Dark Halo and Disk Galaxy Scaling Laws in Hierarchical Universes

11 Pages·0.52 MB·English

by  Julio F. Navarro

DraftversionFebruary1,2008
PreprinttypesetusingLATEXstyleemulateapjv.04/03/99
DARK HALO AND DISK GALAXY SCALING LAWS IN HIERARCHICAL UNIVERSES
Julio F. Navarro1
DepartmentofPhysicsandAstronomy,UniversityofVictoria,Victoria,BCV8P1A1,Canada
and
Matthias Steinmetz2
StewardObservatory,UniversityofArizona,Tucson,AZ85721,USA
Draft version February 1, 2008
0 ABSTRACT
0
We use cosmological N-body/gasdynamical simulations that include star formation and feedback to
0
examinetheproposalthatscalinglawsbetweenthetotalluminosity,rotationspeed,andangularmomen-
2
tumofdiskgalaxiesreflectanalogouscorrelationsbetweenthestructuralparametersoftheirsurrounding
n
dark matter halos. The numerical experiments follow the formation of galaxy-sized halos in two Cold
a
Dark Matter dominateduniverses: the standardΩ=1 CDM scenarioandthe currentlypopularΛCDM
J
model. We find that the slope andscatter of the I-bandTully-Fisher relationare well reproducedin the
4
simulations,althoughnot,asproposedinrecentwork,asaresultofthecosmologicalequivalencebetween

halo mass and circular velocity: large systematic variations in the fraction of baryons that collapse to
2
form galaxies and in the ratio between halo and disk circular velocities are observed in our numerical
v
3 experiments. The Tully-Fisher slope and scatter are recovered in this model as a direct result of the
0 dynamical response of the halo to the assembly of the luminous component of the galaxy. We conclude
0 that models that neglect the self-gravity of the disk and its influence on the detailed structure of the
1 halo cannot be used to derive meaningful estimates of the scatter or slope of the Tully-Fisher relation.
0 Our models fail, however, to match the zero-point of the Tully-Fisher relation, as well as that of the
0 relation linking disk rotation speed and angular momentum. These failures can be traced, respectively,
0
to the excessive central concentration of dark halos formed in the Cold Dark Matter cosmogonies we
/
h explore and to the formation of galaxy disks as the final outcome of a sequence of merger events. Dis-
p appointingly, our feedback formulation, calibrated to reproduce the empirical correlations linking star
- formationrateandgassurfacedensityestablishedbyKennicutt,haslittleinfluenceontheseconclusions.
o
Agreement between model and observations appears to demand substantial revision to the Cold Dark
r
t Matter scenario or to the manner in which baryons are thought to assemble and evolve into galaxies in
s
a hierarchical universes.
: Subject headings: cosmology: theory – galaxies: formation, evolution – methods: numerical
v
i
X
1. INTRODUCTION bations that have been found to be in good agreement
r with the results of numerical experiments (see, e.g., Cole
a The structural parameters of dark matter halos formed
& Lacey 1996, Eke, Cole & Frenk 1996, Eke, Navarro &
in hierarchically clustering universes are tightly related
Frenk 1998, and references therein).
through simple scaling laws that reflect the cosmological
A second example concerns the angular momentum of
context of their formation. These correlations result from
dark halos, which is also linked to halo mass and size
the approximately scalefree process of assembly of colli-
through simple scaling arguments. Expressed in nondi-
sionlessdarkmatterintocollapsed,virializedsystems. An-
mensional form, the angular momentum of dark matter
alyticalstudiesandcosmologicalN-bodysimulationshave
halosis approximatelyindependentofmass,environment,
beenparticularlysuccessfulatunravelingtherelationsbe-
and cosmological parameters, a remarkable result likely
tween halo mass, size, and angular momentum, as well as
due to the scalefree properties of the early tidal torques
the dependence of these correlations on the cosmological
between neighboring systems responsible for the spin of
parameters. Thepicturethatemergesisencouraginginits
individual halos (Peebles 1969, White 1984, Barnes & Ef-
simplicityandinitspotentialapplicabilitytotheoriginof
stathiou 1988, Steinmetz & Bartelmann 1995).
scaling laws relating the structural properties of galaxy
Finally, similarities in the halo formation process are
systems (see, e.g., Dalcanton, Spergel & Summers 1997,
also apparent in the internal structure of dark halos. A
Mo, Mao & White 1998 and references therein).
number of numericalstudies haveconsistently shownthat
Oneexampleistherelationbetweenhalomassandsize–
the shape of the density profiles of dark halos is approx-
adirectresultofthefinite ageoftheuniverse. Thenature
imately “universal”; i.e., it can be well approximated by
of this correlation and its dependence on the cosmologi-
a simple two-parameterfunction whose formulation is ap-
cal parameters is straightforward to compute using sim-
proximatelyindependentofmass,redshift,andcosmology
ple spherical collapse models of “top-hat” density pertur-
1CIARScholar,AlfredP.SloanFellow. Email: [email protected]
2AlfredP.SloanFellow,David&LucilePackardFellow. Email: [email protected]
1
2 Feedback and disk galaxy scaling laws
(Navarro, Frenk & White 1996, 1997, hereafter NFW96 sults in §5, and conclude in §6.
and NFW97, respectively; Cole & Lacey 1996; Tormen,
Bouchet & White 1996,Huss, Jain& Steinmetz 1999a,b). 2. DARKHALOANDDISKGALAXYSCALINGLAWS
Howdothesescalingpropertiesrelatetoanalogouscor- 2.1. Mass, Radius, and Circular Velocity
relationsbetweenstructuralparametersofdiskgalaxysys-
tems? This paper is third in a series where this ques- The radialdistributionofmassindarkhaloshas noob-
tion is addressed through direct numerical simulation of viousedge,sodefininghalo“sizes”isasomewhatarbitrary
galaxy formation in cold dark matter (CDM) dominated procedure. Aplausibleandusefulchoiceistoassociatethe
universes. In spirit, these studies are similar to those of radiusofahalowiththedistancefromthecenteratwhich
Evrard,Summers&Davis1994,Tissera,Lambas&Abadi mass shells are infalling for the first time. This “virial
1997, Elizondo et al. 1999,although are conclusions differ radius” is easily estimated from N-body simulations and
in detail from those reached by them. The first paper in imposes, by construction, a firm upper limit to the mass
our series (Steinmetz & Navarro 1999, hereafter SN1) ex- of the galaxy embedded inside each halo: baryons beyond
amined the origin of the Tully-Fisher relation under the the virialradius havenot hadtime yetto reachthe center
assumption that star formation is dictated by the rate at ofthehaloandthereforecannothavebeenaccretedbythe
which gas cools and collapses within dark halos. We were central galaxy.
able to show that the velocity scaling of luminosity and Numerical experiments show that virial radii depend
angularmomentuminspiralgalaxiesarisenaturallyinhi- sensitively on the enclosed mass of the system, in a way
erarchical galaxy formation models. consistentwiththepredictionsofthesphericaltop-hatcol-
Large discrepancies,however,were observedin the zero lapse model. The “edge” of a halo occurs approximately
point of these correlations, a result we ascribed to the wherethemeaninnerdensitycontrast,∆,isoforderafew
early dissipative collapse of gas into the progenitor dark hundred.3 This result seems to apply equally well to ha-
matter halos and to the subsequent assembly of the final los of all masses, and depends only weakly on cosmology
system through a sequence of mergers (Navarro & Benz through the density parameter, Ω, and the cosmological
1991,Navarro,Frenk&White 1995,Navarro&Steinmetz constant, Λ, (Eke et al. 1996, 1998),
1997, hereafter NS97). We concluded then, in agreement
with a number of previous studies, that agreement be- Ω0.30, if Λ=0;
∆(Ω,Λ)≈178 (1)
tween models and observations requires a large injection (cid:26)Ω0.45, if Ω+Λ=1.
of energy (presumably “feedback” energy from evolving
stars and supernovae)in orderto preventmuchof the gas
from cooling and condensing into (proto)galaxies at early
times, shifting the bulk of star formation to later times
and alleviating the angular momentum losses associated
with major mergers (White & Rees 1978, White & Frenk
1991, Kauffmann, White & Guiderdoni 1993, Cole et al.
1994).
This hypothesis remains to date the most attractive
path to resolve the “disk angular momentum problem”.
On the other hand, further analysis has shown that the
zero-point offset between the model and observed Tully-
Fisherrelationsisactuallyduetothe highcentralconcen-
Fig.1.—TheI-bandTully-Fisherrelationcomparedwiththe
trations of dark matter halos formed in currently popular
resultsofthenumericalsimulations. Dotscorrespondtotheob-
versions of the Cold Dark Matter cosmogony, and there-
servational samples of Mathewson, Ford & Buchhorn, (1992),
fore is unlikely to be reconciled through feedback effects
Giovanellietal(1997), Willick etal(1997), andHan&Mould
(Navarro & Steinmetz 2000, hereafter NS2). We inves-
(1992). Error bars in the simulated magnitudes correspond to
tigate these issues in detail in the present paper, which
adopting a Salpeter or a Scalo IMF.
representsourfirstattemptatimplementingarealisticnu-
merical formulation of feedback within a full-fledged sim-
ulation of the formation of galaxies in CDM halos, and at Oneconsequenceofthisdefinitionisthatallhalosiden-
gaugingitseffectsontheoriginofdiskgalaxyscalinglaws. tified at a giventime have similar densities, fromwhere it
The paper is organized as follows. Section 2 motivates follows on dimensional grounds that the total mass of the
our approach by reviewing the scaling laws linking the halo and its circular velocity at the virial radius must be
properties of dark matter halos and by comparing them tightly related. In terms of the circular velocity, V∆, at
with observational correlations. A brief description of the the virial radius, r∆, halo masses are given by
numerical procedure is included in §3; full details of the
star formation, feedback prescription, and relevant tests 2 1/2 V3 ∆ −1/2
will be presentedin Steinmetz & Navarro(2000,hereafter M∆(V∆,z)= ∆ =1.9×1012
(cid:18)∆(cid:19) GH(z) (cid:18)200(cid:19)
SN3). Section 4 discusses our main results regarding the
slope, scatter and zero-point of the Tully-Fisher relation
3
arontdatoiofnthsepereelda.tioWnebseutwmemenardizieskaanndgduilsacrumssoomuerntmuaminarned- ×HH(0z)(cid:18)200kVm∆s−1(cid:19) h−1M⊙. (2)
3We usethe term “density contrast” to referto densities expressed inunits of the critical density forclosure, ρcrit =3H(z)2/8πG, where
H(z)isthevalueofHubble’sconstant atredshiftz. WeparameterizethepresentvalueofH asH0=H(z=0)=100hkms−1 Mpc−1.
Navarro & Steinmetz 3
This power-law dependence on velocity is similar to that As indicated by eq. 5, reproducing the observed slope
of the I-band Tully-Fisher relation linking the luminosity of the Tully-Fisher relation entails a delicate balance be-
and rotation speed of late-type spirals, tween f , Υ , and the ratio V /V . The simplest
mdsk I rot ∆
possibility is that the three parametersare approximately
V 3 constant in all halos. This is the case arguedby Mo et al.
LI ≈2.0×1010(cid:18)200krmots−1(cid:19) h−2L⊙, (3) (1998),whosuggestthatfmdsk ≈5×10−2,ΥI ≈1.7h,and
V /V ≈ 1.5 are needed in order to reproduce observa-
rot ∆
tionsofgalaxydisks. Althoughplausible,this assumption
(see solid line in Figure 1) a coincidence that suggests a
isatoddswiththeresultsofthenumericalexperimentswe
direct cosmological origin for this scaling law. Eqs. 2 and
presentbelow(§3),soitisworthwhileconsideringasecond
3 can be combined to read,
possibility: that f , Υ , and V /V are not constant
mdsk I rot ∆
−1/2 3 from halo to halo but that their variations are strongly
1 M H ∆ V
L =1.9×1012 disk 0 ∆ correlated,in the manner prescribedby eq. 5. We explore
I
ΥI M∆ H(z)(cid:18)200(cid:19) (cid:18)Vrot(cid:19) now how such correlation may emerge as a result of the
dynamical response of the dark halo to the assembly of a
V 3 disk galaxy at its center.
× (cid:18)200krmots−1(cid:19) h−1L⊙, (4)
where we have introduced the parameter M to repre-
disk
sent the total mass associated with the galaxy disk and
Υ = M /L is the disk mass-to-light ratio in solar
I disk I
units4
Combining eqs. 2, 3, and 4, we find that the fraction of
thetotalmassofthesysteminthegalaxydiskis,atz =0,
1/2 3
M ∆ V
f = disk =8.5×10−3h−1 Υ rot .
mdsk I
M (cid:18)200(cid:19) (cid:18)V (cid:19)
∆ ∆
(5) Fig.2.—Thediskmassfractionversustheratiobetweendisk
This result reemphasizes our implicit assumption that rotationspeedandhalocircularvelocity. Thethickdashedand
within the “virial radius” galaxy systems are dominated solid lines correspond to the constraint imposed on these two
bydarkmatter. Furtherinsightcanbegainedbycompar- quantities by the Tully-Fisher relation (eq. 5) in the ΛCDM
ing M to the total baryonic mass within r , Mmax = and SCDM scenarios, respectively. Dotted lines correspond to
disk ∆ disk
(Ω /Ω )M . Assumingthatthebaryondensityparameter the relation expected for galaxies assembled in NFW halos of
b 0 ∆
is Ω ≈ 0.0125h−2, as suggested by Big Bang nucleosyn- constant“concentration”parameter,aslabeled. Constantdisk
b
thesisstudies oftheprimordialabundanceofthelightele- mass-to-light ratios are assumed throughout; Υ = 2 in the
I
ments (Schramm & Turner 1997), the fraction of baryons upperpaneland Υ =1 in thelower one, respectively.
I
transformed into stars in disk galaxies is given by,
In the interest of simplicity, we shall restrict our anal-
M ∆ 1/2 V 3 ysis to a case where the disk mass-to-light ratio, ΥI, is
f = disk ≈0.85Ω hΥ rot . (6) assumed to be constant, and we shall use the “adiabatic
bdsk Mmax 0 I (cid:18)200(cid:19) (cid:18)V (cid:19)
disk ∆ contraction”approximationtocomputethedependenceof
thevelocityratioV /V onf . Theresultingrelation
Because by definition baryons outside the virial radius rot ∆ mdsk
is sensitive to the detailed mass profile of the dark halo
have yetto reachthe galaxy,Mmax is a firm upper bound
disk and to the assumed structure of the disk. Assuming that
to the baryonic fraction transformed into stars, implying
themodelproposedbyNavarro,Frenk&White (NFW96,
that f <1 (White et al. 1993).
bdsk∼ NFW97) 5 is a reasonable approximationto the structure
The slope and zero point of the Tully-Fisher relation
of the halo, and adopting exponential disk models with
therefore implies that the fraction of the total mass (and
radial scale lengths consistent with the assumption that
of baryons) transformed into stars is a sensitive function
halo and disk share the same specific angularmomentum,
ofthecosmologicalparameters(throughtheproductΩ h,
0 it is possible to derive analytically the dependence of the
and ∆), of the stellar mass-to-light ratio, and of the ratio
velocity ratioon f (see Mo et al 1998for details). We
between the rotation speed of the disk and the circular mdsk
show the results in Figure 2. The thick lines (solid and
velocity of the surrounding halo.
dashed) in this figure correspond to the constraint enun-
ciated in eq. 5 for two different cosmological models (the
2.1.1. Constraints from the slope of the Tully-Fisher
“standard”ColdDarkMattermodel,sCDM:withparam-
relation
eters Ω = 1 and h = 0.5, and a low-density, flat ΛCDM
4Notethatthisdefinitionofthediskmass-to-lightratioincludesstellarandgaseousmass. However,mostofthebaryonsintheTully-Fisher
disks weconsider hereareactually instars (this istruefor observedas wellas forsimulatedgalaxies) sothat thestellarand “baryonic”disk
mass-to-lightratiodifferverylittle. Weshallnotdiscriminatebetweenthemthroughout thispaper. Further,weassumeMI(⊙)=4.15forall
numericalvaluesquotedinthispaper.
5According to these authors, the density profile of a dark halo is well approximated by ρ(r) = δcρcrit(r/rs)−1(1+r/rs)−2, where rs is
a scale radius and δc is a characteristic density contrast. For halos of given mass, M∆, this formula has a single free parameter, which can
be expressedas the“concentration”, c=r∆/rs. Thecharacteristic densitycontrast and theconcentration arerelated bythesimpleformula,
δc=(∆/3)c3/(ln(1+c)−c/(1+c)).
4 Feedback and disk galaxy scaling laws
model, with Ω = 0.3, Λ = 0.7, and h = 0.7). The up- ing f ∝V−3/2 ∝M−1/2. We show below (§4.1) that,
0 0 mdsk ∆ ∆
per and lower panels adopt two different values for the althoughthisdependenceisstrongerthanfoundinournu-
disk mass-to-light ratio, Υ = 2, and 1, respectively. The merical experiments, the overall trend predicted is nicely
I
rightmost point in each of the thick lines corresponds to reproduced in our numerical experiments.
the maximum disk mass fraction allowed by the baryonic Finally, Figure 2 illustrates that the structure and dy-
content of the halo, i.e., fmax = Ω /Ω . Dotted lines in namical response of the halo to the assembly of the disk
mdsk b 0
this figure are the “adiabatic contraction approximation” may be responsible for the small scatter in the Tully-
predictions for different values of the NFW concentration Fisher relation. For illustration, consider two halos of
parameter, c. the same mass, and therefore approximately similar con-
Figure 2 illustrates a few interesting points. The first centration, where the fraction of baryons collected into
one is thatthe disk mass-to-lightratioandthe cosmologi- the central galaxy, f , differs substantially. Provided
mdsk
calparametersdetermineinpracticetherangeofhalocon- that f >0.02, where the “adiabatic contraction” dot-
mdsk∼
centrations that are consistent with the zero-point of the ted curves are approximately parallelto the observational
Tully Fisher relation. Halos formedin the sCDM scenario constraintdelineatedbythethicklines,thesetwogalaxies
musthavec∼<3(5)ifΥI ≈2(1). Thiseffectivelyrulesout will lie approximately along the same Tully-Fisher rela-
the sCDM scenario, since N-body simulations show that tion. Even if the concentration of the two halos were to
halos formed in this cosmology have much higher concen- differ greatly its effect on the scatter of the Tully-Fisher
trations, typically c ∼ 15-20 (NFW96). A similar con- relationwouldberelativelyminor: atfixedf ,V /V
mdsk rot ∆
clusion was reached by van den Bosch (1999), who finds changes by only about 20% when c changes by a factor of
through similar considerationsthat the largeV /V val- two.
rot ∆
uesexpectedinthesCDMscenarioeffectivelyruleoutthis To summarize, assuming that the disk mass-to-lightra-
cosmogony. tio is approximately constant for Tully-Fisher disks, the
The low-densityΛCDMmodelfaresbetter,because the slope of the I-band Tully-Fisher relation may result from
higher value of h and the lower value of ∆ in this model the combinationof three effects: (i) the reduced efficiency
imply smaller f at a given value of the velocity ratio ofcoolinginmoremassivehalos,(ii)themassdependence
mdsk
V /V . However, for Υ = 2 (1), concentrations lower ofhaloconcentrations,and(iii)the dynamicalresponseof
rot ∆ I
than about ∼ 5 (12) are needed. As discussed by NS2, the halo to the disk assembly. This also offers a natural
high-resolution N-body simulations of halo formation in explanation for the small scatter in the observed Tully-
theΛCDMscenarioyieldsconcentrationsoforder∼20,in Fisherrelation. We shalluse ournumericalsimulationsto
disagreement with these constraints unless Υ ≪1. Con- test the verisimilitude of this speculation below.
I
centrations as high as this are similar to those found for
sCDM, andare systematically higher than the values pre- 2.2. Circular Velocity and Angular Momentum
dicted by the approximate formula proposed by NFW97.
Another similarity between the properties ofdark halos
Thisexplainstheapparentdisagreementbetweenthecon-
and galaxy disks concerns their angular momentum. N-
clusions of NS2 and those of semi-analytic models that
body simulationsshowthat, interms ofthe dimensionless
claimreasonableagreementwiththeobservedTully-Fisher
parameter, λ = J|E|1/2/GM5/2, the distribution of halo
relation(Moetal1998,vandenBosch1999): thediscrep- ∆
angular momenta is approximately independent of mass,
ancy can be fully traced to the lower halo central concen-
redshift, and cosmological parameters, and may be ap-
trationsandlowerdiskmass-to-lightratiosadoptedbythe
proximatedbyalog-normaldistributionpeakedataround
latter authors. If concentrations are truly as high as re-
λ ∼ 0.05 (Cole & Lacey 1996 and references therein). (J
ported by NS2, agreement with the Tully Fisher relation
and E are the total angular momentum and binding en-
require Υ ≪ 1, in disagreement with estimates based
I
ergy of the halo, respectively.)
on broad-bandcolors of Tully-Fisher disks (which suggest
Υ ≈2)andwithmass-to-lightratioestimatesofthesolar
I
neighborhood (see NS2 for details).
The second important point that emerges from Figure
2 is that, inorderto matchthe Tully-Fisherrelation,halo
concentrationsmustbeanincreasingfunctionoff (as-
mdsk
suming Υ ≈constant). Indeed, as f increases the
I mdsk
thick solid and dashed lines cross (dotted) curves of in-
creasing c. Since, according to NFW96 and NFW97, con-
centrationdepends directly onhalomass—lowmasshalos
are systematically more centrally concentratedas a result
of earlier collapse times—this is equivalent to requiring
f to be a function of halo mass. This is actually con- Fig. 3.—Specific angular momentum vs circular velocity of
mdsk
sistent with simple disk formation models where the mass model galaxies compared with observational data. Data cor-
of the disk is determined by gas cooling radiatively inside respond to the samples of Courteau (1997), Mathewson et al
adarkhalo,asfirstproposedbyWhite&Rees(1978),and (1992), and the compilation of Navarro (1999). Specific angu-
laterworkedoutindetailbyWhite&Frenk(1991). These larmomentaarecomputedfromdiskscalelengthsandrotation
authors show that, if disks form by gas cooling within an speeds,assuminganexponentialdiskmodelwithaflatrotation
approximately isothermal halo, disk masses are expected curve.
to be roughly proportional to V3/2 (White 1996), imply-
∆
Thebindingenergydependsontheinternalstructureof
Navarro & Steinmetz 5
the halos but the structuralsimilarity between dark halos Fig. 4.—The fraction of the baryons assembled into a disk
established by NFW96 and NFW97 implies that E is to galaxy(fbdsk)versustheratiobetweenthespecificangularmo-
good approximationroughly proportionalto M∆V∆2, with menta of the disk and its surrounding halo (fj). Thick solid
a very weak dependence on the characteristic density of anddashedlinescorrespond totheconstraintsimposedbythe
thehalo. Thespecificangularmomentumofthehalothen Tully-Fisher relation (Figure 1) and by the relation between
may be written as (see Mo et al 1998 for further details), rotation speed and angular momentum (Figure 3). The solid
(dashed)thicklinecorrespondstothesCDM(ΛCDM)scenario,
λ V2 H ∆ −1/2 shown for different values of Υ , as labeled. Symbols corre-
j ≈2 ∆ =2.8×103 0 I
∆ ∆1/2H(z) H(z)(cid:18)200(cid:19) spond to simulated galaxy models as per the labels in Figure
1.
2
V
× ∆ km s−1h−1kpc, (7)
(cid:18)200km s−1(cid:19) OneimportantpointillustratedbyFigure4isthatdisk
galaxiesformedina low-densityuniverse,suchasΛCDM,
where we have used the most probable value of λ = 0.05
need only accrete a small fraction of the total baryonic
in the second equality (see dotted line in Figure 3). The
mass to match the zero-pointof the Tully-Fisher relation,
simple velocity-squaredscaling of this relation is identical
but must draw a much larger fraction of the available an-
tothatillustratedinFigure3betweenthespecificangular
gular momentum to be consistent with the spins of spiral
momentum of disks and their rotation speed,
galaxies. For example, if V = V and Υ = 1, disk
rot ∆ I
masses amount to only about 30% of the total baryonic
V 2
j ≈1.3×103 rot km s−1h−1kpc (8) mass of the halo but contain about 60% of the available
disk (cid:18)200km s−1(cid:19) angular momentum. This is intriguing and, at face value,
counterintuitive. Angular momentum is typically concen-
(solid line in Figure 3), suggestive, as in the case of the
trated in the outer regions of the system (see, e.g., Figure
Tully-Fisherrelation,ofacosmologicaloriginforthisscal-
9 in NS97), presumably the ones least likely to cool and
ing law.
be accreted into the disk, so it is puzzling that galaxies
Combining eqs. 7 and 8, we can express the ratio be-
managetotapalargefractionoftheavailableangularmo-
tween disk and halo specific angular momenta at z = 0
mentumwhilstcollectingasmallfractionofthetotalmass.
as,
The simulations in NS97, which include the presence of a
1/2 2
jdisk ∆ Vrot strong photo-ionizing UV background, illustrate exactly
f = ≈0.45 . (9)
j j (cid:18)200(cid:19) (cid:18)V (cid:19) this dilemma; the UV background suppresses the cooling
∆ ∆
of late-infalling, low-density, high-angular momentum gas
If the rotation speeds of galaxy disks are approximately
and reduces the angular momentum of cold gaseous disks
the same as the circular velocity of their surrounding ha-
assembled at the center of dark matter halos.
los, then disks must have retained about one-half of the
The situation is less severe in high-density universes
available angular momentum during their assembly.
such as sCDM; we see from Figure 4 that disks are re-
The velocity ratio may be eliminated using eq. 6 to ob-
quired to collect similar fractions of mass and of angular
tain a relation between the fraction of baryons assembled
momentum in order to match simultaneously the Tully-
into the disk and the angular momentum ratio,
Fisher and the spin-velocity relations. As a result, any
difficultymatchingtheangularmomentumofdiskgalaxies
1/6 2/3
∆ f
f ≈0.5 bdsk . (10) in sCDM will become only worse in a low-density ΛCDM
j
(cid:18)200(cid:19) (cid:18)Ω0hΥI(cid:19) universe.
Note as well that the problem becomes more severe the
This combined constraint posed by the Tully-Fisher and
lowerthe mass-to-lightratio of the disk. Indeed, from the
the angular momentum-velocity relation is shown in Fig-
point of view of this constraint, it would be desirable for
ure 4 for two different cosmological models. As in Figure
disks formedinthe ΛCDMscenarioto have Υ >2;in this
2,thicksolidlinescorrespondtothe “standard”colddark I∼
case f ∼ f would be consistent with the constraint
mattermodel,sCDM,andthickdashedlinestotheΛCDM j bdsk
posedbyobservedscalinglaws. However,thisistheoppo-
model. Eachcurveislabelledbythevalueadoptedforthe
site of what was requiredto reconcile highly-concentrated
disk mass-to-light ratio, Υ . The precise values of f
I bdsk haloswiththezero-pointoftheTully-Fisherrelation. This
and f along each curve are determined by and the ratio
j conundrumillustrates the fact that accounting simultane-
V /V , and are shown by starred symbols for the case
rot ∆ ously for the mass and angular momentum of disk galax-
V =V and Υ =1.
rot ∆ I ies represents a serious challengeto hierarchicalmodels of
galaxy formation.
3. NUMERICALEXPERIMENTS
3.1. The Code
The simulations were performed using GRAPESPH,
a code that combines the hardware N-body integrator
GRAPE with the Smooth Particle Hydrodynamics tech-
nique (Steinmetz 1996). GRAPESPH is fully Lagrangian
andoptimallysuitedtostudytheformationofhighlynon-
linearsystemsinacosmologicalcontext. Theversionused
6 Feedback and disk galaxy scaling laws
hereincludestheself-gravityofgas,stars,anddarkmatter producesresults thatareintermediate betweenthe “mini-
components, a three-dimensional treatment of the hydro- mal”and“kinetic”feedbackcaseswereporthere. Testson
dynamics of the gas, Compton and radiative cooling, the isolateddisksofvaryingcircularvelocityindicatethatthe
effects of a photo-ionizing UV background (NS97), and a “kinetic feedback” adopted here reduces substantially the
simple recipe for transforming gas into stars. efficiency ofstar formationin systems with circularveloc-
ity below ∼100 km s−1, but has a more modest influence
3.2. The Star Formation and Feedback Algorithm
on the star formation rates in more massive systems, in
The numerical recipe for star formation, feedback and rough agreement with current interpretation of observa-
metalenrichmentissimilartothatdescribedinSteinmetz tional data (Martin 1999).
&Mu¨ller(1994,1995,seealsoKatz1992,Navarro&White
3.3. The Initial Conditions
1993). Fulldetailsonourpresentimplementationarepre-
sented in Steinmetz & Navarro(2000), together with vali- We investigate two variants of the Cold Dark Matter
dating and calibrating tests. A brief description follows. scenario. The first is the former “standard” CDM model,
“Star particles” are created in collapsing regions with cosmological parameters Ω = 1, h = 0.5, Λ = 0,
that are locally Jeans unstable at a rate controlled normalized so that at z = 0 the rms amplitude of mass
by the local cooling and dynamical timescales, ρ˙∗ = fluctuations in 8h−1 Mpc spheres is σ8 = 0.63. Although
c∗ρgas/max(τcool,τdyn). The proportionality parameter, thismodelfailstoreproduceanumberofkeyobservations,
c∗,effectivelycontrolsthedepletiontimescaleofgas,which such as the CMB fluctuations detected by COBE, it re-
in high-density regions, where most star formation takes mainspopularasawell-specifiedcosmologicaltestbedand
place and where τdyn ≫ τcool, is of order τdyn/c∗ ≈ asawellstudiedexampleofahierarchicalclusteringmodel
(4πGρ )−1/2c−1. of galaxy formation. The second is the currently popular
gas ∗
The equations of motion of star particles are only af- low-density CDM model that includes a non-zero cosmo-
fected by gravitational forces, but newly formed stars de- logical constant and which is normalized to match COBE
volve ∼ 1049 ergs (per solar mass of stars formed) about constraints and current estimates of the Hubble constant,
107 yrs after their formation. This energy input, a crude Ω =0.3,Λ=0.7,h=0.7,σ =1.1. Bothmodelsassume
0 8
approximation to the energetic feedback from evolving a value of Ω = 0.0125h−2 for the baryon density pa-
b
massivestarsandsupernovae,islargelyinvestedinraising rameter of the universe, consistent with constraints from
the temperature of the surrounding gas. As discussed by Big-Bang nucleosynthesis of the light elements (Schramm
Katz(1992)andNavarro&White (1993),thisformofen- & Turner 1997).
ergeticfeedbackisratherinefficient;thehighdensitiestyp- In both cosmologies,we simulate regions that evolve to
ical of star forming regions imply short cooling timescales form dark halos with circular velocities in the range (80,
that minimize the hydrodynamical effects of the feedback 350) km s−1 at z = 0. These regions are selected from
energyinput. Thenetresultisthatthestarformationhis- cosmological simulations of large periodic boxes and are
tory ofanobjectsimulatedusingthis “minimalfeedback” resimulated individually including the full tidal field of
formulation traces closely the rate at which gas cools and the original calculation. All resimulations have typically
collapses within dark matter halos (SN1). 32,000 gas particles and the same amount of dark matter
We have generalized this formulation by assuming that particles. The size of the resimulated region scales with
certain fraction of the available energy, ǫ , is invested thecircularvelocityofthe selectedhalo,sothatmostsys-
v
in modifying the kinetic energy of the surrounding gas. temshavesimilarnumbersofparticlesatz =0,regardless
These motions can still be dissipated through shocks but of circular velocity. This is important to ensure that nu-
on longer timescales, leading to overall reduced star for- mericalresolutionisapproximatelyuniformacrossallsys-
mation efficiencies and longer effective timescales for the tems. Gas particle masses rangefrom2.5×106h−1M⊙ to
conversion of gas into stars. 9×107h−1M⊙,dependingonthesystembeingsimulated.
We have determined plausible values for the two pa- Darkmatterparticlemassesareafactor(Ω −Ω )/Ω more
0 b b
rameters, c∗ and ǫv, by comparing the star forming prop- massive. Theirmassislowenoughtopreventartificialsup-
erties of isolated disk galaxy models with the empirical pression of cooling due to collisional effects (Steinmetz &
“Schmidt-law” correlations between star formation rate White 1997). Runs start at z = 21 and use gravitational
and gas surface density reported by Kennicutt (1998, see softenings that range between 0.5 and 1.0h−1 kpc.
SN3 for full details). In the case of “minimal feedback” We have concentrated our numerical efforts on the
(ǫv = 0) low values of c∗, typically ∼ 0.05, are needed to sCDM scenario: about 60 galaxy models satisfy the con-
preventtherapidtransformationofallofthegasinatypi- ditions for analysis outlined below (§3.4). For compari-
calgalaxydiskintostars. Introducingakineticcomponent son,onlysevenΛCDMgalaxymodelsareconsideredhere.
to the feedback prolongs the depletion timescales some- This isbecause ofourrealizationduringthe courseofthis
what, but in all cases, best results are obtained by choos- study that the cosmological scenario has very modest ef-
inglongstarformationtimescales,i.e.,valuesofc∗ ∼0.05. fectsonthescalinglawswediscusshere. Indeed,regarding
Forsuchlowvaluesofc∗,ǫv mustbelessthan0.2inorder disk scaling laws,galaxymodels formedin the sCDM and
to prevent slowing down star formation to rates signifi- ΛCDM scenarios differ mainly because of the adoption of
cantlylowerthanobservedinKennicutt’sempiricalcorre- differentvaluesoftheHubble constant. Wediscussthisin
lations. Similarconstraintsonǫ canbederivedfromhigh more detail below.
v
resolution 1D simulations of supernova remnants (Thorn-
3.4. Identification and Analysis of Model Galaxies
ton et al. 1998). We shall hereafter refer to the (rather
extreme) choice of c∗ = 0.05 and ǫv = 0.2 as the “ki- Model galaxies are easily identified in our runs as star
netic feedback” case. Adopting c∗ > 0.05 and ǫv < 0.2 and gas “clumps” with very high density contrast. We
Navarro & Steinmetz 7
retain for analysis only galaxies in halos represented with ThesymbolswithhorizontalerrorbarsinFigure1show
morethan500darkparticlesinsidethevirialradius;most the numerical Tully-Fisher relation obtained in our simu-
of these systems have more than 1,000 star particles in lations. Solid squares and open circles denote the lumi-
each galaxy. This list is culled to remove obvious ongo- nosities and rotation speeds (measured at r ) of galaxy
gal
ingmergersandsatellitesorbitingthemaincentralgalaxy modelsformedinthesCDMandΛCDMscenarios,respec-
of each halo. The properties of the luminous component tively,underthe “minimalfeedback”assumption. Starred
are computed within a fiducial radius, r = 15(V /220 symbolscorrespondtothe“kineticfeedback”caseapplied
gal ∆
km s−1)h−1 kpc. This radius contains all of the baryonic to the sCDM model. Error bars span the range in lumi-
material associated with the galaxy and is much larger nosities corresponding to assuming either a Salpeter or a
than the spatial resolution of the simulations. The rota- Scalo stellar initial mass function.
tion speeds we quote are also measured at that radius, A few points are clear from Figure 1. (i) The scat-
although in practice the circular velocity in the models ter and slope of the numerical Tully-Fisher relation are in
is rather insensitive to the radius where it is measured: good agreement with observation. (ii) The zero point of
similar results are obtained using r = 3.5(V /220 km the numerical relation is offset by almost 2 (1.25) magni-
gal ∆
s−1)h−1 kpc (see Figure 1 of SN1). tudes for sCDM (ΛCDM) galaxy models. (iii) The effects
Although the resolution of the modeled galaxies is ade- of kinetic feedback on the numerical Tully-Fisher relation
quatetocomputereliablyquantitiessuchasthetotalmass are quite modest. Indeed, only a slight dimming in galax-
orcircularvelocityasafunctionofradius,itisstillinsuffi- ies with V <100 km s−1 is clearly noticeable in Figure
rot∼
cienttogaininsightintothemorphologyofthegalaxy. As 1. We analyze these three results in detail below.
a result, our sample is likely to contain a mixture of Hub-
4.1.1. The Slope
ble types, from S0 to Sd (most of the simulated galaxies
retained for analysis are largely rotationally supported). As discussed in §2.1, agreement between the slopes of
Galaxyluminositiesarecomputedbysimplyaddingthe the numerical and observed I-band Tully-Fisher relation
luminosities of each star particle, taking into account the imply atightrelationbetweenthe fractionofthe massas-
time of creation of each particle and using the latest ver- sembled into the galaxy, f , the disk mass-to-light ra-
mdsk
sionofthespectrophotometricmodelsofBruzual&Char- tio,Υ ,andtheratiobetweendiskandhalocircularveloc-
I
lot (G.Bruzual & S.Charlot 1996, unpublished), see Con- ities, V /V (see eq. 5). Correlations between these pa-
rot ∆
tardo, Steinmetz & Fritze-von Alvensleben (1998) for de- rametersobtainedinour numericalsimulationsareshown
tails. Corrections due to internal absorption and inclina- in Figure 5.
tion are neglected. The IMF is assumed in all cases to be The bottom left panel of Figure 5 is identical to Fig-
independent of time. ure 1, but labels includes the (unweighted) best-fit slope
andrmsdeviation(inmagnitudes)ofeachnumericalrela-
tion. SymbolsareasinFigure1. Asmentionedabove,the
slopes of the sCDM relations are consistent with the ob-
servedone. The sameappliesto the ΛCDMbest-fit slope,
which is virtually indistinguishable from the sCDM one.
This resultis only weaklyaffectedby the narrowdynamic
range coveredby the ΛCDM simulations. Indeed, extend-
ing our analysis to more poorly resolved clumps in the
ΛCDM runs (systems with V <200 km s−1, see dotted
rot
opencirclesinthis panel)showsthatthere is little signifi-
cantdifferencebetweentheeffectiveslopesobtainedinthe
Fig. 5.—Correlations between disk mass, Mdisk, disk rota- sCDMandΛCDMmodels(seealsoFigure2ofNS2). The
tionspeed(measuredatrgal),Vrot,absoluteI-bandmagnitude goodagreementbetweenobservedandnumericalslopesin-
at z =0, MI, and halo circular velocity, V∆, found in the nu- dicate that fmdsk, ΥI, and Vrot/V∆ approximately follow
merical experiments. Symbols are as in Figure 1. Solid line the constraint prescribed by eq. 5. The rest of the panels
in the lower-left panel is the best fit to the observational data in Figure 5 illustrate the nature of this relation.
shown in Figure 1. Solid and dashed lines in the upper left The upper left panel in Figure 5 shows that, to a large
panelcorrespondtoconstantdiskmass-to-lightratio,Υ =2.5. extent, the stellar mass-to-light ratios of galaxy models
I
Solid line in upper-right panel outline the loci of galaxies that are approximately constant; the dotted and dashed lines
haveassembledallavailablebaryons. Slopesquotedinallpan- in this panel correspond to ΥI = 2.5 =constant. On the
elscorrespondtounweightedleast-squaresfits. Thermsscatter other hand, the disk mass fraction is far from constant
values correspond to the y-axis quantity relative to the least- from system to system. As illustrated by the upper right
squares fit, except for theMI-Vrot panel, where it corresponds panel in Figure 5, baryons in low circular velocity halos
to the scatter in M . are much more effective at condensing into central galax-
I
ies than those in more massive systems. Indeed, as V
∆
increases, the numerical results move away from the solid
4. NUMERICALSCALINGLAWS line representing galaxies which have assembled all of the
available baryons. 6 The fraction of baryons assembled
4.1. The numerical Tully-Fisher relation
into the central galaxy varies from ∼ 70% in halos with
6ThesymbolscorrespondingtotheΛCDMmodel(opencircles)havebeenrescaleddownwardsintheupperrightpanelofFigure5sothat
the Mdisk = Mdmisakx solid line is the same as in sCDM. Vertical deviations from this solid line thus indicate in both cases variations in the
fractionofbaryonsassembledintothecentral galaxy.
8 Feedback and disk galaxy scaling laws
V <100 km s−1 to less than ∼ 20% in V ∼ 300 km s−1 and of its response to the assembly of the baryonic com-
∆∼ ∆
halos. This result is unlikely to be an artifact of limited ponent of the galaxy.
numericalresolutionsince,asdiscussedin§3.3,thenumer-
ical resolution of the simulations do not depend system-
atically on halo circular velocity. Furthermore, the same
trend was found in the convergence study of Navarro &
4.1.2. The Zero Point
Steinmetz (NS97, see their Table 1); the observed trend
between f and V must therefore reflect the decreas-
mdsk ∆ In contrast with the good agreement found between
ing efficiency of gas cooling in halos of increasing mass
models and observations for the Tully-Fisher slope, it is
discussed in §2.1.1.
clearfromFigure1thatthereisaseriousmismatchinthe
In spite of the large systematic variations in the disk
zero-point of the numerical and observed Tully-Fisher re-
mass fraction put in evidence by Figure 5, the numerical
lations. This is reflected in Figure 6 as a systematic offset
slope of the Tully-Fisher relation is in good accord with
between the numerical data points and the thick curves
observationsbecausef variationsarecompensatedby
mdsk labeled “sCDM” and “ΛCDM” (analogous to the lines in
correspondingchangesintheV /V ratio. Lowmassha-
rot ∆ Figure 2, but drawn here for Υ =2.5, as appropriate for
loshavehigherf values,butalsohigherV /V ratios I
mdsk rot ∆ our numerical results). At given f , the ratio V /V
than more massive ones (bottom right panel of Figure 5). mdsk rot ∆
is much larger than required by observations, leading to
For constant Υ , agreement with the observed TF slope
I galaxieswhich, atfixed rotationspeed, aretoo faintto be
requires f = M /M ∝ M /V3 ∝ (V /V )3
mdsk disk ∆ disk ∆ rot ∆ consistentwithobservations,asclearlyshowninFigure1.
(see eq. 5), so that if Mdisk ∝V∆α, then Vrot ∝V∆α/3 must Simulated galaxies formed in the sCDM and ΛCDM sce-
be satisfied. Inspection of the exponents (“slopes”) listed narios are physically very similar, since they have similar
inthe right-handpanelsofFigure5demonstratethatthis diskmass-to-lightratiosandsharethesamelocationinthe
condition is approximately satisfied in the numerical ex- f vs. (V /V ) plane. The reason why ΛCDM mod-
disk rot ∆
periments. els appear to be in better agreement with observations in
Figure1islargelydueto theadoptionofahighervalueof
Hubble’s constant for this model. This issue is discussed
indetail by NS2. Fromthe analysisin that paper andthe
discussionin §2.1.1,it is clearthat the zero-pointdiscrep-
ancy is testimony to the large central concentrations of
darkhalosformedinthe twocosmologieswe explorehere.
4.1.3. The Scatter
Fig. 6.—As in Figure 2, but including the results of the
According to eq. 4, the scatter in the numerical Tully-
numerical simulations. The dotted line shows the proportion-
ality fmdsk ∝ (Vrot/V∆)3. The thick dashed line shows the Fisher relationis determined by the variance in the mass-
to-light ratio, Υ , the disk mass fraction, f , and the
“adiabatic contraction” prediction assuming that a fmdsk ∝ I mdsk
V−0.7,andthattheNFWconcentrationparameterisgivenby velocity ratio, Vrot/V∆. These can be derived from the
c∆=20(V∆/100 kms−1)−1/3,asfound inthenumericalexperi- rms values quoted in the right-handpanels and upper-left
panel of Figure 5, respectively, and would imply, taken at
ments. Seetext for furtherdetails.
face value, a large dispersion (>0.6 mag rms) for the nu-
∼
merical Tully-Fisher relation. This is actually about three
Figure 6 illustrates this conclusion more explicitly: timeslargerthanmeasuredinthesimulations(seermsval-
the numerical results follow very closely the fmdsk ∝ ues quoted in bottom left panel of Figure 5). The reason
(Vrot/V∆)3 proportionality outlined by the dotted line. behindthesmallscatterisonceagainlinkedtothedynam-
As discussed in §2.1.1, this proportionality results from icalresponseofthe halotothediskassembly,asdiscussed
the combination of three effects: (i) the decreasing disk in §2.1.1. At fixed halo mass (i.e., fixed V ), galaxies
∆
mass fraction in halos of increasing mass (f ∝ V−0.7 where fewer than average baryons collect into the central
mdsk ∆
approximately, see upper-right panel of Figure 5), (ii) galaxyhavelower than averageV /V ratios,and vicev-
rot ∆
the decrease in concentration of halos of increasing mass ersa. ThisisshowninFigure7,whereweplottheresiduals
(c≈20(V /100 km s−1)−1/3, see NFW96 and NS2), and from the best-fit power laws to the data presented in the
∆
(iii) the response of the dark halo to the assembly of the right-handpanelsofFigure5. Asanticipatedin§2.1.1,the
disk. ThethickdashedlineinFigure6showsthef vs. residuals scale in such a way that variations in the frac-
mdsk
(V /V ) relation that results from these three premises tion of baryons assembled into galaxies scatter along the
rot ∆
using the adiabatic contraction approximation described observed Tully-Fisher relation, reducing substantially the
in §2.1.1; it clearly reproduces the numerical results re- resulting scatter. This helps to reconcile the large scatter
markably well. predictedbyanalyticalestimates(Eisenstein&Loeb1996)
We conclude that the agreement between the slopes of with the results of our numerical simulations. Again, the
the numerical and observed Tully-Fisher relations lends halo structure and response to the assembly of the galaxy
supporttotheviewthattheTully-Fisherslopeisthe(non- are crucial for explaining the observed properties of the
trivial)resultofthe combinedeffects ofthe halostructure Tully-Fisher relation.
Navarro & Steinmetz 9
We conclude that the feedback algorithm explored here
isunabletobringtheangularmomentaofsimulatedgalax-
ies into agreement with observations. Gas must be effec-
tively prevented from collapsing into, or removed from,
dense disks at early times, a requirement that requires a
much more efficient transformation of supernova energy
into gas bulk motions than our algorithm can accomplish
(Weil, Eke & Efstathiou 1998). However, it is difficult to
see howthis canbe attainedwithout violatingconstraints
posed by observed correlations between gas density and
starformationratesinisolateddisks(Kennicutt1998). As
Fig. 7.—Residuals from least squares fits to the numerical discussedin§3.2,the“kineticfeedback”modelparameters
data presented in the right-hand panels of Figure 5. Solid line adopted here are already quite extreme; for example, in-
corresponds to the condition δlogMdisk = 3δlogV∆ required creasing ǫv beyond our choice of 20% in an effort to expel
so that galaxy models scatter along the Tully-Fisher relation, more gas from star forming regions would lead to mod-
reducing substantially its scatter. els that disagree substantially with Kennicutt’s findings
(SN3). Although it cannot be discounted that feedback
4.2. The angular momentum of simulated disks
implementations different from the one adopted here may
In agreement with prior work (see, e.g., NS97 and ref- lead to improved results, we conclude that accounting si-
erences therein), we find that the baryonic components of multaneouslyfortheluminosity,velocity,andangularmo-
the simulatedgalaxymodels arequite deficientin angular mentum of spiral galaxies in hierarchical models remains
momentum. This is easily appreciated in Figure 3, which anunsolvedproblemfortheCDMcosmogoniesweexplore
shows that, at a given V , the specific angular momen- here.
rot
tum of observed disks exceeds that of numerical models
by more than one order of magnitude. This is due to the
5. SUMMARYANDDISCUSSION
transfer of angular momentum from the baryons to the
halo associated with merger events during the formation We reportherethe resultsofnumericalexperiments de-
of the disk, as first suggested by Navarro & Benz (1991), signed to explore whether cosmologically induced correla-
andlaterconfirmedbyNavarro,Frenk&White(1995)and tions linking the structural parameters of cold dark mat-
NS97. Indeed, as shown in Figure 4, the baryonic compo- ter halos may serve to elucidate the origin of disk galaxy
nents of simulated galaxieshave retained, on average,less scaling laws. The numerical experiments include gravity,
than 15% of the specific angular momentum of their sur- pressuregradients,hydrodynamicalshocks,radiativecool-
rounding halos, placing them in the (f ,f ) plane well ing, heating by a UV background,and a simple recipe for
bdsk j
outsideoftheconstraintsimposedbytheobservedscaling starformation. Feedbackfromevolvingstarsis alsotaken
laws between luminosity, rotation speed, and disk size. into account by injecting energy into gas that surrounds
regions of recent star formation. A large fraction of the
4.3. The effects of feedback
energy input is in the form of heat, and is radiated away
Our implementation of feedback seems to have only a quickly by the dense, cool, star-forming gas. The remain-
very modest impact on the results discussed above, even der is introduced through an extra acceleration term that
for the rather extreme “kinetic feedback” case we tried. affects directly the kinematics of the gas in the vicinity of
In the case of the Tully-Fisher relation only systems with star forming regions.
V <100kms−1areaffected(see,e.g.,starredsymbolsin We find that the slope ofthe numericalTully-Fisher re-
rot∼
Figure 1), and the net effect is actually to make the zero- lationisingoodagreementwithobservation,althoughnot,
pointdisagreementworse: therotationspeedsoftheselow- asproposedinpreviouswork,asadirectresultofthecos-
mass galaxiesareroughlyunchanged,but feedback makes mological equivalence between mass and circular velocity
them fainter. This shows yet again that the zero-point of dark halos. Rather, the agreement results from a deli-
problem afflicting the Tully-Fisher relation is not a result catebalancebetweenthe darkhalostructure,the fraction
of processes that affect the baryonic component, such as of baryons collected into each galaxy, and the dynamical
feedback, but of the high central concentration of dark response of the dark halo to the assembly of the luminous
matter near the centers of dark halos and the relatively component.
high disk mass-to-light ratios of our models. Massive halos are significantly less efficient at assem-
Feedback also seems to have a relatively minor impact bling baryons into galaxies than their low-mass counter-
on the “angular momentum problem” as well. Disks sim- parts, a trend that agrees with expectations from theo-
ulated with our “minimum” and “kinetic” feedback im- retical models where the mass of the central galaxy is de-
plementations have angular momenta well below those of termined by the efficiency of gas cooling. As a result of
observedspirals(Figures3and4). Feedbackseemsableto the structural similarity of dark halos, systems that col-
slowdownthetransformationofgasintostarsinlow-mass lect a large fraction of available baryons into a central
systems but fails to prevent most of the gas from collaps- galaxy have their rotation speeds increased substantially
ing into dense disks at early times. The final galaxy is over and above the circular velocity of their surrounding
thus built up as the outcome of a hierarchical sequence halos. The combined effect leads, for Cold Dark Matter
of merger events during which the gas transfers most of halos, to a direct scaling between disk mass and rotation
its angular momentum to its surrounding halo (NS97 and speed, M ∝ V3 , which, for approximately constant
disk rot
references therein). stellar I-band mass-to-light ratio, is in very good agree-
10 Feedback and disk galaxy scaling laws
ment with the observed slope of the I-band Tully-Fisher ternative is quite palatable, since they involve substantial
relation. modificationseithertotheunderlyingcosmologicalmodel,
The scatter in the numerical Tully-Fisher relation (∼ to our understanding of the spectrophotometric evolution
0.25 mag rms, see lower left panel in Figure 5) is sub- of stars, or to our assumptions about the initial stellar
stantially smaller than observed, a somewhat surprising mass function. Furthermore, mass-to-light ratios as low
result given the sizeable dispersions in disk mass-to-light as that would exacerbate the problem posed by the small
ratios and disk mass fractions, as well as in the ratios be- fractionof mass and the large fractionof angular momen-
tween disk rotation speed and halo circular velocity ob- tum collected by the central disk in low-density universes
served in our numerical simulations. The small scatter is (see §2.2 and Figure 4).
mainly due to the fact that, although disk mass fractions Includingtheenergeticfeedbackfromevolvingstarsand
may vary widely from galaxy to galaxy, these variations supernovae does not improve substantially the agreement
arestronglycorrelatedwithvariationsinthediskrotation betweenthe properties ofmodel galaxiesand observation.
speed. Model galaxies therefore scatter along the Tully- This is because, in order to fit the empirical correlation
Fisher relation, minimizing the resulting scatter (Figure linkingstarformationratetogassurfacedensity,ourfeed-
7). This is again a non-trivial result stemming from the back model is only efficient in low-mass systems with cir-
detailed dark halo structure and from its response to the cular velocities below ∼ 100 km s−1. The progenitors of
assembly of the luminous component of the galaxy. massive spirals typically exceed the threshold circular ve-
Does the agreement extend to passbands other than locity at high redshift, leading to efficient gas cooling and
I? Since our galaxy models have approximately constant to early onset of star formation. The effects of the kinetic
mass-to-light ratio in the I-band, the simulated slope will feedback implementation on the Tully-Fisher relation are
be approximatelysimilar in all passbands where mass ap- therefore minor, and only noticeable in systems with low
proximately traces light, such as in the infrared K band. circular velocities. The zero-point disagreement actually
ThisisconsistentwiththeresultsofVerheijen(1997),who worsensatthe faint endas starformationis sloweddown,
find aslope of−7.0inthe I-bandandof−7.9inK,when affecting the absolute magnitudes of model galaxies more
his complete sample of galaxies is analyzed. The K-band thantheirrotationspeeds. Thelimitedimpactofourfeed-
slope is significantly steeper than the I-band’s (−10.4 in back algorithmalso implies that the luminous component
K versus −8.7 in I) only when a restricted sample with ofmodelgalaxiesisassembledthroughasequenceofmerg-
favorableinclination,smoothmorphology,steepHIprofile ers, accompanied by a substantial loss of its angular mo-
edges,andfreefrombarsisconsidered. Thestrongdepen- mentum to the surrounding dark matter. The angular
dence on sample selection reflects the large uncertainties momentum of model galaxies is, as a consequence, about
in the numerical values of the slope that result from the one order of magnitude less that that of observed spirals,
small dynamic range in velocity (less than a factor of ∼3 in agreement with previous work (cf. NS97).
typically) covered by existing Tully-Fisher samples. Tak-
ing this into account, and considering that the selection 6. CONCLUDINGREMARKS
criteria of Verheijen’s complete sample is closer to those The results we discuss here illustrate the difficulties
in our analysis (§3.4), we conclude that our models are facedbyhierarchicalmodelsthatenvisiontheformationof
generally consistent with the slopes of the I and K band disk galaxies as the final outcome of a sequence of merger
Tully-Fisher relations. events. Agreement with observations appears to require
The Tully-Fisher relation is also known to be signifi- two major modifications to our modeling. (i) Dark ha-
cantly shallower in bluer passbands (see, e.g., Verheijen los that are much less centrally concentrated than those
1997) a result that can be traced to systematic variations formed in the two Cold Dark Matter scenarios we explore
in the mass-to-light ratios as a function of disk rotation here. (ii) Feedback effects dramatically stronger than as-
speed. Qualitatively our simulations can reproduce the sumed here, affecting substantially the cooling, accretion,
trend, although it is quantitatively much weaker than ob- and star formation properties of dark halos perhaps as
served (SN1). This is because fewer stars are formed late massive as V ∼200-300km s−1.
∆
in our models than implied by observations (SN3). This The extreme feedback mechanism apparently required
effect may have ledus to underestimate the scatterin ΥI. to bringthe massandangularmomentum ofdisk galaxies
However, considering that the measured scatter is much in accordance with hierarchical galaxy formation models
smaller than the observed one, we still consider our con- should have major implications on further observational
clusionthatslopeandscatterareconsistentwithobserva- clues and traces of the galaxy formation process. Given
tions to be very solid. thepaucityofpresent-dayexamplesofthisprocessatwork
Incontrastwiththissuccess,wefindthatthezero-point (Martin1999),weareledto speculatethatstarformation
of the numerical Tully-Fisher relation is offset by about (andtherefore feedback)episodes were muchmore violent
∼2(∼1.25)magnitudesrelativetoobservationsforgalax- in the past than in the local universe, perhaps as a re-
ies formedinthe sCDM(ΛCDM) scenario. The offsetcan sult of the lower angular momenta and increased surface
be tracedtothehigh-centralconcentrationofdarkmatter density of star forming regions at high redshift. Large
halos formed in these cosmogoniesand, to a lesser extent, scale winds drivenby early starbursts,perhaps associated
to the relatively high stellar mass-to-light ratios found in with the formation of stellar spheroids, may rid sites of
ournumericalmodels(ΥI ≈2.5). The“concentrations”of galaxy formation of early-accreting, low-angular momen-
darkmatterhaloswouldhavetobe reducedbyafactorof tum baryons altogether, allowing higher-than-averagean-
∼3-5or,alternatively,ΥI wouldhavetobereducedtoval- gular momentum material to collapse later into the mod-
ues as low as ∼0.4-0.5in order to restore agreement with estly star-forming, extended disks prevalent in the local
the observed zero-point. As discussed by NS2, neither al- universe.