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A Fractal Galaxy Distribution in a Homogeneous Universe?

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In this letter we present an idea which reconciles a homogeneous and isotropic Friedmann universe with a fractal distribution of galaxies. We use two observational facts: The flat at rotation curves of galaxies and the (still debated) fractal
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  A&Amanuscriptno. (willbeinsertedbyhandlater)  Yourthesauruscodesare:(12.12.1;12.04.1;11.03.1)  ASTRONOMY AND ASTROPHYSICS 12.10.1998  AFractalGalaxyDistributioninaHomogeneousUniverse?    RuthDurrer  1  andFrancescoSylosLabini  1  ; 2 1  Dept.dePhysiqueTheorique,UniversitedeGeneve,24,QuaiE.Ansermet,CH-1211Geneve,Switzerland. 2  INFMSezioneRoma1,Dip.diFisica,Universit a"LaSapienza",P.leA.Moro,2,I-00185Roma,Italy.Received{{{;accepted{{{  Abstract. Inthisletterwepresentanideawhichreconcilesa homogeneousandisotropicFriedmannuniversewithafractaldistributionofgalaxies.Weusetwoobservationalfacts:The atrotationcurvesofgalaxiesandthe(stilldebated)fractaldistributionofgalaxieswithfractaldimension  D  =2.Ouridea canalsobeinterpretedasaredenitionofthenotionofbias. KeyWords: Largescalestructureofuniverse-cosmology:theory-cosmology:darkmattergalaxies:generalItisknownsincetwentyyearsthatthegalaxydistribu- tionexhibitsfractalbehavioronsmallscales(Peebles1980,Mandelbrot,1982).Severalrecentstatisticalanalysesofthree dimensionalgalaxycataloguesindicatethatgalaxiesaredis- tributedfractallywithdimension  D  '  2outtothelargest scalesforwhichstatisticallysignicantdataisavailable, i.e. ,uptoabout100{200  h  ?  1  Mpc(Coleman&Pietronero,1992;SylosLabini etal. ,1998).Intheopinionoftheauthorsofthe presentnote,nodataispointingconvincinglytowardsaho- mogenizationofthegalaxydistribution(  i.e. , D  =3).We believethatthestandardanalyseswhichindicatethatuctu- ationsofthegalaxynumberdensitydecreaseonlargeenough scalesarenotwellsuitedtostudyscaleinvariantstructures,sincetheyaprioriassumetheexistenceofawelldenedav- eragedensityinsidethegivensample(Peebles,1993;Davis 1997).Onthispointthescienticcommunityhasnotreached consensus(Wu  etal. ,1998).Inthisworkweassumewithout furtherargumentsthatthefractalpictureiscorrect.Ourmain pointhereistoshowthatafractaldistributionofgalaxieseven uptotheHubblescalemaybeconsistentwithahomogeneous andisotropicuniverse.Ontheotherhand,manyobservations,mostnotablythe superbisotropyofthecosmicmicrowavebackgroundtogether withitsperfectblackbodyspectrum,givestrongevidence thatthegeometryofourUniverseisveryhomogeneousand isotropiconlargeenoughscales,aso-calledFriedmannmodel.Inthisletterwearguethatthisseeminglyagrantcontra- dictionofdierentpiecesofobservationaldatamayactually bereconciledinarathersimpleway.Ourmainpointisthatafractaldistributionofgalaxies neednotimplyafractalmatterdistribution(asitisalso pointedoutinWu  etal. ,1998).Itisoftenmentionedthatgalaxiesrepresentpeaksinthe matterdistribution.Letuscomparethesetothedistributionof mountainpeaksonthesurfaceoftheearthwhichweknoware fractallydistributedoveracertainrangeofscales.Butitwould befalsetoconcludefromthisfactthattheradiusofthesurface oftheearthfromitscenterisgrosslyvariable;asweknow,this numberisverywellapproximatedbyaconstant.Asimilar eectmayactuallybeatworkinthematterdistributionof theuniverse.WerstinterprettheCOBEDMRresults(Bennett  et al. ,1996)andotherobservationsofCMBanisotropieson smallerangularscales(DeBernardis  etal. ,1997).Theyindi- catethatmatteructuationsarereasonablywelldescribedby aHarrison-Zel'dovichspectrumwith(  M=M  )(    H  (  t  ))  '  10  ?  4  ,where    H  (  t  )denotesthecomovinghorizonscale,   H  (  t  )=      t=a  ,and  a  isthescalefactor.Butsmalldensityperturba- tionsinaFriedmannuniversegrowonlyoncetheuniverse becomesmatterdominatedandeventhenratherslowly(pro- portionaltothescalefactor  a  /  t  2  =  3  ).Densityuctuations todayonagivencomovingscale    shouldthusbeontheor- derof    ?  3  =  2    (    )=10  ?  4  (  t  0  =t  (    ))  2  =  3    10  ?  4  (    0  =  )  2  .Here  t  (    ) denotesthecosmictimeatwhichthescale    entersthehori- zon; t  0  and    0  arethepresentcosmicandconformaltimes respectively.Forexampleonthescale    =200  h  ?  1  Mpcmatter densityuctuationsshouldnotbelargerthan     M M         =200  h  ?  1  Mpc  =(    ?  3  =  2    )  j   =200  h  ?  1  Mpc    (3000  h  ?  1  Mpc  =  200  h  ?  1  Mpc)  2  10  ?  4  =0  : 015 today.(Here  h  parameterizestheuncertaintyofrelatingthere- cessionvelocityofagalaxytoitsdistance, i.e. theuncertainty intheHubbleparameter  H  0  =100  h  km  =  s  =  Mpc.) WenowarguethatafractalgalaxydistributionmaywellbecompatiblewithsuchahomogeneousUniverse.Thiscanbe seenbythefollowingobservation:Itiswellknownthatthe darkmatterdistributionaroundagalaxyleadstoatrotation curves(Rubin  etal. ,1980).Withoutcutosthisyieldsamatter densitydistribution    halo  (  r  )  /  r  ?  2  aroundeachgalaxy.Toobtainthetotalmatterdistribu- tion,weactuallyhavetoconvolve    halo  withthenumberdis- tributionofgalaxies, n  G  .Thefractaldimension  D  =2means thattheaveragenumberofgalaxiesinasphereofradius  r  aroundagivengalaxy,denotedby  N  G  (  r  ),scaleslike  r  2  .This impliesthatthemeannumberdensityofgalaxiesaroundan occupiedpointdecayslike   2  n  G  /  1  r : Ourmainndingisthesimplefactthattheconvolutionof thesetwodensitiesgivesaconstant(uptologarithmiccorrec- tions)    =  n  G      halo  /  1  : (1) Moreprecisely,for  n  G  (  r  )=  C=r  and    halo  (  r  )=  A=r  2  weobtain (with  j y  j  y  and  j x  j  x  )    (  x  )=  AC  Z   d  3  y  1  j y  jj x  ?  y  j 2  =4  AC  Z   R  max  0  min(  x;y  )  xy dy  =4  AC    1  x  Z   x  0  dy  +  Z   R  max  x  dy y    =4  AC  1+ln(  R  max  =x  )] : (2) Thisshowsthatafractalgalaxydistributionwithdimension  D  =2togetherwithatrotationcurves,indicateasmooth matterdistributioninagreementwithourexpectationsfroma Friedmannuniversewithsmalluctuations.Thedarkmatter distributionofatwodimensionalmodelwherethegalaxiesare distributedwithfractaldimension  D  =1isshowninFig.1.Clearly,thisdarkmatterdistributionisveryhomogeneousup tonitesizeeects.Notethattheessentialingredientforthisresultisthat  n  G  /  1  =r    and    halo  (  r  )  /  1  =r    with    +    =3. 0.0 0.5 1.00.00.51.0 Fig.1. Weshowthedarkmatterdistributionofatwodimensionalsetoffractallydistributedgalaxies(  D  =1  ;n  G  /  1  =r  )(lledcir- cles)eachsurroundedbyadarkmatter(dots)halowithdistribution    halo  /  1  =r  .Thehalossumuptoaveryhomogeneousdarkmatter distribution. Itisimportanttoinsistthatthecuto  R  max  islargerthan allthescales  x  considered.Farawayfromgalaxies,forexam- pleinavoid,manygalaxiescontributetothedensityina givenpointandtheimportantmessageisjustthatatrota- tioncurvesindicateactuallythatthetotalresultingdensity mayberatherconstantinvoidsandhasaboutthesamevalue asithasclosetogalaxies.Onemayarguethatthenotionofacertainlumpofmatter 'belonging'toacertaingalaxyisilldenedsucientlyfaraway fromagalaxyandthusthe'halo'densitycannotbedenedon scaleslargerthan,say,halfthedistancetothenextgalaxy.Withthisobjectionweagreeinpractice,itjustmeansthatat asucientdistancefromagivengalaxytheonlymeasurable densityisthetotaldensitywhichcontainsrelevantcontribu- tionsfrommanygalaxies.Butinprincipleitispossibleto assigntoeachgalaxyadensityprole    halo  ,anditisinterest- ingtonotethattheformofthedensityproleindicatedby measurements,whicharepossibleclosetoisolatedgalaxies,is  justsuchastoleadtoaconstanttotaldensityifweconvolve itwiththenumberdensityofgalaxies.Anotherobjectionmaybe,thatwiththisdensityprole thetotalmassofagalaxyisinnite.Butthisisinfactir- relevant,sincenotonlythemassofasinglegalaxygoesto innityas  r  !1  ,butalsothegalaxydensitygoestozero as  r  !1  andthisjustinawaythatthemeasurabletotaldensityisconstant.Besides,theintegralof    halo  shouldnotbe consideredasthe'massofthegalaxy'.Moreprofoundly,the atrotationcurvesareaconsequenceofthefactthatthegrav- itationalpotentialremainsconstantduringlinearclusteringin aFriedmannuniverse,whichholdsbeyondthescaleofsingle galaxies.Clearly,theamplitudes  C  and  A  of  n  G  and    halo  depend onthetypeofgalaxiesconsidered.Galaxiesofdierentabso- luteluminositiesingeneralhavedierentcircularspeedsand dierentabundances.ToquantifyEq.(2)weusetheTully-Fisherrelation(Tully &Fisher,1977),betweenthecircularspeed  v  ofagalaxyand itsluminosity  L  , v  =  v    (  L=L    )  0  : 25  ,with  v    =220  km=s  .The luminosity  L    correspondstoanabsolutemagnitudeof  M    ' ?  19  : 5.Theabundanceof  L    galaxiesasestimatedfromSylos Labini etal. (1998)is  N  G  (  r  )    =  B    r  2  with  B    '  0  : 3  h  2  Mpc  2  : Consideringforthetimebeingjust  L    galaxies,thisgives  n  G  =  C    =r  with  C    '  3  B    =  4    .Todeterminetheamplitudeofthehalodensity,   halo  =  A=r  2  ,weusetheatnessofgalaxyrotationcurveswith(Rubin  etal. ,1976) 1 2 (  v    =c  )  2  '  10  ?  7  =  GM  (  r  )  =r; where  c  denotesthespeedoflight.Combiningthiswith  M  (  r  )=4  Ar  gives  A    = (  v    =c  )  2  8  G : Theconvolutionof  n  G  with    halo  thenleadsto    =  n  G      halo  '  4  C    A    '  3  B    (  v    =c  )  2  8  G : (3) Comparingthiswiththecriticaldensityofauniverseexpand- ingwithHubbleparameter  H  0  ,   c  = 3  H  2 0  c  2  8  G ; (4)   3 weobtainadensityparameteroforderunity,=  =  c  '  v  2    B    =H  2 0    1  : (5) Clearly,thisestimateisverycrudesincenotallgalaxieshave thesamerotationspeedsandtheconstant  B  dependsonthe luminosityofthegalaxy.Butitisareassuringnon-trivial'co- incidence'thatthedensityparameterobtainedinthiswayis oftherightorderofmagnitude.Torenethismodelwehave tondaTully-Fishertyperelationfor  B  (  L  )andintegrateover luminosities.Butsincetheabundanceofgalaxiesdecaysexpo- nentiallywithluminosityabove  L    ,wehave  Z   v  (  L  )  2  dB  (  L  )    v  2    B    ; andreproducetheresult(5).Ourargumentssuggestthatafractalgalaxydistribution maywellbeinagreementwithasmoothmatterdistribution.Neglectinglogcorrections(whichmaybeabsentinamorere- alistic,detailedmodelandwhicharecertainlynotmeasurable withpresentaccuracy),ourmodeldescribesaperfectlyhomo- geneousandisotropicuniverse.Wedonotspecifytheprocess whichhasinducedsmallinitialuctuationsandnallyledto theformationofgalaxies.Wearethusstilllackingaspecic pictureofhowthefractaldistributionofgalaxiesmayhave emerged.PurelyGaussianinitialuctuationsareprobablynot suitedtoreproduceafractalgalaxydistribution.Butcosmic stringsorother'seeds'withlongrangecorrelationcoulddoit.Aworkingmodelremainstobeworkedout.Itmaybeusefultomentionthattheviewpresentedhere actuallyredenesthenotionofbias.Inthestandardscenario,thedarkmatterdensityeld(onlargescales)isGaussianand galaxiesforminthepeaksoftheunderlyingdistributionand theircorrelationfunctionsaresimplyrelated10Herewecon- siderthepossibilitythatthegalaxyanddarkmatterdistribu- tionsmayhavedierentcorrelationpropertiesandhencedif- ferentfractaldimensions,namely  D  =2forthegalaxiesand  D  =3fordarkmatter.EspeciallyinviewofEq.(1),wewant towarnthereaderagainstinterpretingthegalaxydistribution asproportionaltothematterdistributionevenonlargescales.Asimilarideaistheoneofauniversewithafractalgalaxy distributionbutadominantcosmologicalconstant  >  8  G  .ThispossibilityisalsodiscussedinBaryshev  etal. (1998),in connectionwiththelinearityoftheHubblelaw.Moreprecisely,thefractaldimensionofasetofdensity uctuationscandependonthethreshold(seeFig.2).Ina realisticmodel,wewouldexpectthatalsothedarkmatter,aboveacertainthresholdisfractallydistributed.Ingalaxy catalogues,thistendencyisactuallyindicated.Observations showaslightincreaseofthefractaldimensionwithdecreasing absoluteluminosityofthegalaxiesinthesample(SylosLabini etal. ,1998),however,stillwithrelativelymodeststatistics.Suchascenarioisnaturallyformulatedwithintheframework ofmulti-fractals(Falconer,1990;SylosLabini etal. ,1998).Itis,forexample,awellknownfactthatbrightellipticalslie preferentiallyinclusterswhereasspiralgalaxiesprefertheeld (Dressler,1984).Fromtheperspectiveofmulti-fractalitythis impliesthatellipticalsaremoreclusteredthanspirals, i.e. ,theirfractaldimensionislower(Giovanelli etal. ,1986).Ourargumentsindicatethatthevoidsmightbelledby darkmatter.Butsincethisdarkmatterisrelativelysmoothly distributed,itcannotbedetectedbymeasurementssensible  10 −2 10 −1 10 0 r10 2 10 3 10 4       Γ      (        r           ) D=1.1D=1.4D=1.6D=1.8D=1.9D=2 Fig.2. Themeandensityaroundanoccupiedpoint(?(  r  ))isshown forourtwodimensionalmodel.Thegalaxydensitywithfractaldi- mension  D  =1isshownaslledcircles.Asthedensitythreshold decreasesthefractaldimensionapproaches  D  =2.Thechosenover- densitiesforthefractaldemensionsof1.1,1.4,1.6,1.8,1.9and2 are2.1,1.76,1.55,1.34,1.22and1respectively. onlytodensitygradients(like, e.g. ,peculiarvelocities).One needstodeterminethetotaldensityoftheuniverse,forexam- plebymeasuringthedecelerationparameter  q  0  . Acknowledgment: ItisapleasuretothankYuriBaryshev,AlessandroMelchiorri,LucianoPietronero,Pekka TeerikorpiandFilippoVernizziforvaluablediscussionsand comments.Thisworkhasbeenpartiallysupportedbythe EECTMRNetwork"Fractalstructuresandself-organization" ERBFMRXCT980183andbytheSwissNSF. 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