgexp(−µr)
+ = (1+µr), (24)
dt2 r2 m r2
whereq =κ m. Forthestaticsolution(21)q Q /m = αG M,andG = G (1+α),whichyieldsthe
g g g g N N
accelerationlaw:
(cid:26) (cid:20) (cid:18) (cid:19)(cid:21)(cid:27)
G M r
a(r) = − N 1+α 1−exp(−r/r ) 1+ , (25)
r2 0 r
0
wherer =1/µ. Weobservethattheaccelerationofaparticleisindependentofitsmaterialcontent
0
(weakequivalenceprinciple). Theaccelerationlawcanbeextendedtoadistributionofmatter:
a((cid:126)x) = −G (cid:90) d3(cid:126)x(cid:48)ρ((cid:126)x(cid:48))((cid:126)x−(cid:126)x(cid:48))[1+α−αe−µ|(cid:126)x−(cid:126)x(cid:48)|(1+µ|(cid:126)x−(cid:126)x(cid:48)|)]. (26)
N |(cid:126)x−(cid:126)x(cid:48)|3
Wecanwrite(26)as:
(cid:90) ρ((cid:126)x(cid:48))((cid:126)x−(cid:126)x(cid:48))
a((cid:126)x) = − G((cid:126)x−(cid:126)x(cid:48)) d3x(cid:48), (27)
|(cid:126)x−(cid:126)x(cid:48)|3
wheretheeffectivegravitationalcouplingstrengthisgivenby:
G((cid:126)x−(cid:126)x(cid:48)) = G [1+α−αe−µ|(cid:126)x−(cid:126)x(cid:48)|(1+µ|(cid:126)x−(cid:126)x(cid:48)|)]. (28)
N
From Equation (28), we see that G is not, in general, constant. However, for sufficiently
large systems, such as galaxy clusters, it turns out that G behaves as if it is constant.
The STVG fits to galaxy rotation curves, and galaxy cluster dynamics have yielded the best fit
valuesα =8.89±0.34 and µ =0.042±0.004kpc−1[36]. The value for µ corresponds to the vector
field mass m =2.6×10−28eV. The Tully–Fisher law relating the galaxy luminosities to the flat
φ
rotationcurvesisdeducedfromtheSTVGdynamicsinexcellentagreementwiththedata[36]. Inthe
fitstothedynamicsofgalaxyclusters[36],theYukawarepulsiveexponentialtermproducedbythe
vector field φ does not have a significant effect when one uses µ−1 = 24 kpc. As a result, one is
µ
onlyleftwithG = G (1+α),whichisusedtocomputetheκ-convergencelensingmapfortheBullet
N
ClusterandTrainWreckCluster.
2.3. GravitationalLensinginSTVG
From[38],thePoissonequationwiththelensingpotentialgives:
D D 8πG Σ
∇2ψ = L LS NΣ =2 , (29)
θ D c2 Σ
S c
where:
Σ(x,y)
=κ(x,y), (30)
Σ
c
and:
Σ(x,y) = (cid:90) zout ρ(x,y,z)dz, (31)
−zout
(cid:113)
withz = r2 −x2−y2,with:
out out
(cid:34)(cid:18) ρ (cid:19)2/(3β) (cid:35)1/2
r =r 0 −1 , (32)
out c 10−28g/cm3
Galaxies2018,6,41 8of19
where10−28g/cm3isthetotaldensityoftheclusteratr . Onehas:
out
Σ(x,y) = ∑n (cid:90) zout,i ρ (0)(cid:34)1+ xi2+y2i +z2i (cid:35)−3βi/2dz, (33)
i=1 −zout,i i rc2i
whereiistheindexrepresentingthei-thcluster,Σc ≈3.1×109M(cid:12)kpc−2fortheBulletClusterand
Σc ≈ 3.8×109 M(cid:12)kpc−2 fortheTrainWreckCluster. κ isameasureofthecurvatureofspace-time.
InSTVG,wehavethesurfacedensitydefinedas[35]:
(cid:90)
Σ¯(x,y) = (1+α) ρ(x,y,z)dz, (34)
sothatforSTVG:
Σ¯(x,y)
φ(x,y) = . (35)
Σ
c
Inthispaper,Equation(35)isfirstcomputed. Thisisnottheentireκmap,asthisonlyconsiders
thecontributionofthegas. Inordertogettheentireκmap,onemustalsoconsiderthecontributionof
thegalaxies:
Σ¯(x,y)+Σ¯ (x,y)
gal
κ(x,y) = . (36)
Σ
c
Whenwesolve(36)forthegalaxies,weget:
κ(x,y)Σ −Σ¯(x,y)
Σ (x,y) ≈ c . (37)
gal (1+α)
WethencomputethemassofthegalaxiesbyintegratingoverΣ (x,y):
gal
(cid:90) (cid:90)
M = Σ (x,y)dxdy. (38)
gal gal
Inthenextfewsections,theresultsoftheBulletClusterandTrainWreckClusterstudieswill
bediscussed. ThesestudiesweredonebystartingwiththeKing-βmodel. Wewilldiscussthisfirst.
Theκ-mapcomputedinbothcasesusingSTVGisshowntobeconsistentwiththeBulletClusterand
TrainWreckClusterdata.
3. TheKing-βModel
Inthefollowingsections,therecentworkdonetochecktheresultsof[35]willbediscussed,as
wellasworkdoneby[31]. Thisstudywasdonefirstbyassuminganalmostisothermalgasspherefor
theBulletClusterandTrainWreckCluster. Withthisassumption,weresorttotheKing-βmodel[39].
Wethushave:
ρ(r) = ∑n ρ (0)(cid:34)1+(cid:18)ri (cid:19)2(cid:35)−3βi/2, (39)
i
r
i=1 ci
where ρ(0) is the central density for a given X-ray peak and r is the radial distance for that peak.
i
Usingthefactthatgalaxyclustershaveafinitespatialextent,onecanconnecttheKing-βmodelwith
databyintegratingEquation(39)alongtheline-of-sight,givingthetotalsurfacedensity(31):
Σ(x,y) = ∑n Σ (0)(cid:34)1+ xi2+y2i (cid:35)−(3βi−1)/2, (40)
i r2
i=1 ci
Galaxies2018,6,41 9of19
whereΣ (0)isthecentralsurfacemassdensityforaparticularpeak. Weassume:
i
 
x
X = y. (41)
 
z
(cid:113)
Thecentreofthei-thclusterisX andX = X−X ,andr = X2.AderivationofEquation(40)
0,i i 0,i i i
isgiveninAppendixA.
WefitEquation(40)totheΣ-mapfortheBulletClusterdata. Theparametersarethenusedto
computetheκ-map. IncaseswherethegeometryiscomplicatedsuchaswiththeTrainWreckCluster,
wefittheX-raypeakswithsurfacebrightnessprofilesgivenby:
I(r) = ∑n I(0)(cid:34)1+(cid:18)ri (cid:19)2(cid:35)−3βi+1/2, (42)
i
r
i=1 ci
where I(0) is the central brightness for a given X-ray peak. The parameters of this fit are used to
i
computetheκ-mapofthesystem.
4. ResultsandAnalysis
Inthissection,wewilldiscussindetailthestudiesdoneonboththeTrainWreckClusterand
BulletClusterinthecontextofSTVG.Wefirststartbydiscussingtheresultsofthebulletcluster,fitting
thelensingpeaksonlyusingthegasfromthemainclusterandsub-cluster,aswellasthegalaxiesto
thesides.
Finally,wediscussworkdoneontheTrain-WreckCluster. Wefocusonthecentrallensingpeak
(centredonthegas)givenby[32]. Recallthattheresultsfrom[32]showthatthereisalargerthan
expectedlensingpeakatthecentralgas,whereasotherstudies,suchas[34],donotshowthispeak.
Ourintentisnottoresolvethiscontroversy,butrathertodemonstratethatthecentralpeakisnaturally
incorporatedintoSTVG.
RecallthattheYukawarepulsiveexponentialtermproducedbythevectorfieldφ doesnothavea
µ
significanteffectwhenoneusesµ−1 =24kpc. Asaresult,oneisonlyleftwithG = G (1+α),which
N
isusedtocomputetheκ-convergencelensingmapfortheBulletClusterandtheTrainWreckCluster.
4.1. AnalysisofBulletClusterUsingSTVG
TheBulletClusterisamergingsystemlocatedatz = 0.296intheconstellationCarina. Itwas
firststudiedin2006byCloweetal.[30]. Thesystemisadramaticdemonstrationofagravitational
anomalywherealensingpeakisclearlyseparatefromthevisiblegas. Thisdistinctseparationofthe
visiblematterfromthelensingpeaksisconsideredbymanyasevidencethatthelensingpeaksneedto
besourcedwithextramasseveninthecaseofamodifiedgravitytheory. Inthissection,wewillshow
thatthissystemisnaturallyexplainedbyamodifiedgravitytheorywithouttheneedfortheaddition
ofextramass.
Inthefollowing:
1. Wefitthemainandsubcluster(simultaneously)oftheΣ-mapwith(40),usingaleastsquares
fittingroutine. Thisdescribesadouble-βmodel. Fromthefitting,weobtaintheparameters β,
Σ andr .
0 c
2. Wethenusetheparametersfromthedouble-βmodeltocomputethegalaxyamountusing(34)
and(35)toobtainareasonableα.
3. Usingthisα,wecomputetheκ-map.
WecomparetheresultwiththatofBrownstein–Moffat[35]. Wefindthattheamountofgalaxies
accountforamassofabout14.1%usingthedouble-βmodel. Thiswasobtainedusinganα = 2.32.
Galaxies2018,6,41 10of19
Theresultisonly3%differentfromtheestimateobtainedin[35],whichgavea17%estimate(usingan
α =2.82). Relyingontheβ-modelassumessphericalsymmetry. Webelievethereisnoissuewiththis
forthisstudybecausein[35],itwasdemonstratedthatthisassumptionleadstoaverygoodprediction
ofthetemperatureofthemainclusteroftheBC.Theyobtainedaresultof15.5±3.9keV,whichisvery
goodwhencomparedwiththevalueof14.8+2.0keVobtainedempirically. Theircalculatedvaluewas
−1.7
thensuccessfullyusedtoreproducethemassprofileofthemainclusterICM.
4.1.1. Assumptions
Inthisstudy,wemadethefollowingassumptions:
1. ThegashasanisothermaldistributionandcanbemodelledviatheKing-βmodel(fromSection3).
2. AllthevisiblematterinthesystemconsistsonlyoftheX-ray-emittinggasandthegalaxies.
ThefirstassumptionallowsameansoffittingtheX-raydata(Σ-map)withtheKing-βmodel,
whilethesecondassumptionallowsustocomputetheκ-mapfromtheX-raydataonly. Thegalaxies
arethentakentobesufficienttoexplaintheremaininggravity,aswellastheoffsetofthelensingpeaks
fromthegas. Weemphasisethatthefirstassumptionshouldbevalidforthemaingasplasma,forthe
assumptionofgasequilibrium(withtheexceptionoftheshockwavepart)allowsfortheabovestated
verygoodpredictionofthetemperatureofthemainclusterinagreementwiththeobservationaldata.
4.1.2. TheΣ-MapfortheBulletCluster
ThesurfacedensitymapdatafortheBulletClusterisshowninFigure3a. Itisa185×185px2
imagewithamassresolutionof1015M(cid:12)/px2. Thefarthestregionontherightisthesub-cluster,while
theregionontheleftisthemain-cluster. A2DprojectionofFigure3awiththefittingaccordingtothe
King-βmodelisshowninFigure3b. Itspans185pixelsor1572.5kpc(usingthefactthatfortheΣ-map,
thedistanceresolutionisabout8.5kpc/px). Thedataarerepresentedbythebluepoints. Thereare
twopeaks,thewidestbeingthemain-cluster,andthesmallestisthesub-cluster. Theblackcurveis
thefitusingtheBrownstein–Moffatparameters,whicharegivenforasingle-βmodelforthemain
clusterinTable1. Thegreencurveisthefitusingthedouble-βmodelparametersobtainedinthis
study. ThisisshowninTable1forthedouble-βmodel(IM,main+sub-cluster).
1e 1
1.0
Israel-Moffat (IM) parameters
Brownstein-Moffat (BM) parameters
Data
0.8
0.6
c
/
0.4
0.2
0.0
0 200 400 600 800 1000 1200 1400
x (kpc)
(a)Σmapdata (b)STVGprediction
Figure3.SurfacedensitymapoftheBulletCluster(BC)(left)and2Dprojectionofthesurfacedensity
plot(right).Thesubclusteristheshockwavefrontontheright(intopplot),whilethemainclusteris
ontheleft(topplot).Thecolourscalecorrespondstothemassinunitsof1015M(cid:12)andhasaresolution
of8.5kpcpixel−1.ThisisbasedonaredshiftmeasurementoftheBCof260kpcarcmin−1.Inthe2D
projection(bottom),thegreendottedcurveisthefittothemainandsub-clusterdata(bluepoints)
asadouble-βmodel. Theblackdottedcurveisthesingle-βmodelfitusingtheBrownstein–Moffat
parameters.STVG,Scalar-Tensor-Vector-Gravity.