Two phase galaxy formation: The Gas Content of Normal Galaxies
aa r X i v : . [ a s t r o - ph . GA ] N ov Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 21 November 2018 (MN L A TEX style file v2.2)
Two phase galaxy formation: The Gas Content of NormalGalaxies
M. Cook , ⋆ , C. Evoli , E. Barausse , , G.L. Granato , , A. Lapi , , Astrophysics Sector, SISSA/ISAS, Via Beirut 2-4, I-34014 Trieste, Italy INAF, Osservatorio Astronomico di Padova, Vicolo dell’ Osservatorio 5, I-35122 Padova, Italy Dept. of Physics, Univ. di Roma ‘Tor Vergata’, Via della Ricerca Scientifica 1, I-00133 Rome, Italy INAF, Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11, I-34131 Trieste, Italy Centre for Fundamental Physics, University of Maryland, College Park, MD 20742-4111, USA
21 November 2018
ABSTRACT
We investigate the atomic ( HI ) and molecular ( H ) Hydrogen content of normalgalaxies by combining observational studies linking galaxy stellar and gas budgets totheir host dark matter (DM) properties, with a physically grounded galaxy formationmodel. This enables us to analyse empirical relationships between the virial, stellar,and gaseous masses of galaxies and explore their physical origins. Utilising a semi-analytic model (SAM) to study the evolution of baryonic material within evolving DMhalos, we study the effects of baryonic infall and various star formation and feedbackmechanisms on the properties of formed galaxies using the most up-to-date physicalrecipes. We find that in order to significantly improve agreement with observationsof low-mass galaxies we must suppress the infall of baryonic material and exploit atwo-phase interstellar medium (ISM), where the ratio of HI to H is determined bythe galactic disk structure. Modifying the standard Schmidt-Kennicutt star formationlaw, which acts upon the total cold gas in galaxy discs and includes a critical densitythreshold, and employing a star formation law which correlates with the H gas massresults in a lower overall star formation rate. This in turn, allows us to simultaneouslyreproduce stellar, HI and H mass functions of normal galaxies. Key words: cosmology: theory - dark matter – galaxies: formation – galaxies: evo-lution.
Neutral atomic hydrogen is the most abundant element inthe Universe and plays a fundamental role in galaxy for-mation, principally as the raw material from which starsform. Within galaxies, the Interstellar Medium (ISM) acts asa temporally evolving baryonic component; competing pro-cesses cause the accumulation (through external infall fromthe intergalactic medium and stellar evolution) and deple-tion (through star formation and various feedback mecha-nisms) of hydrogen. Thus, observational determinations andtheoretical predictions of the hydrogen budget within galax-ies of various masses and morphologies is of central impor-tance to constraining the physics of galaxy formation (seeKauffmann, White & Guideroni, 1993, Benson et al. 2003,Yang, Mo & van den Bosch, 2003, Mo et al. 2005, Kaufmannet al. 2009) ⋆ E-mail:[email protected] (MC)
Moreover, within the ISM Hydrogen comprises the ma-jority of the cold gas mass, and when non-ionized ex-ists within two-phases, atomic HI and molecular H . Alarge body of observational analysis has shown that withingalaxies, HI generally follows a smooth, diffuse distribu-tion whereas H regions are typically dense, optically thickclouds which act as the birthplaces for newly formed stars(Drapatz & Zinnecker, 1984, Wong & Blitz, 2002, Krumholz& McKee, 2005, Blitz & Rosolowski, 2004, 2006, Wu etal., 2005). Due to the distinct differences in these phases,and the central importance of ISM physics to the evolutionof galaxies, cosmological simulations have begun to includeboth phases (see Gnedin et al. 2009 & references therein),and observations have begun focusing on simultaneous mea-surements of both HI and H (see Obreschkow & Rawlins,2009).The distinction between these two phases has recentlybeen shown to be of crucial importance to constrain thephysics of galaxy formation. in particular resolved spec-troscopy using GALEX showing obscured star forming re- c (cid:13) M. Cook et al.
Figure 1.
Schematic of the model framework, showing how wepartition the evolutionary history into spheroid and disc growthepochs, which follow significantly different evolutionary paths forbaryonic structure growth and result in the bulge-disk dichotomyobserved at z = 0. gions in nearby galaxies (Kennicutt et al. 2003, 2007,Calzetti et al. 2007, Gil de Paz et al. 2007), and variousobservational surveys providing maps of gas in galaxies athigh-resolutions (Walter et al. 2008, Helfer et al. 2003, Leroyet al. 2009), have revealed a deeper level of complexity onsub-galactic scales. These studies allowed theoretical mod-els for the ISM and star formation to be constrained andfurther developed.Furthermore, due to the constant replenishment and de-pletion of Hydrogen in either HI or H phases, and to theirseparate yet interlinked properties, at any epoch, measure-ments of the fraction of HI and H are highly constrainingfor the processes of molecular cloud formation, star forma-tion, baryonic infall and various feedbacks. Therefore, simul-taneously predicting the stellar and gas mass functions ofnormal galaxies is a major challenge for any physically mo-tivated galaxy formation model, requiring an accurate depic-tion of all of the aforementioned processes (see Mo et al. 2005for a detailed discussion). These issues manifest most clearlywithin the largely successful ΛCDM paradigm within thelowest mass systems, where it still remains unclear whetherstrongly non-linear feedback mechanisms, lower star forma-tion efficiencies, or suppression of initial infall onto DM ha-los is the dominant driver for the suppression of luminousstructure formation (Mo et al. 2005). It is more than likelythat a combination of the above-mentioned effects will go a long way to alleviating current tensions between models andobservations, since current semi analytical models (SAMs)incorporate several processes in order to generate a defi-ciency of stellar mass in DM halos; many of which operatemost effectively at low masses (Benson et al. 2003, De Luciaet al., 2004).Observationally, Zwaan et al. (2005) used the catalogueof 4315 extragalactic HI HI Parkes All Sky Survey (HIPASS, Barnes et al. 2001)and obtained the most accurate measurement of the HI mass function of galaxies to date. The HIMF is fitted witha Schechter function with a faint-end slope of − . ± . HI masses of10 . M ⊙ , hence this is most complete analysis so far. Usingthese statistical constraints, it has now become possible tomake stringent comparisons between theoretical models andobservations even in low mass galaxies.The physics of cold gas becomes increasingly relevantfor constraining galaxy formation models at relatively lowmasses (dominated by late-type galaxies), where the pres-ence of gas becomes substantial and therefore may breakthe degeneracies between feedback, star formation, and in-fall processes. Moreover, within the ΛCDM scenario the HI and H mass budgets in galaxies are determined by an intri-cate offset between several competing processes, all of whichhave strong mass dependencies. Thus the present HI and H fractions are strong functions of host DM halo mass and theevolutionary history of each individual galaxy. More specif-ically, the fraction of gas which may be captured by thehost DM halo, and in turn removed by feedback, is expectedto depend strongly on the binding energy of the gas itself,which is principally determined by the DM halo virial massand density distribution. Thus, under this framework theobservational properties of galaxy populations are stronglyinfluenced by their collective host DM halos (White & Rees,1978, see Somerville et al. 2008 for a review).Motivated by the above-mentioned observational ad-vances and theoretical challenges, the primary aim of thiswork is to investigate the physical origins of the relation-ships between HI , stellar, and virial masses of galaxies. Inorder to do this we compare empirical galaxy relations de-rived from observational studies, with a physically motivatedgalaxy formation model. Observationally, we use a numeri-cal approach (described in Shankar et al. 2006) that relieson the assumption of the existence of a one-to-one mappingbetween galaxy properties and host DM halo mass . Weinterpret these results using a physically motivated SAM,(see Cook et al. 2009, hereafter C09, & references therein),which has been shown to reproduce many features of thelocal galaxy population.SAMs provide a powerful theoretical framework withinwhich we can explore the range of physical processes (e.g.accretion mechanisms, star formation, SN feedback, blackhole growth and feedback etc.) that drive the formation andevolution of galaxies and determine their observable proper-ties (see Somerville et al. 2008 and Baugh 2006 for extensive This approach is based also on the assumption on the com-pleteness of the sample over which the mass functions has beenobtained. c (cid:13) , 000–000 wo phase galaxy formation: The Gas Content of Normal Galaxies reviews) whilst remaining computationally feasible, allowingfor statistical samples of galaxies to be generated.As is shown in Fig.1, the backbone to our galaxy forma-tion model is a cosmologically consistent DM halo merger-tree, outlining the merging and accretion histories of DMhalos. For the evolution of baryonic structures within thesemerging trees, we utilize the model of C09, with severalmodifications. This requires us to partition each mass ac-cretion history (MAH) into two phases: A ’fast accretion’ ,merger dominated phase corresponding to spheroid-SMBHco-evolution, followed by a ’slow accretion’ , quiescent phaseallowing for disk structure to form around the pre-processedhalos (see also Zhao et al. 2003, Mo & Mao, 2004), the re-sulting galaxy properties at z = 0 are thus a result of thesub-grid baryonic recipes, and the mass accretion historiesof their host DM halos. Within this work we also expandthe previous model by including several additional effectswhich are thought to be of crucial importance the low-massgas-rich systems, namely, we model the effects of an ioniz-ing ultraviolet (UV) background, cold accretion flows andexploit a two-phase ISM, where star formation is governedby the surface density of H .In summary, we utilise the most up-to-date observationsof the stellar and HI mass functions in order to derive rela-tionships between the host DM halo and the galaxy stellarand gas properties, we interpret these results using a phys-ically grounded SAM, and analyse the nature of the resultsby generating three model realisations, incorporating sev-eral contemporary recipes and highlighting how each helpsalleviate previous model tensions. The outline of the paperis as follows: In § § HI mass functions and use a numerical pro-cedure in order to derive relationships between host DM halomass and the galaxy stellar and HI components. In § § WMAP M = 0 .
27 and Ω Λ = 0 . H = 70 km s − Mpc − . We follow the galaxy evolution using a semi-analytical ap-proach, deriving analytical estimates wherever possible toaid simplicity. We track the evolution of dark matter, hotgas, cold gas and stellar mass using several parameteriza-tions, based on the model described in C09 but with severalimportant modifications.Namely, we now trace the sequential infall of materialonto both spheroid and disc structures and we include theeffects of an ionizing UV background, adiabatic contractionof the DM halo and a two-phase ISM. Here we outline thebasic framework for this model.We initially describe the dark matter halo evolution us-ing an extended Press-Schechter (EPS) algorithm based on
Figure 2.
The derived relationships between virial mass and;stellar (solid/black), HI (dashed/red) and baryonic (dotted/blue)components. Shaded regions corresponding to the 1 − σ gaussianerrors associated with the observational determinations. that developed by Parkinson et al. (2008). These mergertreeshave been tuned in order to reproduce the statistics of halomerger and accretion activity obtained from N-body simu-lations of structure formation (Springel et al. 2005). Withinour model, we utilise these mergertrees by extracting themass accretion history (MAH), i.e. the evolutionary path ofa typical dark matter halo, obtained by tracking the mostmassive progenitor at each fragmentation event whilst mov-ing from z = 0 to progressively higher redshifts (see van denBosch, 2002, C09).Specifying the DM halo properties, we define the virialradius r vir as that of a spherical volume enclosing an over-density in a standard way (Bryan & Norman, 1998). Wecompute the density profile for the dark matter halo usingthe fitting function of Navarro, Frenk & White (1997, here-after NFW): ρ NFW ( r ) = ρ s (cid:16) rr s (cid:17) − (cid:16) rr s (cid:17) − . (1)In order to define the scale radius r s for our NFW profile weintroduce the concentration parameter , which is defined tobe c ( z ) ≡ r vir /r s . This quantity has been studied by severalauthors [see Bullock et al. 2001, Zhao et al. 2003 (hereafterZ03), Maccio et al. 2007] who found large scatter for a fixedhalo mass, but to scale generally with the MAH of the halo.We adopt here the z -evolution of Z03,[ln(1 + c ) − c/ (1 + c )] c − α ∝ H ( z ) α M vir ( z ) − α , (2)where α is a piecewise function which can be found in Z03and where the normalization can be fixed using the expres-sion given by Maccio et al. 2007 at z = 0:log c = 1 . − . (cid:20) log (cid:18) M vir , M ⊙ (cid:19) − (cid:21) . (3)From this, the scale density may be computed for a generalprofile to be ρ s = M vir ( z ) / [4 πr s f ( c )], with f ( c ) = ln(1 + c ) − c c . (4)Finally, we specify the angular momentum properties c (cid:13) , 000–000 M. Cook et al.
Figure 3.
The evolution of typical model galaxies; a low-mass disk-dominated galaxy (left-hand panel), an L ⊙ galaxy with bothbulge and disks present (middle panel), and a high mass, spheroid dominated galaxy (right panel): DM mass (solid/black), hotgas (long-dash/brown), disc gas (short-dash/blue), disc stars (dotted/purple), bulge gas (triple-dot-dashed/green), bulge stars (dot-dashed/aquamarine), BH mass (solid/red) of each halo through the spin parameter , defined to be λ = J vir E / M / G − , where E vir and J vir are the total energyand angular momentum of the halo. Assuming that a DMhalo acquires its angular momentum through tidal torqueswith the surrounding medium, λ remains constant. It hasbeen shown (Cole & Lacey, 1996) that the spin parametervaries little with cosmic epoch, halo mass, or environmentand for a sample of haloes is well fitted by a log-normaldistribution (where ¯ λ = 0 .
04 and σ = 0 . λ = 0 .
04 which remains constant throughout theDM halo evolution.Several studies have shown the concentration evolutionto be strongly correlated to the MAH of the halo (Z03, Li etal., 2006, Lu et al. 2006), finding that DM halos generally ac-quire their mass in two distinct phases; an initial phase char-acterised by rapid halo growth through major merger events,where the halo core structure forms, causing the gravita-tional potential to fluctuate rapidly, followed by a slower,more quiescent growth predominantly through accretion ofmaterial onto the outer regions of the halo. These two differ-ent modes are reflected in the evolution of the concentrationparameter, which remains roughly constant during the ’fastaccretion’ phase and steadily increases during the ’slow ac-cretion’ phase. The transition redshift z t between these twophases can therefore be calculated using the expression forthe concentration parameter evolution given in Z03:[ln(1 + c t ) − c t / (1 + c t )] c − α [ln(1 + c ) − c / (1 + c )] c − α = (cid:20) H ( z t ) H (cid:21) α (cid:20) M vir ( z t ) M vir , (cid:21) − α , (5)where H ( z ) is the Hubble radius, ’0’ denotes quantities eval-uated at the present cosmic time, and c t = 4 and α = 0 . M inf = f coll ˙ M vir . (6)We include the effects of an ionizing radiation backgroundtaking the prescriptions outlined in (Gnedin et al., 2004,Somerville et al., 2008), which is able to partially reduce thebaryonic content in low-mass systems, thus f coll ( M vir , z ) = Ω b / Ω m (1 + 0 . M f ( z ) /M vir ) , (7)where M f ( z ) is the filtering mass at a given redshift, com-puted using the equations in (Kravtsov, Gnedin & Kyplin,2004, Appendix B). A second improvement over our previ-ous model (C09) is to include the effects of cold accretionflows, shown to be the predominant mechanism leading tothe formation of low-mass systems. Below a critical mass M c = M s max[1 , . z − z c ) ] , (8)where M s = 2 × M ⊙ and z c = 3 .
2, we assume that allgas accreted onto DM halos is not shock heated to the virialtemperature of the DM halo, but streams in on a dynamicaltime (see Dekel et al., 2008, 2009, Cattaneo et al, 2006). Wenote that below the shock heating mass scale it has beenshown that rapid cooling does not allow for the formationof a stable virial shock (Keres et al., 2005) resulting in gasflowing unperturbed into the central regions of the DM halo.Thus, in halos below this mass the collapse happens on thedynamical timescale of the system ( t coll = t dyn ), whereas inhalos above this mass t coll = max[ t dyn , t cool ], where the cool- c (cid:13) , 000–000 wo phase galaxy formation: The Gas Content of Normal Galaxies Figure 4.
Stellar mass function output by each of our three modelrealisations compared with the Bell et al. 2003a estimate. We findthat the most sophisticated treatment of the baryonic physics re-sults in agreement to the observations in all but the lowest masshalos ( M s < M ⊙ ), where we find that we overproduce thestellar matter by approximately a factor two. Note that, due tolarge uncertanties in observational mass-to-light ratios, we asso-ciate 0.3dex errors in the stellar mass determinations, and 0.4dexin the value of φ ( M stars ), shown by the shaded area. Model error-bars represent poissonian uncertanties due to the finite samplesize of synthetic galaxies. ing timescale t cool is computed in a standard way, assumingmaterial is shock heated to the virial temperature. The ef-fects of this cold accretion is to moderately enhance starformation at high redshifts relative to the scenario where allmaterial is shock heated, in this way, we model the infalland cooling dynamics using the most current recipes (seeSomerville et al., 2008).In order to calculate the cooling time of the hot gasphase, we assume an isothermal gas in hydrostatic equilib-rium within the NFW profile, such that ρ hot ( r ) = ρ exp (cid:20) − β (cid:26) − ln(1 + r/r s ) r/r s (cid:27)(cid:21) , (9)with β = 8 πµm p Gρ s r s k B T vir , (10)where T vir is the virial temperature, µ is the mean molec-ular mass and ρ is calculated by normalizing to the totalhot gas mass at any given redshift. Also, we assume that thecooling function Λ( T, Z ) is given by the tabulated functionof Sutherland & Dopita, 1993, and assume that the infallingbaryonic material is unprocessed and therefore has primor-dial metallicity, Z = 10 − Z ⊙ . During the high redshift domain, it has been shown thatthe majority of the angular momentum of the collapsingproto-galactic gas dissipates as it collapses and condenses tothe centres of DM haloes (see Navarro & Steinmetz, 2000).Also, within C09 we show that during the early collapsephase, DM halo subunits merge on timescales shorter than the overall dynamical time of the forming halo. Therefore,within the ’fast accretion’ phase we neglect the effects ofangular momentum of cooling proto-galactic gas, which willresult in the formation of a spheroidal gaseous system at arate˙ M coll ( z ) = 4 π Z r vir ( z )0 r ρ hot ( r, z ) t coll ( r, z ) d r , (11)where t coll ( r, z ) is determined by the cold-accretion recipeof Eqs. 8, 9 and 10. We denote this cold gas spheroidal com-ponent (’gaseous bulge’), which act as a reservoir for starformation, by M b,gas ( t ), and stress that besides dissipativecollapse, it can also grow through merger events and disc in-stabilities. Also, our model includes other two components: aspheroidal stellar component (’stellar bulge’) M b,star ( t ) anda low angular momentum cold gas reservoir M res ( t ), whichacts as a source of material eligible to accrete onto a centralblack hole.For simplicity , we assume that the reservoir can bedescribed by an exponential disc surface densityΣ res ( r, z ) = Σ ( z )e − r/r res ( z ) , (12)with the scale radius r res being proportional to the influenceradius of the SMBH ( r res = αGM SMBH /V , with α ≈ a priori information about the geo-metric distribution of baryonic matter within the bulge sys-tem, and since the dynamics and thus evolution of disc struc-ture is correlated to the mass and geometry of the bulgestructure, we assume that the bulge stellar and gaseousmasses settle into a Hernquist density profile ρ ∗ b ( r ) = M ∗ b π r b r ( r + r b ) , ∗ = stars, gas , (13)where the scale radius of this profile is related to the halflight radius by r b = 1 . R eff . Using the fitting of Shenet al. (2003), we take the parametrization as a function ofbulge mass to belog( R eff ) = (cid:26) − .
54 + 0 .
56 log( M b ) [ log( M b ) > . − .
21 + 0 .
14 log( M b ) [ log( M b ) . ψ b d r ( r, t ) = 4 πr ρ b,gas ( r ) t gas ( r ) , (14)where t gas ( r ) is the dynamical time for the gas in the bulge.Therefore in order to compute the total SFR we must inte-grate this expression over all radii.Energetic feedback due to supernova events may trans-fer significant energy into the cold ISM, causing it to bere-heated and ejected from the system. Therefore, by con-sidering energy balance in the ISM, supernovae feedback isable to remove gas from the bulge at a rate: We need to make an assumption about the reservoir geometrybecause that is needed to calculate the velocity of the compositesystem V c [needed e.g. in Eqs. 27 and 32, the adiabatic halo con-traction factor Γ [Eqs. 28 and 29] and the gravitational potential φ of the composite system appearing in Eqs. 15 and 34. However,the geometry of the reservoir is not expected to have a majorimpact on our results, given its small size relative to the othercomponents.c (cid:13) , 000–000 M. Cook et al.
Table 1.
Values of the free parameters of our model. We note that the value of k acc , despite being a free parameter, has a minor impacton our results Description Symbol Fiducial value Reference in the text Impact on this workSN feedback efficiency (bulge) ǫ SN , b . ǫ SN , d . A res − M ⊙ yr − Eq. 18 StrongQSO feedback efficiency f h − Eq. 22, 23 StrongViscous accretion rate k acc − Eq. 20 WeakNote. - A Romano IMF φ ( m ⋆ ) is adopted: φ ( m ⋆ ) ∝ m − . ⋆ for m ⋆ > M ⊙ and φ ( m ⋆ ) ∝ m − . ⋆ for m ⋆ M ⊙ . ˙ M SNb,gas ( t ) = − Z ǫ SN,b E SN η SN d ψ b ( r, t ) / d rφ ( r, t ) d r , (15)where η SN is the number of Type II supernovae expectedper solar mass of stars formed , E SN is the kinetic energyreleased per supernova event, and ǫ SN,b is the efficiency ofsupernovae energy transfer used to remove the cold gas. Fi-nally, φ ( r, t ) is the gravitational potential of the compos-ite system (bulge, reservoir, disc and DM). Therefore, usingthis prescription, we see that supernovae feedback is partic-ularly efficient in smaller halos with shallower gravitationalpotential wells but relatively inefficient in larger halos. Us-ing scaling relations, Granato et al. (2000) were able to showthat stars form faster in larger systems, thus exhibiting theobserved ’anti-hierarchical’ behavior of spheroid galaxies.A growing body of evidence is now showing that theevolution of both the SFR within spheroids and the fuelingof SMBH’s are proportional to one another (Haiman, Ciotti& Ostriker, 2004). A proposed mechanism to account forthis phenomenon has been discussed (Kawakatu, Umemura& Mori, 2003): Radiation drag due to stellar radiation mayresult in the loss of angular momentum at a rate well ap-proximated within a clumpy ISM by:d ln( J )d t ≈ L sph c M b,gas (1 − e − τ ) , (16)where L sph is the total stellar luminosity and τ is the ef-fective optical depth of the spheroid ( τ = ¯ τN int where ¯ τ isthe average optical depth and N int is the average numberof clouds intersecting a light path). Upon loss of angularmomentum this gas may flow into the nuclear region, gener-ating the reservoir of low-J material which fuels BH growthat the rate (Granato et al., 2004)˙ M res ≈ . × − ψ b ( t )(1 − e − τ ) M ⊙ yr − . (17)We note that, within this work we assume that τ is constantfor simplicity, which allows one to rewrite Eq. 17 as˙ M res = A res ψ b ( t ) , (18)where A res is a free parameter. The cold gas stored in thereservoir is then expected to accrete onto the central SMBHwith an accretion rate˙ M bh = min[ ˙ M visc , ˙ M edd ] . (19)In this formula, the viscous accretion rate is given by(Granato et al., 2004) A Romano IMF φ ( m ⋆ ) is adopted: φ ( m ⋆ ) ∝ m − . ⋆ for m ⋆ > M ⊙ and φ ( m ⋆ ) ∝ m − . ⋆ for m ⋆ M ⊙ . This gives η SN = 5 × − M − ⊙ . ˙ M visc = k acc σ G (cid:16) M res M bh (cid:17) , (20)where k acc ≈ − , whilst the Eddington accretion rate issimply ˙ M edd = L edd /ηc , with η ≈ .
15 and L Edd ≈ . × M BH ( t )10 M ⊙ erg s − . (21)QSO activity affects the interstellar medium of thehost galaxy and also the surrounding intergalactic mediumthrough both radiative heating and the kinetic energy inputthrough gas outflows. Assuming that a fraction f h (whichwe treat as a free parameter) of the SMBH luminosity L h istransferred into the cold and hot gas phases, it is possible tocompute the amount of cold and hot gas which is removedfrom the hot gas and gasoeus bulge phases as in Granato etal., 2004:˙ M QSOb,gas = f h L h σ M b,gas M hot + M b,gas , (22)˙ M QSOhot = f h L h σ M hot M hot + M b,gas , (23)where σ = 0 . V vir . This material is assumed to be ejectedfrom the system.For the chemical evolution of the cold bulge gas, we usethe simple instant-recycling approximation (IRA), wherebya fraction of mass is instantly returned into the cold gasphase in the form of processed material . In particular, thisimplies that the effective SFR which enters the evolutionequations for the gas and star bulge masses is given by˙ M SFR b ( t ) = (1 − R ) Z ψ b d r ( r, t )d r , (24)with R = 0 .
25. Also, we assume that M hydrogen =0 . M cold , where the factor takes into account the contri-bution of Helium and other heavier elements. By assuming that material may collapse to form a coolgaseous disc structure during the slow accretion phase, we This value of η corresponds to rapidly spinning SMBH withspin parameter a ≈ . We adopt a Romano et al., 2005 IMF, which has the standardSalpeter slope 1 .
25 in the high mass tail, and flattens to a slope0 . M ⊙ . As shown in Romano et al. (2005), this performsbetter than the Salpeter one in reproducing the detailed chemicalproperties of elliptical galaxies.c (cid:13) , 000–000 wo phase galaxy formation: The Gas Content of Normal Galaxies Figure 5.
Baryonic mass function output by our three modelrealisations as compared to the Bell et al. 2003b estimates. Asis shown, the ’standard’ approach over predicts significantly theamount of baryonic material at all masses, whereas adding theeffects of a UV background and then a two-phase ISM resultsin progressive improvements. The most sophisticated realisationresults in a good fit to the observational mass function across allobservable masses. Note that the shaded area represents the 1 − σ errors, discussed in fig.4. Model error-bars represent poissonianuncertanties due to the finite sample size of synthetic galaxies. may add material to the disc structure at a rate which isgiven, as in the spheroidal case, by˙ M coll ( z ) = 4 π Z r vir ( z )0 r ρ hot ( r, z ) t coll ( r, z ) d r , (25)where t coll ( r, z ) is determined again by the cold-accretiondeterminations (see Eqs. 8, 9 and 10). If we assume a dissi-pationless collapse of material upon cooling within the darkmatter halo, we may relate the dark halo virial radius andspin parameter to the forming disc scale radius. In particu-lar, if we assume an exponential disc surface density profilefor the stellar and gaseous components,Σ ∗ d ( r, z ) = Σ ∗ ( z )e − r/r d ( z ) , ∗ = stars, gas , (26)the disc scale radius r d ( z ) evolves according to the scaling r d ( z ) = (2 π ) − / ( j d /m d ) λr vir ( z ) f ( c ) − / f r ( λ, c, m d j d ). Thefunction f ( λ, c, m d j d ) may be exactly determined through(Mo, Mao & White, 1998) f ( λ, c, m d , j d ) = 2 (cid:20)Z ∞ e − u u V c ( r d u ) V c ( r vir ) d u (cid:21) − , (27)where V c ( r ) is the velocity profile of the composite system(bulge, reservoir, disc and DM) and where m d and j d arethe ratios between the total mass and angular momentumof the disc component and the DM halo mass. More specif-ically, we take m d = ( M starsd + M gasd ) /M vir , and we assume j d = m d (Mo, Mao & White, 1998). In order to account foradiabatic halo response, we take the standard prescriptionof Blumenthal (1986). In particular, denoting by M X ( r ) themass of the a given component ’ X ’ enclosed by a radius r ,from the angular momentum conservation one obtains M i ( r i ) r i = M f ( r f ) r f , (28) where r i and r f are respectively the initial and final radius ofthe shell under consideration, the initial mass distribution M i ( r i ) is simply given by the NFW density profile, while M f ( r f ) is the final mass distribution. Also, mass conserva-tion easily gives M f ( r f ) = M d ( r f ) + M b ( r f ) + M DM ( r f ) + M res ( r f ) = M d ( r f ) + M b ( r f ) + M res ( r f ) + (1 − f gal ) M i ( r i ) , (29)where f gal = M gal /M vir (with M gal = M d + M b + M res ). Byassuming spherical collapse without shell crossing, one canadopt the ansatz r f = Γ r i , with Γ = const (Blumenthal,1986), and Eqs. 28 and 29 may be solved numerically forthe contraction factor Γ.When the surface density of the gaseous disc increases,the cold gas becomes available to form stars. However, atpresent, star formation is poorly understood from both amicroscopic, and large-scales. Therefore we parameterise thestar formation using an empirical Schmidt law (Kennicutt,1998) whereby the star formation rate is related to the sur-face density of cold disc gas:˙Σ sfr ( r, z ) = ǫ sf (cid:20) Σ gasd ( r, z ) M ⊙ pc − (cid:21) n M ⊙ kpc − yr − , (30)where ǫ sf = 2 . × − controls the star formation efficiencyand n = 1 . c ( r ) = σ g κ ( r )3 . GQ , (31)Toomre, 1964. Where σ g = 6km s − is the velocity disper-sion of the gas, Q = 1 . κ ( r )is the epicyclic frequency, given by κ ( r ) = √ V c ( r ) r (cid:18) rV c ( r ) dV c ( r ) dr (cid:19) / . (32)Therefore, the conversion rate of gas mass to stellar mass iscomputed as ψ d ( z ) = 2 πǫ sf Z r c r Σ n d , gas ( r, z )d r, (33)where r c may be calculated by solving Σ gasd ( r c , z ) = Σ c ( r c )for r c .In order to account for feedback due to supernovaeevents, we may compute the amount of cold gas which isejected from the system at each disc radius. In order toremove this cold gas from the disc, the supernovae feed-back must be sufficient to unbind it, therefore we comparethe amount of energy released through supernovae events ateach disc radius with the binding energy at the same radius:˙Σ SN ( r, z ) = − ǫ SN , d E SN η SN ˙Σ sfr ( r, z ) φ ( r, z ) , (34)where φ ( r, z ) again, is the binding energy of the compositesystem (bulge, disc, reservoir and DM). The total amountof cold gas ejected from the system is then given by c (cid:13) , 000–000 M. Cook et al.
Figure 6.
The HI mass function produced by our model as com-pared to the Zwaan et al. 2005 HIPASS galaxy sample. Using thecombined effects of a UV background photo-ionizing radiationand a two-phase ISM we suppress the formation of low mass sys-tems. We find that we reproduce both the normalisation and highmass cutoff accurately, but still slightly overproduce the numbersystems with M HI < M ⊙ , a minor effect discussed in § − σ uncertanties in de-termination of HI masses. Model error-bars represent poissonianuncertanties due to the finite sample size of synthetic galaxies. ˙ M d SN ( z ) = 2 π Z r vir r ˙Σ SN ( r, z )d r (35)For the chemical evolution of the cold disc gas, we useagain the IRA approximation, and the ’effective’ SFR whichenter the evolution equations for the gas and star disc massesis given by˙ M d SFR ( z ) = (1 − R ) ψ d ( z ) , (36)with R = 0 .
25. Also, we assume again that M hydrogen =0 . M cold .Finally, it is known that when discs become self-gravitating they are likely to develop bar instabilities, getdisrupted and transfer material to the spheroidal compo-nent (Christodoulou, Shlosman & Tohline, 1995). We there-fore assume that a stellar or gaseous disk is stable if V c (2 . r d )( GM ∗ disc /r d ) / > α ∗ crit ∗ = stars , gas , (37)where α starscrit = 1 . α gascrit = 0 . Assuming that star formation may only take place insidedense molecular clouds several authors have shown that theSchmidt-Kennicutt star formation law (Eq. 30) may be re-produced within large mass systems by assuming that thestar formation rate is proportional to the molecular cloudmass (Blitz & Rosolowski, 2006, Dutton & van den Bosch,2009), thus: ˙Σ sfr = ˜ ǫ sf Σ mol , HCN (38)Where ˜ ǫ sf = 13Gyr − and Σ mol , HCN = f mol R HCN is themolecular mass surface density as traced by HCN (see Gao& Solomon, 2004, Wu et al. 2005). Thus, calculating theratio of molecular gas to atomic gas allows us to computethe star formation rate at all scales. The fraction of gas indiscs which is molecular has been extensively analysed, andshown to be closely related to the mid-plane pressure withindiscs (Blitz & Rosolowsky, 2006), given by: P mp = π G Σ g (Σ g + ( σ g /σ s )Σ s ) (39)Where, following the detailed prescriptions of Dutton &van den Bosch, 2009, assuming a constant σ g /σ s = 0 .
1. Re-lating the mid-plane pressure to the formation of molecularclouds yields: R mol = Σ mol Σ atom = (cid:18) P mp /k . × (cid:19) . (40)Thus, the molecular fraction is given by f mol = R mol / ( R mol + 1), in order to relate this to the HCN frac-tion, we must further compute the fitting relation of Blitz& Rosolowski, 2006: R HCN = 0 . ∗ (1 + Σ mol / (200 M ⊙ pc − )) . (41)Therefore, we find that, in the high mass (and thus den-sity) galaxies, where the molecular fraction is typically ≈ .
4, whereas in the low-densitygalaxies we asymptote towards an exponent of 2 .
84, sup-pressing star formation in these systems, in accordance withobservations (see Dutton & van den Bosch, 2009 and Blitz &Rosolowski, 2006 for a detailed description). Thus, through-out the evolution of each galaxy, we partition the ISM into HI and H components using the above relations, and com-pute the star formation law (and therefore supernovae feed-back) using equation 42 in order to self-consistently modeleach galaxy under this improved star formation law. HI MASS RELATIONSHIPS
In order to investigate the relationship between the stellarand the gas component in late-type galaxies, we follow theprocedure already exploited by Shankar et al. (2006), anddiscussed in detail in Evoli et al., 2009. We defer the readerto these papers for a more detailed discussion and highlightthe main results here. If two galaxy properties, q and p , obeya one-to-one relationship, we can write φ ( p ) d p d q d q = ψ ( q )d q (42)where ψ ( q ) is the number density of galaxies with mea-sured property between q and q + d q and φ ( p ) is the corre-sponding number density for the variable p . The solution isbased on a numerical scheme that imposes that the numberof galaxies with q above a certain value ¯ q must be equal tothe number of galaxies with p above ¯ p , i.e., Z ∞ ¯ p φ ( p )d p = Z ∞ ¯ q ψ ( q )d q (43) c (cid:13) , 000–000 wo phase galaxy formation: The Gas Content of Normal Galaxies Figure 7.
The H mass function produced by our model as com-pared to the estimations by Keres, 2003. We find that we re-produce both the normalisation, low mass, and high mass cutoffwithin the observational range. Observational error bars are as-sociated to 1 − σ uncertanties in determination of HI masses.Model error-bars represent poissonian uncertanties due to the fi-nite sample size of synthetic galaxies. In the following the HI mass function (HIMF) is givenby; p = M HI and φ ( p ) = HIMF( M HI ), while the variable q isthe stellar mass ( M ∗ ) and ψ ( q ) the corresponding GalacticStellar mass function (GSMF( M ∗ )).Using the HI Parkes All-Sky Survey HIPASS (Meyer etal. 2004) it has become possible to map the distribution of HI in the nearby Universe. The HIMF has been fitted witha Schecter function with a power index of α = − . ± . M ∗ HI /M ⊙ ) = 9 . ± . h − anda normalization of θ ∗ = (6 . ± . × − h Mpc − dex − .Nakamura et al. (2003) estimated the LF in the r ∗ bandfor early- and late- type galaxies separately. Shankar et al.(2006) used the results of Zucca et al. (1997) and Love-day (1998) to extend these results to lower luminosities andgiving a good fit for the late-type galaxy LF in the range10 L ⊙ < L r < · L ⊙ . The GSMF, holding over themass range 10 < M ∗ < is compared with the one ofBell (2003a), and is in good agreement within the uncertain-ties due to the MLR, estimated to be around 30%.The correlation of galaxy properties with the halo mass( M h ) is extremely relevant in the framework of galaxy for-mation theories. To constrain such a relation we used theassociation between the stellar mass and the host halo massderived in Shankar et al., 2006 (Eqn.12 & Fig.1 therein),which has been obtained with the method already mentionedbut using either for the stellar mass function and the galactichalo mass function a fitting of the observations/simulationfor all the galaxies. Finding relations between each of themas components within galaxies is of great importance, be-cause it allows for constraints on the direct outputs frommodels, not requiring significant post processing, and pro-viding useful links between the dark and luminous compo-nents of galaxies. We find that the results of the numericaloutputs may be well approximated by the following analyticfitting functions: M HI . × M ⊙ = ( M ∗ / . × M ⊙ ) . M ∗ / . × M ⊙ ) . (44) M HI . × M ⊙ = ( M vir / . × M ⊙ ) . M vir / . × M ⊙ ) . (45)Fig.2 shows the ratio of the baryonic mass components(stellar, gas and total) to the initial baryon mass associ-ated with each DM halo obtained using the above-mentionedmethods (and explained in Evoli et al., 2009), illustrating theinefficiency of galaxies, especially those of low halo mass, inretaining baryons. These relationships can be used to tightlyconstrain theoretical galaxy formation models in order tointerpret the physical processes relevant to shape these re-lations. Throughout this section we present the results of modelrealisations, each comprising of a sample of ≈ galax-ies in logarithmic virial mass increments in the mass range9 . < log ( M v ( z = 0) /M ⊙ ) < . z = 0 we weight each DM halo with the GHMF,which mirrors the Sheth & Tormen (2002) mass functionwithin most galaxy sized DM haloes, but is derived in orderto account for the increasing probability of multiple galaxyoccupation in the highest mass haloes (see Shankar et al.2006). Throughout this section we also plot the results for’early’ and ’late’ type galaxies, by parting the populationsinto [ M bulge /M total ] > . M bulge /M total ] < . In order to illustrate the general behavior of our galaxy for-mation framework, Fig.3 highlights the evolution of eachmass component from z = 8 up to the present epoch. Inthe left-hand, center and right-hand panels we show a low-mass, intermediate mass and high-mass galaxy respectively,in order to highlight typical model outputs for each system.As can be seen, the evolution of each galaxy differs signifi-cantly due both to scatter at each mass through monte carloselected mass accretion histories, and to the relative differ-ences in efficiencies of the competing processes of infall, starformation and feedback on different mass-scales.We find, in broad agreement with observations, thata typical low-mass galaxy (with M v ( z = 0) ≈ M ⊙ )supports the growth of a disc structure from high redshiftsshowing extended star formation up to z = 0 resulting in agas-rich disc dominated galaxy with a negligible bulge com-ponent. In intermediate mass, L ⊙ galaxies, at high-redshiftsgaseous collapse onto a spheroid system results in the co-evolution of SMBH’s and the spheroidal component result-ing in a gas poor, ’red and dead’ spheroidal stellar compo-nent which acts as the bulge within the resultant formedgalaxy as the disc component grows steadily from z . c (cid:13) , 000–000 M. Cook et al.
Figure 8.
Deriving the relationships in § − σ observational uncertanty. Model error-bars represent poissonianuncertanties due to the finite sample size of synthetic galaxies. to the presence of a large SMBH which acts to effectivelyexpel any residual gas which may infall later, only allow-ing for extremely mild star formation following the maingrowth phase. The resulting galaxy spheroidal componentthus comprises an old stellar population, with little gas andnegligible star formation, and the disc component is overan order of magnitude smaller than the bulge, and gas rich,showing little star formation.These results are consistent with the evolutionary histo-ries tuned to match the chemical properties of local galax-ies of different morphological type and different host halomasses (Calura, Pipino & Matteucci, 2008 & referencestherein) which have also been shown to be consistent withphotometry within the local Universe (Schurer et al. 2009)and are generally in agreement with observations of statis-tical samples of galaxies (Driver, et al. 2006), however, ascan also be seen, due to the stochastic nature of the model,fluctuations in galaxy properties are expected, and thus wemust revert to statistically representative samples of galaxypopulations in order to make more robust comparisons.Within these systems in-falling baryonic material isshock heated and therefore may form a static atmosphereof hot-gas. Feedback from a growing SMBH may thus haltthe cooling of this gas, quenching star formation and sweep-ing out the ISM (See Granato et al., 2004). Within thesesystems it is thought that baryonic material is shock heatedand therefore cooling and feedback from a growing SMBHmay quench star formation by sweeping out the ISM (seeGranato et al. 2004) The local stellar and baryonic mass functions provide a pow-erful constraint on theoretical models of galaxy formation:Encompassing much of the relevant physical processes whichdetermine the assembly of baryons within DM halos.In order for models to reproduce observational results,it has become clear that physical processes of gas accretion,supernovae feedback and photo-ionization are most effec-tive in the lowest mass DM halos where the shallow poten-tial wells are inefficient at trapping and holding baryonicmaterial. Thus these processes drive the evolution of thefaint-end slope of the mass functions (Benson et al., 2002),whereas the brightest galaxies (above L ∗ ) are embeddedwithin large DM halos which effectively trap baryonic ma-terial. Within these systems in-falling baryonic material isshock heated and therefore may form a static atmosphere ofhot-gas. Feedback from a growing SMBH may thus halt thecooling of this gas, quenching star formation and sweepingout the ISM (See Granato et al., 2004). Coupled with theincreasing subhalo contribution (through the cluster massfunction) whereby in DM halos with M v ( z = 0) > theprobability of a single galaxy occupation is low, gives rise tothe relatively sharp cutoff of stellar mass within the largestDM halos (see Shankar et al. 2006 & Somerville et al. 2008,& contained references).Rather than computing the spectral energy distribu-tion (SED) assuming any mass-to-light ratio for each modelgalaxy (relying upon further model assumptions such asthe dust-to-gas ratio, molecular cloud structure and opti-cal depth etc) by comparing the stellar and total baryonbudgets within each DM halo, we provide the most directanalysis of model outputs. However, we also note that un-certainties in the observational conversion of luminosity tostellar (and total baryonic) mass are systematically relatedto the spectral energy SED fitting methods used in order toextract physical quantities from the multi-wavelength obser-vations, and on the quality of the observations themselves,this is known to have large uncertanties and we hope in asubsequent work, to utilize synthetic spectra using the de-tailed star formation histories, galaxy geometries, gas anddust content, in order to self-consistently model the multi-wavelength SED and make comparisons with the luminousproperties of galaxies. In Fig.4 we show the Schecter func-tion fit to the Bell et al. (2003a) estimate for the local stellarmass function, in order to generate this mass function theauthors utilize a large sample of galaxies from the Two Mi-cron All Sky Survey (2MASS) and the
Sloan Digital Sky Sur-vey (SDSS) converting galaxy luminosity into stellar massusing simple models to convert the optical and near infraredobservations into stellar masses.As we have shown in Fig.4, using three levels of so-phistication we are able to highlight the differences be-tween; a ’standard’ model, whereby we ignore the effects of aphoto-ionizing background and employ a standard Schmidt-Kennicutt (1998) star formation law, a ’UV’ model wherebywe include the suppression of mass flowing into galaxiesthrough photo-ionization (see Eqn. 6 & 7), and a ’sophisti-cated’ model, which combines the effects of a photo-ionizingbackground with a two-phase ISM and modified star forma-tion law. We will refer to these model names throughout thenext sections. c (cid:13) , 000–000 wo phase galaxy formation: The Gas Content of Normal Galaxies More specifically, using a ’standard’ model we signifi-cantly over-produce the number density of low mass galax-ies. This occurs even when we increase supernovae feedbackefficiencies to extremely high levels (see Mo et al. 2005 for adiscussion), further pushing the parameter to higher valuesthan ǫ sn = 0 . . Secondly, increasingthe supernovae feedback efficiencies whilst lowering the starformation efficiencies simply has the effect of reducing thestellar mass but not the gaseous mass, resulting in gas frac-tions which increase with increasing supernovae efficiencyand thus are in disagreement with observations (see Mo etal., 2005 for a detailed discussion).By suppressing the initial in-fall of material due to anionizing UV background radiation, we are able to improveagreements with the mass function. However, we achieve thebest reproduction of the stellar mass function through theadditional reduction in star formation efficiency when em-ploying the ’sophisticated’ star formation law, which is de-temined by the amount of H gas present in galaxy discs.Alone, the stellar mass function provides a crucial ob-servation for any physical galaxy formation model to re-produce, but by simultaneously comparing both the stel-lar and gaseous properties of galaxies, we are able to breakdegeneracies between gas infall and cooling, star formationand feedback processes, resulting in a significantly improvedconstraint on the theoretical framework. This has led pre-vious authors to claim that the standard model, wherebylow mass systems support efficient gas cooling leading tolarge gaseous rotationally supported discs, which then un-dergo mild star formation , significantly over-predicts galaxy HI masses when compared to observations (see Mo et al.2005) causing serious tension between theory and observa-tion.In Fig.5 we compare model results with the Bell et al.(2003b) baryon mass function. We find that suppressing theinfall of baryonic gas due to an ionizing UV backgroundsignificantly improves agreement between observation andmodel output, since material is prevented from in-fallinginto the halos initially and thus requiring less feedback inorder to gain agreement of the low stellar and gas fractionsin these halos. A final, and further improvement betweenmodel and observation is achieved by partitioning the ISMinto neutral and molecular gas, which has the effect of reduc-ing the SFR efficiency preferentially in the lowest mass DMhalos. We find that the best agreement between model andobservation naturally results from using the most sophisti-cated treatment of the ISM physics, and the initial infallof gas. We do however, still find discrepancies in the lowestmass halos ( M star ( z = 0) < ), over-predicting the baryonand stellar content within these DM halos, and slightly over-producing the number density of the most massive galaxies.This issue is discussed in § We note also that we use a universal stellar initial mass func-tion (IMF) which has been constrained in order to reproducethe chemical properties of galaxies accurately, increased super-novae rates are achievable using a more top-heavy IMF, howeverthis has a strong chance to offset other properties of the formedgalaxies with respect to observations (see Romano et al. 2005 fora discussion.)
Figure 9.
The fraction of baryonic matter in the form of HI relative to the cosmological fraction showing that using the ’so-phisticated’ model we are able to successfully reproduce the de-rived relationships in § HI mass by a factor of two withinthe lowest mass halos, we discuss the implications of this in § − σ observational uncertanty.Model error-bars represent poissonian uncertanties due to the fi-nite sample size of synthetic galaxies. However, the observations (with their large uncertain-ties), remain relatively insensitive to the detailed propertiesof the galaxies at different masses, a strong motivation forthe method adopted in §
3, whereby linear scales amplify thediscrepancies between models. HI and H mass functions Due to the implemented star formation law within the ’so-phisticated’ model, whereby we partition the total cold gasmass into atomic and molecular Hydrogen , where f mol = R mol/ ( R mol +1) and R mol is given by Eq.40 at each time-increment in order to self-consistently generate a two phaseISM, we are able to make simultaneous predictions for boththe HI and H mass functions. In Fig.6 we compare ourmodel results to the observational estimates of Zwaan et al.2005. The sample, comprising of 4315 extragalactic emis-sion line estimated HI masses from the HIPASS catalogue,is complete down to M HI ≈ M ⊙ , which allows for an un-precedented determination of the low mass slope. We findthat above M HI ≈ M ⊙ we accurately reproduce the formand normalisation of the mass function, but we do signifi-cantly over-produce the HI masses at the low-mass extreme.Secondly, in Fig.7 we compare the H mass function asderived from the CO-mass function determination by Kereset al. 2003 based on the FCRAO extragalactic CO survey of200 galaxies (Young et al. 1995). The conversion from CO to H is achieved through a ’ χ -factor’ which proves to be a del-icate and difficult task to determine, we utilize the H massfunction derived by Obreschkow & Rawlins, 2009 as thisappears to be the most robust determination, and also plot Taking M H = f mol M hydrogen and M HI = (1 − f mol ) M hydrogen c (cid:13) , 000–000 M. Cook et al. the H mass function derived within the originalKeres et al.,2005 work (using a constant χ -factor). We refer the readerto their work for further details. We find that within obser-vational errors, we reproduce this function over the entireobservational range, however, we note that the large uncer-tanties within the determination of the precise value of the χ -factor means that we are relatively loosely constrained,and we view this result as a general prediction of our model,rather than a constrained observational match.Physically interpreting these results, we may concludethat utilising a two-phase ISM whereby star formation scalesonly indirectly with the total gas mass, but directly with themolecular H mass, we find that we may accurately repro-duce both the HI and H mass functions of galaxies apartfrom an over-prediction of neutral HI within the lowest masshalos. This discrepancy also appears within the stellar massfunction (Fig.4) and the baryonic mass function (Fig.5), al-though it is greatly improved using the ’sophisticated’ ISMtreatment. Exploiting the methods outlined in § f b = Ω b / Ω m = 0 .
16. These relation-ships provide direct observational constraints onto galaxyformation models and allow for a direct analysis of the rel-evant processes occurring over a range of mass scales (seeShankar et al. 2006). As can be seen, the lowest mass systems( log ( M v /M ⊙ ) <
11) are strongly dark matter dominated,due to the inefficiency of weak gravitational potentials inthese systems being able to capture and contain baryonicmaterial and then efficiently form stars. Typically these sys-tems are strongly disc-dominated, late-type galaxies.Intermediate mass systems (11 < log ( M v /M ⊙ ) < log ( M v /M ⊙ ) > H mass, we are able to re-produce the observations to a high accuracy throughout theentire mass range due to the relative decrease in SF effi-ciency, with the only discrepancies at the lowest masses. Itis important to note here that due to the sequential build-upof matter within halos from high redshifts to low, the im- Figure 10. HI -to-stellar mass ratio as a function of the stellarmass. Using derived relationships from § HI and stellar mass, finding that usinga two-phase ISM and related SFR computations we accuratelyreproduce the observations across the entire observational range,unlike the more simplistic frameworks, where we systematicallyconvert too much cold gas into stellar material. The shaded regionrepresents the 1 − σ observational uncertanty. Model error-barsrepresent poissonian uncertanties due to the finite sample size ofsynthetic galaxies. portance of ionizing backgrounds and star formation on lowmass systems manifests throughout the entire mass rangeat z = 0 (see Somerville et al. 2008 for further discussionof this). Under the standard ΛCDM scenario whereby smallstructures collapse first and form progressively larger sys-tems, large structures at z = 0 are therefore significantly in-fluenced by small scale processes at higher redshifts, wheretheir constituent parts were forming and evolving, thus thelow mass behavior is of global importance to galaxy forma-tion theory. In analogy to the previous section, in Fig.9 we plot the frac-tion of HI to DM halo mass we are able to see that thegas fractions are highly sensitive to the different physicalprescriptions, the scatter is attributed to the fact that thecold gas (and thus HI mass) component at any time is con-trolled by the competing processes of infall, star formation,feedback and recycling. We also find that, due to the uncer-tainties and intrinsic dispersion in the relationship betweenthe dark matter and gaseous matter, the observational trendis not as constrained for the gaseous as for the stellar com-ponent. Observationally, the lowest mass DM halos becomeprogressively more depleted in HI , due to suppression of in-fall and increased efficiency in generating gaseous outflows,intermediate mass DM halos (11 > ( log ( M v /M ⊙ ) > . HI mass, due to the global maxima in trap-ping and containing baryonic material in DM halos (or con-versely, a global minima in feedback processes), and at highmasses we see a progressive loss of HI abundance in halos,signifying an increase in removal of gaseous material throughcooling inefficiencies and nuclear feedback processes.In Fig.9 we show the outputs for each of the three re- c (cid:13) , 000–000 wo phase galaxy formation: The Gas Content of Normal Galaxies alisations, as can be clearly seen, the only successful repro-duction of the majority of the observations is the ’sophisti-cated’ model, whereby we model both the effects of a two-phase ISM and of an ionizing background radiation. There-fore modeling the star formation rate depending linearly onthe surface density of molecular gas in the disc componentresults in a correct conversion of cold gas to stellar mate-rial (as can be seen from simultaneous fits to both Fig.8& Fig.9), using the standard Schmidt-Kennicutt (1998) starformation law (and their constrained normalisation) we con-vert too much material from gas into stars, resulting in anover-production of stellar mass and an underproduction ofgas mass in all DM halos, whereas the modification outlinedin Dutton & van den Bosch (2009) allows for the correctfraction of virial mass in the HI phase, apart from in thelowest mass systems (11 > ( log ( M v /M ⊙ ), where we find ob-servation and model predictions in alarming disagreement.As with previous works, but to a lesser extent, we findthat we over-produce the amount of gas in the lowest massDM halos, and more worryingly, we do not reproduce theobserved down-turn in gaseous mass within the lowest masssystems ( log ( M v /M ⊙ ) < M v ( z = 0) < M ⊙ . We my attribute this effect with ourlack of description of environmental effects, since the lowestmass halos are more probable to be located within largerover-densities and therefore subject to external forces (seethe § It is suspected that within the lowest mass systems, gascollapse, star formation and supernovae feedback result ina self-regulated conversion of gas into stars, thus these ef-fects can be seen to produce precisely the correct fractionof stars to gas in Fig.10. Within our model the cold gascomponent fluctuates more than any other galaxy compo-nent since it is constantly being replenished due to infall andstellar recycling, exhausted through star formation and ex-pelled through feedback processes. Therefore, directly com-paring each galaxies gaseous and stellar properties providesa stringent comparison between observation and model. In-terestingly we find that plotting these quantities, we repro-duce a tight correlation with relatively little scatter.We find an overall agreement to the data across the en-tire mass range under investigation using the ’sophisticated’model, showing that the HI fraction becomes increasinglylarge with decreasing stellar mass, having roughly equal stel-lar and HI masses at M s ≈ M ⊙ increasing to a factorof ≈
100 times more HI than stellar mass within systemswith M s ≈ M ⊙ . Conversely, without using the effects ofa two-phase ISM and UV background radiation, we find ageneral offset in all masses, with an over-efficient conversionof cold gas to stars (as may also be seen in Figs.8 & 9).We also find that this result is relatively robust againstparameter choices; only having a significant dependence onthe star formation efficiency parameter (which is constrainedobservationally). We may attribute this to the fact that starformation is modeled as a function of the surface density of H in galaxies and within the low mass disc dominated sys- Figure 11.
The predicted SFR as function of the stellar mass at z = 0 for our three realisations compared to Elbaz et al. 2007. Ascan me seen, the star formation rates in all three models are sim-ilar, all showing good agreements in the low/intermediate masssystems but models overpredicting the SFR in the highest masssystems. The shaded region represents the 1 − σ observational un-certanty. Model error-bars represent poissonian uncertanties dueto the finite sample size of synthetic galaxies. tems is relatively inefficient (due to the low surface densitiesand relatively high star formation thresholds) resulting in awell defined conversion of cold gas to stars and hence, HI tostellar mass ratio. This conclusion also has important conse-quences when considering the overproduction of both stellarand HI masses in the lowest mass DM halos, since we ac-curately reproduce the self-regulation properties of galaxies(as the correct balance between star formation and feedbackis required to simultaneously reproduce the correct gas andstellar mass budgets in DM haloes). Overproduction in thelow mass regions within Fig.8 & Fig.9 therefore, must be at-tributed to the accretion of too much material, a conclusionwhich is confirmed by several other authors (see Mo et al.2005 & references therein).We would like to plot also the H mass counterpartsfor figures 8 & 9, however, due to the unconstrained massfunctions, as discussed in § A final useful diagnostic to be used to constrain galaxyformation models is the instantaneous star formation rate(SFR) which is found to vary significantly within galaxiesof different stellar mass. In Fig.11 we compare each modelrealisation with the observational SFR estimates as a func-tion of stellar mass by Elbaz et al. 2007, who used a largesample of SDSS galaxies with spectroscopic data in order toaccurately determine the SFR (see Brinchmann et al. 2004).We find little difference between the SFR predicted byeach model realisation as expected since each star formationrate prescription is constrained by z = 0 galaxy proper-ties. Moreover, we generally find that within the low mass, c (cid:13) , 000–000 M. Cook et al. disc-dominated region ( M s < M ⊙ ) we obtain a goodagreement between model and observation but we signifi-cantly over-predict the star formation rates in the largestgalaxies. We may attribute this to a lack of AGN quenchingwithin the disc component of the largest mass systems atlate times, which has been applied in several models as arather ad-hoc ’radio-mode’ feedback (see Croton et al. 2006,Bower et al. 2006). Since, within our models we do nothave any suppression of the growth of discs around largepre-formed bulges, aside from the late transition from thespheroid formation epoch to the disc formation epoch. Wehope to investigate the effects of energetic feedback froma formed SMBH-spheroid system on the late properties ofthe disc since we hypothesise that; despite having a negli-gible effect on the growth of a pre-formed SMBH, star for-mation is prevented within the spheroid component at latetimes since even arbitrarily low accretion rates onto the cen-tral SMBH results in energetic feedback capable of heatingthe ISM, however, within our framework, during the quies-cent disc growth phase no material is assumed to collapseonto the spheroid structure, even with a slight adjustmentto our model we may allow for some material to collapseat late times onto the spheroid-SMBH system, resulting inthe quenching of star formation in the larger systems andnaturally generating Seyfert-type galaxies. However, for sim-plicity we have neglected this effect within this work, andhope to investigate the physical mechanisms capable of gen-erating this self-consistently, within a subsequent work. As a main advance of this model over current SAMs, wedetail the properties of the ISM by modeling the formation ofmolecular clouds ( H regions) through pressure argumentswithin the disc component. This enables us to modify thestar formation law, and thus allows us to gain insights intothe more detailed gas-properties of normal galaxies underthe semi-analytical framework.Within this work we highlight the importance of mod-eling the ISM in two-phases advocating it as a simple, yetimportant advance over current frameworks; whereby theformation of H regions is determined by the planar pres-sure within the gaseous disc structure. This added ingredientis important for two main reasons: Firstly the decline in H regions within low mass systems results in a higher fractionof cold gas in the form of neutral HI which is thus detectablethrough conventional 21-cm line surveys (see Barnes et al.2001), whereas H mass estimates prove to be significantlymore difficult, relying on uncertain conversion factors be-tween H and CO-lines, therefore, assuming a single con-version between total cold gas and HI , as is commonlydone in SAMs provides inaccurate outputs. Secondly, thestar formation properties of galaxies have been shown torelate explicitly to the detailed properties of the internalstructure of the ISM (see Krumholz et al. 2009, Gnedin etal. 2009, Obreschkow & Rawlings, 2009), and therefore thisadded layer of complexity should now be embedded withincurrent SAMs.In Fig.12 we show the fraction of cold gas which is inthe form of H within the disc component as a functionof total stellar mass (disc and bulge masses). As can beseen, there is a tight relationship with little scatter in the Figure 12.
The molecular fraction of gas within galaxy disks.Showing that in low surface density galaxies, the formation ofmolecular clouds is suppressed, and thus star forming regionsare diminished, and within the high-mass systems, the spheroidcomponent dominates reducing the surface density in disks andtherefore the molecular gas fraction rapidly declines. We hope tocompare this result to observational studies as they become avail-able. Model error-bars represent poissonian uncertanties due tothe finite sample size of synthetic galaxies. lowest mass systems due to the overall dominance of disccomponents and the molecular fraction decreases steadilywith decreasing stellar mass, we find a peak in molecularfraction corresponding to approximately 80% H at M s ≈ M ⊙ where the disc mass reaches a maximum. Above thismass the spheroid component dominates and the disc surfacedensity thus drops, lowering the efficiency of molecular cloudformation rapidly. Prompted by several theoretical attempts to model the low-mass end of the stellar mass function (Somerville et al. 2008& references therein) and significant observational effort toconstrain the HI mass function using large surveys of galax-ies (Barnes et al. 2001, Zwaan et al. 2005), within this workwe have developed a physically motivated model in orderto explain the inefficiencies of low-mass DM halos in trap-ping baryonic material and forming stars. Motivated primar-ily by the importance of physical descriptions within smallscale systems which form the building blocks of larger sys-tems within a hierarchically clustered Universe, also becausetheoretical models have either neglected, or find significanttroubles in simultaneously matching both the stellar and HI mass functions within the low mass end (see Mo et al. 2005).Therefore, within this paper we have utilized well con-strained observations of the stellar (Bell et al. 2003a, 2003b)and HI (Zwaan et al. 2005) mass functions and employed anumerical technique (Shankar et al. 2006) in order to deriverelationships between galaxy properties and their host DMhalos. Assuming a one-to-one mapping of these systems, weare thus able to make detailed comparisons between modelsand observations.In order to interpret these phenomenological relation- c (cid:13) , 000–000 wo phase galaxy formation: The Gas Content of Normal Galaxies ships physically, we develop a cosmologically groundedgalaxy formation model outlined in C09 but with severalsignificant modifications. Under this framework we followthe development of baryonic material as it accumulates andevolves within growing DM halos and is subject to cooling,heating, possible angular momentum losses, star formationand recycling, and feedback through both supernovae eventsand the growth of a central supermassive black hole. Bycomparing three model realisations with varying sophisti-cation; one using a ’standard’ approach, whereby the cos-mological baryonic fraction f b = Ω b / Ω m is allowed to ac-crete within halos throughout their lifetimes and the starformation in discs (and thus the vast majority of low masssystems) is given by the Schmidt-Kennicutt (1998) star for-mation law, determined by the total cold gas mass, a ’UV’model, whereby the infall of baryonic material within lowmass DM halos is suppressed due to the presence of an ion-izing UV background (see Eqn.6, 7), and finally a ’sophisti-cated’ model with a modified infall due to UV radiation anda modified star formation law which requires a two-phaseISM, comprising of neutral HI and molecular H and typi-cally results in a lower star formation rate within low masssystems at early times.These realisations clearly demonstrate that using a’standard’ approach, the most simple case is not able to si-multaneously match both the stellar and HI mass functions,significantly over-producing low mass galaxies (see Figs.4, 5,6, 8 & 9), whereas the use of a ’sophisticated’ approach, weare able to match observations reasonably accurately to rel-atively low masses, finding discrepancies within the lowestmass systems. As an additional consequence of the modifiedstar formation law in the ’sophisticated’ model realisation,we naturally partition the cold gaseous ISM into HI and H components, allowing for predictions of both of these quan-tities without the need for further assumptions, finding thatwe are able to match both components simultaneously to agood accuracy over the entire mass range when comparingto the H results of (Keres et al. 2003, Obreschkow & Rawl-ings, 2009, Fig.7), and only over-producing the amount of HI mass in the lowest mass galaxies (see Zwaan et al. 2005& Fig.6).Finally, analysing several properties of galaxies againsttheir stellar masses, we find that, unlike the simple ap-proaches, the HI -to-stellar mass ratio is accurately repro-duced using the sophisticated treatment (see Fig.10), indi-cating that the self-regulation of star formation allows forthe correct conversion of material, is relatively insensitiveto the infall and supernovae feedback rates within physicallimits. Comparisons of the star formation rates within thesemodels however shows little difference at z = 0 as expected,and indicates a secondary problem with our simple physi-cal model, over-predicting the SFR in the largest systems(with M v > M ⊙ ), however, these systems are typicallyspheroid-dominated and are therefore relatively insensitiveto the details of the disc formation recipes.It is clear to assess the limitations and the manifesta-tions of this relatively simplistic approach to galaxy forma-tion modeling, under our improved framework we still re-quire large feedback efficiencies in order to sufficiently sup-press star formation within the lowest mass systems: As hasbeen studied by Mo et al. 2005, preheating through pre-virialised structure formation may further reduce the bary- onic infall onto the lowest mass halos, further reducing theneed for such high SN efficiencies, other environmental ef-fects such as tidal stripping and harassment may help tofurther improve the theoretical framework. However, despitethese further degrees of freedom, we also note that changesin the initial mass function towards something more top-heavy will naturally have a higher supernovae fraction perstellar population, this therefore further reduces the need forhighly efficient supernovae feedback within these models. Inorder to assess these additional effects we would need to ac-count for all the environmental effects associated with galaxyevolution, expanding our modeling from a single mass accre-tion history to a full merger-tree framework, however, thisadds a great deal more complication and uncertain physics(such as the evolution of satellite structures within DM ha-los, the merger rates of galaxies, the outcome of galaxy com-ponents in mergers of different ratios), this will be the sub-ject of a further analysis in a subsequent paper.A second, minor shortcoming of our modeling appearsto come from the complete separation of spheroid and discgrowth within the two epochs of DM halo growth, wherebythe disc and spheroid only share material through disc in-stabilities which are generally quite rare at late times dueto the stabilization generated by the pre-formed spheroid.Despite showing that our quasi-monolithic scenario for thegrowth of galaxies to show promising results, we also hopeto include explicitly the effects of merger events and envi-ronmental effects in a future work.In conclusion, focusing mainly on the low mass galaxypopulation, adopting several theoretical improvements over’standard’ SAMs, we are able to simultaneously match boththe stellar, HI and H mass budgets within DM halos,and the star formation properties of galaxies within the ob-servational ranges. This promising result indicates that atpresent, the ’standard’ approach to modeling the low massevolution of galaxies is somewhat over-simplified within cur-rent SAMs which only have a single-phase ISM, and, due tothe hierarchical nature of structure formation, may manifestas significant tensions in progressively larger systems withina full merger-driven framework (see Somerville et al. 2008for a discussion). We therefore advocate the use of moresophisticated treatments of the interstellar medium withincurrent and future SAMs.Assessing the limitations of our framework, we concludethat further suppression of infall onto the lowest mass sys-tems would allow for a further reduction in the need forstrong supernovae feedback and should further ease ten-sions between models and observations, this could only comethrough environmental effects such as tidal shocks or grav-itational pre-heating (Mo et al. 2005), however this effecthas not been studied in detail through hydrodynamic sim-ulations and remains to be fully investigated. By adding achannel whereby even small amounts of gaseous materialmay be transferred to the spheroid component during latetimes, small amounts of ’radio mode’ AGN activity may betriggered, little affecting the spheroid or the SMBH masses,but significantly lowering the SFR in the discs, preferentiallyat large masses, naturally resulting in Seyfert-type activegalaxies and reducing the SFR in these large discs, hopefullybringing Fig.11 into better agreement with observations, wealso hope to investigate the pan-redshift galaxy populationunder this framework (Cook et al., 2009b submitted). c (cid:13) , 000–000 M. Cook et al.
Interestingly however, within this relatively simplisticframework we are able to self-consistently reproduce severalof the key observations, it is therefore clear that mergers,to some degree, are not the dominant driver for the globalevolution of the galaxy population. It will therefore providea useful exercise to mount our physical prescriptions onto afull merger-tree DM background which should allow us tomodel environmental effects consistently. The main resultsfrom this paper indicate however, that using a relatively sim-ple framework, we find an reasonable agreement to the stel-lar, HI and H mass functions of galaxies arising naturally,thus, we advocate all current SAMs to begin to incorporatetwo-phase ISM physics into their frameworks. ACKNOWLEDGMENTS
We thank P. Salucci for providing the initial seeds for thiswork, and also to F. Shankar and A. Schurer for stimu-lating discussions which helped the progress of this work.MC thanks L. Paulatto for considerable computational as-sistance, and we thank A. Ferrara for careful reading of themanuscript. MC has been supported through a Marie Curiestudentship for the Sixth Framework Research and Train-ing Network MAGPOP, contract number MRTN-CT-2004-503929. E.B. acknowledges support from NSF Grant No.PHY-0603762.
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