Dark halo properties from rotation curves

17 Pages·0.67 MB·English

by  Raul Jimenez

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed1February2008 (MNLATEXstylefilev2.2)
Dark halo properties from rotation curves
Raul Jimenez1, Licia Verde1,2 and S. Peng Oh3
1Department of Physics and Astronomy, Rutgers University,136 Frelinghuysen Road, Piscataway, NJ 08854–8019, USA.
2Princeton UniversityObservatory, Princeton, NJ 08544, USA.
3Theoretical Astrophysics, MailCode 130-33, Caltech, Pasadena, CA 91125, USA.
2
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1February2008
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n
a ABSTRACT
J
We study a largesetof highspatialresolutionopticalrotationcurves ofgalaxieswith

1 the goalofdetermining the modelparametersfora disk embedded within acolddark
2
matter (CDM) halo that we model either with a Navarro, Frenk & White (NFW)

profile or pseudo-isothermal profile. We show that parameter degeneracies present in
1
lowerresolutiondata are lifted at these higher resolutions.34%of the galaxiesdo not
v
have a meaningful fit when using the NFW profile and 32% when using the pseu-
2
5 doisothermal profile, however only 14% do not have a meaningful fit in either model.
3 In both models we find correlationsbetween the disk baryonfraction fd and the spin
1 parameterofthehaloλ′,betweenfd andthedarkhalomassM200,andbetweenM200
0 andtheconcentrationparameterc.Weshowthatthedistributionoftheconcentration
2 parameters c, for a NFW halo, is in good agreement with CDM predictions; no sig-
0 nificant galaxy population is found with very low values of c. The overalldistribution
h/ of λ′ is in good agreement with theoretical predictions from hierarchical tidal torque
p theory. The whole sample is also well fitted by a pseudo-isothermal dark halo with a
- core, but the size of the core is rather small (6% of the virial radius or smaller; for
o
70% of the sample the core size is less than 2 kpc). Thus we conclude that the profile
tr of dark matter is steep (r−1 or steeper) down to this radius; large dark matter cores
s
(andthereforeverylowdarkmattercentraldensities)seemtobeexcluded.LSBstend
a
: to have higher values of λ′ for a given fd and lower values of c for a given mass than
v HSBs.Inanappendix wegivesomeusefulformulaforpseudo-isothermalprofilehalos
i
X and discuss in detail the issue of parameter degeneracies.
r Key words: cosmology:theory—galaxies:formation—galaxies:spiral—galaxies:
a
kinematics and dynamics
1 INTRODUCTION density cusp predicted by CDM (e.g., McGaugh & de Blok
(1998);Mooreetal.(1999);Dalcanton&Bernstein(2000)).
Furthermore,numericalSPHsimulationsofdiskforma-
The Cold Dark Matter (CDM) paradigm for structure for- tion within dark halos fail to match the zero-point of the
mation has proved remarkably successful in explaining the observed Tully-Fisher. Onepossible solution is to lower the
observed large-scale properties of the universe, such as the central concentration of the dark matter halos (Navarro &
abundance and clustering of galaxies and clusters (Peacock Steinmetz2000b).
et al 2001; Verde et al. 2001; Lahav et al. 2001), the sta- Theseproblemshavestimulatednumerousproposedas-
tistical properties of the Lyα forest (e.g., Phillips et al. trophysical solutions, as well as modifications of the fun-
(2001)), and the power spectrum of the cosmic microwave damental CDM paradigm itself (e.g., Spergel & Steinhardt
background anisotropies (e.g., Jaffe et al (2001)). However, (2000); Kamionkowski & Liddle (2000); Colombi, Dodelson
a numberof puzzling discrepancies remain when CDM pre- &Widrow(1996);Bode,Ostriker&Turok(2001);Goodman
dictionsareextrapolatedtosmallscales.Chiefamongthem (2000); Hu,Barkana & Gruzinov (2000); Cen (2001)).
are: (i) The substructure problem. CDM over-predicts the Inthispaperweonceagain examinethedensityprofile
numberofsatellitesaroundaMilky-Waysizedgalaxybyan problembyfittingCDMmodelstoobservedrotationcurves
order of magnitude (Klypin et al. 1999; Moore et al. 1999). of spiral galaxies. Our study differs from previous work in
(ii)Thedensityprofileproblem.Observedrotation-curvesof two respects. Firstly, we use optical rotation curves rather
dwarfandlowersurfacebrightnessgalaxiessuggestthatthe than the HI rotation curves used in previous studies which
inner regions have a constant density core rather than the areplaguedbybeamsmearing(Swatersetal.2000;vanden
2 Jimenez, Verde & Oh
Bosch et al. 2000). Even if the effects of beam smearing halo and within a pseudo-isothermal profile halo (i.e., an
are neglected, therelatively large errors and limited spatial isothermalspherewithaconstantdensitycore).Thechoice
sampling of HI rotation curves imply that they cannot be of a pseudo-isothermal halo is motivated by recent claims
usedtodiscriminatebetweenconstantdensitycoresandr−1 in theliterature that thedark matter profileof low–surface
cusps(vandenBosch&Swaters2001).Bycontrast,theop- brightness galaxies (LSB, e.g., de Blok et al. (2001)) and
tical rotation curves we use are free from beam smearing, some high–surface brightness galaxies (HSB, e.g., Salucci
have smaller errors and higher spatial resolution. We show (2001);Borriello&Salucci(2001);Salucci&Burkert(2000))
that, with this superior data, various parameter degenera- is better represented by a flat core rather than the steeper
cies present when fitting HI rotation curves can be lifted. profile found in CDM N-bodysimulations.
This allows ustodistinguish betweencore andcusp-likein-
ner profiles. Secondly, while many studies have focused on
relativelysmall samplesofdwarfs (butseeNavarro(1998)), 2.1 The observational sample
we usea large sample (400 galaxies) spanninga wide range
We use three sets of observed rotation curves. The first is
in luminosity and surface brightness.
thecatalogcompiledbyCourteau(1997)(Courteausample)
Our goal is to determine the best fitting model pa-
whichconsistsofopticalH long–slitrotationcurvesfor300
rameters for a disk within the CDM profile proposed by α
Sb–ScUGCgalaxiesandR-bandphotometry.Thesecondis
Navarro Frenk and White (1997; NFW) and within a
asampleof74spiralgalaxies(Sa–Sd)observedagaininthe
pseudo-isothermal profile halo and study the distributions
optical (H ) by Palunas & Williams (2000) (PW sample).
of the recovered disk parameters. α
For this sample I–band photometry and the velocity field
WefindthattheNFWprofileprovidesagoodfitto66%
in two dimensions is available. Finally, we also consider the
of the galaxies in the sample, with a distribution of recov-
recently published optical (H ) rotation curves for 26 Low
eredconcentrationparametersandspinparametersbroadly α
SurfaceBrightness(LSB)galaxiesbyMcGaugh,Rubin&de
consistent with that predicted by CDM numerical simula-
Blok (2001). These consist of long slit H observations for
tions. When the sample is fitted with an isothermal profile α
26galaxiesandB-bandphotometryforasmallersample.In
with a core, 68% of thegalaxies are well fitted whithin this
total we have400 optical rotation curves.
model and the best fit parameters favor cores with small
All three samples provide inclination corrected curves
sizes (below 6% of the dark halo virial radius). Many rota-
withtheircorrespondingerrors.TheCourteauandPWsam-
tion curves that have no meaningful fit in one dark matter
pleshavesurface photometry and givevaluesfor thestellar
profile are well fitted by the other. There is no meaningful
diskscale lengths,butwedonothavediskscale lengthsfor
fit in either model for only for 14% of the sample. This is
theLSB sample.
consistent with the inner dark matter profiles being steep
The sample of galaxies studied here is somewhat het-
(slope of -1 or steeper) down to radii that are few percent
erogeneous:weuseLSBandHSBgalaxies;somecurveshave
of the virial radius.
muchsmallererror-barsthanothers,andthesamplingrange
In both models the recovered baryonic mass to light
oftherotationcurvevariessignificantly:inparticular,some
ratios are broadly in agreement with predictions from syn-
curvesdonotextendtolarge enoughradiitoshowtheflat-
thetic stellar populations.
tening of the rotation curve. The diversity of the sample is
This paper is organized as follows. In section 2 we
both astrength as well as a weakness: it allows us toprobe
present the set of rotation curves we analyze and the two
general statistical trends across a wide range of galaxy lu-
models we use to fit the data (exponential disk embedded
minosities and surface brightness.
in a dark matter halo with a NFW profile or a pseudo-
isothermalprofile).Thesetwomodelshavefourfreeparam-
eters, but since for most of the galaxies the scale length of
2.2 The models
the disk is given, we report the analysis for four free pa-
rameters in appendix II and present in the main text the Many authors have investigated galactosynthesis models
resultsforthefitwiththreefreeparameters.Insection3we (Dalcanton, Spergel & Summers 1997; Jimenez, Heavens,
illustrate and discuss the correlations we find between the Hawkins&Padoan1997; Mo,Mao&White1998; Jimenez,
best fit parameters for thetwomodels. Wediscuss thegen- Padoan, Matteucci & Heavens 1998; Somerville & Primack
eralimplicationsofthesefindingsinsection4wherewealso 1999;vandenBosch2000;Avila-Reese&Firmani2000;Fir-
compare the derived baryonic mass-to-light ratios with the mani & Avila-Reese 2000; Navarro & Steinmetz 2000b) in
range allowed by stellar populations. Finally, we conclude which the properties of disk galaxies are determined pri-
and summarize our results in section 5. In appendix I we marily by the mass, size, and angular momenta of the ha-
developtheexpressionfortherotationcurveofanexponen- los in which they form, and which may contain the effects
tialdiskembeddedinapseudoisothermalhalo.Inappendix of supernova feedback, adiabatic disk contraction, cooling,
III we study the degeneracies among the parameters of the merging,andavarietyofstar-formation recipes. Inthispa-
models. per,wearemainlyconcernedwiththedynamicalproperties
of the disk/halo system which affect the rotation curve, in
the spirit of Dalcanton et al. (1997); Jimenez et al. (1997),
andMo,Mao&White(1998)(hereafterMMW).Wemodel
2 ROTATION CURVES
spiral galaxies as exponential disks (i.e., disks with surface
Westudya large set of observedrotation curvesof galaxies density profiles Σ(r) = Σ0exp(−r/Rd), where Rd is the
with the aim of determining the model parameters for a scalelengthofthedisk)embeddedwithinCDMhaloswhich
disk within a NFW (Navarro, Frenk & White 1997) profile we model either with NFW profiles or pseudo-isothermal
Dark halo properties 3
Figure 1.NFWmodel.Thelocationofthesampleofgalaxiesindifferentprojectionsofthethreedimensionalparameterspace(M200,
fd,c)whenfittedwithaNFWdarkhalomodel.ThediamondscorrespondtogalaxiesfromCourteau(1997)andthetrianglesarefrom
Palunas & Williams (2000). The solid and dashed lines in the top-right panel are the N-body simulations predictions for the c-M200
correlation(seetextformoredetails).Alsoplotted(bottom-rightpanel)isthededucedvalueforλ′vs.fd.Thesolidlineistheleastsquare
fit from 14 dwarfs by Burkert (2000), in good agreement with the correlation found here from 362 galaxies. Note also the correlation
betweenfd andM200.
2 2
spheres.Inbothmodelsweassumethatthebaryonsinitially Vd(r)=4πGΣ0Rdy [I0(y)K0(y)−I1(y)K1(y)], (2)
follow thesame densityprofileas thedark matterandthen
coolandsettleintoanexponentialdiskinadynamicaltime. with y = r/(2Rd), Σ0 = fdM200/(2πRd2), In and Kn de-
notingthemodified Bessel functions of thefirst and second
We assume that the baryons that do not settle in the disk
kind. The contribution to the rotation curve from the dark
follow the dark matter distribution. The disks have a mass
matter is:
fractionfd ≡Mdisk/M200 6Ωb/Ωm.HereMdiskdenotesthe
baryonicmassthatsettleintoanexponentialdisk,M200 de- VC2DM(r)=GMCDM(r)/r, (3)
notesthemasswithinaradiusR200thatenclosesanaverage
density 200 times the critical density, Ω denotes the den- where M(r) denotes the mass enclosed within the radius r.
m
sity parameter and Ω the baryon contribution to the den- Foramodel tobefully specified,we requireexpressionsfor
b
sityparameter.Throughout,weassumeaΛCDMcosmology thedisk scale length Rd and for M(r).
given by:(Ωm,ΩΛ,Ωb,h,σ8h−1)=(0.3,0.7,0.039,0.7,1.0). TheNFWfittothedensityprofileofdarkmatterfound
In these models, therotation velocity is given by: in collisionless N-bodynumerical simulations is:
δ0
Vc2(r)=Vd2+VC2DM (1) ρ(r)=ρcrit(r/r )(1+r/r )2 (4)
c c
where the baryonic disk rotation velocity is (Binney & where ρcrit is the critical density (ρcrit = 277.3h2 M⊙
Tremaine (1987), p.77) kpc−3) and r is roughly the break radius at which the
c
4 Jimenez, Verde & Oh
slope of the profile changes from −1 to −3. The con-
centration parameter c is defined as c = R200/rc, where
R200 = (M200/(4/3π200ρcrit))1/3 and M200 is assumed to
bethetotalmass ofthedarkmatterhalo. Thedarkmatter
mass within an enclosed radius r is:
3 1
M(r)=4πρcritδ0rc 1+cx −1+ln(1+cx) , (5)
h i
where x=r/R200. From equation (5), therelation between
δ0 and cis:
200 c3
δ0 = . (6)
3 ln(1+c)−c/(1+c)
Disk formation affects the dark matter profile by the adia-
batic contraction it inducesin theinnerregions of thedark
matter halo. As in MMW, we model this by assuming that
the halo responds adiabatically to the slow contraction of
thedisk,andtherefore thattheangularmomentumofdark Figure 2. NFW model. The distribution of c (left panel) and
matter particles is conserved. This yields the relation be- λ′ (rightpanel)deducedfromNFWhalosfitstorotationcurves.
tween the initial radius r and final radius r where a dark Also over-plotted as continuous lines are the theoretical predic-
i
matter particle ends up: tions for each of these parameters. Note that the derived values
of c are broadly inagreement withthe predictions from N-body
GMf(r)r=GM(ri)ri (7) simulations(seetextformoredetails).
where M(r ) is given by equation (5), and M (r) =
i f
Σ exp(−r/R )+M(r )(1−f )isthetotalfinalmasswithin
o d i d A fitting formula for the disk scale length R as a function
d
r (see MMW for details). of M200,c, fd and λ′ for thisprofile is given in AppendixI.
Assuming angular momentum conservation, MMW de-
Anyrotation curvecanbefittedwiththisprofileusingfour
velopedafittingformulaforRd(c,M200,λ′,fd)whichweuse parameters: λ′, M200, rc, fd.
(see their equations (28) and (32)). Note that we use the
For each galaxy rotation curve we explore the whole
modified spin parameter λ′ ≡ (j /f )λ, where j ≡ J /J
d d d d likelihoodsurfaceandthebestfittingmodelisthenobtained
and Jd,J are the total angular momentum of the disk and using a standard χ2 minimization.
halo respectively. If the specific angular momentum of the
We discuss in detail the values for the disk parameters
disk is equalto that of thehalo, then j =f and λ′ =λ.
d d found for these two models (NFW and pseudoisothermal)
In the NFW model, for a given M200, the overdensity when using all four free parameters in Appendixes II and
δ0 and thus the concentration parameter c depends on the III.In this section we eliminate one of thefour free param-
collapse redshift. There is therefore a correlation between c eters (λ′) by using the observed scale lengths of the expo-
andM200,sincemoremassivehaloscollapseatlowerredshift nential disks R . Then the only remaining free parameter
d
thanless massiveones. However,thereisafairly widescat-
for thebaryonicdisk is f . Sincewe donot haveR for the
d d
terinthedistributionofcforfixedM200 (e.g.,Jing(2000)). LSBsampleandforsomeofthegalaxiesinthePWsample,
Furthermore, there is growing observational evidence that
theanalysis with threefree parameters is performed on the
the distribution of concentration parameters might not be
CourteauandPW samples(intotal362galaxies). Thedisk
as predicted by CDM (e.g., Navarro & Steinmetz (2000b);
parameter distributions for the four parameters fit is quite
Keeton (2001)); one of the goals of this paper is to explore
similar to that for the three parameter fit. We discuss the
thisquestionfurther.Wetherefore keepcasafreeparame-
locusofLSBgalaxiesintheparameterspaceinAppendixII
ter. Thus, there are four free parameters in our model: the
andshow that theyfollow thesame trendsas theCourteau
spin of the dark halo λ′, the mass of the dark halo M200, and PW samples.
the concentration parameter c and the fraction of the total
mass in thedisk f .
d
Forthepseudo-isothermalspheretheprofileofthedark
matter is: 3 BEST FIT MODELS
r 2 −1 3.1 Fits to disk models within NFW profiles
ρ=ρ0 1+ (8)
(cid:20) (cid:16)rc(cid:17) (cid:21) Sinceweusetheobserved(stellar)scalelengthsofthebary-
where ρ0 denotes the finite central density. We have devel- onic disk (Rd),thefree parameters in ourmodel are: M200,
oped a similar analytic model to compute the properties of f and c. For each galaxy rotation curveweexplore theen-
d
an exponential disk in this profile (see Appendix I). In this tirelikelihoodsurfacewithinthefollowingboundaries:109 <
case, we do not include the adiabatic compression of the M200/M⊙ <7×1013, 0.1<c<100 and 0.001<fd <0.36.
halo due to the baryons, since we want to use a constant Thechoiceoftheboundariesisjustifiedasfollows: forM200
density inner core for the final dark matter distribution by by the typical velocities at large radii in the sample, for c
fiat.Inthepseudo-isothermalprofilecase,theCDMrotation bytheoreticalansatzsgivenbyN–bodysimulationsandan-
velocity is: alyticpredictions(e.g.,MMW);andfinallyforf arealistic
d
VC2DM =4πGρ0rc2h1− rrcarctan(cid:16)rrc(cid:17)i. (9) uonpptehrebboaurynodncfaranctbioengfivde∼n 0b.y2 (neu.gcl.eJoasyffnetehtesails(c2o0n0s1t)r;aEinftss-
Dark halo properties 5
Figure 3. Pseudo-isothermal model. Same as fig. 1 but when using the pseudo-isothermal dark matter halo. Note the similaritywith
fig.1.
tathiouetal.(2001));arealisticlowerboundismoredifficult 34%) of the galaxies in the sample the model with a NFW
to set, but it is reasonable to believe that at least few per- profile for the dark matter fails to reproduce the observed
centofthebaryonsinthehalocooltoformthedisk.Weset rotation curve. This is in agreement with a previous study
a higher value f =0.36 and a lower value of f =0.001 to by van den Bosch & Swaters (2001). They analyze (with
d d
exploreparameterspacemorefully.Thebestfittingmodelis a different procedure and different modelling assumptions
thenobtainedbyusingastandardinversevarianceweighted from ours) 20 rotation curves of late type dwarfs and find
χ2 minimization1.Wecomputethevalueofχ2 ontheentire thatfor6ofthesegalaxies (i.e.,30%) nomeaningfulfitcan
gridofpointstoensurethatweconvergeonaglobal,rather be obtained. It is important to note that our analysis uses
thanlocal,minimuminχ2.Thediscretenessofpointsinour a different approach, a much larger sample of galaxies that
solution reflects the finitespacing of our grid. is not limited to dwarfs, and still concludes that the same
Figure 1 shows the position of each galaxy for all the percentage of galaxies are well fitted bya CDM profile.
projectionsofthethreedimensionalparameterspace(M200, There are some correlations among disk parameters
fd andc).Inaddition,forthebestfittingvaluesofM200,fd worth commenting on. The top-left panel of fig. 1 shows
and cFowre12h2avgealcaoxmiepsuotuetdoλf′thaend36p2lottheitχa2gmaiinnsitmfizda.tionpro- oafnaavnotiid-caonrcreelfaotriolnowbemtwaesesnesMan20d0laonwdsfpdi,nswi(twheadcislecaursszotnhee
cedure does not find a global minimum within the bound-
significanceandpossiblephysicaloriginofthiszoneofavoid-
ariesforthediskparameters.Thismeansthatfor∼1/3(i.e.,
ance in Verde,Oh & Jimenez (2002)).
Thetop-right panelof fig.1 shows thatlow mass halos
1 Thebestfitting modeltoall400rotation curvescanbefound haveonaveragehigherconcentrationparametersthanmas-
at http://www.physics.rutgers.edu/∼raulj/rot/roti.ps sivehalos,inagreementwiththeoreticalpredictions.Thisis
(rotipseu.ps) not surprising since in hierarchical CDM models low mass
6 Jimenez, Verde & Oh
halos form at higher redshift than massive halos. The solid portion oftheplot maybeunstabletobulgeformation and
and dashed lines in the top right panel of fig. 1 show the becomebulge-dominated;disksmayform preferentially out
medianM−crelationpredictedfromnumericalsimulations of low angular momentum gas, which settles in the center
(Bullock et al. (2001) figs. 4 and 5) for low density and first; finally for some combinations of the parameters disks
highdensityenvironmentsrespectively.Althoughtherecov- mightfailtotriggerstarformation.InVerde,Oh&Jimenez
ered distribution of c vs. M200 seems to show a somewhat (2002) we discuss in more detail the origin of the zone of
steeperdeclineofcwithincreasingM200 (inagreementwith avoidance.
the findings of van den Bosch, Burkert & Swaters (2001); In Appendix II, we explore the degeneracies between
van den Bosch & Swaters (2001) for 14 dwarf galaxies), it model parameters. Although we find that there should be
is difficult to draw a definitive conclusion due to the scat- no significant degeneracies between model parameters for
terinthetheoryprediction(simulationsshowa30%scatter most of these high-quality rotation curves, we find that for
around this median relation) and in the recovered correla- the lowest quality rotation curves, significant degeneracies
tion. The overall distribution of halo concentrations is in develop.Thedegeneracies havealmost exactly thesame di-
reasonableagreementwithΛCDMpredictions.Thisismore rection as the 3 correlations recovered from the data. We
clearly seen in the left panel of fig. 2 where the theoretical therefore caution the reader that the correlations between
predictionsbyJing(2000)areoverplottedasathicklineon model parameters seen above maybe the result of parame-
thedistributionofc(obtainedfromthosegalaxiesforwhich terdegeneraciesiftherearesystematicerrorsintherotation
a meaningful fit was found). Maybe more interesting is the curvesor thequoted errors havebeen underestimated.
fact thatveryfewgalaxies arefoundtohavevaluesofcbe-
lowthetheoreticalpredictions.Thesmalltailof20galaxies
withc<0.3ismostlydueto14galaxiesforwhichf >0.2,
d 3.2 Fits to disk models within a
i.e., above the value allowed by nucleosynthesis. For these
Pseudoisothermal halo
galaxiesthebestfitparametercombinationisthusunphysi-
cal.In§4wewillarguethatthismightbeduetolimitations Wefitthepseudoisothermal halo modelof§2.1 tothesame
of our model for therotation curves. set of galaxies as above using the same inverse variance
The bottom-left panel shows that there is no evidence weighted χ2. Our aim is two-fold: first to see if the same
for a correlation between fd and c. On the other hand correlations amongtheparametersfoundfortheNFWpro-
(bottom-right panel) thereis a clear correlation between λ′ file still hold for this new profile and second, to investigate
and fd. This is in agreement with that found by Burkert thedistribution of core sizes.
(2000) (the solid line shows his best least square fit to the Wefindthat,forthisdarkmatterprofilemodel,for114
data)andvandenBosch,Burkert&Swaters(2001) butfor galaxiesoutofthe362(i.e.32%)theχ2minimizationproce-
only 14 dwarf galaxies. This is puzzling because cosmologi- duredoesnotfindaglobalminimumwithintheboundaries.
calmodelsdonotpredictanycorrelation betweenλ(where Figure 3 shows the location of galaxies for the projec-
λ′=jd/fdλ) and fd. tions of the three dimensional parameter space (M200, fd,
Burkert(2000)pointsoutthatthiscorrelationcanarise c), where c ≡ R200/rc. As for the NFW profile, there is a
because of a correlation between the specific disk angular clear anti-correlation between fd and M200 and the same
momentum (jd/fd) and fd. If baryons that make up the zone of avoidance. In Verde, Oh & Jimenez (2002) we dis-
disk cooled from the inside out, the outermost rings of the cuss in detail the physical origin of this zone of avoidance.
disk, that contain most angular momentum, cooled later; Hereweonlypointoutthatthiszoneofavoidanceseemsto
thuspredictingacorrelation with aslope veryclose tothat beindependent of theshape of thedark matter profile.
observed.Inthisscenariothetotalspecificangularmomen- Thereisnoevidenceforacorrelationbetweenf andc.
d
tum of the disk should be significantly smaller than that of AsfortheNFWprofilewecan nowusethebestfittingval-
the dark matter, since most of the galaxies have fd below uesof(M200,fd andc)tocomputethevalueofλ′usingeqs.
nucleosynthesis value, but this seems not to be thecase for 19 and20 from AppendixI.Thereis a clear correlation be-
the14galaxiesusedintheanalysis(vandenBosch,Burkert tweenf andλ′(bottomrightpaneloffig.3),verysimilarto
d
& Swaters 2001; Burkert 2000). that found for the NFW model (fig. 1). The distribution of
On theright panel of fig. 4we show thedistribution of λ′ forthewholesampleisshownontherightpaneloffig.4;
λ′ for galaxies of our larger sample for which a meaningful the thick solid line is the theoretically predicted value. Al-
fit was found. Also plotted as a solid thick line is the theo- thoughthetheoreticalpredictionisobtainedfromnumerical
reticalprediction(Bullocketal.2001)forthedistributionof simulations, where the dark matter halo profile is closer to
λfoundinN-bodysimulations(whichisverysimilartothat the NFW, there is reasonable agreement. This is not sur-
predicted by linear theory (Heavens & Peacock 1988)). Al- prising since theN-body predictions are verysimilar to the
thoughtheagreementisreasonable,thereisatailofgalaxies lineartheoryones,andifhaloswereisolated(i.e.,nofurther
for which therecovered λ′ is below thepredictions. merging after formation) they would be well approximated
Burkert (2000) did not find this, but this may be due byisothermal spheres (e.g., Gunn & Gott (1972)).
tothefactthatthegalaxiesinthesampleofvandenBosch, The upper-right panel shows that most galaxies in the
Burkert & Swaters (2001) are LSB’s so they are selected to samplehavec=R200/rc largerthan20,alreadypointingto
havehigh spin (Jimenez et al. 1998). small sizes of thecores. This is betterillustrated in theleft
We argue that the most significant feature of the panel of fig. 4, where we show the histogram of R200/rc for
bottom-right panel of figure 1 is the lack of galaxies above thegalaxies for which meaningful fit are obtained. There is
thesolid line(whatwecallzoneofavoidance).Manydiffer- a sharp cut–off at R200/rc ∼ 16, with 70% of the galaxies
ent effects might be at play here: disks in the bottom right having R200/rc > 100. This implies that the size of dark
Dark halo properties 7
thesecorrelations andthezoneofavoidanceinthedifferent
projectionsofthethreedimensionalparameterspace(M200,
c, f ) seem to be independent from the shape of the dark
d
matterprofile.Thediskparametercorrelationswefindfrom
this large (362 galaxies) and diverse sample of galaxies are
inagreementwiththosefoundfrompreviousstudiesthatin-
volvedmuchsmallersamples(∼20)andonlydwarforLSBs
galaxies (Burkert 2000; van den Bosch, Burkert & Swaters
2001).
Both NFW and pseudoisothermal profiles are good fits
to about 70% the rotation curves. For the NFW profile we
find no meaningful fit for 34% of the galaxies (122 out of
362),whileforthepseudoisothermalprofilewefindnomean-
ingfulfitfor32%(114outof362).Wedonotexpecttofind
that the model provides a good fit for all the galaxies in
the sample: galaxies might be more complicated than our
Figure 4. Pseudo-isothermal model. Left panel: distribution model for the rotation curves. In fact some rotation curves
of the concentration parameter c = R200/rc for the pseudo- showstrongevidenceforspiralarmsandbarsintherotation
isothermal dark halo. Note the sharp cut-off at c = 16, 70% pattern, and we have not attempted to model any bulge or
galaxies in the sample have c > 100, i.e. 70% of galaxies have barcomponent.However,wefindthatmanyrotationcurves
cores with sizes below 1% of the dark halo virial radius. Right that have no meaningful fit for one dark matter profile are
panel:thedistributionofλ′.Thecontinuouslineisthenumerical well fitted by the other: for only 52 galaxies (14% of the
simulationsprediction(seetextformoredetails).
sample) there is no meaningful fit in either model. These
correspond to the poorest quality rotation curves (low spa-
tialresolutionandpoorspatialcoverage,i.e.noflatorinner
rotation curve).
Fig. 5 shows that the core radius in the pseudo-
isothermal profile (rcISO) is much smaller than the break
radiusintheNFWprofile(rcNFW):rcISO isabout10times
smaller than rcNFW.Inotherwords, best fittingmodels for
the dark matter in this sample of galaxies are those with
steepdarkmatterprofiles(r−1 orsteeper),atleastdownto
6% of thevirial radius.
Theaboveconsiderationsleadustoconcludethatlarge
dark matter cores (and therefore low central dark matter
densities) seem to beexcluded. In fig. 6 we show thedistri-
bution of central dark matter densities recovered from the
bestfitstotherotation curves.FortheNFWprofilemodel,
sincethecentraldensityisnotfinite,thedensitieshavebeen
evaluatedatr=0.5kpc.Thisiscomparabletothesmallest
radiusatwhichtherotationcurvecanbereliablymeasured
in the samples used in this papers.
The arrow shows the central density (at r = 0.5 kpc)
inferred byBorriello &Salucci(2001) (seetheirfig.5)from
9 galaxies.
We have also selected 37 galaxies (out of the 48 LSBs)
Figure 5. The size of the core radius in the pseudo-isothermal
analyzedbydeBlok etal. (2001), thatshowindication of a
model compared to the break radius in the NFW model. Note
flat core, and computed the distribution of inferred central
thattheformeris10timessmallerthanthelatterandthat70%
densities(the9galaxieswhichshownoflatteningofthepro-
ofthesamplehascoresbelow2kpc.
file should have higher central densities). This distribution
is shown as thedotted line in fig. 6.
matter cores is modest, about 6% of the virial radius or There are a few features worth commenting on. First,
smaller. themeandensityforthepseudo-isothermalprofileisslightly
lower than that for the NFW profile at 0.5 kpc. The aver-
age value of derived central densities for the 27 core galax-
ies selected from de Blok et al. (2001) is not significantly
4 GENERAL CONSIDERATIONS
lowerthantheaveragevalueforthepseudo-isothermalpro-
From the recovered best fittingdisk parameters in the con- file.Therefore,thereisquiteareasonableagreementforthe
text of the 2 models (NFW profile and pseudo-isothermal central densities derived from our model and those derived
profile) we can draw some general conclusions. First of all, invertingthePoissonequationintheaboveLSBgalaxies.Of
the correlations between c and M200, fd and M200 and fd course, the NFW densities will be much higher at radii be-
and λ′ are quite similar for the two dark matter profiles: low 0.5 kpc. Since densities for smaller radii have not been
8 Jimenez, Verde & Oh
Figure6.Thedistributionofcentraldarkmatterdensityforthe Figure 7. NFW model. The predicted (M/L) in our analysis,
twoprofilesatadistancefromthecenterof0.5kpc.Thedashed usingtheNFWdarkhaloprofile,fortheCourteauandPWsam-
linecorresponds to pseudo-isothermal darkhalos whilethe solid ples.TheleftpanelcorrespondstotheCourteausample.Theright
linecorrespondstoNFWdarkhalos.Notethemodestdensityof panel showsthe histogramforthegalaxies inthe PWsampleas
dark matter. The arrow shows the central density (at r = 0.5) asolidline. The dashed lineshows the resultingdistribution for
inferredby Borriello& Salucci (2001) and the dotted lineis the the (M/L) ratio assuming a maximum disk model (Palunas &
distributionofcentraldensitiesfor27galaxiesfromdeBloketal. Williams2000).Thegrayareascorrespondtotheallowedvalues
(2001)thatshowaflatcore. fromsynthetic stellarpopulations.
directly derived in a reliable way, it is difficult to assess
whether or not thehigh dark matter densities predicted by
the NFW profile are excluded. We simply point out that
physicalmechanismsthatremoveefficientlymostofthedark
matterinsidethecentralkpcfrom asteepdarkmatterpro-
filearecurrentlyunderinvestigation (Merrittetal.2002,in
preparation; Milosavljevic et al. 2002, in preparation).
4.1 M/L of the sample
A good independent test of the values of the best fit disk
parameters(inparticularoff )recoveredfrommodellingof
d
the rotation curves is a comparison of the baryonic mass-
to-light ratios (M/L) inferred from therecovered diskmass
M = fdM200 and measured luminosity (L; from photom-
etry) with values predicted by synthetic stellar population
Figure8.Pseudo–isothermalmodel.Thepredicted(M/L)inour
models.Stellarsynthesismodelsassumeaninitialmassfunc-
analysisandinsynthetic stellarpopulations. Thelinestyles and
tion (IMF) and a star formation rate, age and metallicity shadedareashavethesamemeaningasinfig.7.
to uniquely predict the M/L. The IMF is an ansatz which
can be guided by observations in the solar neighborhood
(e.g.,Kroupa(2001)),whilethestarformationrate,ageand profile (for the 240 galaxies for which meaningful fits were
metallicity can beinferred with help of thespectrum of the found)andthepublishedvaluesforthereddening-corrected
stellarpopulation.Inourcase,becausespectraarenotavail- integrated photometry for the Courteau and PW samples
able, we have to make reasonable guesses for these param- (Courteau1997;Palunas&Williams2000);Fig8showsthe
eters. Once they have been fixed, current stellar synthesis distribution of M/L ratios recovered by using the pseudo-
models agree, quite remarkably, in their predictions about isothermalprofilemodel.Inbothcases,therecovereddistri-
the M/L ratio (e.g., Bell & de Jong (2001)). However, un- bution of M/L is fairly broad.
certainty in the assumptions ultimately limit the precision In both figures, the allowed range of M/L using the
in the M/L predictions from synthetic stellar population synthetic stellar population models developed in Jimenez
models, leaving a 20-40 % uncertainty. Taking these uncer- et al. (1998) are shown as gray areas. This range is ob-
taintiesintoaccount,alargedisagreementbetweentheM/L tained by varying B − R (between 0 and 1.5), the IMF
recovered from the rotation curve modelling and permitted (Salpeter; Miller–Scalo) and the star formation history of
values from stellar synthesis models, would imply that the galaxies (a star formation recipe with a star formation rate
modelling of the rotation curves is probably incorrect. M˙∗ ∝ exp(−t/t∗), where 3 Gyr 6 t∗ 6 20Gyr). These as-
Fig. 7 shows thebaryonic M/L ratios derived byusing sumptions allow 0.1 < (M/L) < 3 where the subscript
R,I
the best fitting model values for M200 and fd for the NFW R, I refers to the M/L in the R and I bands; the assumed
Dark halo properties 9
ageof thestellar population hasthegreatest impact on the We do not have extended information about the environ-
derivedM/Lrange,withlowvaluesofM/Lassociatedwith mentofthegalaxiesfromoursample,butverylowvaluesof
progressively youngerstellar populations. f are found also from analysis of LSB’s galaxies (e.g. van
d
The M/L derived from the NFW profile show a some- den Bosch, Burkert & Swaters (2001)) that are not found
what broader range than allowed: for the Courteau sample orbiting clusters. Thus the ram pressure stripping mecha-
4% of thegalaxies haveuncomfortably lowM/L while 16% nismseemsnottoberesponsibleforthelowvaluesoff we
d
have M/L larger than allowed from synthetic stellar pop- recover.
ulations; for the PW sample 14% have low M/L and 7% These verylow valuesof f arealso responsible for the
d
have too high M/L. For the pseudoisothermal profile the tail of low M/L galaxies in figures 7 and 8.
situation isverysimilar: forthetheCourteau sample2% of Atpresent,weregardtheselowvaluesoff aspointing
d
galaxies have low M/L and 17% have too high M/L while toward some inadequacy of the simple disk-halo model, or
forthePWsample5%havetoolowM/Land10%toohigh. the assumed form of the dark matter profile. On the other
It iscommonly claimed intheliterature thatNFW fits hand,thehighmassendofthegalaxymassfunctionisdom-
require M/L ratios which are too low; this is then used as inated by elliptical galaxies and it is not inconceivable that
anargumentagainstdarkmatterprofileswithcentralcusps we are selecting only those dark halos that manage toform
(deBlok,McGaugh&Rubin2001).Themajorityofstudies a disk.
have however focused on LSBs; we find that for HSBs only Sincethevalueoff mainlyaffectstheinnerpartofthe
d
a small fraction of galaxies have excessively low M/L. Low rotationcurve,wemayask:howshouldthedarkmatterpro-
valuesofM/Lmayplausiblybeassociated withveryyoung filebemodifiedsothattherecoveredf ismoremeaningful?
d
stellar populations (e.g., if the last burst of star formation It is clear that dark matter density in the center needs to
is ∼ 1 Gyr old, then (M/L)R,I ∼ 0.1. Tighter constraints bereduced.However,asdiscussedabove,thebestfitmodel
on M/L are only possible with multi-color photometry or forthepseudoisothermalprofilehasverysmall cores(anda
ideally, spectroscopy). Note that the distributions of M/L steeper –r−2– slope outside).
ratios derived from the pseudoisothermal profile are very We are investigating possible physical mechanisms
similartothosederivedusingtheNFWprofile(fig.7and8); withintheCDMparadigmthatcanachievesuchresult,thus
removingthecentraldarkmattercuspdoesnotchangeM/L decreasing substantially thedark mattercontentof a NFW
significantly.Thusweconcludethatthederiveddiskmassis halo within the inner kpc (Merritt et al. 2002, in prepara-
fairlyindependentofthedarkmatterprofileweassumeand tion; Milosavljevic et al. 2002, in preparation).
in broad agreement with valuesallowed bysyntheticstellar
population models.
A possible criticism of studying HSB galaxies is that,
5 CONCLUSIONS
in contrast to LSBs, the rotation curves of these galaxies
are not dark matter dominated. In an extreme case, this We have analyzed a large set of high spatial resolution ro-
would correspond to a maximum disk model, in which the tation curves of galaxies with the goal of determining the
circular velocity is attributable solely to the disk. In this model parameters for a disk embedded within a cold dark
case,wewouldbecompletelyinsensitivetothedarkmatter matter (CDM) halo that we have modeled in two ways: ei-
profile. The dashed line histograms in the right panels of ther with a NFW profile or a pseudo-isothermal profile. To
figures 7 and 8 show the M/L ratio derived by Palunas this aim we have developed the expression for the rotation
& Williams (2000) fitting a maximum disk model to their curve of an exponential disk embedded in a pseudoisother-
sample.Notsurprisingly,theirderivedaverageM/Lislarger mal halo (AppendixI).
than that of both the NFW and pseudoisothermal profile; Our study differs from previous work in two respects.
however,itisalsolargerthanallowedbythesyntheticstellar Firstly, we use optical rotation curves rather than HI ro-
population models. The Sa-Sd disk galaxies in the Palunas tation curves, so the rotation curves we use are free from
& Williams (2000) sample have B−R < 1.2, which fairly beam smearing and have smaller errors and higher spatial
robustly implies (M/L)R,I < 2.5. The M/L ratios implied resolution.Secondly,whilemanystudieshavefocusedonrel-
bythemaximumdiskmodelsaremuchlarger:theyaremore atively small samples of dwarfs, we use a large sample (400
typically associated with ellipticals. galaxies) spanning a wide range in luminosity and surface
brightness.
WefindthattheNFWprofileprovidesagoodfitto66%
4.2 Low recovered disk mass fractions f
d of the galaxies in the sample (in agreement with previous
It is worth commenting on the low values of f we ob- studies, van den Bosch & Swaters (2001)) that were based
d
tain. For both the NFW and pseudo-isothermal fits, we onfew(20)dwarfgalaxies),withadistributionofrecovered
often obtain values of f as low as f ∼ 10−3, even for concentrationparametersandspinparametersbroadlycon-
d d
M200 ∼ few×1012M⊙. This is implausibly low: it implies sistent with that predicted by CDM numerical simulations.
only∼1%ofthebaryonsinthehalocooltoformthedisk.It Whenthesample isfittedwith anisothermal profilewith a
is difficult tothink of anyfeedback mechanism (e.g., super- core, 68% of the galaxies are well fitted within this model
novae explosions) which could have such an extreme effect and the best fit model favors cores with small sizes (almost
ininhibitingcoolingorexpellingbaryonsfromthedisk(e.g., all have rc < 0.06R200 and ∼ 70% have rc < 0.01R200).
Efstathiou (2000)). A possible mechanism that can consid- However, we find that many rotation curves that have no
erably reduce the amount of gas in disk galaxies is RAM meaningful fit for one dark matter profile are well fitted by
pressure stripping (Quilis, Moore & Bower 2000). However the other: for only 52 galaxies (14% of the sample) there is
this mechanism works only on galaxies orbiting in clusters. nomeaningfulfitineithermodel.Theselargelycompriselow
10 Jimenez, Verde & Oh
qualityrotationcurves.Ourfindingsareconsistentwiththe cannotbefitbyNFWprofiles,∼57%asopposedto∼33%
innerdarkmatterprofilesbeingsteep(slopeof-1orsteeper) for the entire sample (those which do fit have somewhat
downtoradiithatarefewpercentofthevirialradius.Large lowbuttheoretically plausibleconcentrationparametersc).
dark matter cores (and therefore low central dark matter This difference may be a 1 σ statistical fluctuation (there
densities) seem thusto be excluded. are only 26 LSBs in the sample), it may reflect the greater
Numerical SPH simulations of disk formation within fidelity of LSB rotation curves to the true underlying dark
darkhalosfindthatdiskslooseasignificantfractionoftheir matter distribution, or it may perhaps be due to a system-
angular momentum if: 1) the dark matter profile is steep atic difference in thedark matter profile of halos which un-
in the center and 2) the baryons settle in right after virial- dergo greater tidal torquing. On the other hand, for the
ization (Navarro & Steinmetz 2000a). On the other hand, pseudoisothermal model, only one LSB has fd > 0.2. This
werecoversteepdarkmatterprofilesandnosignificantloss seems to indicate that LSBs are indeed fitted better with
of angular momentum. There are several routes to explain a dark matter core. For the LSB sample, 9 galaxies have
this. Some authors have pointed out that if baryons are al- dark matter cores with sizes below 1.5 kpc and 16 below
lowed to settle well after virialization of the halo (e.g. S´aiz 3 kpc. This is in fair agreement with our previous findings
et al. (2001)), then there is no significant loss of angular that the size of the dark matter core is modest. It is also
momentum. Another possibility to consider is that current in agreement with the dark matter core size inferred from
hydrodynamical simulations lack the resolution to simulate inverting the Poisson equation using the observed rotation
amulti-phaseinterstellarmediumandalsotofollowthede- curve (de Blok et al. 2001). Ultimately, our sample choice
tailedformationofgiantmolecularclouds,andthereforethe involves a trade-off: we gain increased precision in the ro-
sitesofstarformation,withinthesettlingdisk.Itistherefore tation curve (which breaks parameter degeneracies) and a
notclearthatcurrentsimulationscontainallthephysicalin- much larger sample encompassing a much broader class of
gredientsneededtosimulatethecomplexprocessofbaryons galaxies, but also acquire a somewhat larger uncertainty in
cooling down into a disk. thedisk contribution to therotation curve.
In both models, for the galaxies that have meaningful In general, LSB galaxies tend to have higher values of
fit to the rotation curves, the recovered baryonic mass to λ′ foragivenfd andlowervaluesofcforagivenmassthan
light ratios are broadly in agreement with predictions from HSBgalaxies(seeAppendixII).Thissupportstheviewthat
syntheticstellar populations. thedarkmatterprofileofLSBsmightreflectanenviroment
Fromthislargesampleofgalaxieswefindseveralcorre- in which dark halos undergo a highertidal torque.
lations among the best fitting model parameters and some IftheM/Lofthediskwaswellconstrained,thisambi-
regions in the parameters space that are not populated, we guitycanbeeliminated:itwouldbepossibletosubtractthe
refer to those regions as zone of avoidance. These correla- disk contribution to therotation curveand recover directly
tions an zone of avoidance are remarkably similar in both the inner profile of the dark halo by inverting the Poisson
profilesand donotchangewhen therotation curvesarefit- equation (e.g., deBlok et al. (2001)). TheM/L can becon-
ted byamodel with threefree parameters or with four free strained from multi-color photometry or even better, high
parameters. In particular we find a strong correlation be- S/N spectraofthestellarpopulationingalaxies(extending
tween the disk mass fraction and the spin parameter for ascloseaspossibletotheKband)togetherwithhigh-spatial
different kind of galaxies (HSBs and LSBs), in good agree- resolutionrotationcurvesespeciallyoftheinner1-2Kpc.A
mentwithfindingsofpreviousstudiesthatinvolvedonlyfew more detailed treatment of this issue will be presented in a
dwarf LSBs galaxies. These correlations may be a result of forthcoming paper.
parameter degeneracies if errors in the rotation curve have
been underestimated;otherwise, theycould haveaphysical
origin.
ACKNOWLEDGMENTS
When fitting the sample with the NFW profile we ob-
tain that for the 66% of galaxies for which meaningful fits We thank David Spergel for insightful comments and stim-
totherotationcurvesarefound,thedistributionofconcen- ulatingdiscussions. Wealso thankPovilasPalunasandTed
trartion parameters is in good agreement with predictions Williams for making available their rotation curve sample.
fromN-bodysimulations.Also,thedistributionofdiskspin LV is supported in part by NASA grant NAG5-7154. SPO
parameters is in broad agreement with the distribution of issupportedbyNSFgrantAST-0096023. LVandRJthank
spinparametersforthedarkmatterpredictedfrom N-body the TAPIR group at Caltech for hospitality. This research
simulations and linear theory, but with a small tail at low hasmadeuseofNASA’sAstrophysicsDataSystemAbstract
spins. Service.
The primary motivation in previous work of studying
the rotation curves of dwarfs and/or LSB’s was that they
wouldbedarkmatterdominated.Althoughitisquiteplau-
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