MAMPOSSt: Modelling Anisotropy and Mass Profiles of Observed Spherical Systems. I. Gaussian 3D velocities

0.58 MB·English

by  Gary A. Mamon

Mon.Not.R.Astron.Soc.000,000–000(2012) Printed7January2013 (MNLATEXstylefilev2.2)
MAMPOSSt: Modelling Anisotropy and Mass Profiles of Observed
Spherical Systems. I. Gaussian 3D velocities
Gary A. Mamon1,2, Andrea Biviano3 and Gwenae¨l Boue´4,5,6,1
3 1Institutd’AstrophysiquedeParis(UMR7095:CNRS&UPMC),98bisBdArago,F–75014Paris,France,e-mail:[email protected],
1 2Astrophysics&BIPAC,DepartmentofPhysics,UniversityofOxford,OxfordOX13RH,UK
0 3INAF/OsservatorioAstronomicodiTrieste,viaG.B.Tiepolo11,34143Trieste,Italy
2 4AstronomieetSyste`mesDynamiques,IMCCE-CNRSUMR8028,ObservatoiredeParis,UPMC,77Av.Denfert-Rochereau,75014Paris,France
n 5CentrodeAstrof´ısicadaUniversidadedoPorto,RuadasEstrelas4150-762Porto,Portugal
a 6DepartmentofAstronomyandAstrophysics,UniversityofChicago,5640SouthEllisAvenue,Chicago,IL60637,USA
J

4
Accepted2012Dec05.Received2012Dec05;inoriginalform2012Jul20

]
O
C ABSTRACT
Mass modelling of spherical systems through internal kinematics is hampered by the mass
.
h / velocity anisotropydegeneracyinherentin the Jeans equation,as well as the lack of tech-
p
niquesthatarebothfastandadaptabletorealisticsystems.Anewfastmethod,calledMAM-
-
o POSSt,isdevelopedandthoroughlytested.MAMPOSStperformsamaximumlikelihoodfit
r of the distribution of observed tracers in projected phase space (projected radius and line-
t
s of-sight velocity). As in other methods, MAMPOSSt assumes a shape for the gravitational
a potential(orequivalentlythetotalmassprofile).However,insteadofpostulatingashapefor
[ the distributionfunctionin termsof energyandangularmomentum,orsupposingGaussian

2 line-of-sight velocity distributions, MAMPOSSt assumes a velocity anisotropy profile and
v a shape for the three-dimensional velocity distribution. The formalism is presented for the
5 caseofaGaussian3Dvelocitydistribution.Incontrasttomostmethodsbasedonmoments,
5 MAMPOSStrequiresnobinning,differentiation,norextrapolationoftheobservables.Tests
4 on cluster-mass haloes from ΛCDM dissipationless cosmological simulations indicate that,
1 with500tracers,MAMPOSStisableto jointlyrecoverthevirialradius,tracerscale radius,
2. darkmatterscaleradiusandouterorconstantvelocityanisotropywithsmallbias(<10%on
1 scale radii and <2%on the two otherquantities) and inefficienciesof 10%, 27%, 48%and
2 20%, respectively. MAMPOSSt does not perform better when some parameters are frozen,
1 and even particularly worse when the virial radius is set to its true value, which appears to
: betheconsequenceofhalotriaxiality.TheaccuracyofMAMPOSStdependsweaklyonthe
v
i adoptedinterloperremovalscheme,includinganefficientiterativeBayesianschemethatwe
X introducehere, whichcan directly obtainthe virialradius with as goodprecisionas MAM-
r POSSt.Additionaltestsaremadeonthenumberoftracers,thestackingofhaloes,thechosen
a
aperture, and the density and velocity anisotropy models. Our tests show that MAMPOSSt
withGaussian3Dvelocitiesisverycompetitivewithothermethodsthatareeithercurrently
restricted to constant velocity anisotropy or 3 orders of magnitude slower. These tests sug-
gestthatMAMPOSStcanbeaverypowerfulandrapidmethodforthemassandanisotropy
modelingofsystemssuchasclustersandgroupsofgalaxies,ellipticalanddwarfspheroidal
galaxies.
Key words: methods:analytical – galaxies: kinematics and dynamics– galaxies:haloes –
galaxies:clusters:general–darkmatter
1 INTRODUCTION betterEinasto(Navarroetal.2004)densityprofilesderivedindis-
sipationlesssimulationsofasingledarkmattercomponentonone
Thedeterminationofmassprofilesisoneofthefundamentalissues hand,andthe1/r2 densityprofilesfoundfortheDMinhydrody-
of astronomy. Subtracting the mass density profile of the visible namicalcosmologicalsimulations(Gnedinetal.2004).Moreover,
component, one deduces the dark matter (hereafter, DM) density the knowledge of the total density profile serves as a fundamen-
profile,whichcanbeconfrontedtothepredictionsfromcosmolog- tal reference, relative to which one can scale various astronomi-
icalN-bodysimulations.Thisisespeciallyrelevantgiventhedif- caltracerssuchasthemassdensityprofilesofthestellar,gasand
ferencesbetweenthetotalNFW(Navarro,Frenk,&White1996)or
(cid:13)c 2012RAS
2 Mamonet al.
dustcomponents,aswellastheluminosityindifferentwavebands. Thetraditionalwaytoanalyzethedistributionofparticlesin
Thesestudiescanbeperformedasafunctionofsystemmassand PPSistoassumeaformforthesix-dimensionaldistributionfunc-
otherattributessuchasgalaxycolour(e.g.,Wojtak&Mamon2013 tion(DF)intermsofenergy(E)andangularmomentum(J)and
andreferencestherein). fit the triple integral of equation (5) below, using this DF for f,
Massprofilescanbederivedfrominternalmotions,oralter- tothedistribution of particlesin PPS.Theworry isthat wehave
nativelyfromX-rayorlensingobservations.Thispaperfocuseson nogoodaprioriknowledgeoftheshapeoftheDF,f(E,J).One
massprofilesfrominternalkinematics.Inthisclassofmass mod- cleverideaistothroworbitsinagravitationalpotential,sinceeach
eling,onehastodealwithadegeneracybetweentheunknownra- orbit isa Dirac delta function in energy and angular momentum.
dialprofilesoftotalmassandofthevelocityanisotropy(hereafter Onethenseeksalinearcombinationoftheseorbits,withpositive
‘anisotropy’) coefficients, to match the data. This orbit model (Schwarzschild
1979;Richstone&Tremaine1984;Syer&Tremaine1996)isvery
σ2(r)+σ2(r)
β(r)=1 θ φ (1) powerful (and can handle non-spherical gravitational potentials),
− 2σr2(r) buttooslowtoobtainmeaningfulerrorsontheparameters.Asimi-
(notethat,insphericalsymmetry,onemusthaveσ =σ ).While lar,andinprinciplefaster,techniqueistoassumethattheDFisthe
φ θ
radialouterorbitsareexpectedforstructuresinanexpandinguni- linearcombination(againwithpositivecoefficients)ofelementary
verse (e.g. Ascasibar & Gottlo¨ber 2008 for dark matter particles DFs(Dejonghe1989; Merritt&Saha1993;Gerhardetal.1998),
and Ludlow et al. 2009 for subhaloes), the dissipative nature of butonlyonesuchstudyhasbeenmade(Kronawitteretal.2000),
thegasdynamicsisexpectedtoproducetangentialorbitsinthein- anditisnotclearwhethertheelementaryDFs,althoughnumerous,
nerregionsofsystemsformedfromgas-richmergersorcollapse. constituteabasisset.
Therefore, lifting the Mass - Anisotropy Degeneracy can provide An important step forward has been performed by Wojtak
usefulconstraintsontheformationofthestructureunderstudy. etal.(2008),whoanalyzedthehaloesinΛCDMcosmologicalsim-
Acommonmethodtoextractthemassprofileistoassumethat ulations,toshowthattheDFcanbeapproximatedtobeseparable
theline-of-sight(hereafter,LOS)velocitydistribution,atgivenpro- inenergyandangularmomentum,withasimpleanalyticalapprox-
jectedradius,isGaussian(Strigarietal.2008;Battagliaetal.2008; imationfortheangularmomentumterm.Inasequel,Wojtaketal.
Wolfetal.2010).Thesemethodsperformadequatelyonthemass (2009)haveshownthatitisfeasibletofitthedistributionofparti-
profile, but provide weak constraints on the anisotropy (Walker clesinPPSwithequation(5),usingtheapproximationoftheDF
etal.2009).Merritt(1987)pointedoutthatanisotropicmodelshave foundbyWojtaketal.(2008).
non-GaussianLOSvelocitydistributions.Therefore,theobserved However, it is not yet clear whether self-gravitating quasi-
kurtosisofthedistributionofLOSvelocitiesservesasapowerful sphericalastrophysicalsystemshavetheDFof ΛCDMhaloes:In
constrainttotheanisotropy(Gerhard1993;vanderMarel&Franx particular,ifthedynamicalevolutionofthesesystemsisinfluenced
1993; Zabludoff, Franx, & Geller 1993). In fact, if one assumes bythedissipationoftheirgaseouscomponent,theDFmaynotbe
that the anisotropy is constant throughout the system, the fourth separableintermsof energyandangularmomentum. Dissipation
orderJeansequationcanbeusedtoexpresstheLOSvelocitykur- isnotexpectedtoaffectmuchtheinternalkinematicsoflargesys-
tosisasanintegralofthetracerdensity,anisotropyandtotalmass tems such as galaxy clusters1, but is expected to be increasingly
profiles (Łokas 2002). Moreover, Richardson & Fairbairn (2012) importantinsmallersystemssuchasgalaxygroups,andespecially
wereabletogeneralizetheexpressionfortheLOSvelocitykurto- galaxiesthemselves.Forthisreason,itisusefultoconsideramass-
sis for radially varying anisotropy in the framework of separable modelingmethodthatisindependentofthedependenceoftheDF
augmenteddensityand4thorderanisotropyequal tothestandard onenergyandangularmomentum.
anisotropy(thelatterappearstobeanexcellentapproximationfor Inthiswork, wepresent analternativemethod, inwhichwe
ΛCDMhaloes,seeFig.10ofWojtaketal.2008).Onecanthenper- fitthedistributionofparticlesinPPSmakingassumptionsonthe
formajointfitoftheobservedLOSvelocitydispersionandkurtosis radial profiles of mass and anisotropy as well as the radial vari-
profiles.Thiswasfoundto(partially)liftthemass-anisotropyde- ationsof thedistributionof space-velocities. Wecallthismethod
generacywhenappliedtodwarfspheroidalgalaxies(Łokas2002) Modelling of Anisotropy and Mass Profiles of Observed Spheri-
andtheComacluster,(Łokas&Mamon2003):thejointconstraint cal Systems, or MAMPOSStfor short.2 TheMAMPOSStmethod
ofLOSvelocitydispersionandkurtosisprofilesallowstheestima- isdescribed inSect.2.1, itsGaussian approximation isdescribed
tionofboththemassprofile(i.e.,normalizationandconcentration) inSect.2.2.Testsonhaloesderivedfromacosmological N body
andtheanisotropyofthecluster,contrarytothecasewhentheLOS simulationarepresentedinSect.3.AdiscussionfollowsinSect.4.
kurtosisprofileisignored.
An interestingroute istoperform non-parametric inversions
ofthedataassumingeitherthemassprofiletoobtaintheanisotropy
2 METHOD
profile(anisotropyinversion,pioneeredbyBinney&Mamon1982)
ortheanisotropyprofiletoobtainthemassprofile(massinversion, 2.1 Generalmethod
independentlydevelopedbyMamon&Boue´ 2010andWolfetal.
TheobservedtracerpopulationofasphericalsystemhasaDF
2010).Theseinversionmethodsarepowerfulinthattheyare non-
parametric,buttheysufferfromtheirrequiringtheuserto binthe
f(r,v)=ν(r)f (vr), (2)
v
data,smoothit,andextrapolateitbeyondtherangeofdata. |
Hence,onewouldliketogoonestepfurtherandconstrainthe
full information contained in the observed projected phase space
1 However,thejointX-rayandlensinganalysis ofaclusterbyNewman
(projectedradiiandLOSvelocities,hereafter,PPS)ofLOSveloc-
etal.(2009)revealsashallowerinnerdensityprofilethanNFW,suggesting
ities as a function of projected radii. In other words, rather than
thatdissipationisalsoimportantinclusters.
usingthe0th,2ndandpossibly4thmomentsoftheLOSvelocity 2 MAMPOSSt should evoke the mass analog of a lamppost, and mam-
distribution,wewishtousethefullsetofevenmoments. poster´ıainSpanishmeansmasonry,hencethebuildingblocksofstructures.
(cid:13)c 2012RAS,MNRAS000,000–000
ModellingAnisotropyandMass Profiles 3
whereν(r)isthetracernumberdensityprofile. Wojtaketal.2009),weanalyticallyderiveh(v R,r)fromequa-
z
|
MAMPOSStfitsthedistributionofobjectsinPPS(projected tion (3) for known 3D velocity distributions. With the analytical
radiusRandLOSvelocityv ),assumingparametrizedformsfor formofh(v R,r),equation(4)providesthesurfacedensitydis-
z z
|
tributionoftracersinPPSthroughasingleintegral.Note,however,
(i) thegravitationalpotential(orequivalentlyatotalmassden-
thatanothersingleintegral isrequiredbecause theexpression for
sityprofile,throughthePoissonequation),
h(v R,r)willinvolveσ (r)(seeeqs.[25]and[26],below,forthe
(ii) theanisotropyprofile(eq.[1]) z| r
Gaussian case), which is obtained by solving the spherical Jeans
(iii) thedistributionof3Dvelocities,f (vr).
v | equation
Consider a point P at distance r from the centre, O, of the
spherical system, with projected radius R 6 r and consider the d νσr2 +2β νσr2 = ν(r)GM(r) (8)
spherical coordinates wheretheunit vectors er andeθ areinthe (cid:0)dr (cid:1) r − r2
planecontainingOPandtheLOS,whilee isperpendiculartothis whereβistheanisotropyof(eq.[1])forourgivenchoicesoftotal
φ
plane.Consideralsothecylindricalcoordinatesystem(vz,v⊥,vφ), mass and anisotropy profiles. Wethus need toinsert the solution
whereez istheaxisalongtheLOSande⊥ istheaxisperpendic- (vanderMarel1994;Mamon&Łokas2005)
ulartotheLOS,butintheplanecontainingOandPandtheLOS. 1 ∞ s dt GM(s)
TheJacobian of thetransformation fromthespherical coordinate σ2(r)= exp 2 β(t) ν(s) ds, (9)
r ν(r) t s2
systemtothenewoneisunity,henceonecanwrite Zr (cid:20) Zr (cid:21)
in the expression for h(v R,r) (eq. [3]) to derive g(R,v ), via
d3N z| z
fv(vz,v⊥,vφ|{r,R}) ≡ (cid:18)dvzdv⊥dvφ(cid:19)r,R teiqouna(t7io)n.I(n4)e,qwuhateiroenβ((9t)),Misg(isv)en=,w(sh2il/eGν)(rd)Φis/dobstiasinthedewraidtihaelqpuroa--
d3N fileof thetotal mass (thisisthe only instancewhere thegravita-
= tionalpotentialentersMAMPOSSt).Foragivenchoiceofparame-
dv dv dv
(cid:18) r θ φ(cid:19)r,R ters,thesingleintegralofequation(4)mustbeevaluatedforevery
≡ fv(vr,vθ,vφ|{r,R}). data point (R,vz), whereas the other integral (eq. [9]) for σr(r)
needonlybeevaluatedonce,onanadequategridofr.
The distribution of LOS velocities at P is then obtained by
Note that for projected radii extending from R to R
integrating velocities over the two perpendicular axes (dropping min max
andabsoluteLOSvelocitiesextendingfrom0toamaximumveloc-
r,R fromf forclarity):
v
{ } ity,whichforprojectedradiusRistheoreticallyequaltov (R)=
esc
dN +∞ +∞ 2Φ(R),andinpracticeispossiblyspecifiedbyacutofobvi-
h(vz|R,r)≡(cid:16)dvz(cid:17)r,R =Z−∞ dv⊥Z−∞ fv(vz,v⊥,vφ)dvφ.(3) pous−velocityinterlopers,vcut(R),onecanwrite
Notethatdynamical systemshavemaximumvelocitiessetbythe Rmax vcut(R) Rmax
2πRdR g(R,v )dv = 2π RΣ(R)dR
escape velocity, 2Φ(r) (where Φ(r) is the gravitational po- z z
tential), on one han−d, and by the maximum allowed (observable) ZRmin Z−vcutR) ZRmin
absoluteLOSvelpocityontheotherhand.Inwhatfollows,wewill = ∆Np, (10)
neglectbothlimits,unlessexplicitlymentionedotherwise. whereweusedequation(4)forg(R,v ),assumedthath(v R,r)
z z
|
Thesurfacedensityofobservedobjects(thetracer)inPPSis isnormalised,reversedtheorderoftheintegralsin r andv ,and
z
thenobtainedbyintegratingalongtheLOS whereN (R)isthepredictednumberofobjectswithinprojected
p
radius R, while∆N = N (R ) N (R ). Equation (10)
g(R,vz) = Σ(R) hh(vz|R,r)iLOS thenimpliesthatthepprobabiplitymdaexnsi−tyopfobmseirnvinganobjectat
∞
rν(r) position(R,v )ofPPSis
= 2 h(v R,r)dr (4) z
z
√r2 R2 |
ZR∞ rdr− +∞ +∞ q(R,vz)= 2πR∆gN(R,vz)
= 2ZR √r2−R2 Z−∞dv⊥Z−∞f(r,vz,v⊥,vφ)dvφ.(5) = 4πR p∞ rν(r) h(v R,r)dr (11)
where ∆Np ZR √r2−R2 z|
+∞ ∞ rν(r)dr R2 ∞ R
Σ(R)=Z−∞ ν(r)dz=2 ZR √r2−R2 (6) = ∆Nprν3Z0coshuν(cid:16)rν coshu(cid:17)h(vz|R,Rcoshu)du,
(12)
isthetracersurfacedensityatprojectedradius R.Equation(5)is e
e
equivalenttoequation(2)ofDejonghe&Merritt(1992). whereequation(11)arisesfromequation(4),whileequation(12)
Ifthetracernumberdensityprofile ν(r),appearinginequa- isobtainedbywritingr = Rcoshu.Here,N (R)isthenumber
p
tion(4),isnotknownandiftheincompletenessofthedataisin- oftracersinacylinderofprojectedradius R,thetermsν andN
p
dependent of theprojected radius, thenone can estimate ν(r)by aregivenby
AbelinversionofΣ(R)ofequation(6):
N(r ) r e e
∞ ν(r)= ν ν , (13)
1 dΣ dR 4πr3 r
ν(r)= . (7) ν ν
−π dR √R2 r2 (cid:16) R(cid:17)
Zr − Np(R)=N(rν) Nep r , (14)
Butthisisnotnecessary,asweshallseebelow. ν
(cid:16) (cid:17)
In MAMPOSSt,rather than replace thevelocities by energy whereN(r)isthecumulativetracernumberdensityprofile,while
e
and angular momentum and numerically solve the triple integral r isthecharacteristicradiusofthetracer.Oneeasilyverifiesthat
ν
ofequation(5)(asfirstproposedbyDejonghe&Merritt,seealso q(R,v )dRdv =1.ThevaluesofR andR appearing
z z min max
RR
(cid:13)c 2012RAS,MNRAS000,000–000
4 Mamonet al.
in∆N (eq.[10])canbehardlimits,oralternativelytherespective (iii) We terminate the LOS integration in equation (11) at
p
minimumandmaximumprojectedradiioftheobservedtracers if roughly15virialradii,3 r ,insteadofinfinity,astheHubbleflow
v
nohardlimitsarespecified. pushes the velocities of the material beyond this distance to val-
Wefittheparameters(massscaleorconcentrationandpossi- uesover3σ abovethemeanofthesystem(seeMamon,Biviano,
v
blynormalization,anisotropylevelorradius,aswellasthetracer &Murante2010,hereafterMBM10).TheLOSintegrationvaries
scaler –ifnotpreviouslyknown)thatenterthedeterminationof onlyveryslightlywiththenumberofvirialradii,soaslongasthe
ν
g(R,v ) to the observed surface density, using maximum likeli- virialradiusiscorrecttoafactoroftwo,thischoiceofintegration
z
hoodestimation(MLE),i.e.byminimizing limitisnotanissue.
n Wenowneedtochooseamodelfortheshapeofthe3Dveloc-
ln = lnq(Ri,vz,iθ), (15) itydistribution.WhileMAMPOSSt,can,inprinciple,berunwith
− L − |
Xi=1 anymodel,thesimplestoneisthe(possiblyanisotropic)Gaussian
fortheN-parametervectorθ,wherenisthenumberofdatapoints, distribution,whichwedescribeinSect.2.2below.
withqgivenbyequation(11).
Writing θ = r ,η , where η is the vector of the N 1
{ ν } − 2.2 Gaussian3Dvelocitydistributions
parametersotherthanr ,onehas
ν
Thesimplestassumptionforthe3Dvelocitydistributionisthatit
q(R,v r ,η)=p (Rr ) p(v R,r ,η), (16)
z| ν 0 | ν × z| ν isGaussian:
where 1 v2 v2+v2
f (v ,v ,v )= exp r θ φ , (21)
p0(R|rν)= Np(Rma2xπ|rRν)Σ−(RN|prν(R)min|rν) (17) wvherrethθeveφlocity(d2iπs)p3e/r2siσornσsθ2σiare(cid:20)f−un2cσtir2on−sofr2.σTθ2his(cid:21)Gaussian
and distribution assumes no streaming motions: e.g. no rotation, and
g(R,v θ) no mean radial streaming, which is adequate for Rmax < rv in
p(vz|R,η)≡hh(vz|R,r)iLOS= Σ(Rrzν|) . (18) high-mass haloes (i.e. groups and clusters) and Rmax < 4rv in
| galaxy-mass haloes (Cuesta et al. 2008). Inserting equation (21)
Combiningthelastequalityofequation(18)withequation(5),in- intoequation(3)andintegratingoverv leadsto
φ
tegratingoverLOSvelocities,reversingtheorderofthetwoouter
iytniioteenlgdirssalaslrpoe(favtdhzye|Rkren)soduwvltnzin,=gthqe1un.a,dSarcouc,poilrfedtiihnnetgesgtcoraaell,eqauonafdtitouhnseisntr(ga1ce5eq)ruaadntidisotr(ni1b(6u6)-), h(vz|R,r)=Z−+∞∞2π√11−βσr2 exp(−(cid:2)(12−(β1−)vβr2)+σr2vθ2(cid:3))dv⊥.
R (22)
maximizingthelikelihoodamountstominimizing
Callingθtheanglebetweentheline-of-sight(directionz)andthe
ln ′ = lnp(v R,r ,η). (19)
− L − z| ν radialvectorr,onehas
Now, if rν isXnot known, then one may be tempted to solve for vr = vz cosθ+v⊥ sinθ, (23)
itbyminimizing ln = lnp (Rr ),andthenproceed
withequation (19−) tomLi0nimiz−eforthe0N | 1ν remaining parame- vθ = −vz sinθ+v⊥ cosθ, (24)
ters,η.However,since ln =P ln ′ −ln 0(fromeqs.[15] withwhichtheintegraloverv⊥inequation(22)yieldsaGaussian
− L − L − L
and[16]),themostlikelysolutionforθthatminimizes ln will distributionofLOSvelocitiesatpointP:
not in general be that which minimizes at the same tim−e Lln ′
and ln .Moreover, ifoneseekstoobtainthedistribut−ionsLof h(v R,r)= 1 exp vz2 , (25)
param−eterLs0ηandrν consistentwiththeMLEsolution(forexam- z| 2πσz2(R,r) (cid:20)−2σz2(R,r)(cid:21)
plewithMarkovChainMonte-Carlotechniques),thejointanalysis ofsquareddisppersion
ofequations(11)and(15)isrequired.Ontheotherhand, ifr is
ν
knownfromotherdata,whilethecurrentdatasetisknowntohave σ2(R,r)= 1 β(r) R 2 σ2(r). (26)
acompleteness,C(R),thatisafunctionofprojectedradius,then z − r r
onecouldindeedminimizeln ′ ofequation(19).Theproperso- (cid:20) (cid:16) (cid:17) (cid:21)
L Theintegralof h(vz R,r)alongtheLOSisobtained fromequa-
lutionisthentominimize ln weightingthedatapointsbythe |
− L tions(4)and(25):
inversecompleteness,i.e.minimizing
n lnq(R ,v θ) g(R,v ) = 2 ∞ rν 1−βR2/r2 −1/2
−lnL′′ =− C(iRiz),i| . (20) z rπ ZR √r2−R2 (cid:0) σr (cid:1)
Xi=1 exp vz2 dr. (27)
Forcomputationalefficiency,weperformthefollowingtasks: × −2 (1 βR2/r2) σ2
(cid:20) − r(cid:21)
(i) Foreachrunofparameters,wefirstcomputelogσr(rj)from Accordingtoequations(18)and(27),theprobabilityofmeasuring
equation(9)onalogarithmicgridofrj,andcomputecubic-spline avelocityvz atgivenprojectedradiusRis
coefficients at these radii. Then, when we compute the LOS in-
g(R,v )
tegralofequation (4)foreach (R ,v ),weevaluateσ (r)with p(v R) = z
i z,i r z| Σ(R)
cubicsplineinterpolation(inlog-logspace,usingthecubicspline
coefficientsdeterminedatthestart).
(ii) Forsimpleanisotropymodels,theexponentialterminequa- 3 Thevirial radii are loosely defined hereas theradius wherethemean
tion(9)isgivenbyequations(A2)and(A3). densityofthehalois200timesthecriticaldensityoftheUniverse.
(cid:13)c 2012RAS,MNRAS000,000–000
ModellingAnisotropyandMass Profiles 5
= 1 0∞(ν/σz) exp −vz2/ 2σz2 dz . (28) 3 TESTS
√2π ∞νdz
R 0 (cid:2) (cid:0) (cid:1)(cid:3) 3.1 Simulatedhaloes
R
We remind the reader that β is a chosen function of r, ν is TotestMAMPOSSt,weusecluster-masshaloesextractedbyBor-
a function of r given by equation (7), while σr is a function of gani et al. (2004) from their large cosmological hydrodynamical
r given by equation (9). For isotropic systems (β = 0), equa- simulationperformedusingtheparallelTree+SPHGADGET–2code
tion(27)leadstoaGaussiandistributionofLOSvelocities.How- ofSpringel et al.(2005). Thesimulationassumesacosmological
ever, foranisotropic velocity tensors, the distribution of LOS ve- model with Ω = 0.3, Ω = 0.7, Ω = 0.039, h = 0.7, and
0 Λ b
locitieswillgenerallynotbeGaussian(asMerritt1987foundwhen σ =0.8.TheboxsizeisL=192h−1Mpc.Thesimulationused
8
startingfromdistributionfunctionsinsteadofGaussian3Dveloc- 4803 DMparticlesand(initially)asmanygasparticles,foraDM
ittoietsh)e.pHoepnuclea,rtahsesuGmaputsisoinanthnaattgur(eRo,fvzh)(visz|GRa,urs)siaisnnoontvezq:ueivvaelnenift tpoar2ti2c.l5ehm−a1sscoofm4o.v6i2ng×k1p0c9uhn−ti1lMz⊙=.T2haensdoffitexneidngaflteenrwgtahrdwsa(si.see.t,
h(vz R,r) isaGaussian at point P, itsintegral along theLOS is 7.5h−1 kpc). The simulation code includes explicit energy and
|
notGaussian,unlessβ=0andσrisconstant. entropyconservation,radiativecooling,auniformtime-dependent
IfoneoftheparameterstodeterminewithMAMPOSStisthe UV background (Haardt & Madau 1996), the self-regulated hy-
normalization of the mass profile, one should not be tempted in brid multi-phase model for star formation (Springel & Hernquist
expressingtheradiiintermsofthevirialradiusrv,thevelocitiesin 2003),andaphenomenologicalmodelforgalacticwindspowered
termsofthevirialvelocityvv,thetracerdensitiesintermsofwhat byType-IIsupernovae.
wewish(astheyappearinboththenumeratoranddenominatorof DMhaloeswereidentifiedbyBorganietal.(2004)atredshift
eq.[28]).Doingso,equation(28)becomes z = 0 with a standard Friends-of-friends (FoF) analysis applied
to the DM particle set, with linking length 0.15 times the mean
p(v R) v p(v R) inter-particledistance.AftertheFoFidentification,thecentreofthe
z v z
| ≡ |
e e = √12π 0∞(ν/σz) e∞xpνd−zvz2/σz2 dz , (29) hovaleordwenassistyetctroitethrieopnowsiatisotnheonfaitpspmlioedsttbooduentderpmairntiec,lefo.rAeaspchhehraicloal,
R 0 (cid:2) (cid:3) ourproxyforthevirialradius,r200,wherethemeandensityis200
e e e e e
timesthecriticaldensityoftheUniverse.
R
where the quantities with tildes are in virial units. Equation (29)
e e Tosavecomputingtime,weworkedonarandomsubsample
indicates that when one varies the rv (and the virial velocity in ofroughlytwomillionparticlesamongthe4803.Wehaveextracted
proportion as vv = 2/∆H0rv), the highest probabilities are 11cluster-masshaloesfromthesesimulations, among which, ten
reached for the highepst normalizations: vz becomes very small, are about logarithmically spaced in virial radius, r200, while the
whileσz isunaffected tofirst order. Thisunphysical resultisthe 11th haloisthemostmassiveintheentiresimulation.Theirprop-
consequenceofusingaparameter(thevirialradius)aspart ofthe
e erties are listed in Table 1. We made no effort to omit irregular
datavaeriable.Ontheotherhand,usingequation(28),oneseesthat haloes, but among the list of 12 irregular haloes out of 105 ex-
thehighestprobabilitiesp(vz R)arereachedatintermediatevalues tractedbyMBM10fromthesamesimulation,2areinoursample
|
ofthenormalization.
(haloes17283 and434).Welistthecharacteristicradii r ,r ,r
s H B
Takingthesecondmomentofthevelocitydistributionofequa- ofthreemodelsfittedbyMLEtothemassdensityprofilesofthe
tion(28)leadstotheequationofanisotropicprojectionyieldingthe particledata(from0.03to1r ),namely:
200
LOSvelocitydispersion,σ (R):
z
(i) theNFWdensityprofile
+∞
Σ(R)σz2(R) = vz2g(R,vz)dvz ρ(r) r−1 (r+r )−2 , (32)
s
Z−∞ ∝
∞
= 2 νrdr where rs ≡ r−2 is the radius of slope −2 in the mass density
rπ ZR σz(R,r)√r2−R2 profile,relatedtotheconcentrationc≡r200/r−2;
+∞ v2 (ii) theHernquistdensityprofile(Hernquist1990)
v2exp z dv
×Z−∞ z (cid:20)−2σz2(R,r)(cid:21) z ρ(r) r−1 (r+rH)−3 , (33)
∞ R2 rdr ∝
= 2 νσr2 1−β(r)r2 √r2 R2 . (30) whererH =2r−2,
ZR (cid:20) (cid:21) − (iii) theBurkertdensityprofile(Burkert1995)
Equation(30)recoverstheequationofanisotropickinematicpro-
jection,firstderivedbyBinney&Mamon(1982). ρ(r)∝(r+rB)−1 r2+rB2 −1 , (34)
If interlopers are removed with a velocity cut v (R), then
theexpressionforh(vz R,r)becomes cut whererB≃0.657r−(cid:0)2. (cid:1)
|
Denotingthescalesr ,r andr bythegenericr ,themasspro-
h(v R,r)= exp −vz2/ 2σz2(R,r) .(31) filesofthesemodels(sreqHuiredfoBreq.[9])canbewρritten
z
| √2πσ (R,r)erf v (R)/ σ (R,r)√2
z (cid:8) cu(cid:2)t z(cid:3)(cid:9)
M(r/r )
Insummary,MAMPOSSTw(cid:8)ithGaussi(cid:2)an3Dvelocit(cid:3)ie(cid:9)scom- M(r)=M(rρ) M(1)ρ , (35)
puteslikelihoodsfromequations(15),(11)or(12),(25),(26),and b
(9),inthatorder. where
b
(cid:13)c 2012RAS,MNRAS000,000–000
6 Mamonet al.
Table1.Propertiesof11cosmologicalhaloes
rank ID r200 rs rH rB A rβ A∞
1 18667 0.789 0.179 0.401 0.117 1.14 0.276 1.33
2 21926 0.842 0.123 0.342 0.085 1.34 0.050 1.73
3 30579 0.890 0.189 0.443 0.120 1.33 0.053 1.69
4 25174 0.956 0.144 0.377 0.099 1.23 0.162 1.36
5 3106 1.010 0.297 0.661 0.166 1.05 2.384 1.09
6 8366 1.076 0.434 0.819 0.249 1.11 0.689 1.29
7 13647 1.151 0.227 0.536 0.151 1.19 0.265 1.41
8 1131 1.174 0.197 0.499 0.133 1.18 0.352 1.34
9 17283 1.298 0.505 1.009 0.277 1.04 0.727 1.05
10 434 1.374 0.317 0.699 0.210 1.30 0.165 1.70
11 5726 1.660 0.407 0.921 0.249 1.42 0.050 2.20
Stack 1.09±0.08 0.26±0.04 0.60±0.08 0.17±0.02 1.21±0.04 0.26±0.08 1.45±0.10
Notes:Propertiesobtainedfromfitstotheparticledataof11haloes.Cols.1and2:clusteridentification;col.3:virialradiusr200;col.4:scaleradius(=r−2)
oftheNFWmassdensityprofile(eq.[32]);col.5:scaleradius(=2r−2)oftheHernquistmassdensityprofile(eq.[33]);col.6:scaleradius(≃0.657r−2)
oftheBurkertmassdensityprofile(eq.[34]);col.7:meananisotropy(A = σr/σθ)withinr200;col.8:anisotropyradiuswiththeMLanisotropymodel;
col.9:asymptoticanisotropy(A∞ =σr/σθ)atinfiniteradiuswiththeTanisotropymodel.Radiiareinunitsofh−1Mpc.Themeasuredanisotropiesdo
notincorporatestreamingmotions.
x
ln(x+1) , (NFW)
− x+1

M(x)= x+x 1 2 , (Hernquist) (36)
(cid:16) (cid:17)
b ln (x+1)2(x2+1) 2 tan−1x. (Burkert)
TheNFW(cid:2) model has long(cid:3)−been known to fit well the den-
sity profiles of ΛCDM haloes (Navarro et al. 1996), and while
Navarroetal.(2004)foundthatEinastomodelsfitthemevenbet-
ter,MBM10 found that theNFWmodel describestheouterLOS
velocitydispersion profileof theDM component oftheirstacked
cluster-mass halo in Borgani et al.’s hydrodynamical cosmologi-
cal simulation even (slightly) better than the Einasto model. The
HernquistmodeldiffersfromtheNFWonebecauseithasasteeper
logarithmicslopeatlargeradii,γ dlnρ/dlnr= 4ratherthan
≡ −
3.TheBurkertmodel,ontheotherhand,hasthesameasymptotic
−
γ = 3astheNFWmodel,butacoreatthecentre,γ =0,rather
−
thanacusp(γ = 1inboththeNFWandHernquistmodels).
−
InTable1,wealsolistthevaluesoftheparameterscharacter-
izingdifferentvelocity-anisotropymodels,namely:
(i) theconstantanisotropymodelσ /σ = (1 β)−1/2 =
r θ
− A
(‘Cst’modelhereafter),whereweassumesphericalsymmetryand Figure1. Velocity anisotropy profiles ofthe 11haloes (broken coloured
thereforeσθ =σφ; lines).ThesmoothblackcurveistheMLanisotropymodelwithrβ =r−2
(ii) the model (‘ML’ model hereafter) of Mamon & Łokas (or,equivalently,theTanisotropymodelwithβ∞=0.5).
(2005);
1 r
β (r)= , (37)
ML 2 r+r
β
characterizedbytheanisotropyradiusr ;
β
Note that the ML and the T models used here are identical for
(iii) ageneralizationoftheMLmodel,whichisalsoasimplified
versionofthemodelofTiretetal.(2007),isotropicatr = 0and β∞ = 0.5andrβ = r−2.Withthesevalues,theMLandTmod-
elsprovideagoodfittotheaverageanisotropy profileofaset of
withanisotropyradiusidenticaltor−2(hereaftercalled‘T’model):
cluster-masscosmologicalhaloes(MBM10).
r
βT(r)=β∞ r+r−2 , (38) filesaInndFtihge.1M,wLeanshisoowtrothpeyimndoidveidluwailthharlβo=velro−ci2ty(oarn,iesqoutrivoaplyenptrloy-,
characterizedbytheanisotropyvalueatlargeradii, β∞.InourT theTanisotropymodelwithβ∞ =0.5).Thereisahugescatterin
model,theanisotropyradiusissettothescaleradiusofthe mass theβ(r)oftheindividualhaloes,asalreadyobservedby,e.g.,Wo-
densityprofile.Notealsothatinthefollowingweprovidetheval- jtak et al. (2008), especially at r > 0.3r , while 8 out of 11
200
uesof ∞ (σr/σθ)∞ =(1 β∞)−1/2,ratherthanβ∞. haloeshaveβ(0)=0 0.15.
A ≡ − ±
(cid:13)c 2012RAS,MNRAS000,000–000
ModellingAnisotropyandMass Profiles 7
3.2 Observingconesandinterloperremoval tracers.Allthesemodelsarecharacterisedbythetwofreeparame-
ters,the‘virial’radiusr andacharacteristicscale-radius(r ,r ,
TotestMAMPOSSt,weselect500DMparticlesaroundeachhalo, 200 s H
andr fortheNFW,Hernquist,andBurkertmodels,respectively).
outtoamaximumprojecteddistance R fromthehalocentre, B
max Herafter,wegenericallyuser torefertothischaracteristicscale-
for which we consider three values: r 0.66r , r , and ρ
500 ≃ 200 200 radiusofthemassdensityprofile.
r 1.35r .Weanalyzethreeorthogonalprojectionsforeach
h1a0lo0–≃thesear2e0i0nfactconeswithanobserveratD=90h−1Mpc We use the NFW model, in projection (Bartelmann 1996;
Łokas & Mamon 2001), to fit the projected number density pro-
away,buttheopeninganglebeingverysmallhasnonoticeableef-
file of the tracer. Note that the normalization of this profile does
fectonourresults.TheparticlesintheseconesareusedbyMAM-
notentertheMAMPOSStequations,sotheonlyfreeparameteris
POSStastracersofthehalogravitationalpotential.
However, these500-particle samplesinclude interlopers, i.e. r−2. Herafter we call this parameter rν, to avoid confusion with
thecharacteristicradiusoftheNFWmassdensityprofile.Weonly
DMparticlesthatarelocatedinprojectionatR 6 R ,butare
max consideronemodelforthenumberdensityprofileofthetracer,be-
effectivelyoutsideR inreal(3D)space,i.e.withr>R .It
max max cause this is a direct observable, unlike the mass density profile.
isimpossibletoremovealltheseinterlopersintheobservedredshift
Whileoneshouldnotbetoorestrictiveinthemodelchoiceforthe
space,whereonly3ofthe6phase-spacecoordinatesofthetracers
mass density profile, the observer isgenerally able to choose the
areknown(e.g.,MBM10).Moreover,sincetheLOSvelocitydis-
best-suited model for the tracer number density profile by direct
tributionofinterlopersinmockconesaroundΛCDMhaloesisthe
examination of the data before running MAMPOSSt. We choose
sumofaGaussiancomponentandauniformone(seeMBM10for
the NFW model because it provides a reasonable description of
aquantifiedview),andsincetheGaussianoneresemblesthatofthe
the number density profiles of the DM particlesin our simulated
particlesinthevirialsphere,itisimportanttoremovetheflatLOS
haloes.
velocitycomponent,atleastathighabsoluteLOSvelocity,where
Forthevelocityanisotropyprofile,weconsiderthethreemod-
itdominates.Itispossibletoremovethesehigh v objectswith
| z| elsdescribedabove,Cst,ML,andT,eachcharacterisedbyasin-
suitableinterloperremovalalgorithms.
To see how MAMPOSSt depends on the choice of the in- gle anisotropy parameter, A,rβ, and A∞, respectively. In equa-
terloperremovalalgorithm,wehereconsiderthreedifferentalgo- tion(38),weuser−2=rρ.
To search for the best-fit solution, we run the MAMPOSSt
rithms.
algorithmincombinationwiththeNEWUOA4 minimizationrou-
The first one is a new, iterative algorithm, that we name
tineofPowell(2006).Forestimatingerrorbarsonthebestfitpa-
“Clean”,whichisfullydescribedinAppendixB.Cleanfirstlooks
rameters, as well as confidence contours on pairs of parameters,
forgapsintheLOSvelocities,thenestimatesthevirialradiusr
200 wefitourmodelparametersusingtheMarkovChainMonteCarlo
from the aperture velocity dispersion, assuming an NFW model
(MCMC)technique(e.g.,Lewis&Bridle2002).InMCMC,thek-
with ML anisotropy with an anisotropy radius rβ = r−2 and a dimensionalparameterspaceispopulatedwithproposals,foreach
concentrationdependingontheestimateofr viatherelationof
200 ofwhichthelikelihoodiscomputed.Thenewproposalisaccepted
Maccio`,Dutton,&vandenBosch(2008),thenonlyconsidersthe
iftheratioofnewtopreviouslikelihoodiseithergreaterthanunity
galaxies within2.7σ (R)from the median LOS velocity, and fi-
z orelsegreaterthanauniform [0,1]randomnumber.Theproposal
nallyiterates.Ourassumedanisotropyprofilefitsreasonablywell
isfoundbyassumingak-dimensionalGaussianprobabilitydistri-
theanisotropyprofilesofDMhaloes(MBM10),asisclearforour
butionaroundthepreviousproposal.Weadoptthepubliclyavail-
11haloes(seeFig.1).Thefactor2.7wasfoundbyMBM10tobest
able CosmoMC code by A. Lewis.5 We run 6 chains in parallel
preservethelocalLOSvelocitydispersionfortheassumeddensity
usingMessageParsingInterface(MPI),andthecovariancematrix
andanisotropymodels.
isusedtoupdatetheparametersoftheGaussianproposaldensity
We also consider two other interloper removal algorithms,
toensurefasterconvergence.
namely:
Fig. 2 illustrates the MAMPOSSt analysis via MCMC for
(i) themethod(hereafter,dHK)ofdenHartog&Katgert(1996), thegeneralcasewithfourfreeparameters,usingtheNFWmodel
a widely used procedure that works reasonably well on cluster- for the mass density profile, and the constant (free parameter)
mass haloes from cosmological simulations (Biviano et al. 2006; anisotropymodel.Inparticular,itshowsthatthedifferentparame-
Wojtaketal.2007),despiteitscrudeunderlyingphysics; tersarenotcorrelated,exceptforapositivecorrelationbetweenrρ
(ii) themethod(hereafter,KBM)ofKatgert,Biviano,&Mazure and .
A
(2004, seetheirAppendix A),inwhich agalaxy isflagged asan Our FORTRAN code takes roughly 1 second to find the
interloperunder thecondition v /σ > 1.85(R/r )−0.3, with MAMPOSSt 4-parameter solution for a 500 particle sample run
z z 200
r derivedfrom σ usingeq.(8)ofCarlberg,Yee,&Ellingson in scalar on a decent desktop or laptop computer, and 4 minutes
200 z
(1997). This method was invented as a poor-man’s proxy for the toproduce confidencelimitsforthissolutionwiththeCosmoMC
dHKmethodwhentheobservationalsamplingofthehaloprojected (Lewis&Bridle2002)MCMCcode,with6chainsof40000ele-
phase-spaceispoor. mentsruninparallel(MPI)onaPCequippedwitha4-core8-thread
IntelCore-I72600processor.
Theresultsforthedifferentinterloperrejectionmethods,mass
densityandvelocityanisotropymodels,andforthedifferentmax-
3.3 Thegeneral4-parametercase
imumprojectedradiiusedintheselectionofthe500particlesare
There is no a priori limitation on the number of free parameters listed in Table 2 and displayed in Fig. 3. We list and show the
thatcanbeusedinMAMPOSSttocharacterisethemassandve- biweight measures (see, e.g., Beers, Flynn, & Gebhardt 1990) of
locityanisotropyprofiles.Withsamplesof6500tracers(assumed
masslessthroughout thesetests)itisappropriatetoconsider 4
∼
freeparameters,twoforthemassdistribution,oneforthevelocity 4 NEWUOAisavailableathttp://plato.asu.edu/ftp/other software/newuoa.zip
anisotropy distribution, and one for the spatial distributionof the 5 CosmoMCisavailableathttp://cosmologist.info/cosmomc/.
(cid:13)c 2012RAS,MNRAS000,000–000
8 Mamonet al.
Figure2.IllustrationofMAMPOSStanalysisforthegeneralcase,withβindependentofradius,fora500particlesamplefromaxisxofhalo25174(grey
brokenlineinFig.1),usingMCMC(with6chainsof40000elements).Thecontoursare1,2,and3σ.Theredarrowsandstarsindicatethemaximum
likelihoodsolution,whilethegreenarrowsandcrossesshowthetruesolution(Table1).ThepriorsfortheMCMCwereuniformwithintheboxesofeach
panelandzerobeyondtheboxes.
meananddispersionoflog(o/t)whereoistherecoveredvalueof senanisotropymodel.Onaverage,thevaluesofther parameter
200
theparameterandtitstruevalue,because,accordingtoourtests, arerecoveredwithalmostnobias(from 1to+4%)andwithonly
−
theyperformbetterthanstandardstatisticalestimatorsoflocation 10%inefficiency.Ther parameterestimatesarealwaysslightly
ν
∼
andscalewhentheparentdistributionsarenotpureGaussians.We positivelybiased(4–7%),andarerecoveredwith 25%efficiency.
∼
call‘bias’and‘inefficiency’themeananddispersionoflog(o/t). Also,ther parameterestimatesgenerallydisplayaslightpositive
ρ
Ifthedispersionintruevaluesofagivenparameterissmall,one bias,except fortheKBM interloperremoval method, and overall
canspuriouslyobtainlowvaluesofthelog(o/t)dispersionwhen thebiasrangesfrom 2to+15%,whiletheefficiencyrangesfrom
−
theMAMPOSStandtruevaluesshownocorrelation.Wetherefore 50to 90%.
∼ ∼
alsolist the Spearman rankcorrelation coefficient between o and Asfarastheanisotropyparameterisconcerned,theMLmodel
t,markinginboldfacethosecorrelationsthataresignificantatthe behaves very differently from the Cst and T models, in that it is
99%confidencelevel.Welisttheresultsforallinterloperrejection virtually impossible to constrain the anisotropy radius of the for-
methodsandallanisotropymodelsonlyfortheNFWmassdensity mer, r , while it is possible to obtain quite good constraints on
β
model and for the R 6 r200 radial selection. For simplicity, we the anisotropy parameters of the other two models, and ∞.
onlyshowalimitedsetofresultsfortheothermassdensitymodels Moreprecisely,ther estimatesarealwaysnegativelyAbiasedA(by
β
andfortheotherradialselections. 60%)andareaffectedbyahugedispersion(almostoneorderof
∼
Remarkably,asseeninTable2,theresultsforthefourparam- magnitude).Ontheotherhand, and ∞arealmostunbiased(the
A A
etersarealmost independent of the interloperremoval algorithm, biasrangesfrom 10to+5%)andtheyareaffectedbydispersions
−
theCleanandKBMalgorithmperformingslightlybetterthandHK. of,typically, 20%,ifweconsidertheCleanandKBMinterloper
∼
Theresultsforr ,r ,andr alsodependverylittleonthecho- removal algorithms. So,apparently, itismuch easiertoconstrain
200 ν ρ
(cid:13)c 2012RAS,MNRAS000,000–000
ModellingAnisotropyandMass Profiles 9
Table2.MAMPOSStresultsfordifferentinterloperremovalalgorithms,densitymodels,apertures,andnumberofparticles
N Rmax Membership ρ(r) β(r) r200 rν rρ anisotropy
bias ineff. corr. bias ineff. corr. bias ineff. corr. bias ineff. corr.
500 r200 Clean NFW Cst 0.004 0.040 0.909 0.027 0.102 0.835 0.032 0.217 0.578 0.007 0.073 –0.255
500 r200 Clean NFW ML –0.003 0.040 0.904 0.024 0.104 0.832 0.057 0.229 0.601 –0.221 0.887 –0.172
500 r200 Clean NFW T –0.006 0.040 0.903 0.026 0.103 0.838 0.039 0.169 0.709 0.007 0.085 0.621
500 r200 dHK NFW Cst 0.018 0.042 0.885 0.027 0.099 0.838 0.051 0.319 0.406 0.004 0.147 –0.215
500 r200 dHK NFW ML 0.012 0.041 0.909 0.028 0.100 0.840 0.059 0.286 0.611 –0.161 0.904 –0.118
500 r200 dHK NFW T 0.012 0.044 0.902 0.027 0.100 0.844 0.022 0.199 0.636 –0.045 0.264 0.464
500 r200 KBM NFW Cst 0.005 0.038 0.906 0.018 0.100 0.850 –0.006 0.218 0.535 0.020 0.078 –0.198
500 r200 KBM NFW ML –0.003 0.039 0.908 0.020 0.100 0.851 –0.005 0.232 0.557 –0.191 0.795 0.101
500 r200 KBM NFW T –0.006 0.038 0.911 0.020 0.099 0.856 –0.010 0.184 0.689 0.018 0.094 0.595
500 r200 Clean Her T 0.002 0.039 0.909 0.026 0.102 0.835 0.039 0.132 0.755 0.014 0.086 0.546
500 r200 Clean Bur T 0.000 0.039 0.910 0.047 0.196 0.377 0.048 0.145 0.704 –0.019 0.071 0.603
500 r500 Clean NFW T –0.004 0.048 0.877 0.089 0.115 0.902 0.016 0.143 0.744 0.009 0.108 0.232
500 r100 Clean NFW T –0.014 0.035 0.905 0.039 0.179 0.420 0.093 0.210 0.538 –0.001 0.090 0.436
100 r200 Clean NFW Cst –0.001 0.058 0.844 0.033 0.201 0.537 –0.133 0.341 0.342 0.003 0.119 –0.053
100 r200 Clean NFW ML –0.011 0.061 0.834 0.034 0.199 0.539 –0.087 0.336 0.484 –0.137 0.925 –0.245
100 r200 Clean NFW T –0.008 0.058 0.850 0.032 0.200 0.532 –0.108 0.277 0.436 0.014 0.143 0.249
Notes:Theseresultsarefor11haloeseachobservedalong3axes,general4free-parametercase.Col.1:Numberofinitiallyselectedparticles(beforeinterloper
removal);col.2:Maximumprojectedradiusfortheselection,wherer500 ≃0.65r200andr100 ≃1.35r200;col.3:Interloper-removalmethod(dHK:den
Hartog&Katgert1996;KBM:Katgertetal.2004;Clean:App.B);col.4:massdensitymodel(NFW:Navarroetal.1996;Her:Hernquist1990;Bur:Burkert
1995);col.5:anisotropymodel(Cst:β = cst;ML:eq.[37],Mamon&Łokas2005;T:eq.[38],adaptedfromTiretetal.2007);cols.6–8:virialradius;
cols.9–11:tracerscaleradius;cols.12–14:darkmatterscaleradius;cols.15–17:velocityanisotropy(i.e.,AfortheCstmodel,rβ fortheMLmodel,and
A∞fortheTmodel).Thecolumns‘bias’and‘ineff.’respectivelyprovidethemeanandstandarddeviation(bothcomputedwiththebiweighttechnique)of
log(o/t),whilecolumns‘corr.’listtheSpearmanrankcorrelationcoefficientsbetweenthetruevaluesandMAMPOSSt-recoveredones(valuesinboldface
indicatesignificantcorrelationsbetweenoandtvaluesatthe>0.99confidencelevel).
the‘normalisation’ofagivenanisotropyprofile,thantoconstrain fortheTiretanisotropymodel.Theresultsareverysimilar forthe
thecharacteristicradiusatwhichtheanisotropychanges, particu- NFWandHernquistmodels.ResultsaresimilaralsofortheBurk-
larlysoifthischangeismild,asintheMLmodel.Note,however, ert model, except for the scale r of the number density profile,
ν
that the difficulty of MAMPOSSt in constraining the anisotropy forwhichthebiasandinefficiencyarebothhigherthanthose ob-
parameterof theML model does not meanthat theMLmodel is tainedusingtheNFWandHernquistmassmodels.Sincethemodel
apoorrepresentationofreality,andinfactFig.1suggests theop- weuseforthenumberdensityprofilehasnotchanged(aprojected
posite.Moreover,constraintsobtainedonthe r ,r ,andr pa- NFW),thisresultsuggeststhatitisdifficulttoaccomodateatracer
200 ν ρ
rametersareequallygoodwiththeCstandTanisotropymodels,as withacentralcuspyspatialdistributioninapotentialwithacentral
withtheMLone. core.
As seen in Table 2, correlations between recovered and ob- Alltheresultsdescribedsofarwereobtainedforaselection
servedvaluesoftheparametersr200,rν,andrρarealmostalways of500particleswithinRmax =r200.ChangingthevalueofRmax
signficant.Thisisalsotrueforthe A∞ anisotropyparameter,but isnot without effectson theresults. Theinefficiency on r200 de-
notforAandrβ.InFig.4,weshowthecorrelationsexistingbe- creaseswhenRmax isgraduallyincreasedfrom r500 tor100.The
tweenthetrueandrecoveredvaluesofthedifferentparameters,us- inefficiencyonanisotropyislargestforthesmallestR ,andsta-
max
ingtheTanisotropymodel,forthe11haloesalongthe3different tisticallysimilarforthetwolargervalues.Increasingtheaperture
orthogonalprojections.Projectionseffectsrenderthedetermination tothevirialradiusoraboveincreasesthenumberoftracersnearthe
ofthemassandanisotropyprofileofasingle500-particlehalovery virialradiuswherer isestimated.Moreover,increasinglylarger
200
uncertain.However,Fig.4showsthat 500tracersaresufficient apertureswillcapturebettertheasymptoticvalueoftheanisotropy
∼
torankhaloesforeachofthedifferentparametersconsideredhere, profile (r). In contrast, r is less efficiently determined when
ρ
A
i.e.bymass(r200),scaleradiusofthetracerdistribution(rν)andof Rmaxisincreasedfromr500 tor100,whilerν hasitsworstineffi-
thetotalmassdistribution(rρ),andoutervelocityanisotropyA∞. ciencyforRmax =r100,withstatisticallyequivalentvaluesforthe
The importance of projection effects is also very clear from twosmallerapertures.Thismightbeduetotheincreasingfraction
Fig.5, wherewedisplay theratioof therecovered totruevalues ofunidentifiedinterlopers,and/ortothepresenceof(sub)structures
oftheparametersforeachhaloalongthethreedifferentprojection atlargerradii.
axes.Thisfigurealsoshowsthatthereisnotrendofunder-orover- ToassessthesensitivityoftheMAMPOSSttechniquetothe
predictingtheparametervalueswithhalomass. numberoftracers,besidesthe500-particleselection,wehavealso
All the above considerations apply for the NFW mass pro- considered samples of 100 particle tracers, randomly extracted
file. Our tests with the Hernquist and Burkert mass profiles give fromthesameprojectionsofthesame11cosmologicalhaloes.Re-
similarresults,ascanbeseeninTable2,whereforsimplicity,we sultsoftheMAMPOSStanalysisarelistedatthebottomofTable2
onlylisttheresultsfortheCleaninterloper-removalalgorithmand anddisplayed inFig.3.Alsointhiscase, forsimplicity,weonly
(cid:13)c 2012RAS,MNRAS000,000–000
10 Mamonet al.
Figure4.CorrelationofMAMPOSStandtruevaluesofthe4jointly-fitpa-
rameters(CaseGen),withthe‘T’anisotropyprofile,foreachofthe3×11
haloeswith500tracers.Eachpanelcorrespondstoadifferentparameter,as
labelled(unitsofradiiareinh−1Mpc).Differentsymbolsidentifydiffer-
entprojections, x-axis:blackdiamonds,y-axis:redsquares,z-axis:blue
circles.
removal algorithm, and we only consider the NFW mass density
model,forsimplicity.
Figure3.MAMPOSStresiduals,log(o/t),fortheMAMPOSStparame- Specifically,weconsiderthefollowingCases:
ters(top:virialradius,2ndpanel:tracerscaleradius,3rdpanel:DMscale
radius,bottom:velocityanisotropy)forthedifferentschemesofinterloper A) General [Gen]: r , r , r , and the anisotropy parameter
200 ν ρ
removal(seetext).Themean(dots)anddispersion(errorbars)oflog(o/t) (oneamongthefollowing: ,rβ, ∞,dependingontheanisotropy
arerespectivelyillustratedasfilledcirclesanderrorbars,forthe33sam- modelconsidered)areallfAreeMAAMPOSStparameters.Thisisthe
plesof500(‘Clean’,dHKandKBM)and100(‘Clean’(N=100))particles. caseconsideredsofar.
ResultsfortheanisotropymodelsCst,T,andMLareshownlefttorightin
B) Generalwithr fittedoutsideMAMPOSSt[Split]:thefree
green,blue,andred,respectively. ν
parameters are the same as in the Gen case, but r is fitted out-
ν
sideMAMPOSSt,viaMLE.Wethussplittheminimizationofthe
parametersintotwoparts.
displayalimitedsetof results.Whencompared totheresultsfor
C) Known virial mass or radius [KVir]: r is fixed and as-
the500-particlesamples, thereisnosignificantchange intheav- 200
sumed to be exactly known, r and the anisotropy parameter are
erage values of the bias with which the different parameters are ρ
freeparametersinMAMPOSSt,r isanexternal freeparameter,
recovered, except for r ,wherethebiasbecomes negative, while ν
ρ asintheSplitcase.
itwasmostlypositiveforthe500-particlesamples.Ther param-
ρ D) Estimatedvirialmassorradius[EVir]:similartoKVir,ex-
eter value underestimation is not very severe, however, 6 25%.
cept that r is not the true value, but the value estimated from
Theefficiencieswithwhichthedifferentparametersareestimated 200
theLOSaperturevelocitydispersion(afterinterloperremoval,see
aresignificantlyaffectedbythereductioninnumberoftracers.The
AppendixB).
dispersionincreasesfrom 10to15%forr ,from 25to60%
∼ 200 ∼ E) ΛCDM:r isestimatedfromr usingthetheoreticalrela-
forr ,from 60to100%forr ,andfrom 20to33%for ρ 200
ν ∼ ρ ∼ A tionbetweenthesetwoquantitiesprovidedbyMaccio`etal.(2008);
and A∞. Thereis no significant change in the dispersion for rβ, r and the anisotropy parameter are free parameters in MAM-
butthiswasalreadyextremelylargeforthe500-particlesamples. 200
POSSt,r isanexternalfreeparameter,asintheSplitcase.
ν
F) MassfollowsLight[MfL]:r andtheanisotropyparameter
200
arefreeparametersinMAMPOSSt,r isanexternalfreeparame-
3.4 Caseswithconstraintsonparameters ν
ter,asintheSplitcase,andr isassumedtobeidenticaltor .
ρ ν
Whatistheeffectofreducingthenumberoffreeparametersonthe G) TiedLightandMass[TLM]:r andtheanisotropyparam-
200
performanceoftheMAMPOSStalgorithm?Toassessthispointwe eter are free parameters in MAMPOSSt, while r and r are as-
ρ ν
considerseveralcasesthatreproducewhatobserversdoinpractice sumedtobeanidenticalfreeparameter.
whenfacedwiththeproblemofdeterminingtheinternaldynamics H) Isotropic [β-iso]: = 1 is assumed, r ,r are free pa-
200 ρ
A
ofcosmologicalhaloes.Inallcases,weconsider500particlesse- rametersinMAMPOSSt,r isanexternalfreeparameter,asinthe
ν
lectedwithinr ineachhalo.WeonlyapplytheCleaninterloper Splitcase.
200
(cid:13)c 2012RAS,MNRAS000,000–000