A Slight Excess of Large Scale Power from Moments of the Peculiar Velocity Field

0.33 MB·English

by  Edward Macaulay

Mon.Not.R.Astron.Soc.000,000–000(2010) Printed28January2011 (MNLATEXstylefilev2.2)
A Slight Excess of Large Scale Power from Moments of the Peculiar
Velocity Field
1 E. Macaulay1⋆, H. Feldman2, P. G. Ferreira1, M. J. Hudson3,4,5, R. Watkins6
1
0 1Astrophysics,UniversityofOxford,DenysWilkinsonBuilding,KebleRoad,OxfordOX13RH,UnitedKingdom.
2 2DepartmentofPhysicsandAstronomy,UniversityofKansas,Lawrence,KS66045,USA.
3DepartmentofPhysicsandAstronomy,UniversityofWaterloo,Waterloo,ONN2L3G1,Canada.
n
4Institutd’AstrophysiquedeParis-UMR7095,CNRS/Universite´PierreetMarieCurie,98bisboulevardArago,75014Paris,France.
a
J 5PerimeterInstituteforTheoreticalPhysics,31CarolineSt.N.,Waterloo,ON,N2L2Y5,Canada.
6DepartmentofPhysics,WillametteUniversity,Salem,OR97301,USA.
7
2

]
O 28January2011
C
.
h ABSTRACT
p
- The peculiar motions of galaxies can be used to infer the distribution of matter in the
o
Universe.Ithasrecentlybeenshownthatmeasurementsofthepeculiarvelocityfieldindicates
r
t an anomalously high bulk flow of galaxies in our local volume. In this paper we find the
s
a implications of the high bulk flow for the power spectrum of density fluctuations. We find
[ thatanalyzingonlythedipolemomentofthevelocityfieldyieldsanaveragepowerspectrum
amplitudewhich is indeedmuch higher than the ΛCDM value. However,by also including
3
shearandoctupolemomentsofthevelocityfield,andmarginalizingoverpossiblevaluesfor
v
the growth rate, an average power spectrum amplitude which is consistent with ΛCDM is
1
5 recovered.Weattempttoinfertheshapeofthematterpowerspectrumfrommomentsofthe
6 velocityfield,andfindaslightexcessofpoweronscales∼1h−1Gpc.
2
Key words: cosmology: large scale structure of the universe – cosmology: observation –
.
0 cosmology:theory–galaxies:kinematicsanddynamics–galaxies:statistics
1
0
1
:
v
1 INTRODUCTION the simplest statistic: the bulk flow. The results were surprising:
i
X thereisclearevidenceforalargebulkflow,whichisveryunlikely
Peculiarvelocitiesareusefulcosmologicalprobes.Inprinciple,the
r withinthecurrentpreferredmodelofstructureformation,i.e.aflat,
peculiarvelocityfieldisanunbiasedtraceroftheunderlyingmatter
a Friedman-Robertson-Walker universewithanappreciablecosmo-
distribution, and should be sensitive to structure on scales larger
logical constant permeated by Gaussian, scale-invariant perturba-
thanthenominalsizeofthesurvey.Thestudyoflargescaleflows
tions(knownastheΛCDMmodel).ThefindingsofWatkinsetal.
hasalongandhallowedtradition;throughoutthe1980sandearly
(2009);Feldmanetal.(2010)leadstothequestion:isthereexcess
1990s,measurementsoflargescaleflowsweredeployedtonotonly
poweronlargescales?Asyetwedonothaveadirectmeasurement
constrain the fractional energy density in matter, Ωm but also to ofthepowerspectrumofdensityfluctuationsonthesescalesinthe
identifypossiblesourcesofgravitationalattractionwhichmightlie
nearbyUniverse;thecosmicmicrowavebackgrounddoesprobea
outsidecurrentgalaxysurveys(Strauss&Willick(1995)).
widerangeofscalesbutathighredshift.Couldthelargebulkflow
Velocitycataloguesarehardtoconstruct. Whilecurrentred-
beasignatureforlargescalefluctuationsatlowredshift?Itisthis
shift surveys have of order 105 objects, velocity catalogues are
questionwewishtoaddressbyusingthemeasurementofthebulk
typically restricted to of order 103 galaxies. Yet over the past
flowandafewoftheotherlowermomentstoestimatethepower
three decades, enough measurements of peculiar velocities have
spectrumofdensityfluctuations.
been accrued to be able to construct a reasonably complete cat-
alogue out to a maximum distance of about 100h−1 Mpc. In a
recentseriesofpapers(Watkinsetal.2009;Feldmanetal.2010),
Jaffe&Kaiser(1995)inferredconstraintsonthematterpower
the authors showed that a collection of peculiar velocity surveys
spectrum parameterized by σ and shape parameter Γ with data
8
couldbecombinedtoconstructareliable,well-behaved“compos-
from Lauer&Postman (1994), and seperately with data from
ite”catalogue. Thiscompositecataloguewasthenusedtoextract
Riessetal.(1995).Theyfoundthatincludingonlythebulkflowof
theLauer&Postmancataloguefavouredamodelwithhighlarge
scalepower,whichbecamemorereasonablewhentheshearofthe
⋆ email:[email protected] velocityfieldwasincluded,suggestingaverylargescaleexcessof
2 E. Macaulayet al.
power.However,thedatafromRiessetal.(1995)didnotsuggest The approach we describe here isto combine individual peculiar
anylargescaleexcess. velocity measurements to estimate the velocity field of our local
TheshearanddipolemomentsusedbyJaffe&Kaiser(1995) volumebyweighingeachmeasurementaccordingtohowsensitive
are slightly different to the minimum variance shear and dipole itistoeachcomponentoftheunderlyingvelocityfield.Wecande-
moments weusehere, because theJaffe&Kaiser (1995) method scribethevelocityfieldv(r),wherevisthethreecomponentsof
assumes that the non-modeled velocity is noise. Thus the dipole thepeculiarvelocityfieldatpositionr.Foreachgalaxy(labelled
moment from the Jaffe&Kaiser (1995) method will depend on n), we measure the line of sight component of this field, S, i.e.
whethertheshearisincluded,whereastheminimumvariancefor- Sn(r) = v(rn)·ˆrn whereˆrn is the unit vector pointing in the
malismweemployhereestimateseachmomentindependently of r direction.WeassumethismeasurementisdrawnfromaGaus-
n
the total number of moments analyzed. Further, Jaffe&Kaiser sian distribution with variance σn2. An additional term, σ∗2, is in-
(1995) compare very different catalogues with different geome- cludedinthevariance,toaccountfornonlinearflows.Themethod
tries, i.e., densities, distribution, etc. The minimum variance isfairlyinsensitivetotheparticularchoiceofσ∗2,asthecombined
methodcombinesindividualmeasurementstoestimatethevelocity uncertaintyofσn2 +σ∗2tendstobedominatedbythemeasurement
momentsforanidealizedgeometry,andtheresultsarethusinde- uncertainty,σ2.
n
pendentofthegeometryofanyparticularsurvey. Thesimplestresultwecanquoteforasurveyofpeculiarve-
Kolatt&Dekel (1995) inferred constraints on large scale locitiesisthe‘bulkflow’vector,u,theaveragevelocitymagnitude
structure from the POTENT velocity field. Uncertainty in the anddirectionofthegalaxiesinasurvey.Wemustbecarefulwhen
growth of structurerate, f, wasincluded byreporting thedegen- combiningpeculiarvelocitymeasurementstoincludetheeffectof
erate combination of P(k)f2. With current improved constraints only measuring the line of sight component. This is achieved by
onthegrowth rate, andmorepeculiar velocitydata available,we multiplyingeachcomponentofSn(inthecoordinatesystemofthe
attempttoconstraintheunbiasedmatterpowerspectrumfrommo- bulk flow) by a weight wa, which essentially depends on the or-
mentsofthepeculiarvelocityfield. thogonalityofS tothecoordinatesystemofthebulkflow,sothat
RecentlySongetal.(2010)appliedamethodtoestimatebulk ua = nwa,nSn.Kaiser(1988)showedthattheweightsforthe
flows from two point correlations in galaxies distributions, and bulkfloPware
fewoveuenrs,detehneloairteeavrnidtahleyantsciteshoefonarlnyaopmlraoragbloeeusbssulcylaklheflisgohuwpbiutnolkd0a.fl0toa3wfhroMismpmcSo−Ds1tS,sSwe.nhHseiroteiwvaes- wi,n =A−ij1Xm σxˆn2j+·rˆσn∗2 (1)
toscales>0.01hMpc−1. where
msuertehmToedhnoitsloipgnaytpoetaronisinesscttoriurmpcatoutreraetodefaaosnufroinllldooicwvailsd.upIaenlcuspeleicactriuolvniaer2lovwceietlyorcefiivteiyledwm.Wethaee- Aij =Xm (xˆi·σrˆm2m)+(xˆσj∗2·rˆm) (2)
review the method of Kaiser (1988) to weight peculiar velocity Because of sparse sampling, bulk flow results between
measurementstoestimatethebulkflow.Wealsoreviewthemethod different surveys are not necessarily immediately comparable.
of Watkinsetal.(2009) tousepeculiar velocity measurements to Watkinsetal.(2009)devisedamethodtoweightthebulkflowofa
estimatethebulkflowofourlocalvolume,sothatresultsfromdif- particularsurveytoestimatethebulkflowofahypotheticalspher-
ferentsurveyscanbedirectlycompared.Wedescribethepeculiar ically symmetric survey, where the density of galaxies n falls of
velocitycataloguescompiledbyWatkinsetal.(2009).Wethenre- as n(r) ∝ exp(−r2/2RI2), where RI2 isthe characteristic depth
viewtheworkofFeldmanetal.(2010)toincludehighermoments ofthesurvey.Thisallowsbulkflowsfromdifferentsurveystobe
ofthevelocityfield.Insection3wedescribehowwerelatethese directly compared, and combined to give us a better estimate of
momentsofthevelocityfieldtoconstraintsonlargescalestructure. thebulkflowofourlocalvolumeofspace.Westartwiththefor-
Wepresentourresultsinsection4. malism from Kaiser (1988), and then introduce the method from
Watkinsetal.(2009).
Theideaisthatthebulkflowwemeasure,u,isessentiallya
convolutionofthepeculiarvelocityfieldvwiththewindowfunc-
2 COMBININGPECULIARVELOCITY tionofthesurveyW,
MEASUREMENTS
Toestimatethepeculiar velocity of galaxiesat cosmological dis- ui(r0)=Z d3rWij(r)vj(r+r0) (3)
tances,weneedboththemeasuredredshiftofthegalaxy,andanin-
Thisallowsustocalculatethevarianceofudirectlyfromthecon-
dependentdistancemeasure,suchastheluminositydistance.From
volutiontheorem
thedistancemeasure, wecanestimatewhat wewould expect the
redshifttobesolelyduetotheHubbleflow.Wecanthenattribute huiuji= d3kPv(k)Wi2j(k) (4)
thedifferencebetweenthisexpectedredshiftandthemeasuredred- Z
shifttothepeculiarvelocityofthegalaxy(Peebles(1993)).There wherethewindowfunctionisgiven(inrealspace)by
aretwokeydifficultieswiththemethod.Thefirstisthattheuncer-
2ta0in%tyfoornTthuellylu-Fmiisnhoesri,tyFadbiesrta-Jnaccekissonty,poircaFlulyndraatmheerntlaalrgPel:a∼ne1m0etao- Wij(r)=A−ij1Xn δ(r−rn)σn2rˆi+rˆjσ∗2 (5)
surements,and∼5%forsupernovae.Theseconddifficultyismore Itisextremelyusefultosplithuiujiintotwoterms:oneterm,
inherent:thismethodonlyprovidesthelineofsightcomponentof R(v),duetomeasurementvariance,andanoiseterm,R(ǫ) dueto
ij ij
thepeculiarvelocity.Theimportanceofthiseffectcanbeillustrated
ifweconsiderahypotheticalbulkflowfromanarbitrarynorthto thevariancefromnonlinearflows,sothathuiuji=Ri(jv)+Ri(jǫ).
Wethenhavethat
thesouth.Asurveyofgalaxiestoournorthorsouthwouldbesen-
sitivetothisflow,whereasasurveytooureastorwestwouldnot. R(ǫ) =A−1 (6)
ij ij
LargeScalePower fromPeculiarVelocityMoments 3
and vi(r)=Ui+Uijrj+Uijk(rjrk−Λjk)+... (16)
Ri(jv) =Z d3kWi2j(k)Pv(k) (7) whereΛjkisgivenby
where Λjk = rjrkd3r (17)
Z
Wi2j =Wil(k)Wj∗m(k)kˆlkˆm (8) inordertVoremoveoverlappingcomponents.Thelineofsightcom-
Wewillreturntoequations6and7whenwecometoestimatelarge ponentisthen
scalestructureinsection3.
Theapproachof Watkinsetal.(2009)istofindweightsthat s(r)=Uirˆi+Uijrrˆirˆj+Uijk r2rˆirˆjrˆk−Λjkrˆi +... (18)
(cid:0) (cid:1)
minimizetheaveragevariancebetweenaparticularbulkflowvec- Feldmanetal. (2010) have calculated the minimum variance
tor u and the average bulk flow of our hypothetical survey, U, weightsforthe19componentsofthethirdorderexpansion,overk
(ui−Ui)2 .Ifweassumethemeasurementerrorisuncorrelated rangesfrom0.002to0.196hMpc−1.Wecanthinkoftheresulting
w(cid:10)iththebulk(cid:11)flow,wecanexpand (ui−Ui)2 as setof19components ofthedipole,quadropole andoctupoleasa
(cid:10) (cid:11) formofdatacompressioncontainingthehighestsignaltonoisein-
(ui−Ui)2 = wi,nwi,mhSnSmi+ Ui2 −2 wi,nhUivni(9) formationinagivenpeculiarvelocitysurvey.Indeed,thatwillbe
(cid:10) (cid:11) Xn,m (cid:10) (cid:11) Xn our philosophy - tousethisformof thedatatoinferinformation
Beforeminimizingthisexpressionwithrespecttowi,n,weenforce abouttheunderlyingdensityfield.
theconstraintthatWi2j(k)→1/3ask →0,sothattheweighting We study the ‘COMPOSITE’ peculiar velocity catalogue
isequalforeachdimensionofthepeculiarvelocityfield.Wethus compiledbyWatkinsetal.(2009),andalsousedinFeldmanetal.
havetominimize (2010).Thecatalogueconsistsof4,536peculiarvelocitymeasure-
wi,nwi,mhSnSmi+ Ui2 ments,withacharacteristicdepthof34h−1Mpc.Thecharacteristic
depthisgivenbytheaveragedistancetoeachgalaxy,weightedby
Xn,m (cid:10) (cid:11)
theinversesquareofthepeculiarvelocityuncertainty.
1
−2 wi,nhuiUiiλ(Pnmwi,nwi,m− ) (10)
3
Xn
whereλisaLagrangemultiplierand 3 RELATINGVELOCITYTOMATTER
Pnm = d2kˆ rˆn·kˆrˆm·kˆ (11) Wehavepresentedamethodtocombinelineofsightestimatesof
Z 4π (cid:16) (cid:17) thepeculiarvelocityofindividualgalaxiesintoanestimateofthe
Differentiatingequation10,andsettingtheresultequaltozero momentsdescribingthevelocityfieldofourlocalvolume.Wenow
gives wanttobeabletocomparethismeasurementtoexpectationsfrom
ourcosmology.Thebasicideaisthat,inthelinearregime,galaxies
(hSnSmi+λPnm)wi,m =hSnUii (12) flow towards local over-densities of matter, so the velocity field
Xm v(r)isgivenby
I1wt2ii,simne,maasnaitedrriQxtofiofsroomrlv.heSSunfobUrsittiihtugetiivwnegesiGghtfso,rwhSi,mnS,mifiw,Pe rfeowrrPitneme,qwuaitifoonr v(r)= f4Hπ0 Z d3r′δ(r′)|(rr′′−−rr|)3 (19)
f istheperturbationgrowthrate,∂lnδ/∂lna,andδisthematter
(G+λP)wi =Qi (13)
densitycontrast.ThematterdensitycontrastismodelledasaGaus-
whichisnoweasytosolvefortheweightswi. sian random field with a power spectrum defined as h|δ˜(k)|2i =
wi =(G+λP)−1Qi (14) (2π)3P(k), where δ˜is the Fourier transform of the density con-
trastinrealspace.
Aswellasthebulkflowof thesurvey,wecanalsoconsider AusefulwayofparametrizingP(k)isintermsofbandpow-
higher moments of the peculiar velocity field if we consider the ers:
fieldasaTaylorexpansion,asin
P(k)= Pα kα <k<kα+1 (20)
vi(r)=Ui+Uijrj+Uijkrjrk+... (15) (cid:26) 0 otherwise
Ui is the bulk flow vector - also known as the ‘dipole moment’. Toestimatethemostlikelymatterpowerspectrum,basedon
Uij isthe‘sheartensor’orthe‘quadrupolemoment’,andprovides the peculiar velocity data, our approach is to construct, and then
information about the distance at which the bulk flow attractor minimize,thelikelihoodfunctionLasafunctionofPα:
islocated.The‘octupoletensor’U providesinformationabout
thevelocityfieldonscalessmalleritjhkanthesurvey.Feldmanetal. −2lnL(Pα)∝ln|C|+uTC−1u (21)
(2010)haveextendedtheworkofWatkinsetal.(2009)toinclude whereuarethevelocitymomentcomponents.Thecovariancema-
thesehigherordermoments.Ifweassumethatthepeculiarveloc- trix,C,isderivedfromequation4
ityfieldisentirelyduetogravitationalinfall,weexpectthefieldto
becurl-free,andconsequentlyUijandUijktobesymmetric.Thus Cpq =Rp(vq)+Rp(ǫq) (22)
thepeculiarvelocityfieldcanbedescribedtothirdorderbythe19
indepBenefdoernetwveelporcoitcyeemdo,mweenhtasv,eutio. becarefulwiththedefinitionof lwohcietryemthaetr‘iexr’roRrp(mvq)atirsixg’ivRenp(ǫq)byisegqiuvaetniobny7e.qWuaetiroenla6te,athnedvtheelo‘cviety-
covariancematrixtothepowerspectrumbyequation7,sothat
theoctupoletensor,becausesomecomponentsofthetensoroverlap
withthedipolemoment.Assuch,wemodifytheexpansionofthe f2
R(v) = dkP(k)W2 (k) (23)
velocityfieldtobe pq 2π2 Z pq
4 E. Macaulayet al.

Simulation
Dipole ΛCDM
1 OΛCctDupMo±le10% 3pc)]
M
e]0.8 −1P(k) [(h 104
ativ
Rel0.6
d [
oo 10−2 10−1
h
eli
Lik0.4
Dipole
0.3 Quadrupole
Octupole
0.2
W0.2
0.1
0
102 103 104 105 106
Average P(k) [(h−1 Mpc)3] 0 10−2 10−1
k [h Mpc−1]
Figure1.LikelihooddistributionsfortheaverageP(k)valueinthekrange
ofourwindowfunction,inferredfromasimulated peculiarvelocitycata-
Figure2.Intheupperfigureisplottedthepowerspectrumshape,inferred
logue.Whetherweincludeonlythebulkflowofthevelocityfield(labelled
onlyfrompeculiarvelocitydatageneratedfromasimulatedcatalogue.We
‘Dipole’),oralsoincludetheshearandoctupolemoments(labelled ‘Oc-
tupole’),werecoveravaluewhichiscompatiblewiththefiducialΛCDM havechosenthreebands,evenly spacedinlogk.Theshadedregionrep-
resents the marginalized 1σ uncertainty on the power in each band. For
value. Forclarity, thelikelihood functions havebeenscaledarbitrarily to
themaximumvalueofthedipolefunction.TheΛCDMvaluehasbeenil- comparison,thetheoreticalΛCDMpowerspectrumhasbeenoverplotted.
Theresults shown here include the dipole, shear, and octupole moments
lustratedhereandinfurthersuchplotsasaGaussiandistributionwithan
arbitrarystandarddeviationof10%ofthefiducialvalue. ofthesimulatedvelocityfield.Asdiscussedlater,theuncertaintiesineach
bandhavealsobeenmarginalizedoverthegrowthrate.Inthelowerpanel
areplotted theaverage values ofthedipole, shear, andoctupole window
functions.Thedipoleismostsensitiveonscalesk.0.01hMpc−1.
Wecanchoose thewidthof eachband-power byintegrating
the window function (which is independent of Pα) over the bin
rangekαtokα+1.Wecanthenfactorintheaveragepowerampli-
tudeinthisrange,sothat issimilartotheeffectobservedbyJaffe&Kaiser(1995)whenin-
cludingtheshearofthepeculiarvelocityfield,andalsonoticedby
Rp(vq) ≃ 2fπ22 Xα PαKpαq (24) FeldmInanFiegtuarle.(32,0t1h0e).growthratewasfixedatthefiducialΛCDM
value,toillustratetheimportanceofincludinghighermomentsof
wherethekernelKisgivenby
the velocity field. However, presently the best constraints on the
kα+1 growth rate at low redshift are from Peacocketal. (2001)’smea-
Kα = dkW2 (k) (25)
pq pq surement of the redshift space distortion compression parameter.
Z
kα
Thisconstrainsthegrowthratetof = 0.49±0.14.Wenowcon-
We thus have a likelihood function for the velocity moments in sider the effect of marginalizing over the growth rate, with this
termsofthegrowthrateandpowerspectrumband-powers. priorapplied.Likelihoodcontoursforthegrowthrateandaverage
Wehavetestedourmethodologyonsimulatedpeculiarveloc- powerintheCOMPOSITEsurveyareshowninFigure4.Whenwe
itycatalogues,generatedfromaΛCDMpowerspectrum.Wegen- marginalizeoverthegrowthrate,weseeinFigure5thattheaver-
erateaseriesofmockcataloguesfromasetofN-bodysimulations, agepowerdoesnotexcludethefiducialΛCDMvalue.Howeverin
estimatethecorrespondingdipole,quadropolesandoctopolesand thecontextofΛCDMmodels,thegrowthrateisnotafreeparame-
then minimize the corresponding likelihood to recover the input ter,ratheritisdeterminedprimarilybyΩm(whichalsoaffectsthe
power spectrum.Weillustrateourresultsforonesuchrealization shape of thepower spectrum and theaverage power). Inorder to
inFigures1and2,andconfirmthatourmethoddoesnotgenerate obtain consistency with the predicted average power, on requires
anyspuriouslargescalepower. f ∼ 0.7, whichismuch largerthanthefiducial ΛCDMvalueof
0.48.
Inadditiontothetotalaveragevalue,wecanattempttoinfer
theshapeofthepowerspectrum.Thewindowfunctionwasdivided
4 RESULTSANDDISCUSSION
intothreeranges, evenlyspacedinlogk.Thelikelihoodfunction
Wefirstconsider themostlikelytotalaveragepower of oursam- wasthenevaluatedintermsofthepowerineachofthethreebins,
ple,overtheentirekrangeofourwindowfunction,k = 0.002to Pα,andthegrowthrate.Asbefore,aprioronthegrowthratewas
0.196hMpc−1.WeseeinFigure3(a)thatwhenweincludeonly applied,andthisparameterwasmarginalized.Whenweonlycon-
thedipolemomentofthevelocityfield,theaveragepowerforeach siderthebulkflow,wefindthatthelikelihoodissufficientlybroad
surveyismuchhigherthantheΛCDMvalue.Thisisentirelycon- andcorrelatedthatitis,inpractice,impossibletopindownthree
sistent with the results found in Watkinsetal. (2009). When we independentestimatesofPα.Thisisdisappointing–themeasure-
alsoincludetheshearandoctupolemoments,wefindmuchbetter ment of the dipole alone does not allow us to identify if there is
agreementwiththeΛCDMvalue,aswecanseeinFigure3(b).This largescalepower.
LargeScalePower fromPeculiarVelocityMoments 5

COMPOSITE 0.9
1 ΛCDM
0.8
0.8 0.7
elihood [Relative]0.6 Growth Rate, f000...456
Lik0.4
0.3
0.2 0.2
0.1
0
104 105 106 104 105
Average P(k) [(h−1 Mpc)3] Average P(k) [ (h−1 Mpc)3]
(a)
Figure4.Likelihood contours fortheCOMPOSITEsurvey,withdipole,
shear, and octupole moments included, with contours spaced at1, 2and
3σconfidencelevels.Wearenowconsideringtheaveragepowerandthe
COMPOSITE perturbationgrowthrateasfreeparameters.Aprioronthegrowthratehas
1 ΛCDM
beenapplied.Forcomparison,the1dimensionallikelihooddistributionfor
theCOMPOSITEsurveyinFigure3(b)isaslicethroughthis likelihood
0.8 contouratthefiducialΛCDMvalueofthegrowthrateatz = 0,i.e.f =
0.48.Ascanbeseenhere,includingobservationallyconstrainedvaluesof
e]
ativ thegrowthrateallowsforawiderrangeofallowedPα.
Rel0.6
d [
o
o
h
eli
Lik0.4
COMPOSITE
1 ΛCDM
0.2
0.8
0 e]
104 Average P(k) [(h−1 Mpc)3] 105 Relativ0.6
(b) od [
o
h
eli
Figure3.Intheupperpanelisthelikelihooddistributionfortheaverage Lik0.4
amplitudeofthepowerspectrum,inferredonlyfromthedipolemoment.
To account for the anomalously high bulk flow, we have to conclude a
0.2
powerspectrumamplitudewhichisincompatibly higherthantheΛCDM
value.Inthelowerpanel,weshowtheeffectofincludingtheshearandoc-
tupolemomentsofthevelocityfield.Bynowincludingthehighermoments,
0
theresultsaregenerallymorecompatiblewiththeΛCDMvalue.Here,the 103 104 105 106
growthrate,f,iskeptfixedatthefiducialΛCDMvalue.Inbothfigures,the Average P(k) [(h−1 Mpc)3]
shadedareaisthe1σconfidenceregion.Theshadedareaoftheupperfigure
appearsdisproportionatelywidebecausethelargetailofthedistributionis Figure5.AveragepowerfortheCOMPOSITEsurvey,withdipole,shear,
distortedbythelogarithmicscaleonthehorizontalaxis. andoctupole moments included, now marginalized overthe growth rate.
Theshadedregionindicatesthe1σuncertaintyboundariesonthelikelihood
function.TheexpectedΛCDMvalueisnowincludedinthisrange.
However, when we also include the shear and octupole mo-
ments of thevelocity field, wefind that the band powers are vir-
tuallyuncorrelated,andwecanobtainthreeindependentmeasure- are,inprinciple,severelyaffectedbyedgeeffects.Indeed,there-
ments.InFigure6weplotthemarginalized1σuncertaintyforeach centanalysisoftheSDSSdatainSylosLabinietal.(2009)should
Pα fortheCOMPOSITEcatalogue.Thereisaslightdetectionof makeonebewaryaboutoverinterpretingestimatesofstructureon
excesspoweronthelargestscales. thelargestscalesofthesurvey.Yetthefactthatthesamelevelof
We find that our estimate of the excess power seems to re- fluctuationsisobtainedfromapeculiarvelocitysurvey,subjected
flectthelargescaleestimatesofP(k)foundintheestimatefrom toacompletelydifferentanalysismaybeanindicationthatredshift
both the SDSS main galaxy power spectrum from Percivaletal. surveyestimatesofstructureonthelargestscalemustbetakense-
(2007) and the 2dFGRS power spectrum from Coleetal. (2005). riously. Furthermore, an excess of clustering on similar scales at
Oneshouldbecarefulaboutover-interpretingthesesimilarities,as higherredshift wasrecentlyfoundbyThomasetal.(2010)inthe
both the SDSS and 2dFGRS estimates of power on large scales MegaZDR7photometricredshiftsurvey.
6 E. Macaulayet al.
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Figure6.Powerspectrumshapeinferredfromdipole,shear,andoctupole
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momentsfortheCOMPOSITEcatalogue. Theshadedregionsare1σun-
PriceI.,SutherlandW.,TaylorK.,2001,Nature,410,169
scleirgthaitnetixecse,smsaorfgpinoawliezredonovsecraltehseogfro∼wt1hhr−at1eGanpdc.thTehoestheearrPeαth.eWsceafilensdaat PeeblesP.J.E.,1993,PrinciplesofPhysicalCosmology.Prince-
tonUniversityPress
whichwearemostsensitive. Thisexcessofpowerappearstoagreewith
thelargestscalesinthe2dFGRSandSDSSsurveys,whichareplottedhere Percival W. J., Nichol R. C., Eisenstein D. J., Frieman J. A.,
as inPercivaletal. (2007)andhavebeen deconvolved fromtheir survey Fukugita M., Loveday J., Pope A. C., Schneider D. P., Szalay
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BahcallN.A.,BrinkmannJ.,ConnollyA.J.,MeiksinA.,2007,
ApJ,657,645
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Whatshouldwemakeofthisexcessofpower,andhowdoes L91
itrelatetoourinitialquestion:theanomalouslyhighbulkflow?To SongY.,SabiuC.G.,NicholR.C.,MillerC.J.,2010,JCAP,1,
startwith,wehavedemonstratedthat itisincompletetoinfer re- 25
sultsfrommeasurementsofthebulkflowalone,asincludinghigher StraussM.,WillickJ.,1995,PhysicsReports,261,271
momentsofthevelocityfieldyieldssignificantlydifferentresults. SylosLabiniF.,VasilyevN.L.,BaryshevY.V.,Lo´pez-Corredoira
Asnoted in Feldmanetal. (2010), the low shear moments of the M.,2009,AstronomyandAstrophysics,505,981
velocityfieldsuggestthatthesourcesresponsibleforthebulkflow ThomasS.A.,AbdallaF.B.,LahavO.,2010,ArXive-prints
are very far away. In the formalism we have adopted here, only WatkinsR.,FeldmanH.A.,HudsonM.J.,2009,MNRAS,392,
oneofthebulkflowmomentsisanomalouslyhigh-theotherbulk 743
flowmoments,andtheshearandoctupolemoments,areconsistent
withΛCDM.Fromoneperspectiveonecouldarguethattheextra
freedomfromincluding19momentsofthevelocityfield,andthe
growth rate, as opposed to just three moments of the dipole mo-
ment, provide a way to interpret the anomalously high bulk flow
whichisconsistentwithΛCDM.Ontheotherhand, allmoments
are not equivalent. The dipole moments are special in the sense
thattheyprobethelargestscales(andarealsothemomentsthatare
most robustly measured). Consequently the inferred shape of the
powerspectrumisdifferentfromΛCDM.
5 ACKNOWLEDGMENTS
WewouldliketothankJoeZuntzandAndrewJaffeforusefuldis-
cussions.PGFacknowledges support fromSTFC,BIPACandthe
OxfordMartinSchool.HAFhasbeensupportedinpartbyanNSF
grantAST-0807326andbytheUniversityofKansasGeneralRe-
search Fund. MJH has been supported by NSERC and acknowl-
edgesthehospitalityoftheInstitutd’AstrophysiquedeParis,and
thefinancialsupportoftheIAP/UPMCvisitingprogrammeandthe
FrenchANR(OTARIE).