On The Survival and Abundance of Disk-dominated Galaxies
aa r X i v : . [ a s t r o - ph ] N ov D RAFT VERSION O CTOBER
28, 2018
Preprint typeset using L A TEX style emulateapj v. 10/10/03
ON THE SURVIVAL AND ABUNDANCE OF DISK-DOMINATED GALAXIES J UN K ODA , M ILOŠ M ILOSAVLJEVI ´C , AND P AUL
R. S
HAPIRO Draft version October 28, 2018
ABSTRACTWe study the formation of disk-dominated galaxies in a Λ CDM universe. Their existence is considered tobe a challenge for the Λ CDM cosmology, because galaxy mergers isotropize stellar disks and trigger angularmomentum transport in gas disks, thus fostering the formation of central stellar spheroids. Here, we postulatethat the formation of stellar spheroids from gas-rich disks is controlled by two parameters that characterizegalaxy mergers, the mass ratio of merging dark matter halos, and the virial velocity of the larger merginghalo. We utilize merger histories generated from realizations of the cosmological density field to calculate thefraction of dark matter halos that have avoided spheroid formation, and compare the derived statistics withthe spheroid occupation fractions in surveys of nearby galaxies. We find, for example, that the survival rateof disk-dominated galaxies in Λ CDM is just high enough to explain the observed fractional representation ofdisk-dominated galaxies in the universe if the only mergers which lead to central spheroid formation are thosewith mass ratios M / M > . V vir , >
55 km s - . We discuss the physical origin of thiscriterion. Subject headings: cosmology: theory — galaxies: formation — galaxies: spiral INTRODUCTION
The existence of disk-dominated galaxies, with littleor no bulge, is frequently cited as a challenge to the Λ CDM cosmology (e.g., Kautsch et al. 2006; Kormendy2007). Apart from the question of whether or notthe theory of galaxy formation in Λ CDM can suc-ceed in making these observed rotationally-supported diskgalaxies in the first place (D’Onghia & Burkert 2004;Abadi, Navarro, & Steinmetz 2003), the survival of suchdisks, once formed, is our focus here. The mergers thatevery galaxy-hosting dark matter halo experiences can trig-ger angular momentum transport in the interstellar mediumof the merger remnant. If a substantial amount of gas istransported into the central kiloparsec of the remnant, thegas can fuel a starburst forming a central stellar system—a “classical” bulge (i.e., self-gravitating, baryon-dominatedstellar system supported by random motions rather thanrotation). By contrast, “pseudobulges” can also arise insome disk galaxies if they have not experienced recent ma-jor mergers, by the secular transport of angular momen-tum (e.g., by galactic bars, Jogee, Scoville, & Kenney 2005).These pseudobulges (sometimes called “disky bulges”) aresupported more by rotation than random motion, however(e.g., Kormendy & Kennicutt 2004, and references therein). About one third of all disk galaxies in the local universe do notcontain bulges or pseudobulges (Kautsch et al. 2006) and an- Department of Physics, University of Texas, 1 University Station C1600,Austin, TX 78712. Department of Astronomy, University of Texas, 1 University StationC1400, Austin, TX 78712. Pseudobulges are to a larger degree supported by rotation than classi-cal bulges and can have rotational velocity-to-1D velocity dispersion ratios V max / σ >
1. Their velocity dispersions are smaller than expected from theFaber-Jackson relation. Pseudobulges tend to have flattened isophotes andsurface brightness profiles close to exponential. The classical bulges andpseudobulges can be distinguished from a third class, the boxy or peanut-shaped bulges, which are bars seen edge-on (Bureau & Freeman 1999;Athanassoula 2005). The classification of galaxies by their bulge content is usually done usingsimple bulge-disk or concentration models, which may not be accurate in ex-treme disk-dominated galaxies (e.g., Böker, Stanek, & van der Marel 2003). other third contain only pseudobulges. Allen et al. (2006) car-ried out a Sérsic spheroid and exponential disk decompositionon a large sample of galaxies and find that 30% of exponentialdisks have small bulge-to-total ratio B / T < .
2. Barazza et al.(2007) report that 20% of disk galaxies can be visually classi-fied as bulgeless. For our purposes here, we shall assume that the survivalof disk-dominance means that no classical bulge is formed.Although the precise characteristics of mergers that formclassical bulges remain unknown, disk-dominated galaxiesmust have avoided major mergers during and after forma-tion. Just how large the mass ratio of the merging galactichalos must be in order to induce bulge formation is some-what uncertain, however. Mergers of similar-mass galaxieshave been shown to trigger starbursts and result in ellipti-cal galaxies, by gas dynamical and N -body simulations forgalaxy halo masses M & M ⊙ , merging at relative veloc-ities of the order of the virial velocity (Mihos & Hernquist1996; Cox et al. 2007). In that case, a bulge forms fromthe momentum-exchange and compression of gas in mergershocks and the outward angular momentum transport inducedby merger torques. Pure N -body simulations of such merg-ers find that the pre-existing stellar disks are mixed anddestroyed (e.g., Naab & Burkert 2003), leaving an ellipti-cal galaxy. For minor mergers, N -body and gas dynami-cal simulations with halo masses M & M ⊙ indicate thatdisks survive but bulges can also grow (Mihos & Hernquist1994; Cox et al. 2007; Eliche-Moral, Balcells, & Aguerri2006, see also D’Onghia et al. 2006). However pure N -body simulations of this process find that these bulgesare pseudobulge-like (i.e., partially supported by rotation),rather than classical, for mass ratios 0 . . M / M . . V max .
70 km s - ) and gas-rich galax-ies has not yet been simulated. Pure disk galaxies also contain nuclear star clusters (e.g., Böker et al.2002; Walcher et al. 2005) which could be products of secular gas transport(Milosavljevi´c 2004), but these star clusters are tiny by comparison with thebulges described above.
THE ABUNDANCE OF DISK-DOMINATED GALAXIESSemianalytic models of galaxy formation(e.g., Kauffmann, White, & Guiderdoni 1993;Baugh, Cole, & Frenk 1996) assume they can track themorphological type of galaxies by converting a disk com-ponent into a spheroidal component in mergers with massratios µ ≡ M / M greater than some threshold. Thesemodels successfully reproduce the distribution of earliermorphological types by tuning the critical mass ratio fordisk destruction, and adopting a critical bulge-to-total massratio that discriminates broadly between disk and ellipticalgalaxies. Recent semianalytic models employing mergertrees extracted from hydrodynamic N -body simulations yielda similar result (Maller et al. 2006). However, those studieswhich focus on the survival of disks generally do not addressthe abundance of disk-dominated galaxies. Existing ab initiocosmological simulations have yielded disks with bulges(e.g., Abadi, Navarro, & Steinmetz 2003), but currently lackthe dynamic range to explore a large enough volume to sam-ple the statistics of galaxy morphology while simultaneouslyresolving the formation and mergers of individual galaxies.The purpose of this work is to compare the predicted disksurvival probabilities during hierarchical merging in a Λ CDMuniverse with the observed statistics of galaxies at the end ofthe Hubble sequence. This comparison is then used to placeconstraints on the physics of bulge-forming mergers. In § 2.1,we discuss the role of mergers in bulge formation. In § 2.2,we describe the effect of cosmic reionization and Jeans-massfiltering on bulge formation. In § 2.3, we present a calcula-tion of bulge formation and disk survival probabilities derivedfrom galactic halo merger trees generated from realizations ofcosmological density fields. In § 2.4, we compare observeddisk galaxy statistics with these merger tree results and placeconstraints on the properties of bulge-forming mergers. In§ 3, we summarize our main conclusions. Standard cosmo-logical parameters consistent with the
Wilkinson MicrowaveAnisotropy Probe (Spergel et al. 2007) are assumed through-out. DISK SURVIVAL IN Λ CDM COSMOLOGY
Mechanisms for Bulge Formation in Mergers
The merging of gas-rich galaxies fosters bulge formationdirectly and indirectly. Directly, the time-dependent gravita-tional potential of the two merging components deflects someof the gas into the center of the merger remnant, where it getscompressed in shocks and fuels a starburst. Indirectly, thegravitational tidal field of the merging components excitesnonaxisymmetric perturbations inside the merging galaxies(bars, spirals, etc.) which then torque disk gas into the cen-ter of the galaxy (e.g., Combes 1998, and references therein).In the center, again, shocks are ubiquitous and play a rolein angular momentum transport. The indirect channel shouldbe important in minor mergers, especially where the smallergalaxy loses its gas to ram pressure stripping in the earlystages of the merger.The strength of direct merger torques is a function of themass ratio of the host dark matter halos of the merging galax-ies, µ ≡ M / M . The strength of nonaxisymmetric distor-tions in minor mergers with µ ≪ The strength of merger shocks is char-acterized by the Mach number M sh , which is the ratio of theshock velocity V sh to the sound speed of the warm neutral gas c s ∼
10 km s - . Merger-driven strong shocks are radiative,because the post-shock cooling time is much shorter than thedynamical and sound crossing times of the H I disk. In thislimit the shocks are isothermal and the shock compression is ∼ M .In major mergers, we expect V sh ∼ V gal , where V gal is therelative velocity of the two galaxies, while in minor mergers V sh . V gal , although the forcing of the gas can be strong wherethe gravitational torque is amplified locally by a resonance.In view of these considerations, we postulate that, besidesthe mass ratio µ , the efficiency of gas transport in mergersis controlled by a second parameter, the merger Mach number M mer ≡ V vir , / c s ∼ β - V gal / c s , where V vir , is the virial veloc-ity of the larger merging halo, and β ∼ V gal of the two galaxies (i.e., merg-ing dark matter halo centers) at the small radii that are relevantto bulge formation ( r ∼ few kpc) to the virial velocity V vir , . The relative velocity of gas disks could be smaller thanthe virial velocity, i.e., β <
1, if the merger starts affect-ing the gas only when the distance between the halo cen-ters has become much smaller than the scale radii r s in theNavarro, Frenk, & White (1997, NFW) profiles of the two ha-los. The circular velocity of the NFW dark matter densityprofile ρ ∝ ( r / r s ) - (1 + r / r s ) - at small radii r ≪ r s equals V circ ≈ . V vir ( r / r s ) / , where the factor 1 . c ≡ r / r s = 10. The factor depends only weaklyon c ; it is only slightly smaller ( ≈ .
1) for c = 5. The circularvelocity V circ reaches its maximum at r ≈ . r s . The scaleradius equals r s = 75 kpc c - (cid:18) M M ⊙ (cid:19) / [ Ω m (1 + z ) + Ω Λ ] - / h - / . (1)With this we find that β ≈ (0 . + . z ) (cid:16) c (cid:17) / (cid:18) r (cid:19) / (cid:18) M M ⊙ (cid:19) - / , (2)implying that at low redshifts, the relative orbital velocity ofthe galaxy centers could be half the virial velocity of the largermerging halo when the separation is r ∼ M sh directly with β M mer , becausewhile M mer ≫ µ ( β M mer ) α ≥ f crit , where f crit is a threshold and α is a power, e.g., α = 1 (the “linear hypothesis”) and α = 2 (the“compression ratio hypothesis”). For angular momentum transport by spiral shocks, see, e.g.,Rozyczka & Spruit (1993), Savonije et al. (1994), and Goodman & Rafikov(2001). Here and throughout the letter, V vir ≡ ( GM / r ) / , where M isthe mass inside a sphere with radius r centered on the halo within whichthe mean density equals 200 times the critical density of the universe. ODA ET AL. 3An important distinct possibility is that in which bulgeformation is driven cumulatively , rather than induced in asingle merger (Bournaud et al. 2007). The central densityof gas in the disk of a late-type disk galaxy could increasegradually due to slow, continuous radial gas inflow. Evi-dence for such inflow can be found in the “central light ex-cess” (above the exponential law) in pure disk galaxies (e.g.,Böker, Stanek, & van der Marel 2003). The inflow couldbe excited by perturbations associated with minor mergers.Through their differential gravitational perturbations, manyconsecutive minor mergers can induce a slow, secular driftin the angular momentum distribution of the disk fluid, whichcould lead to central accumulation without leaving any char-acteristic signatures of merger-driven evolution (dynamicallyhot stellar components, etc.). The inflow could also be drivenby nongravitational processes, such as the magnetorotationalinstability in the gas disk (Milosavljevi´c 2004, see also, e.g.,Piontek & Ostriker 2004). The resulting increase of the cen-tral surface density brings the galaxy closer to the thresh-old for gravitational instability. The bulge or pseudobulgeformation-triggering merger then must only nudge the galaxyover the threshold for, e.g., nuclear bar formation, where thegalaxy is already marginally unstable.For systems which are gas-poor, collisionless mergers ofstars can result in an elliptical galaxy or a classical bulge, ifand only if the µ is sufficiently large (e.g., Naab & Burkert2003; Bournaud, Jog, & Combes 2005). This bulge forma-tion can be characterized by merger ratio µ , but it is indepen-dent of V vir , because the gravitational dynamics without gasis scale free. However, there will still be a dependence on V vir , through the requirement that the galaxies prior to theirmerger were able to form long-lived (i.e. low-mass) stars (seebelow). The Critical Virial Velocity for Bulge Formation afterReionization and Jeans-Mass Filtering
In order for the merger of two halos to have produced abulge, the halos must have contained a substantial amount ofgas, or else stars already formed from collapsed gas. Thegas content of small-mass halos, however, was affected bythe reheating of the intergalactic medium (IGM) out of whichthe gas inside those halos collapsed, by cosmic reionization,a phenomenon known as Jeans-mass filtering (Shapiro et al.1994). The gas pressure of the reheated intergalactic mediumcompetes with gravitational instability, in that case, to sup-press structure formation in those baryons which would oth-erwise have formed galaxies with virial velocity below somethreshold. The Jeans length in the IGM for a gas photoheatedto ∼ K corresponds to a halo mass after collapse and viri-alization for which the circular velocity is V circ = 55 ( T IGM / K) / km s - (3)(Iliev et al. 2007). The actual threshold virial velocity is un-certain, because one must account for the time-dependentgrowth of fluctuations in an evolving background and becausethe formation of dark matter halos affects the baryons in anonlinear way. Estimates of the value of the velocity thresholdwhich results range from about 30 to 80 km s - (Efstathiou1992; Thoul & Weinberg 1996; Navarro & Steinmetz 1997;Kitayama & Ikeuchi 2000). Whatever the precise valueshould be, this would impose a lower limit to the critical virialvelocity of merging halos capable of producing a bulge, as de-scribed above, i.e., V crit , min ∼ -
80 km s - . Since the virial velocity threshold which results from Jeans-mass filtering depends primarily on the temperature, othersources of IGM heating could have a similar effect. The su-pernova explosions associated with massive star formation,for example, could also heat the intergalactic gas. Such feed-back could also have unbound the interstellar gas from thegalaxies which formed these stars, if the galaxy virial veloci-ties were small enough. Merger Histories of Low-Mass Galaxies
To explore the sensitivity of the fractional abundance ofdisk-dominated galaxies produced during structure formationto the critical values of µ and V vir , , and thus to place con-straints on the values of these two parameters that are com-patible with the observed statistics, we generate merger his-tories of low-mass, disk-galaxy hosting halos and study disksurvival in this population of halos. We utilize those mergerhistories to calculate the abundance of disk-dominated galax-ies as a function of µ crit and V vir , crit . We compare the result-ing abundances with the incidence of late-type galaxies in the Tully Galaxy Catalog (§ 2.4).The merger histories are generated from the nonlinear evo-lution of the initial, linearly perturbed cosmological den-sity field using the publicly-available Lagrangian perturbationcode PINOCCHIO (Monaco et al. 2002). The code generatesa Gaussian-random field on a cubic mesh, distributes particleson the mesh, and determines the collapse time of each parti-cle using an ellipsoidal collapse criterion. The “collapsed”particles are moved by Lagrangian perturbation theory andrelated to virialized objects, which are the dark matter halos,by a linking criterion. We employed 512 particles with cos-mological parameters Ω m = 0 . Ω Λ = 0 . σ = 0 .
74, and h = 0 .
73, in a cubical box with 50 comoving Mpc on a side.The mass of an individual particle was m part = 3 . × M ⊙ ,and halos with more than 10 particles were selected for in-clusion in the merger tree. For a given redshift, PINOCCHIOprovides a list of all of the halos with mass M > m part whichformed inside the comoving box at this or any earlier redshift,and a complete list of their merger events. Each merger eventis characterized by the merger redshift and the masses of thehalos participating in the merger.We compute the fraction of halos containing disk-dominated galaxies as a function of the threshold for spheroidformation that is parametrized by the critical mass ratio µ and critical virial velocity V vir , of the larger halo at the timeof each merger. Specifically, we assume that a merger with µ > µ crit will create a central stellar spheroid if the halo hasa virial velocity V vir , > V vir , crit . We follow the most massiveprogenitor branch of the merger history of each halo and iden-tify the resulting z = 0 halo as containing a disk-dominatedgalaxy if no spheroid has yet formed in the halo based on thedefined criterion.There are rare cases in which a progenitor mass is so smallat high redshift that bulge-forming mergers are not resolvedby our numerical results. We have estimated the number ofsuch cases and confirmed that it is negligible. For 1 / <µ crit < /
3, the fraction of current halos in the mass rangewe will describe below, in which bulge-forming mergers oc-cur with a halo containing less than 50 particles is less than3%. For µ crit = 1 /
2, the fraction increases to 8%, but the totalmass of the merger remnant is small for those mergers with µ > /
2. If the smaller halo contains 50 particles, the mergerremnant with µ > / M min , the minimum mass of interest for our THE ABUNDANCE OF DISK-DOMINATED GALAXIEScomparisons with present-day galaxies.We consider halos with present masses in the range M min < M < M max , where M min = 5 × M ⊙ (corresponding to V vir , min ≈
60 km s - ) and M max = 10 M ⊙ (corresponding to V vir , max ≈
160 km s - ); the resulting galaxy statistics are com-pared with the observed galaxy statistics in the same approx-imate mass range. [Since the halo mass and the maximumcircular velocity of the galaxy disk are not known for most ofthe galaxies in each observed sample, we use the Tully-Fisherrelation to estimate halo masses for the observed galaxies.]The present total number density of halos in the above rangeis 0 .
021 Mpc - . Disk-dominated galaxy abundances thus cal-culated will not be strongly dependent on the specific choiceof M max because halos with masses M < M max dominate thenumber density of halos in the universe today. However, theabundances will be sensitive to M min . In §2.4, we explore thesensitivity to the choice of M min . An approach more accuratethan the one employed here would dispense with M min andwould consider halos of all masses and then match the frac-tional disk and irregular galaxy abundances as a function ofhalo mass.We ignore the finite duration of the merger, which is thetime elapsed between the halo contact and the final bulge for-mation. Indeed, the delay accounting for a finite merger dura-tion will affect only the low-redshift, disk-destroying mergerswhich are in the minority ( z <
1, Fig. 4). The dynamical fric-tion time scale was recently calibrated in N -body simulations(Jiang et al. 2007), t dyn = 0 . ǫ . + . .
86 1 µ ln(1 + µ - ) r vir V circ , (4)where ǫ is the circularity parameter. Setting ǫ = 0 . V circ = V , this simplifies to t dyn ∼ . µ + µ - ) H ( z ) - , (5)where H ( z ) - is the Hubble time at halo merger. If a pair ofhalos with mass ratio µ > . z = 1, the galax-ies in the halos merge by z = 0. In the halo mass range corre-sponding to disk-dominated galaxy hosts, the present fractionof mergers in progress is only .
5% for mass ratios µ > . .
10% for µ > . Results
Figure 1 shows the fraction of galaxies without classi-cal bulge as a function of the bulge formation criterion( V vir , crit , µ crit ), which is the result of the model described in§ 2.3. In order to compare these theoretical contours withthe observed abundance of disk-dominated galaxies, we select2281 galaxies in the nearby universe from the Tully GalaxyCatalog that have blue magnitudes in the range - < M B < -
17 and are located at distances D < h - Mpc at whichthe catalog is reasonably complete. This luminosity range ischosen to render the number density of galaxies equal to thedensity 0 .
021 Mpc - of halos that we synthesize (see § 2.3).The corresponding circular velocity range calculated from theTully-Fisher relation (e.g., Kannappan, Fabricant, & Franx2002) is 60 km s - < V c <
160 km s - . This range is consis-tent with the range of virial velocities in our theoretical halosample, which is a self-consistency check of our association http://haydenplanetarium.org/universe/duguide/exgg_tully.php F IG . 1.— The fraction of disk-dominated galaxies (galaxies without classi-cal bulges) that results from bulge formation criteria ( V vir , crit , µ crit ) character-ized by the critical merger mass ratio, µ crit , and the critical virial velocity ofthe larger halo at merger, V vir , crit .F IG . 2.— Same as in Fig. 1, but the contours are labeled by the cumulativefraction of morphological types in the Tully sample of galaxies. Assum-ing that the type Sc and later do not contain classical bulges, the formationcriteria along the Sc contour are compatible with the observed fraction ofdisk-dominated galaxies in the sample. of galaxies with dark matter halos in our numerical halo cata-log for Λ CDM.The assumption that the Tully-Fisher relation can be used torelate the luminosities of galaxies in the Tully catalog to theirhalo virial velocities (and, hence, to their masses, M ) worksbest for the spiral galaxies but is less certain for the ellipticaland S0 galaxies. The Tully-Fisher relation for S0 galaxies isshifted to lower luminosity by about M B ∼ + . V max ,and the scatter is larger compared to spirals (Bedregal et al.2006). This estimate is uncertain because the Tully-Fisher re-lation or virial mass-to-light ratio is unknown for S0s at smallmass near M min . The virial mass-to-light ratio for ellipticalgalaxies could be a factor of 10 larger relative to spirals in theODA ET AL. 5 B -band (Hoekstra et al. 2005; Guzik & Seljak 2002). If weshift the luminosity range of the subsample that correspondsto M min < M < M max by + + . M > M max ) and the number of S0s will increase by 30%,but the sum of E and S0 will only decrease from 25% to 20%,and the Sc contour move from 34% to 36%. Hence, uncertain-ties regarding the virial velocities of the ellipticals and S0s inthe sample do not significantly affect our determination of thedisk-dominated portion.Figure 2 is the same as Figure 1; it shows the fractionof disk-dominated galaxies for various bulge formation cri-teria. Each pair of morphological type and fraction printed onthe contour indicates that that morphological type and latertypes occupy the corresponding fraction in the subsample ofthe Tully catalog. If we choose to assume that a particularmorphological type (e.g., Sd, Sc, or Sb) and all later morpho-logical types are disk-dominated galaxies, while the earliermorphological types are galaxies with classical bulges, thenthe fraction of disk-dominated galaxies in the subsample isexplained by the parameters ( V vir , crit , µ crit ) along the contourlabeled by the chosen transitional morphological type. Theclassification by morphological type does not precisely sepa-rate galaxies that have classical bulges from those do not; nev-ertheless we assume that Sc and later type galaxies are eitherbulgeless or have pseudobulges, while Sbc and earlier typescontain classical bulges. Assuming this correspondence, thecontour “Sc” in the figure shows the parameter space locusyielding the abundance of galaxies without classical bulges.While the criterion for the survival in mergers of a givenmorphological type must lie on the appropriate contour, fromthe statistics alone it cannot be determined which specificvalue of ( µ crit , V vir , crit ) along the contour is the true physi-cal criterion for bulge formation. For example, the observedabundance of disk-dominated galaxies is consistent with thehypothesis that mergers with µ > . V vir , >
55 km s - create classical bulges, while mergers that do notsatisfy these criteria do not. This hypothesis is not unique; asomewhat larger µ crit and somewhat smaller V vir , crit , and viceversa, would be equally plausible on the basis of the statisticsalone. Were the critical velocity for bulge formation above ∼
65 km s - , however, then the relative abundance of disk-dominated galaxies would be greater than observed for allmass ratios µ . In that case, there would be too few galax-ies with bulges relative to their observed abundance. A sim-ilar upper limit to V vir , crit results from the fact that classicalbulges are unlikely to form from mergers that are too minor.If V vir , crit ∼
65, mergers only produce enough disk-dominatedgalaxies if µ crit ≪
1, which may be implausibly small. Onthe other hand, the critical virial velocity cannot be much lessthan the minimum value imposed by Jeans-mass filtering dis-cussed in §2.2. This means that V vir , crit cannot be much less than ∼
60 km s - , either. To identify the true, unique crite-rion for bulge formation, one must resort to physical insightto exclude implausible, extreme criteria that are still allowedby the statistics.If we assume that µ crit lies between 0 . .
2, then thecritical merger Mach number M mer , crit ≡ V vir , crit / (10 km s - )lies between 5 and 6. When the two halo centers have ap-proached to within kiloparsecs of each other, the true Machnumber of the gravitational perturbation will be reduced bythe value of β ∼ . Therefore, the question of the phys-ical plausibility of the criterion can be rephrased: In view F IG . 3.— The redshift distribution of first mergers that create classicalbulges. We assumed µ crit = 0 . V vir , crit = 55 km s - as the critical param-eters for parameters of bulge formation, which is consistent with the abun-dance of disk-dominated galaxies in the local universe (see Fig. 2). of the gravitational hydrodynamics of the gas disks in merg-ing galaxies, is it physically plausible that a merger with µ ∼ . - . M mer ∼ - µ M α mer does lead to centralgas inflow and bulge formation? For the ad hoc choice α = 1and assuming β ∼ , this yields a criterion µ crit ( β M vir , crit ) α ≡ f crit ∼ . - M min , we varythis parameter from the fiducial cutoff at 5 × M ⊙ to alower cutoff at 3 . × M ⊙ . The number density of ha-los increases by 50% to 0 .
034 Mpc - . To compensate forthe change in the number density of galaxies, we move thelower luminosity cutoff for selection from the Tully sample to M B < - .
5. The observed fraction of disk-dominated galax-ies (Sc or later morphological type) remains unchanged at ≈ µ tendto occur at smaller V vir , in smaller halos. The criterion, e.g.,with ( µ crit , V vir , crit ) = ( ,
57 km s - ) on the Sc contour in Fig-ure 2 shifts only a small amount, to ( ,
50 km s - ), as the masscutoff is lowered to 3 . × M ⊙ . Any lower cutoffs than thisare not appropriate, because at luminosities M B . -
15, diskgalaxies give way to irregulars as the most common galaxytype (e.g., Binggeli, Sandage, & Tammann 1988).In Figure 3, we plot the redshift distribution of the earli-est bulge-forming mergers of z = 0 halos for µ crit = 0 . V vir , crit = 55 km s - . The distribution is insensitive to thechoices of µ crit and V vir , crit , as long as the two parametersremain on the same contour in Figure 2. This shows thatbulge-forming mergers generally took place long after reion-ization was completed (i.e., z rei & F IG . 4.— Fraction of halos that have not experienced a merger with massratio M / M or larger since redshift z . ical collapse epoch for these merging halos was at z < V vir , crit imposed byJeans-mass filtering is applicable.In Figure 4, we plot the fraction of halos that have ex-perienced a bulge-forming merger after redshift z . The fig-ure shows that 60% of the halos with masses in the range(0 . - × M ⊙ have not experienced mergers with µ & .
05 after z = 1, and 30% have not experienced mergers with µ > . z = 2. Toth & Ostriker (1992) placed constraintson the mergers that could have taken place during the lifetimeof a galactic disk by quantifying the role of mergers in theheating and thickening of the Milky Way’s disk. The Galaxycould not have accreted more than 5% of its present massduring the past 5 Gyr, they found. Subsequent work refinedthe estimates of disk heating, resulting in less stringent con-straints on the merger history (e.g., Velazquez & White 1999; Benson et al. 2004). Kauffmann & White (1993) generatedmerger histories of Milky Way-sized halos using the Press-Schechter excursion set theory and found that in an open uni-verse with Ω m = 0 . Ω Λ = 0, the abundance of disks is con-sistent with the Toth-Ostriker constraint. We considered haloswith masses smaller than that of the Milky Way, which in thestandard Λ CDM universe merge less frequently (70% havehad no merger with µ > .
05, and 80% have had no mergerwith µ > . z = 1) than the halos in earlier studies (30%for µ > .
05 and 50% for µ > . CONCLUSIONS
In order to explain the observed space density and fractionof disk-dominated galaxies within the Λ CDM cosmology, wepropose a bulge-forming criterion such that only those halomergers with mass ratio greater than µ crit ∼ . V vir , crit ∼
55 km s - formed classical bulges, while other mergers did not. Thiscriterion has some degeneracy between µ crit and V vir , crit , but V vir , crit cannot be larger than about 65 km s - without under-producing the galaxy fraction with bulges, or much smallerthan ∼
60 km s - since Jeans-mass filtering after reionizationinhibits such small-mass halos from acquiring and retainingbaryons or forming stars. This bulge-forming criterion alsogives a reasonable dimensionless condition, µ crit β M mer , crit ∼
1, for the impact of merger on the gas disk from the pointof view of angular momentum transport. The validity of thisbulge formation criterion needs to be confirmed by further an-alytic calculation or hydrodynamical simulations of mergersin the halo mass range V vir , crit ∼
60 km s - .We would like to thank Shardha Jogee for detailed com-ments, and John Kormendy for inspiring and illuminating dis-cussions. This work was supported in part by NSF grant AST-0708795 to MM, and NASA ATP grants NNG04G177G andNNX07AH09G and NSF grant AST-0708176 to PRS. REFERENCESAbadi, M. G., Navarro, J. F., Steinmetz, M., & Eke, V. R. 2003, ApJ, 591,499Allen, P. D., Driver, S. P., Graham, A. W., Cameron, E., Liske, J., & dePropris, R. 2006, MNRAS, 371, 2Athanassoula, E. 2005, MNRAS, 358, 1477Barazza, F. D., Jogee, S., & Marinova, I. 2007, ApJ, submittedBaugh, C. M., Cole, S., & Frenk, C. S. 1996, MNRAS, 283, 1361Bedregal, A. G., Aragón-Salamanca, A., & Merrifield, M. R. 2006, MNRAS,373, 1125Benson, A. J., Lacey, C. G., Frenk, C. S., Baugh, C. M., & Cole, S. 2004,MNRAS, 351, 1215Binggeli, B., Sandage, A., & Tammann, G. A. 1988, ARA&A, 26, 509Böker, T., Laine, S., van der Marel, R. P., Sarzi, M., Rix, H.-W., Ho, L. C., &Shields, J. C. 2002, AJ, 123, 1389Böker, T., Stanek, R., & van der Marel, R. P. 2003, AJ, 125, 1073Bournaud, F., Jog, C. J., & Combes, F. 2005, A&A, 437, 69Bournaud, F., Jog, C. J., & Combes, F. 2007, preprint (arXiv:0709.3439)Bureau, M., & Freeman, K. C. 1999, AJ, 118, 126Combes, F. 1998, Starbursts: Triggers, Nature, and Evolution, Les HouchesSchool, 175Cox, T. J., Jonsson, P., Somerville, R. S., Primack, J. R., & Dekel, A. 2007,preprint (arXiv:0709.3511)D’Onghia, E., & Burkert, A. 2004, ApJ, 612, L13D’Onghia, E., Burkert, A., Murante, G., & Khochfar, S. 2006, MNRAS, 372,1525 Efstathiou, G. 1992, MNRAS, 256, 43PGoodman, J., & Rafikov, R. R. 2001, ApJ, 552, 793Goldreich, P., & Tremaine, S. 1980, ApJ, 241, 425Guzik, J., & Seljak, U. 2002, MNRAS, 335, 311Eliche-Moral, M. C., Balcells, M., Aguerri, J. A. L., & González-García,A. C. 2006, A&A, 457, 91Hoekstra, H., Hsieh, B. C., Yee, H. K. C., Lin, H., & Gladders, M. D. 2005,ApJ, 635, 73Iliev, I. T., Mellema, G., Shapiro, P. R., & Pen, U.-L. 2007, MNRAS, 376,534Jiang, C. Y., Jing, Y. P., Faltenbacher, A., Lin, W. P., & Li, C. 2007, preprint(arXiv:0707.2628)Jogee, S., Scoville, N., & Kenney, J. D. P. 2005, ApJ, 630, 837Kannappan, S. J., Fabricant, D. G., & Franx, M. 2002, AJ, 123, 2358Kauffmann, G., & White, S. D. M. 1993, MNRAS, 261, 921Kauffmann, G., White, S. D. M., & Guiderdoni, B. 1993, MNRAS, 264, 201Kautsch, S. J., Grebel, E. K., Barazza, F. D., & Gallagher, J. S., III 2006,A&A, 445, 765Kormendy, J., & Kennicutt, R. C., Jr. 2004, ARA&A, 42, 603Kormendy, J. 2007, preprint (arXiv:0708.2104)Kitayama, T., & Ikeuchi, S. 2000, ApJ, 529, 615Maller, A. H., Katz, N., Kereš, D., Davé, R., & Weinberg, D. H. 2006, ApJ,647, 763Mihos, J. C., & Hernquist, L. 1994, ApJ, 425, L13Mihos, J. C., & Hernquist, L. 1996, ApJ, 464, 641