Scaling Laws in the Distribution of Galaxies

52 Pages·2004·1.26 MB·English

Scaling Laws in the Distribution of Galaxies
BernardJ.T.Jones∗
KapteynInstitute,UniversityofGroningen,P.O.Box800,9700AVGroningen,TheNetherlands
VicentJ.Mart´ınez†
Observatori Astrono`mic de la Universitat de Vale`ncia, Edifici d’Institutsde Paterna, Apartat de Correus22085,
46071Vale`ncia,Spain
EnnSaar‡
TartuObservatory,To˜ravere,61602Estonia
VirginiaTrimble§
4 Astronomy Department, University of Maryland, College Park MD 20742, USA
0 PhysicsDepartment,UniversityofCalifornia,IrvineCA92697USAMaryland,USA
0
2 (Dated:June3,2004)

n
ResearchdoneduringthepreviouscenturyestablishedourStandardCosmologicalModel.Therearemanydetails
u stilltobefilledin,butfewwouldseriouslydoubtthebasicpremise. Pastsurveyshaverevealedthatthelarge-
J scaledistributionofgalaxiesintheUniverseisfarfromrandom:itishighlystructuredoveravastrangeofscales.

3 Surveysbeingcurrentlyundertakenandbeingplannedforthenextdecadeswillprovideawealthofinformation
aboutthisstructure. Theultimategoalmustbenotonlytodescribegalaxyclusteringasitisnow,butalsoto

explainhowthisaroseasaconsequenceofevolutionaryprocessesactingontheinitialconditionsthatweseein
1 theCosmicMicrowaveBackgroundanisotropydata.
v Inordertoachievethiswewillwanttodescribecosmicstructurequantitatively: weneedtobuildmathematically
quantifiabledescriptionsofstructure. Identifyingwherescalinglawsapplyandthenatureofthosescalinglaws
6
isanimportantpartofunderstandingwhichphysicalmechanismshavebeenresponsiblefortheorganizationof
8
clusters,superclustersofgalaxiesandthevoidsbetweenthem. Findingwherethesescalinglawsarebrokenis
0
equallyimportantsincethisindicatesthetransitiontodifferentunderlyingphysics.
6
Indescribingscalinglawswearehelpedbymakinganalogies withfractals: mathematical constructs thatcan
0
possessawidevarietyofscalingproperties. Wemustbeware,however,ofsayingthattheUniverseisafractal
4
onsomerangeofscales: itmerelyexhibitsaspecifickindoffractal-likebehavioronthosescales. Weexploit
0 the richness of fractal scaling behavior merely as an important supplement to the usual battery of statistical
/ descriptors.
h
Wereview the history of how we have learned about the structure of the Universe and present the data and
p
methodologies that are relevant to the question of discovering and understanding any scaling properties that
-
o structuremayhave.Theultimategoalistohaveacompleteunderstandingofhowthatstructureemerged.Weare
r gettingclose!
t
s
a
v: PACSnumbers:98.62.Py,89.75.Da,98.65.Dx,98.65.-r,98.62.Ve,98.80.Es
i
X
r Contents B. Galaxiesas“IslandUniverses” 6
a C. Earliestimpressionsongalaxyclustering 6
I. PHYSICALCOSMOLOGY 2 D. Hierarchicalmodels 6
A. Cross-disciplinaryphysics 3 1. Charlier’sHierarchy 7
B. Statisticalmechanics 3 2. Carpenter’slaw 7
C. Scalinglawsinphysics 3 3. DeVaucouleurshierarchicalmodel 8
E. Thecosmologicalprinciple 8
D. Somepsychologicalissues 4
IV. DISCOVERINGCOSMICSTRUCTURE 9
II. THECOSMICSETTING 4
A. Earlycatalogbuilders 9
A. Keyfactors 5
1. TheLicksurvey 9
B. Somecaveats 5
2. PalomarObservatoryskysurvey 10
3. AnalysisofPOSSclusters 10
III. EARLYIDEASABOUTTHEGALAXYDISTRIBUTION 5
B. RedshiftSurveys 11
A. Cosmogony 5
1. Whydothis? 11
2. Redshiftdistortions 11
3. Flux-limitedsurveysandselectionfunctions 12
4. Correctionstoredshiftsandmagnitudes 12
∗Electronicaddress:[email protected] C. Thefirstgenerationofredshiftsurveys 13
†Electronicaddress:[email protected] 1. CfAsurveys 13
‡Electronicaddress:[email protected] 2. SSRSandORS 13
§Electronicaddress:[email protected] 3. Stromlo-APMandDurham/UKSTredshiftsurveys 13
2
4. IRASredshiftsamples:PSCz 14 5. Nonlinearenhancements 41
5. ESODeepSliceandtheLasCampanasredshiftsurvey 14 E. Nonlineardynamicmodels 41
D. Recentandon-goingSurveys 14 1. AdhesionApproximations 42
1. 2dFgalaxyredshiftsurvey 14 2. TheRandomHeatEquation 42
2. Sloandigitalskysurvey 14 3. TheSolutionoftheRHequation 42
3. 2MASSand6dF 16 4. StatisticalMoments 43
4. Deepspectroscopicandphotometricsurveys 16 5. TheSchrodingerEquation 43
E. Theradio,X-rayandγ-rayskies 16 6. GeneralComments 43
F. DistributionofquasarsandLy-αclouds 17
G. Thecosmicmicrowavebackground 17 VIII. CONCLUDINGREMARKS 44
1. Structurebeforeoureyes 17 A. Aboutscaling 44
2. Definingthestandardmodel 18 B. Futuredatagathering 44
3. Initialconditionsforgalaxyformation 18 C. Understandingstructure 45
D. Aboutsimulations 45
V. MEASUREMENTSOFCLUSTERING 19 E. Wherewestandontheory 45
A. Thediscoveryofpower-lawclustering 19 F. Andfinally... 45
B. Thecorrelationfunction:galaxies 19
1. Definitionsandscaling 19 Acknowledgments 46
2. Estimators 21
3. Recentdeterminationsofthecorrelationfunction 22 References 46
4. Correlationdimension 22
5. Correlationlengthasafunctionofsampledepth 23
C. Galaxy-galaxyandcluster-clustercorrelations 23
I. PHYSICALCOSMOLOGY
1. Analysisofrecentcatalogs 24
2. Theoreticalexpectations 24
3. Richnessdependenceofthecorrelationlength 25 WiththediscoveryoftheCosmicBackgroundRadiationby
D. Thepairwisevelocitydispersion 26 Penzias and Wilson (1965), cosmology became a branch of
E. Lightdoesnottracemass 27 physics: therewasawelldefinedframeworkwithinwhichto
1. Massdistributionandgalaxydistribution:biasing 27
formulatemodelsandconfrontthemwithobservationaldata.
2. Massandlightfluctuations 27
Priortothattherehadbeenafewimportantobservationsand
VI. FURTHERCLUSTERINGMEASURES 28 afewimportantsolutionstotheEinsteinField Equationsfor
A. Higherordercorrelationfunctions 28 General Relativity. We suspected that these were somehow
B. Three-pointcorrelationfunctions 28
connected: that the Friedman-Lemaitresolutionsof the Ein-
C. Thepowerspectrum 28
stein field equationsdescribedthe cosmologicalredshiftlaw
D. Thebispectrum 29
E. Fractaldescriptorsofclustering 30 discoveredbyHubble.
1. Acautionaryword 30 With the discovery of the background radiation we were
2. Structurefromcountsincells 30 left in no doubt that the Universe had a hot singular ori-
3. Scalingpropertiesofcountsincells 30
gin a finite time in our past. That important discovery also
4. Quantifyingstructureusingmultifractals 31
showed that our Universe, in the large, was both homoge-
5. Intermittency 32
6. Multifractality 32 neous and isotropic, and it also showed the appropriateness
oftheFriedman-Lemaitresolutions.
VII. CLUSTERINGMODELS 33
The establishment of the “Big Bang” paradigm led to a
A. Cosmologicalsimulations 33
search for answers, in terms of known physical laws, to key
1. Aarseth 33
2. Subsequentdevelopments 34 questions: why was the Universe so isotropic, how did the
3. Confrontingwithreality 34 structureweobserveoriginate?andsoon.Cosmologistsbuilt
4. Scalingindarkmatterhalos 35 models involving only known physics and confronted them
5. Scalingingalaxyproperties 35
with the data. Cosmology became a branch of physics with
B. Statisticalmodels 36
a slight difference: we cannot experiment with the subject
1. Neyman-Scottprocesses 36
2. Simplefractalmodels 36 of our discussion, the Universe, we can only observe it and
3. Morecomplexclusteringmodels 36 modelit.
4. Voronoitesselations 37
With the current round of cosmic microwave background
5. Lognormalmodelsandthelike 37
anisotropymapsweare abletosee directlytheinitialcondi-
6. Saslaw-Shethmodels 37
7. BalianandShaeffer 37 tionsforgalaxyformationandfortheformationoflarge-scale
C. Dynamicalmodels 38 structure.Thatobservedstructureisthoughttoreflectdirectly
1. Stableclusteringmodels 38 thefluctuationsinthegravitationalpotentialthatgavebirthto
2. BBGKYhierarchy 38
cosmicstructureanditisaconsequenceofthephysicsofthe
3. Pancakeandadhesionmodels 38
earlyuniverse.Thegoalistolinkthoseinitialconditionswith
4. Renormalizationgroup 39
5. ThehalomodelandPThalomodel 39 whatweseetoday.
6. Moreadvancedmodels 39 The aim of this article is to show how the “homogeneous
D. Hydrodynamicmodelsforclustering 40 and isotropic Universe with a hot singular origin” paradigm
1. Cosmologicalgasdynamics 40
hasemerged,and to explainhow, withinthis framework,we
2. ThecosmicBernoulliequation 40
canquantifyandunderstandthegrowthofthelargescalecos-
3. Zel’dovichapproximation 41
4. Super-Zel’dovichapproximations 41 micstructure.
3
A. Cross-disciplinaryphysics Onemajorproblemwashowtodescribethisstructure. By
1980, it was known that the two-point correlation function
Gravitation is the drivingforceof the cosmosand so Ein- lookedlikeapowerlawonscales1<10h−1Mpc. Itwasalso
stein’sGeneralTheoryofRelativityistheappropriatetoolfor knownthatthe3-pointfunctiontoohadapowerlawbehavior
modelling the Universe. However, that alone is not enough: and that it was directly related to sums of products of pairs
other branchesof physicshaveplayed a keyrole in building oftwo-pointfunctions(ratherlike the Kirkwoodapproxima-
whathasemergedasa“StandardModel”forcosmology. tion).However,N-pointcorrelationfunctionswerenotreally
Nucleosynthesis played an early role in defining how the evocativeoftheobservedstructureandweredifficulttomea-
light elements formed (Alpheretal., 1948): the abundances surepastN =4.
ofHeliumandDeuteriumplayavitalpartinconfrontingour Two suggestions for describing large scale cos-
models with reality. In following how the cosmic medium mic structure emerged: void probability functions
cooled sufficiently to enable gravitational collapse to form proposed by White (1979) and measured first by
galaxiesandstarsweneedtounderstandsomeexoticmolec- MaurogordatoandLachieze-Rey (1987) and multifractal
ularchemistry. measures (Jonesetal., 1988), the latter being largely mo-
Today, our understanding of high energy physics plays a tivated by the manifest scaling behavior of the lower order
keyrole:someevendefinedanewdisciplineandrefertoitas correlationfunctionsonscales< 10h−1 Mpc. Bothofthese
“astro-particlephysics”.Wehavestrongevidencethatthereis descriptorsencapsulatethebehaviorofhighordercorrelation
asubstantialamountofdarkmatteringalaxiesandclustersof functions.
galaxies.Sofarwehavenotbeenabletosaywhatisthenature
ofthisdarkmatter.Thereisalsogrowingevidencethattheex-
pansionoftheUniverseisaccelerating:thiswouldrequirean
C. Scalinglawsinphysics
all-pervadingcomponentof matter or energythat effectively
hasnegativepressure. Ifthisweretruewewouldhavetores-
The discovery of scaling laws and symmetries in natu-
urrectEinstein’scosmologicalconstant,orinvokesomemore
ral phenomena is a fundamental part of the methodology of
politicallycorrect“fifthforce”conceptsuchasquintessence.
physics. Thisisnotnew: we canthinkofGalileo’sobserva-
tionsoftheoscillationsofapendulum,Kepler’sdiscoveryof
theequalarealawforplanetarymotionandNewton’sinverse
B. Statisticalmechanics squarelawofgravitation. Someauthorsclaimthattheactual
discoveryof the scaling laws is attributable to Galileo in the
The statistical mechanics of a self-gravitating system is a contextof the strength of materials as discussed in his book
totally nontrivial subject. Most of the difficulty arises from TwoNewSciences(Peterson,2002).
thefactthatgravitationisanalways-attractiveforceofinfinite
Theestablishmentofascalingrelationshipbetweenphysi-
range: thereisnoanaloguetotheDebyeshieldinginplasma
calquantitiesrevealsanunderlyingdrivingmechanism. Itis
physics. Perhaps the most outstanding success was the dis-
thetaskofPhysicstounderstandandtoprovideaformalism
coverybyJeansinthe1920’sofequilibriumsolutionstothe
forthatmechanism.
Liouvilleequationforthedistributionfunctionofacollection
Theself-affineBrownianmotionisagoodexampleforvi-
ofstars(theJeansTheorem).Thishasledtoawholeindustry
sualillustrationofascalingprocess(seeFig.1). Inthiscase
ingalaxydynamics,butithashadlittleornoimpactoncos-
scalingisnon-uniform,becausedifferentscalingfactorshave
mologywherewe mightlike toview theexpandinguniverse
to be applied to each coordinateto keep the same visual ap-
withgalaxiescondensingoutasaphasetransitioninaction.
pearance.
This has not deterred the brave from tackling the statisti-
Thebreakingofsymmetriesandofscalinglawsisequally
calmechanicsorthermodynamicsofself-gravitatingsystems,
importantandhasplayeda keyrole in20thcenturyphysics.
butitisperhapsfairtosaythatsofartherehavebeenfewout-
Scale invariance is typically broken when some new force
standingsuccesses.ThediscussionbyLynden-BellandWood
or phenomenoncomes into play, and the result can look far
(1968)oftheso-calledgravo-thermalcollapseofastellarsys-
moresignificantthanitreallyis. DubrulleandGraner,1994;
teminaboxisprobablyascloseasanyonehascome. Itwas
GranerandDubrulle, 1994 have suggested that this may be
only in the 1970’s that cosmologists “discovered” the two-
the case for the Titius–Bode law (which is, of course, not a
point clustering correlation function for the distribution of
law,andcanbetracedbackbeforeTitiusandBodeatleastto
galaxiesanditwasnotuntilthelate1980’swiththediscovery
DavidGregoryin1702). Theirpointisthat,iftheprimordial
by deLapparentetal. (1986) of remarkable large scale cos-
proto-planetarydiskhada power-lawdistributionofdensity
mic structurethat we evenknew whatit was we were trying
and angular momentum then any process that forms planets
todescribe.
willgivethemsomethingliketheTitius–Bodedistributionof
TheearlyworkofSaslaw(1968,1969)on“Gravithermody-
namics” predated the knowledge of the correlation function.
Following the discovery of the correlation function we saw
theworkofFallandSeverne(1976),Kandrup(1982),andFry
1Thenaturalunitoflengthtodescribethelargescalestructureisthemega-
(1984b),providingmodelsfortheevolutionofthecorrelation parsec(Mpc): 1Mpc=106pc≃ 3.086×1022m≃ 3.26×106 light
functioninvariousapproximations. years.histheHubbleconstantinunitsof100Mpc−1kms−1.
4
FIG.1 Scalingrelationsinone-dimensionalBrownianmotionx(t).Insuccessivezoomstheverticalcoordinate(x)ismultipliedby2,while
thehorizontalcoordinate(thetimet)ismultipliedby4toproperlyrescalethecurve.
orbitsizes. Thusthedistributioncannotbeusedasa testfor D. Somepsychologicalissues
anyparticularformationmechanism.
Cosmology presents physics with a formidable challenge.
Within cosmology, some of the examples of quan-
tized redshifts reported over the years (Burbidge, 1968; TheUniverseisnotaboundedandisolatedsystem. TheUni-
verseisfarfrombeinginanyformofdynamicalequilibrium.
BurbidgeandBurbidge, 1967; Tifft, 1976) may have been
analogous cases, where the “new phenomenon of physics” Thegravitationalforceisofinfiniterangeandalwaysattrac-
tive. Nor can we experiment on the subject of interest, we
was observational selection effects resulting when strong
emission lines passed into and out of the standard observed are mere observers. Thus the usualconcepts fromstatistical
physicscannotbesimplyimported,theyhavetoberedefined
wavelengthbands.
tosuitthesespecialcircumstances.
As we shall see, there are important scaling relationships
Thisprocessofredefinitionisapttomisdirectthestruggle
in the spatial distribution of galaxies. This scaling is almost
for understanding the issues involved and is inevitably frus-
certainlyaconsequenceoftwofactors:thenatureoftheinitial
trating to those who work in statistical physics or who seek
conditionsforcosmicstructureformationandthefactthatthe
to use techniquesfromstatistical physics. Indeedthere have
gravitationalforcelawisitselfscale-free.
beenoccasionswherethenotionsofthestandardmodelhave
This scaling is observed to break down at very large dis- beenabandonedsimplyinordertoexploitstandardconcepts
tance.Thisbreakdownisaconsequenceofthelarge-scaleho- thatwouldotherwisebeinvalid(eg.: modeluniverseshaving
mogeneityoftheUniverseandofthefactthattheUniversehas onespatialdimensionormodeluniversesthathavezeromean
a finite age: gravitational agglomeration of matter has only density in the large). Those papers may be interesting, but
been able to spread over a limited domainof scales, leaving theyhavelittleornothingtodowiththeUniverseasweknow
thelargestscalesunaffected. it.
The scaling is also expected to break down for small ob-
jects wherenon-gravitationalforceshave playeda role: gas-
dynamicprocessesplayanimportantroleinthelaterstagesof II. THECOSMICSETTING
galaxy formation. There are important scaling relationships
among the properties of galaxies which provide clues to the Theestablishmentofa definitivecosmologicalpicturehas
mechanisms of their formation. We do not deal with these beenoneofthetriumphsof20thCenturyphysics. FromEin-
in detail here, although the main scaling laws in the galaxy stein’sfirstinvestigationsintorelativisticcosmologicalmod-
propertiesaresummarizedinSect. VII.A.5. els, through Hubble’s discovery of the cosmic expansion, to
5
the discovery of the Cosmic Microwave Background Radia- stituentsoftheUniverse.Itistruethatwecanobservecosmic
tion in 1965, most physicists would now agree on the basic structureoveranenormousrangeoftheelectromagneticspec-
ingredientsofwhatmightaswellbecalled“theStandardCos- trum,butneverthelesswefacetheprospectthatabout85%of
mologicalModel”. Theastrophysicsofthe21stcenturywill what there is out there may forever remain invisible except
consist largely of filling in and understanding the details of indirectlythoughitsgravitationalinfluence.
thismodel: anontrivialprocessthatwillconsumesubstantial Fortunately, we can directly study the gravitational influ-
human,technicalandfinancialresources. enceofthedarkcomponentinanumberofways. Ifitisuni-
While there are suggestions that the standard model may formly distributed it has an influence on the overall cosmic
notbecomplete,thedataasawholedonotasyetdemandany expansionand on the physicsof the early Universe. We can
furtherparametrizationsuchas“quintessence”. Ofcourse,as detectitsinfluencebystudyingthecosmicexpansionlaw,or
ourunderstandingof fundamentalphysicsdeepens,the stan- bystudyingthenatureofthespatialinhomogeneitiesseenin
dard model might be recast in a new wider, more profound, thecosmicmicrowavebackgroundradiation. Ifitisnotuni-
frameworksuchasthatofferedbybranecosmologies. formlydistributedit willinfluencethe dynamicsof the large
scalestructureasseeninthevelocitymapsforlargesamples
ofgalaxiesanditmayrevealitselfthroughstudiesofgravita-
A. Keyfactors tionallensing.
Ournumericalsimulationsoftheevolutionofstructurecan
There are several important factors to support our current
inprincipletakeaccountofseveralformsofmatter.Whilethis
viewofcosmicstructureformation:
hasbeenasuccessfulprogram,thelackofdetailedknowledge
ThediscoverybyHubblein1928ofthelinearvelocity- about the nature of the dark matter is nevertheless a serious
•
distancerelationshipforgalaxies(Hubble,1929). This impediment. Some astrophysicists would turn the problem
relationshipwas sooninterpretedbyRobertson(1928) around and argue that those simulations that best reproduce
as being due to the expansion of the Universe in the whatisseenwillprovideimportantinformationaboutthena-
mannerdescribedbytheFriedman-Lemaitrecosmolog- tureofthedarkmatter.
ical solutions of the Einstein Field equationsfor grav-
itation. Thesesolutionsdescribeda homogeneousand
isotropicUniverseemergingfromasingularstateofin-
III. EARLYIDEASABOUTTHEGALAXYDISTRIBUTION
finitedensity: theBigBang. Lateron,BondiandGold
(1948)andHoyle(1948)providedanalternativehomo-
A. Cosmogony
geneousandisotropicexpandingmodelthatavoidedthe
initialsingularity:theSteadyStateTheory.
Inthe4th. CenturyBCE,Epicurustaughtthattherearean
Thediscoveryin1965oftheCosmicMicrowaveBack- infinite numberof worldslike (and unlike)ours, while Aris-
•
groundRadiation tells us the cosmologicalframework totletaughtthatthereisonlyone.Neitherhypothesiscancur-
withinwhichwehavetowork. OurUniverseis, inthe rentlybefalsified,andindeedwemayseethecontinuationof
large, homogeneous and isotropic; it was initially hot this metaphysical battle in the so-called inflationary cosmo-
enough to synthesize the element Helium. This is the logicalmodels.
Hot Big Bang theory promoted early on by Gamow. Philosophers since Anaximander (Kahn, 1994) have long
ThisdiscoverysignaledtheendoftheSteadyStateThe- debated the true nature of the Universe, presenting often re-
ory. markablyprescientideasnotwithstandingthelackofanyreal
data. Given the lack of data, the only basis for constructing
The observation in 1992 by the COBE satellite of the
• a Universe was symmetry and simplicity or some more pro-
large-scalestructureoftheUniverseatveryearlytimes
foundcosmologicalprinciple.
provides us with precise information about the initial
The ancients saw nested crystalline spheres fitting neatly
conditionsforstructureformation. Thisisongoingre-
intooneanother: thiswasapartofthethencultureofthink-
searchthat will lead to detailed knowledgeof the fun-
ing of mathematics (i.e. geometry in those days) as being
damentalparametersofourStandardModelandtode-
somehowafundamentalpartofnature2. Laterthinkerssuch
tailed knowledge of the initial conditions in the Big
asSwedenborg,KantandDescartesenvisionedhierarchiesof
Bangthatresultedinthecurrentlyobservedstructure.
nested whirls. While these ideasgenerallyexploitedthe sci-
We know a great deal about our Universe. Studies of cos- entific trends and notions of their time, none of them were
micstructuremustfallwithinthepreceptssetbyourStandard formulatedin terms of physics. Manyare reviewedin Jones
Modelortheywillsimplybedismissedatbestasbeingaca- (1976) where detailed references to the classical works are
demiccuriositiesoratworstasbeingtotallyirrelevant. given.
B. Somecaveats
2Einstein’s greatintellectual coupwastogeometrize theforceofgravity:
Themostimportantcaveatinallofthisisthefactthatwhen
wearegovernedonlargescalesbythegeometryofspace-timemanifesting
studyingcosmicstructureweobserveonlytheluminouscon- itselfastheforceofgravity.
6
Perhapsthefirstdetailedpresentationofcosmogonicideas that galaxies were the building blocks of the Universe (eg:
inthemodernveinwasduetoPoincare´ inhisLec¸onssurles McCrea(1964)andAbellinundergraduatelecturesatUCLA
Hypothe`sesCosmogoniques(Poincare´,1894),someofwhich 1961-1963).
wastobeechoedbyJeansinhistextsonAstronomyandCos- Infact,mostgalaxiesareclustered. Thisisimplicitinim-
mogony(Jeans,1928). Jeans’workissaidtohavehadapro- agestakenwithsmallertelescopeshavinglargerfields(Shap-
foundeffectonHubble’sownthoughtsaboutgalaxyevolution leyoftensaidthatlargetelescopeswereover-rated(Shapley,
andstructureformation(Christianson,1995). 1932), perhaps in part because he had deliberately cut him-
self off from them by moving to Harvard) and explicit in
the remarksof Zwicky (1938, 1952) who had begunto look
at the Universe throughSchmidt-colouredglasses. (The 18”
B. Galaxiesas“IslandUniverses”
SchmidttelescopeonPalomarMountaincameintouseacou-
pleofyearsbefore).
Once upona time there was a single galaxy. William and
CarolineHerschelhaddrawnamapoftheGalaxy(Herschel,
1785) on the basis that the Sun was near the center of the
Galaxy, and this image persisted into the 20th Century with
C. Earliestimpressionsongalaxyclustering
the “KapteynUniverse”(Kapteyn, 1922) which depicted the
the MilkyWay asa relativelysmallflattenedellipsoidalsys-
Inthe19thcenturyWilliamHerschelandCharlesMessier
temwiththeSunatitscenter,surroundedbyahaloofglobular
notedthattheamorphousobjectstheyreferredtoas“nebulae”
clusters. Trumpler(1930)recognizedtheroleplayedbyinter-
weremorecommoninsomepartsoftheskythanothersand
stellarabsorption;heprovidedafarlargerviewoftheGalaxy
inparticularintheconstellationofVirgo.
andmovedtheSunoutwardsfromthecenteroftheGalaxyto
apositionsome30,000lightyearsfromtheGalacticCenter. However, clusters of galaxieswere not describedin detail
Competing with this view was the hypothesis of Island until the work of Wolf (1924) who described the Virgo and
Universes, though at least some astronomers 100 years ago Comaclustersofgalaxies. Itwasnotknownatthattimethat
thoughtthat had been completely ruled out. Remember that thenebulae,astheywerethencalled,wereinfactextragalac-
100 years ago it was not knownthat the “nebulae” were ex- ticsystemsofstarscomparablewithourownGalaxy.
tragalacticsystems: theywerethoughtofaswhirlpoolsinthe Hubble, using the largest telescopes, noted the remark-
interstellarmedium. able overall homogeneity and isotropy of the distribution of
ThecontroversybetweentheGreatGalaxyandIslandUni- galaxies. The first systematic surveysof the galaxydistribu-
verseviewsculminatedinthegreatdebatebetweenCurtisand tionwereundertakenbyShapleyandhiscollaborators(often
Shapley in 1920 (Hoskin, 1976). Shapley, who had earlier uncitedandunder-acknowledgedwealthyBostonianwomen).
placedourSunintheouterreachesoftheGreaterGalaxyby This lead to the discovery of numerous galaxy clusters and
observingthe distributionofglobularclusters3, defendedthe evengroupsofgalaxyclusters.
Great Galaxy hypothesis and won the day for all the wrong
reasons.
However, it was left to Edwin P. Hubble to settle the is-
D. Hierarchicalmodels
sueinfavouroftheIslandUniverseswhenhefoundCepheid
variablesinthegalaxyNGC6822andtheAndromedanebula
(Hubble,1925a,b). The clustering together of stars, galaxies, and clusters of
galaxiesinsuccessivelyorderedassembliesisnormallycalled
Therewasoneanomalythatpersistedintotheearly1950’s:
a hierarchy, in a slightly different sense of the dictionary
ourGalaxyseemedtobethelargestintheUniverse.Thiswas
meaning in which there is a one-way power structure. The
resolvedbyBaadewhorecognizedthattherewereinfacttwo
technically correct term for the structured universes of Kant
populations of Cepheid variables (Baade, 1956). This dou-
andLambertismultilevel.Acompletemultileveluniversehas
bledthedistancestotheexternalgalaxies,therebysolvingthe
three consequences. One is the removal of Olbers paradox
problem.
(the motivation of John Herschel and Richard Proctor in the
ForHubbleandmostofhiscontemporarieswhathadbeen
19thcentury). Thesecond,recognizedbyKantandLambert,
foundwere“fieldgalaxies”largelyisolatedfromoneanother.
is that the universe retains a primary center and is therefore
Thiswasinpartdueto thesortsoftelescopeandtheirfields
nonuniform on the largest cosmic scales. The third, recog-
ofviewthatHubblewasusing(Hubble,1934,1936)andalso
nizedbytheIrishphysicistFournierd’AlbeandtheSwedish
in partdue to the lingeringeffectsof the phrase“Island uni-
astronomerCarlCharlierearlyinthe20thcenturyisthatthe
verse” which evoked images of isolation. Indeed, as late as
totalamountofmatterismuchlessthaninauniformuniverse
the 1960’s, astronomers who should have known better said
with the same localdensity. D’Albeputforwardthe curious
additionalnotionthatthevisibleuniverseisonlyoneofase-
riesofuniversesnestedinsideeachotherlikeChineseboxes.
This is not the same as multiple 4-dimensional universes in
3Weshouldrecallthatataboutthis timeLinblad(1926)andOort(1928)
higherdimensionalspaceanddoesnotseemtobeaforerun-
showedthatthestarsintheGalaxywereorbiting aboutadistantcenter,
thusclearlyplacingtheSunelsewherethanatthecenter. nerofanymodernpicture.
7
1. Charlier’sHierarchy
The idea that there should be structure on all scales up to
thatoftheUniverseasa wholegoesbacktoLambert(1761)
whowastryingtosolvethepuzzleofthedarknightskythatis
commonlycalled “Olber’s paradox”. (It was not formulated
by Olbers and it is a riddle rather than a paradox (Harrison,
1987)). Simply put: if the Universe were infinite and uni-
formly populated with stars, every line of sight from Earth
wouldeventuallymeetthesurfaceofastarandtheskywould
therefore be bright. The idea probably originated with John
Herschelin a review ofHumboldt’sKosmoswherethe clus-
teringhierarchyissuggestedasasolutiontoOlber’sParadox
asanalternativetodustabsorption.
At the start if the 20th century, The Swedish astronomer
Carl Charlier provided a cosmological model in which the
galaxies were distributed throughoutthe Universe in a clus-
tering hierarchy (Charlier, 1908, 1922). His motivation was
toprovidearesolutionforOlber’sParadox. Charliershowed
thatreplacingthepremiseofuniformitywithaclusteringhi-
erarchywould solve the problemprovidedthe hierarchyhad
aninfinitenumberoflevels(seeFig.2).
Charlier’sideawasnotnew,thoughhewasthefirstperson
toprovideacorrectmathematicaldemonstrationthatOlber’s
Paradox could indeed be resolved in this way. It should be
recalledthathewasworkingatatimebeforeanygalaxieshad
measuredredshiftsandlongbeforethecosmicexpansionwas
FIG.2 Hierarchicaluniverseswereverypopular attheendofthe
known.
19thcenturyandthefirsthalfofthe20thcentury. Reproducedfrom
ItisinterestingthattheCharliermodelhaddeVaucouleurs
Harrison(2000),Cosmology,CambridgeUniversityPress.
asoneofitslongstandingsupporters(deVaucouleurs,1970).
More recently still there have been a number of at-
tempts to re-incarnate such a universal hierarchy in terms recognize within its confines. This gave him a sample of 7
of fractal models. Fractal models were first proposed clusters with similar data, all from Mt. Wilson plates (5 in
by Fournierd’Albe (1907) and subsequently championed theMt.Wilsondirector’sreportfor1929-30andonethenjust
by Mandelbrot (1982) and Pietronero (1987). Several found by Lundmark). He was inspired to graph log(N) vs.
attempts have been made to construct hierarchical cos- the linear sizes of the clusters (Carpenter, 1931) and found
mological models (a Newtonian solution was found by a straight line relation, that is, a power law in N(diameter),
Wertz(1971),general-relativisticsolutionswereproposedby nowherenearassteep asN D3 orN proportionaltovol-
∼
Bonnor (1972); Ribeiro (1992); Wesson (1978)). All these ume. The then known globular cluster system of the Milky
solutions are, naturally, inhomogeneouswith preferredposi- Way (with about 35 clusters within 105 pc) also fit right on
tion(s) for the observer(s), and thus unsatisfactory. So the hiscurve.
presenttrendtoconciliatefractalmodelswithcosmologyisto Carpenter later considered a larger sample of clusters and
usethemeasureoflastresort,andtoassumethatalthoughthe foundthatasimilarcurvethenactedasanupperenvelopeto
matter distribution in the universe is homogeneous on large thedata(Carpenter,1938). Ifhisnumbersaretransformedto
scales, the galaxy distribution can be contrived to be fractal the distance scale with H = 100 km s−1 Mpc−1, then the
0
(Ribeiro, 2001). Numerical models of deep samples contra- relationsare(deVaucouleurs,1971)
dictthisassumption.
logN(max)=2.5+1.5logR(Mpc) (1)
or
2. Carpenter’slaw
logN(max)=2.19+0.5logV(Mpc3) (2)
EdwinF.CarpenterspenthisearlydaysatStewardObser-
vatory(ofwhichhewasdirectorformorethan20years,from andthemaximumnumberdensityingalaxiesperMpc3isalso
1938)scanningzone plates to pick outextragalacticnebulae proportionalto 0.5log(V). De Vaucouleurs called this Car-
forlater study. In 1931,he founda new cluster in the direc- penter’s law, though the discoverer himself had been some-
tionofCancer(independentlydiscoveredbyHubbleatabout what more tentative, suggesting that this sort of distribution
the same time.) He measured its size on the sky, estimated (which we would call scale free, though he did not) might
itsdistance,andcountedthenumberofgalaxies,N,hecould meanthattherewasnofundamentaldifferenceamonggroups,
8
clusters, and superclusters of galaxies, but merely a non-
random,non-uniformdistribution,whichmightcontainsome
information about the responsible process. It is, with hind-
sight, not surprisingthat the first few clusters that Carpenter
(1931)knewaboutwerethedensestsort,whichdefinetheup-
perenvelopeofthelargerset(Carpenter,1938).Theideasofa
numberofotherproponents,bothobserversandtheorists,on
scale-free clustering and hierarchical structure are presented
(nonetoosympathetically)inChapter2ofPeebles(1980).
3. DeVaucouleurshierarchicalmodel
De Vaucouleurs first appears on the cosmological stage
doubting what was then the only evidence for galaxy evolu-
tion with epoch, the Stebbins-Whitford effect, which he at-
tributed to observational error (deVaucouleurs, 1948). He
was essentially rightaboutthis, butwidely ignored. He was
atothertimesasupporterofthecosmologicalconstant(when
it was not popular) and a strong exponent of a hierarchical
universe,inwhichthelargeststructuresweseewouldalways
haveasizecomparablewiththereachofthedeepestsurveys
(deVaucouleurs,1960,1970,1971). Hepointedoutthatesti-
matesoftheageoftheuniverseandofthesizesofthelargest
objects in it had increased monotonically (and perhaps as a
sortofpowerlaw)withtimesinceabout1600,whiletheden-
sitiesofvariousentitiesvs.sizecouldallbeplottedasanother
powerlaw,
ρ(r) r−x,withxbetween1.5and1.9. (3)
∼
FIG. 3 In this idealized diagram de Vaucouleurs shows two hier-
By putting “Carpenter’s Law” into modern units, de Vau- archical frequency distributions of the number of clumps per unit
couleurs showed that it described this same sort of scale- volume. Inthetoppaneltherearenocharacteristicscalesinthedis-
tribution. Thisisthemodel proposedbyKiangandSaslaw(1969).
freeuniverse. Aslightlymorecomplexlaw,withoscillations
The bottom panel shows a more sophisticated alternative in which
arounda mean, falling line in a plot of density vs. size (see
theoveralldecreaseofthenumber ofclumpsperunitvolumedoes
Fig.3),couldhavegalaxies,binaries,groups,clusters,andsu-
not behavemonotonically withthescale, butitdisplaysaseriesof
perclustersas distinct physicalentities, withoutviolatinghis
localmaximacorresponding tothecharacteristicscalesofdifferent
mainpointthatwhatyouseeiswhatyouareabletosee.
cosmicstructures: galaxies,groups,clusters,superclusters,etc. Re-
De Vaucouleurssaid that it would be quite remarkable if, producedfromdeVaucouleurs(1971),AstronomicalSocietyofthe
justatthemomenthewaswriting,centuriesofchangeinthe Pacific.
best estimate for the age and density of the universe should
stop their precipitous respective rise and fall and suddenly
leveloffatcorrect,cosmicvalues. Thusheseemedtobepre- E. Thecosmologicalprinciple
dictingthatevidenceforauniverseolderthan10-20Gyrand
for structures larger than 100 Mpc should soon appear. (He ThenotionthattheEarthisnotatthecenteroftheUniverse
heldfirmlytoavalueofH near100kms−1Mpc−1formost isgenerallyreferredtoasthe“CopernicanPrinciple”,though
0
ofhislatercareer,exceptforthe1960paperwhereitwas75, it traces its originsback to Aristarchuswho thoughtthat the
but thought of local measurements of H as being relevant Sun and the stars were in fact fixed, with the stars being at
0
onlylocally). greatdistances.
Remarkable, but apparently true. Instead of taking off The modern notion that the Universe on the very largest
again, estimates of the age of the universe made since 1970 scalesshouldbehomogeneousandisotropicappearstohave
fromradioactivedecay of unstablenuclides, fromthe evolu- originatedwithEinstein(1917). Atthattimetherecouldhave
tionoftheoldeststars,andfromthevalueoftheHubblecon- beennoobservationalbasisforthisassumption.However,ho-
stant,increasinglyconcur.Andgalaxysurveyshavenowpen- mogeneityisaconsequenceofthenotionthatwearenotina
etrated a factor 10 deeper in space than the Shane-Wirtanen special place in the Universe and the assumptions of homo-
and Harvard counts in which de Vaucouleurssaw his super- geneityandisotropyprovideforeasysolutionsoftheEinstein
clusters. field equations. The first cosmological models of Einstein
9
andofde Sitter werebasedonthisprinciple. Robertsonand IV. DISCOVERINGCOSMICSTRUCTURE
WalkerderivedtheirfamoussolutionoftheEinsteinequations
usingonlythatprinciple. A. Earlycatalogbuilders
It was frequentlystated in the years thatfollowedthat the
Observational cosmology, like most other physical sci-
Universeinthelargelookedhomogeneousandisotropic.The
ences, is technology driven. With each new generation of
firstsystematicstudywasHubble(1926)whousedasample
telescopeandwitheachimprovementinthephotographicpro-
of 400 galaxies with magnitudes, the sample was thoughtto
cess,astronomersprobedfurtherintotheUniverse,catalogu-
becompletetomagnitude12.5.Hefoundhiscountsfittedthe
ingitscontents.
relationship
Early on, Edward Fath used the Mount Wilson 60” tele-
scope to photograph Kapteyn’s selected areas. That survey
showed inhomogeneities that were later analyzed by Bok
logN(<m)=0.6m+constant (4)
(1934) and Mowbray (1938) who demonstrated statistically,
usingcountsincells,thatthegalaxydistributionwasnonuni-
form. About this time, Carpenter (1938) noticed that small
andconcluded,importantly,that“Theagreementbetweenob-
objects tend to be dense while vast objects tend to be tenu-
servedandcomputedlogN overarangeofmorethan8mag.
ous. He plotteda remarkablerelationshipbetweenscale and
isconsistentwiththedoubleassumptionofuniformluminos-
densityrangingallthewayfromtheUniverse,throughgalax-
ityanduniformdistributionor,moregenerally,indicatesthat
iesandstellarsystemstoplanetsandrock,asithasbeenex-
thedensityfunctionisindependentofthedistance.” Hegoes
plainedinSect.III.D.2. Thiswasperhapsthefirstexampleof
ontolookatsystematicsintheresidualsinthisplotandcon-
ascalingrelationshipincosmology.
cludes that they may be due to “... clustering of nebulae in
By 1930, the Shapley/Ames catalog of galaxies revealed
thevicinityofthegalacticsystem. TheclusterinVirgoalone
theVirgoclusterasthedominantfeatureinthedistributionof
accountsforanappreciablepart.”
bright galaxies. It was already clear from that catalog that
Hubble only had data to magnitude 12. Anyone look- the Virgo Cluster was part of an extended and rather flat-
ing at the considerably fainter Shane and Wirtanen’s iso- tened supercluster. This notion was hardly discussed except
plethicmapsofgalaxycountsbasedontheLickSkySurvey bydeVaucouleurswhothoughtthatthiswasindeedacoher-
(ShaneandWirtanen (1967)), or the more recent Center for entstructurewhoseflatteningwasduetorotation.
Astrophysics (CfA-II) slices data (GellerandHuchra, 1989) TheLickSurveyoftheskyprovidedextensiveplatemate-
might be forgiven for questioning the homogeneity conjec- rial that was later to proveone of the key data sets for stud-
ture! ies of galaxy clustering. The early isoplethic maps drawn
byShaneandWirtanen(1954)providedthefirstcartographic
Thefirstdemonstrationofhomogeneityinthegalaxydistri- viewofcosmicstructure.Theircountsofgalaxiesincellswas
butionwasprobablytheobservationbyPeeblesthatthe(pro- toprovideRubin(1954)andLimber(1954)withthestimulus
jected)two-pointcorrelationfunctionestimatedfromdiverse to introducethe two pointclusteringfunctionasa descriptor
catalogs probing the galaxy distribution to different depths ofcosmicstructure.
followed a scaling law that was consistent with homogene-
But it was the Palomar Sky Survey using the new 48”
ity. The adventofautomatedplate-measuringmachinespro-
Schmidttelescopethatwastoprovidethekeyimpetusinun-
vided deeper and more reliable samples with which to con-
derstandingtheclusteringofgalaxies.Zwickyandhiscollab-
firmtheuniformdistributionnumber-magnituderelationship.
orators at Caltech systematically cataloged the position and
However,atthe faintestmagnitudelevels, thesecountsshow
brightness of thousands of brighter galaxies on these plates,
significantsystematicdeviationsfromwhatisexpectedfrom
creating what has become known as the “Zwicky Catalog”.
auniformdistribution: thesedeviationsareduetotheeffects
Abell (1958) made a systematic survey for rich clusters of
ofgalaxyevolutionatearlytimesandtheirinterpretationde-
galaxies and drew up a catalog listing thousands of clus-
pends on models for the evolution of stellar populations in
ters. This has becomesimply known as the “Abell catalog”.
galaxies. Recent, very deep studies (Metcalfeetal. (2001))
Fig. 4 shows a modern image of the cluster Abell 1689 ob-
showconvincingly“... thatspacedensityofgalaxiesmaynot
tainedbythe ACScameraaboardoftheHubbleSpaceTele-
havechangedmuchbetweenz =0andz =3”.
scope (HST). A catalog of galaxy redshifts noting the clus-
ters to which galaxies belonged was published in 1956 by
The first incontrovertible proof of cosmic isotropy came
Humasonetal.(1956).
only as recently as early 1990s from the COBE satellite
all-sky map of the cosmic microwave background radiation
(Smootetal., 1992). This map is isotropic to a high degree,
with relative intensity fluctuations only at the level of 10−5. 1. TheLicksurvey
Withthisobservation,andwiththereasonablehypothesisthat
the Universe looks the same to all observers (the Coperni- The first map of the sky revealing widespread clustering
can Principle) we can deduce that the Universe must be lo- and super-clustering of galaxies came from the Lick survey
cally Friedman-RobertsonWalker, ie: homogeneousas well of galaxies undertaken by ShaneandWirtanen (1967) using
asisotropic(Ehlersetal.,1968). large field plates from the Lick Observatory. This was, or
10
decadesandmore4.
2. PalomarObservatoryskysurvey
ThetwomaincatalogsofclustersderivedfromthePalomar
Observatory Sky Survey (POSS) were that of Abell (1958)
and that of Zwicky and his collaborators (Zwickyetal.,
1961–1968).
Abellwentonimmediatelytosaythattherewassignificant
higherorderclusteringinhisdata,giving,in1958,ascalefor
superclusteringof 24 (H /180)−1 Mpc. In 1961 at a meet-
0
ingheldinconnectionwiththeBerkeleyIAUAbellpublished
(Abell, 1961) a list of these “super-clusters”, dropped the
Hubbleconstantto75kms−1Mpc−1andestimatedmassesof
1016 1017M withvelocitydispersionsintherange1000-
⊙
3000−kms−1. Ataboutthesametime,vandenBergh(1961)
remarks that Abell’s most distant clusters (distance class 6
havingredshiftstypicallyaround50,000kms−1)showstruc-
tureontheskyonascaleofsome20◦,correspondingto100
Mpc, for his H = 180 km s−1 Mpc−1, or about 300 Mpc
0
usingcurrentvalues.
FIG.4 TheclusterofgalaxiesAbell1689atredshiftz=0.18seen Zwicky explicitly and repeatedly denied the exis-
bytheHSTwithitsrecentlyinstalledAdvancedCameraforSurveys tence of higher order structure (ZwickyandBerger, 1965;
(ACS).Thearcsobservedamongsthundredsofgalaxiesconforming ZwickyandKarpowicz, 1966; ZwickyandRudnicki, 1963,
theclusteraremultipleimagesoffar-awayindividualgalaxieswhose
1966). Some of his “clusters” were on the order of 80 Mpc
lighthasbeenamplifiedanddistortedbythetotalclustermass(vis-
across (for H less than 100), had significant substructure,
ible and dark) acting as a huge gravitational lens, (image courtesy 0
and would to any other person have looked like superclus-
of NASA, N. Benitez (JHU), T. Broadhurst (The Hebrew Univer-
ters! Herzog, one of Zwicky’s collaborators in the cluster
sity), H. Ford (JHU), M. Clampin (STScI), G. Hartig (STScI), G.
catalog,foundlargeaggregatesofclustersinthecatalogand
Illingworth (UCO/Lick Observatory), and the ACS Science Team,
andESA). had the temerity to say so publicly in a Caltech astronomy
colloquium. He was offered “political asylum” at UCLA
byGeorgeAbell. Karachentsev(1966) also reportedfinding
largeaggregatesintheZwickycatalog.
anyhowshouldhavebeen,thedefinitivedatabase. Itwasthe
subjectofstatisticalanalysisbyNeymanetal.(1953),which
wasamajorstartingpointforwhathavesubsequentlybecome 3. AnalysisofPOSSclusters
knownasNeyman–Scottprocessesinthestatisticsliterature.
Ironically,althoughtheseprocesseshavebecomeadiscipline Up until about 1960 most of those involved seemed to
in their own right, they have since that time played only a envisage a definite hierarchy of structures: galaxies (per-
minorroleinastronomy. haps binaries and small groups), clusters and superclusters.
Kiangremarkedthattheexistingdatawerebestdescribedby
ScottintheIAUSymposium15(Scott,1962)mentionsthat
continuous, “indefinite”, clustering: quite different from the
thereareclearlylargerstructurestobeseeninthesecounts,as
clustering hierarchyas understoodat the time (Kiang, 1961;
ShaneandWirtanen (1954) had already noted. They spoke
KiangandSaslaw,1969). Kiang,incidentally,bridgedacriti-
of “larger aggregations” or “clouds” as being rather general
caleraindataprocessing,using“computers”(i.e.,poorlypaid
features. TheLicksurveywaslatertoplayanimportantrole
non-PhDlabour,mostlywomenafterthestyleofShapley)and
inPeebles’systematicassaultontheproblemofgalaxyclus-
lateronrealcomputers(Atlas). Flinetal.(1974)cameinde-
tering. PeeblesobtainedfromShanethenotescontainingthe
pendently to the same conclusion, and in his presentation at
originalcountsin10’x10’cellsandcomputerizedthemforhis
IAUSymposium63wasscoldedbyKiangfornothavingread
analysis. Thecountsin1 degreecellshadbeenusedfirstby
theliterature.
VeraCooper-Rubin(asVeraRubinwasthenknown)tostudy
galaxyclusteringintermsofcorrelationfunctions,a taskset
byheradviserGeorgeGamow. Rubindidthisatatimewhen
there were no computers. It was TotsujiandKihara (1969)
4BJ“discovered” thispaperatthetimeofwritinghisReview ofModern
who first did this on a computerand publishedthe first two-
Physicsarticle(Jones,1976)whileperusingthePublicationsoftheAstro-
pointcorrelationfunctionaswe nowknowitwiththepower
nomicalSocietyofJapanintheInstituteofTheoreticalAstronomyLibrary
lawthathasdominatedmuchofcosmologyforthepastthree inCambridge.Theredonotappeartobeanycitationspriortothattime.