Experimental limits to the density of dark matter in the solar system

inthesolarsystem

ØyvindGrøn

OsloCollege,FacultyofEngineering,CortAdelersGate30,N-0254Oslo,NorwayInstituteofPhysics,UniversityofOslo,P.O.Box1048,N-0316Blindern,Oslo3,Norway

arXiv:astro-ph/9507051v1 13 Jul 1995HaraldH.Soleng

CERN,TheoryDivision,CH-1211Geneva23,Switzerland

(16November1994;revised8June1995)

Abstract

Onthescalesofgalaxiesandbeyondthereisevidenceforunseendarkmatter.Inthispaperwefindtheexperimentallimitstothedensityofdarkmatterboundinthesolarsystembystudyingitseffectuponplanetarymotion.SubjectHeadings:Gravitation—DarkMatter(solarsystemtests)PACSnumbers:04.20.-q95.35.+d

ToappearintheAstrophysicalJournal

TypesetusingREVTEX

1

1.INTRODUCTION

AccordingtoNewton’sinversesquareforcelaw,thecircularspeedaroundanisolatedobjectofmassMshouldbe

vc=

󰀇

r

.

Indiskgalaxieswedo,however,observethatthecircularspeedsareapproximatelyindepen-dentofratlargedistances.Thestandardexplanationisthatthisisduetohalosofunseenmatterthatmakesuparound90%ofthetotalmassofthegalaxies(Tremaine1992).Thesamepatternrepeatsitselfonlargerandlargerscales,untilwereachthecosmicscaleswhereabaryonicdensitycompatiblewithsuccessfulbigbangnucleosynthesisislessthan10%ofthedensitypredictedbyinflation,i.e.thecriticaldensity.

Theflatrotationcurvesofgalaxies,takenatfacevalue,implythattheeffectivegravi-tationalforcefollowsa1/rlawatlargescales.ThiscouldeitherbeduetodarkmatterortoadeparturefromNewtoniandynamicsatsmallaccelerations(Milgrom1983;Bekenstein1992)orlargescales(Sanders1990).Aneffectivegravitationalaccelerationlawoftheform

√g=−

ratsmallaccelerationsa≪a0hasbeenreported(Kent1987;Milgrom1988;Begeman,Broeils,&Sanders1991)tobesuccessfulinreproducingtheobservationsofgalacticsys-tems.1Theconstanta0hasbeendeterminedbystudiesofgalaxyrotationcurvesanditsvaluehasbeenfoundtobea0≈10−8cms−2.AsnotedbyMilgrom(1983),thisvalueofa0≈cH0.

Withsucha1/rforcelawthecircularspeedwouldapproachvc=(GMa0)1/4.IftheluminosityLofagalaxyisproportionaltoitsmassM,thenthisrelationwouldexplaintheinfraredTully–Fisherlaw(Tully&Fisher1977)whichstatesthatcircularspeedsingalaxiesscaleasvc∝L1/4.

Thetheoreticalunderpinningforthe1/reffectiveforcelawisnotyetfirmlyestablished.ItmightbeduetoamodificationofgravityalongthelinesofMilgrom(1983),butitseemstobedifficulttoconstructaviablerelativistictheoryofthiskind(ZhytnikovandNester,1994).Accordingly,thestandardviewisthattheeffectivegalactic1/rforcelawiscausedbydarkmatter.Atthispoint,itisworthmentioningthatalarge-distanceforcelawofthistypecanbereproducedwithinstandardgeneralrelativitytheorywithaverysimple,butperhapsunrealistic,mattersource(Soleng1993,1995).Ourkeypointisthatgeneralrelativityisquitecapableofexplainingtheobservedgravitationalpropertiesoftheuniverseprovidedwegiveittherightinput.Mostlikelythedarkmatterisamixtureofseveralcomponents,suchasweaklyinteractingparticles,blackholes,browndwarfs,neutronstars,aswellasenergystoredinhigh-frequencyoscillationofNewton’sgravitationalcoupling(Accetta&Steinhardt1991;Steinhardt&Will1994).Whatevertheoriginofthe1/rforcelawmight

be,itsreportedexperimentalsuccessforcesustotakeitseriously.Accordingly,wethinkthatitisparticularlyimportanttocomparethedensitiesofdarkmatterinferredfromlargescaledynamicswithexperimentallimitsfromlocaltests.Ifdarkmatterexistsintheformofmicroscopicobjects,onewouldexpectthatthisunknownformofenergypenetratesintogalaxiesandalsoenterthesolarsystem.

Braginsky,Gurevich,andZybin(1992)havestudiedtheeffectofdarkmatterboundinthegalaxybutunboundtothesolarsystem.Suchunbounddarkmatterwouldproduceananisotropyinthegravitationalbackgroundofthesolarsystem.Theresultingtidalforcesinduceanadditionalperihelionprecession.Assumingρd=0.3GeV/cm3=5.4×10−25g/cm3theycomputedthemagnitudeoftheresultingsecularorbitdistortion.Theeffectmaybeobservedbyreasonableimprovementsofpresentobservationaltechniques(KlionerandSoffel1993;Braginsky1994).ApossibleinfluenceofdarkmatterontheEarth–MoonsystemhasbeenconsideredbyNordtvedt(1994)andbyNordtvedt,M¨uller,andSoffel(1995).

Inthispaperwefocusonadarkmattermodelinwhichthedensityofdarkmattervariessoslowlywithinthesolarsystemthatitcanconsideredconstant.ThisisareasonableassumptionifdarkmatterinthesolarsystemreallyisinthemuchdeeperpotentialofthegalaxywiththeSuncausingonlyalocaldensityperturbationinthegalacticdarkmatterbackground.Itwillalsobeassumedthattheequationofstateoftheunseenmatterisalmostdust-like,thatis,thepressurewillbeassumedtobemuchlessthantheenergy-density.Basedonthismodelwecalculateanupperlimittothedensityofdarkmatterbyconsideringitseffectupontheperihelionprecessionoftheplanets.Wehavealsocarriedoutsimilarcomputationswithadarkmatterdensityproportionalto1/r4and1/r2usingtheresultsinSoleng(1994)and(1995),respectively.Thecorrespondingexperimentalboundsdonotvarymorethanoneorderofmagnitude.ThisweakdependenceonthedistributionfunctioncorroboratestheclaimofAndersonetal.(1989)andshouldbeexpectedbecause(tolowestorder)theperihelionprecessioncausedbydarkmatterisgivenbytheintegrateddarkmattermassatagivenorbitalradius.

2.SOLARSYSTEMWITHDARKMATTER

Inordertostudythegravitationaleffectsofhypotheticaldarkmatteronplanetarymotion,weneedasolutionofEinstein’sfieldequationsforastatic,sphericallysymmetricspace–timeandagivendistributionofdarkmatter.Theline-elementforastatic,sphericallysymmetricgravitationalfieldcaningeneralbewrittenas2

ds2=−e2µ(r)dt2+e2λ(r)dr2+r2dΩ2.

(1)

Weshallassumethatthedarkmatterhasaconstantdensityρ0(withinthesolarsystem).AtasurfacewherethedarkmatterpressureequalsthegalacticdarkmatterpressurepG≈10−7ρG(characterizedbyvelocitiesof220km/s),wematchthegravitationalfieldofthesolarsystemtotheexteriorfieldofthegalaxy.Weshallassumethatgtt≈−1atthisdistance

(thisassumptionisalwaysusedinlocalgravitationalproblems).Thenthett-componentofthefieldequationstakestheform

d

r

−

8π

M+4π

=−(ρ0+p)2drr−2r(M+

4π

M+4π

=−(ρ0+p).drr(r−2M)

Integrationleadsto

p=

K11−

2M

2M

−1

󰀉−16M2πρ0/3

exp−

󰀋

2π

󰀄

e−

r

2π

dr

=−(ρ0+p)

dµ

ρ0+p4

󰀎2

whereK2isanewintegrationconstant.Insertingthepressurefromequations(3)leadsto

gtt=−K21−

󰀆

2M

3

ρ0r2

.

TheconstantK2canbedeterminedbydemandingthatgtt=−1atthesurfacer=rmatchwherethepressureequalsthegalacticdarkmatterpressurepG.Thisradiusisgivenby

rmatch=

andhence

K2=exp−

󰀆

󰀁

3

ρ0+pG

󰀎󰀂1/2

󰀎2

(4)

4π

ρ0+p0

≈1

accordingtotheassumptionthatp≪ρ0.

Ourmodelofthedarkmatterfilledspace–timeinthesolarsystemisthusrepresentedbytheapproximateline-element

ds=−1−

2

󰀆

2M

3

ρ0r

2

󰀉

dt+

2

dr2−8πr

3

ρ0r2isasmallnumber.Atlarger

scaleswherethedarkmatterpressureequalsitsgalacticvalue,thegravitationalfieldisofcoursedeterminedbythemassdistributionofthegalaxy.

3.PERIHELIONPRECESSION

TheLagrangefunctionforatestparticlemovingintheθ=π/2planeinthegeometryspecifiedbyequation(5),is

2L=−1−

󰀆

2M

3

ρ0r

2

󰀉

˙2+t

r

r˙2

−

8π

rr

+−

4π8π

gtt

+

p2r

gφφ5

=−1.

TofirstorderinMandρ0,onefinds

r˙=1−

2

󰀆

2M

3

ρ0r

2

󰀉󰀌

−1−

p2φ

dφ2

+u=

M

3

ρ0

2

Pφ

+3Mu20+

4π

2

u30pφ

.(9)

Substitutingu0(1+ε)foruintoequation(8)withε≪1,andusingequation(9)and

calculatingtofirstorderinε,weget

dε

2u40pφ

2

󰀎

ε.

TheEinsteinprecessioncomingfromthesolarmassM=M⊙is

∆φ0=6πM⊙u0.

Inaddition,thereisadarkmatterinducedprecession

∆φdark=−4π2

ρ0

(10)

ρ0

3

2π

5

M⊙p2φu0

|δφobs|

2GM⊙

2π

6

∆φ0

.

wherewehaveinsertedNewton’sconstantGandthespeedoflightcinordertosimplifythenumericalcalculations.TheperihelionprecessionoftheasteroidIcarusisknownwithabout8%accuracy.ItsdistancefromtheSunis1.076A.U.WithM⊙=2×1033gandr0=1.076A.U.=1.61×1013cmweobtain

ρ0󰀁1.8×10−16g/cm3.

(13)

Thisvalueisaboutsevenordersofmagnitudeabovethemeangalacticmassdensity,anditshowsthatmeasurementsoftheperihelionprecessionoftheplanetsdonotputstrictlimitsonthedensityofbounddarkmatter.Notethatwiththisdensity,themassofsphericallysymmetricbounddarkmatterofconstantdensity(13)withintheorbitofPlutoislessthan2×10−5M⊙andthusinagreementwithourassumptions.WithinUranuswefindalimitof

Mdark(rU)󰀁2×10−6M⊙≈0.6M⊕

whereM⊕istheEarth’smass.ThisboundisofthesamesizeastheboundfoundbyAndersonetal.(1989)bynumericallyanalyzinghowtheorbitofUranuswouldbeaffectedbydarkmatter.Recently,theboundobtainedbysuchmethodshasbeenstrengthenedtoaround0.2M⊕(Anderseonetal.1995),butasimilarimprovementshouldalsobepossibleusingimprovedperihelionprecessionobservations.

Letusfinallychecktheassumptionthatthedarkmatterhasapressurewhichismuchlessthanitsenergydensity.Themaximumpressureisatthecenterofthesolarsystem.Now,ifwerequirethatthemodelisvalidouttotheOortcloudatseveralthousandA.U.andthatmacthingtothegalacticdarkmatterdistrubutiontakesplacehere,wefindthatthepressuregradientnecessaryforhydrostaticequilibriumisverysmall.Usingequation(4)andmatchingat,say,rmatch=5000A.U.gives

p0−pG

darkmatterisnon-baryonic.Wedonotknowwhatthenon-baryonicdarkmatteris.Onepossibility(Accetta&Steinhardt1991;Steinhardt&Will1994)isthatthedarkmatterinpartisoscillationenergycausedbyrapidoscillationsofNewton’sconstant.AneffectiveBrans–Dickefieldisaconsequenceofmanyunificationschemesandalsoaningredientofextendedinflationarymodels(La&Steinhardt1989).Extendedinflationwoulddrivethescalarfieldawayfromtheminimumofitspotential,andthefieldwouldthenstarttooscillatewheninflationends.Thispossibilityonlyillustratesthatdarkmattermaybehaveratherdifferentlyfromnormalmatterandthatnostoneshouldbeleftunturnedinthesearchforthemasswhichseemstomakeupmostoftheuniverseweinhabit.

ACKNOWLEDGMENTS

ItisapleasuretothankSlavaG.Turyshevforpointingoutsomeusefulreferences.Wearealsoindebtedtoananonymousrefereeforconstructivecriticism.

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