Size Evolution of Spheroids in a Hierarchical Universe
Francesco Shankar, Federico Marulli, Mariangela Bernardi, Simona Mei, Alan Meert, Vinu Vikram
aa r X i v : . [ a s t r o - ph . C O ] S e p Mon. Not. R. Astron. Soc. , 1– 21 (2012) Printed 4 December 2018 (MN L A TEX style file v2.2)
Size Evolution of Spheroids in a Hierarchical Universe
Francesco Shankar ⋆ , Federico Marulli , Mariangela Bernardi , Simona Mei ,Alan Meert and Vinu Vikram Max-Planck-Instit¨ut f¨ur Astrophysik, Karl-Schwarzschild-Str. 1, D-85748, Garching, Germany Dipartimento di Astronomia, Universit´a degli Studi di Bologna, via Ranzani 1, I-40127 Bologna, Italy Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd St, Philadelphia, PA 19104 GEPI, Observatoire de Paris, CNRS, Univ. Paris Diderot; Place Jules Janssen, 92190 Meudon, France
ABSTRACT
Unveiling the structural evolution of spheroids, and in particular the origin of thetight size-stellar mass relation, has become one of the hottest topics in cosmology inthe last years and it is still largely debated. To this purpose, we present and discussbasic predictions of an updated version of the latest release of the Munich semi-analytic hierarchical galaxy formation model that grows bulges via mergers and discinstabilities. We find that while spheroids below a characteristic mass M s ∼ M ⊙ grow their sizes via a mixture of disc instability and mergers, galaxies above it mainlyevolve via dry mergers. Including gas dissipation in major mergers, efficiently shrinksgalaxies, especially those with final mass M s . M ⊙ that are the most gas-rich,improving the match with different observables. We find that the predicted scatterin sizes at fixed stellar mass is still larger than the observed one by up to . z = 2 progenitors of massive galaxies with M star ∼ (1 − × M ⊙ and B/T > . z = 0, are found to be mostly disc-dominatedgalaxies with a median B/T ∼ .
3, with only ∼
20% remaining bulge-dominated. Themodel also predicts that central spheroids living in more massive haloes tend to havelarger sizes at fixed stellar mass. Including host halo mass dependence in computingvelocity dispersions, allows the model to properly reproduce the correlations withstellar mass. We also discuss the fundamental plane, the correlations with galaxy age,the structural properties of pseudobulges, and the correlations with central black holes.
Key words: galaxies: structure – galaxies: formation – galaxies: evolution – cosmol-ogy: theory
One of the most important and still debated problems inCosmology is the formation and evolution of galaxies. To-day we see galaxies having a variety of morphologies, rang-ing from less massive, pure stellar discs, to intermediatemass bulge plus disc galaxies, to more massive, spheroidalsystems. The origin of this transition is still for many re-spects unclear. More specifically, while angular momentumconservation may explain many properties of discs (e.g.,Governato et al. 2007), the origin of bulges is still largelyunsolved and debated. Why do some galaxies show bulges ⋆ E-mail: [email protected] while others don’t? Or, in other words, what is the origin ofthe gradual conversion from discs to spheroids?It is clear that if we want to understand galaxy forma-tion we need to observe the high-redshift Universe. Deepobservations in the last decades or so have however un-veiled a full complex zoology of high-redshift (proto)galaxiesthat makes even more puzzling - but also more excit-ing - assessing the actual routes chosen by nature tobuild the galaxy populations we observe today. Alongwith starforming discs and dust-enshrouded galaxies (e.g.,Blain et al. 2002; Magdis et al. 2011), deep optical andnear-infrared surveys have in fact discovered the pres-ence of numerous extremely compact and passively evolv-ing galaxies up to z & c (cid:13) F. Shankar et al. − M ⊙ , with a factor of ∼ . − − z >
5, while larger galax-ies at fixed stellar mass are generally younger. Along sim-ilar lines and extending the analysis to other high-redshiftgalaxy populations, Mosleh et al. (2011) concluded that thestructure of galaxies is somewhat correlated to their activity,i.e., the sizes of galaxies at a given stellar mass is somewhatcorrelated to its star formation rate level, similarly to whatis observed in the local Universe. At lower redshifts it hasbeen shown that the size-age relation at fixed stellar mass issimilarly shaped for lenticulars (Shankar & Bernardi 2009;van der Wel et al. 2009), i.e., older systems are more com-pact, but becomes rather flat for bulge-dominated galaxies(Shankar et al. 2010b; Bernardi et al. 2010; Trujillo et al.2011). Thus whatever process formed massive ellipticals, itmust have been fine-tuned to bring all young and old high-redshift massive spheroids on the same local size-mass rela-tion.Understanding the evolutionary link these compact andlarge high-redshift galaxies might have with the variety ofstarforming galaxies at similar redshifts and stellar mass, ifany, and with the local early-type galaxy population remainsan open debate (e.g., Cole et al. 2000; Benson et al. 2003;Granato et al. 2004; Menci et al. 2004; Baugh et al. 2005;Bower et al. 2006; Granato et al. 2006; De Lucia et al. 2006;Menci et al. 2006; De Lucia & Blaizot 2007; Monaco et al.2007; Fan et al. 2008; Somerville et al. 2008; Dekel et al.2009b; Khochfar & Silk 2009; Neistein & Weinmann 2010;Bournaud et al. 2011b; Gonz´alez et al. 2011).According to the standard cosmological paradigm ofstructure formation and evolution, dark matter haloes havegrown hierarchically, through the continuous merging ofsmaller units into larger systems. In this scenario, galax-ies form inside this hierarchically growing system of haloes(e.g., Cole et al. 2000; De Lucia et al. 2006). However, theactual role played by mergers (major, minor, wet, and dry),in the structural evolution of massive spheroids is still un- certain (e.g., Hopkins et al. 2010, and references therein).Some models of galaxy formation (e.g., Eggen et al. 1962;Merlin et al. 2012) envisage that most of the mass in lo-cal massive spheroids was formed and assembled in a strongand rapid burst of star formation at high redshifts, and theremnants evolved almost passively thereafter, without beingstrongly affected by late merging events.Galaxy formation models built on top of large N-bodydark matter numerical simulations or analytic merger trees,claim instead that although the stars of the most massivespheroids are the oldest being formed at very high redshifts,they have assembled a large fraction of their final stellarmass only at relatively late times via a sequence of minor andmajor merger events (e.g., Baugh et al. 2005; De Lucia et al.2006; De Lucia & Blaizot 2007; Khochfar & Silk 2006a;Gonz´alez et al. 2011). It has long been known that binarymergers between discs can indeed produce spheroidal galax-ies and also explain many of their structural properties (e.g.,Barnes 1992; Hernquist 1992; Robertson et al. 2006), thoughseveral issues remain to be solved in this basic scenario (e.g.,Naab & Ostriker 2009). In high-redshift and gas-rich discgalaxy mergers, however, gas dissipation inevitably formscompact spheroids (e.g., Naab et al. 2006; Robertson et al.2006; Hopkins et al. 2009). Hierarchical models then natu-rally explain the evolution of the size (and mass) of mas-sive compact spheroids as a sequence of “dry” (gas-poor)and mainly minor mergers that puff up the outskirts ofthe galaxy leaving the central regions of the galaxy al-most intact (e.g., Naab et al. 2007, 2009; Ciotti 2009). Nu-merical simulations have however shed doubts on the co-herence with which mergers can bring galaxies along thetight structural relations observed in the local Universe (e.g.,Ciotti & van Albada 2001; Nipoti et al. 2009).Another class of models explains size evolution ofearly-type galaxies via a quasi-adiabatic expansion phaseconsequent to the blow-out of substantial amounts ofmass via quasar and/or stellar feedback (Fan et al. 2008;Damjanov et al. 2009; Fan et al. 2010). Initial numerical ex-periments to test the latter proposal as a viable expla-nation to size evolution have been recently performed byRagone-Figueroa & Granato (2011).Bulges could also be formed via in-situ processes thatare broadly classified as disc instabilities. In unstable self-gravitating discs, the instability may drive the formationof a bar with mass transferred from the disc into a cen-tral bulge (e.g., Cole et al. 2000). The degree of mass trans-ferred to the bulge varies from one model to the other.Some models consider the instability quite a violent processcapable of transferring most of the disc into a bulge andalso induce a starburst (Bower et al. 2006). Besides bars,other types of instabilities could contribute to the forma-tion of bulges. Observations from the deep SINS survey of z ∼ c (cid:13) , 1– 21 ize Evolution mandatory. The aim of this work is to explore the full pre-dictions of a state-of-the-art semi-analytic model (SAM) ofgalaxy formation that evolves massive spheroids in a hierar-chical fashion. More specifically, we will present and discussbasic predictions on the size-mass relation and its evolutionwith redshift of relatively massive spheroids. We will mainlyfocus on their size evolution at fixed stellar mass and in dif-ferent environments, the structural properties of their pro-genitors, but also touch upon several other related issues.Our objective is not to prove that the model discussed hereis the correct one, but rather to lay out the successes andfailures of a detailed hierarchical model against the wealthof data now becoming available from large and deep surveys.We will also stress that several outcomes of the model con-sidered here are shared by many other hierarchical SAMs,making most of our conclusions of particular interest to thefield of galaxy formation, but also discuss key differences.The paper is organized as follows. In Section 2 we in-troduce the hierarchical model considered for this work, fo-cussing on the features relevant to this work. In the samesection we present a large sample of well studied early-typegalaxies taken as term of comparison for model outputs. Sec-tion 3 contains a discussion of key issues such as the size(and velocity dispersion)-stellar mass relation, its scatterand evolution with redshift, and the role of environment. Wediscuss in Section 4 the evolutionary features of spheroids,the analogies and differences with other SAMs, and waysto further constrain galaxy formation models. We then con-clude in Section 5. In the Appendices we will also brieflydiscuss a number of related topics, such as the distinctionbetween classical and “pseudobulges”, the connection withblack holes and with galaxy ages, the fundamental plane andits evolution with time. All of the results presented in this work are the outcomeof running the original numerical source code by Guo et al.(2011), i.e., the latest rendition of the semi-analytic model(SAM) developed at the Max Planck Institute for Astro-physics. As detailed below, with respect to the originalGuo et al. (2011) model, we have modified the computationof bulge radii exploring a variety of possibilities, and addedthe calculation of the coupled velocity dispersions. Note thatrunning the code and producing a new galaxy catalog eachtime, is different from studying the online catalogs as it al-lows a self-consistent thorough study of the structural evolu-tion of galaxies in the SAM. We stress that the modificationsapplied to the original Guo et al. (2011) model do not affectany other galaxy property except for sizes. Thus the model(we checked) maintains the same exact performance withrespect to the observables (e.g., the stellar mass function)as presented in Guo et al. (2011). Before discussing bulgesizes in detail, we first provide below a brief overview of themodel.The Munich SAM aims at providing a comprehensivepicture of the evolution of galaxies and their central super-massive black holes within the hierarchical structure andmerging of dark matter haloes and subhaloes within the concordance ΛCDM cosmology. To this purpose, it is imple-mented on top of the large, high-resolution cosmological N-body
MILLENNIUM I (Springel et al. 2005) and
MILLENNIUMII (Boylan-Kolchin et al. 2009) simulations. Given that inthis paper we are mainly interested on the structural proper-ties of the most massive galaxies in the local Universe, all ofthe results presented here have been obtained by running thecode on the significantly larger
MILLENNIUM I simulation.The latter simulation follows the evolution of N = 2160 dark matter particles of mass 8 . × h − M ⊙ , within acomoving box of size 500 h − Mpc on a side, from z = 127to the present, with cosmological parameters Ω m = 0 . b = 0 . h = 0 .
73, Ω Λ = 0 . n = 1, and σ = 0 . minor merger ( M /M < . major mergers( M /M > .
3) instead disrupt any stellar disc present andproduce a spheroidal remnant, which contains all the oldstars present in the progenitor galaxies and all the new starsformed out of the burst triggered by the merger.Bulges can be formed even via secular evolution in themodel. The type of disc instabilities considered in this modelare secular processes that transfer only the portion of thestellar mass necessary to keep the disc marginally stable(see details in Section below, and in Guo et al. 2011). Thisway of modelling disc instabilities is different from, e.g.,Bower et al. (2006) that instead assume the entire mass ofthe disc is transferred to the bulge during the instability,with any gas present assumed to undergo a starburst. Morein general, the present model lacks at the moment any bulgeformation via strong gas rich disc instabilities and/or clumpaccretion under dynamical friction.Guo et al. (2011) have shown that their model is capa-ble of reproducing the size distribution of local discs rea-sonably well, and additional comparisons can be found in,e.g., Fu et al. (2010); Kauffmann et al. (2012). In this pa-per we will mainly focus on the predicted structural proper-ties of massive spheroids and their evolution with redshift,and only briefly touch, where relevant, on the evolution ofdiscs that will be addressed elsewhere. In particular, our c (cid:13) , 1– 21 F. Shankar et al.
Figure 1. left panel : Predicted median 3D half-mass radius versus stellar mass for different cuts in
B/T , as labelled.
Right panel :median 2D projected half-light effective radius R e as a function of stellar mass for a subsample of SDSS early-type galaxy sample. Itis clear that the model is at variance with the data, predicting much flatter size-mass relations below a “characteristic mass scale” of M star ∼ M ⊙ , especially for higher B/T galaxies. This characteristic mass is completely absent in the data. primary interest in this work are bulge-dominated galax-ies with bulge-to-total ratio
B/T > .
7, a threshold cho-sen because, as discussed below, the contribution of pseu-dobulges (Appendix B) in this regime is negligible. This inturn allows us to properly discern the actual role played bymergers in building their structural properties we observetoday. Where necessary, we will also devote some attentionto bulges grown via disc instabilities, though we refer thereader to Appendix B for some more specific discussion, andto the separate work by Shankar et al. (2012) for additionaland complementary analysis of this issue.
The bulge 3D half-mass radius R H of a merger remnant iscomputed from energy conservation. Following Cole et al.(2000), the model assumes that when two virialized galacticsystems merge, their R H is given by E fin = E int , + E int , + E orb . (1)The left-hand side of Eq. (1) is the self-binding energyof the remnant, defined as E fin = − G ( M + M ) R H , (2)with R H the half-mass radius of the remnant.The terms E int , , are the self-binding energies of themerging progenitors, and are usually expressed as E int , i = − GM i R i , (3)with M i the total stellar (including any stars formed duringthe merger) plus cold gas masses.The orbital energy E orb is usually expressed in terms ofthe internal energy of the system at the radius of minimalseparation E orb = − f orb c GM M R + R , (4)where we initially set f orb = 1 and c = 0 . d could also be computed in the center of mass system of ref-erence as E orb = µV / − GM M /d , in terms of the re-duced mass µ = M M / ( M + M ), and relative velocity V = k ~V − ~V k . However, while this expression might havethe advantage of not being directly dependent on f orb , itis still model-dependent. In fact, satellite galaxies strippedaway of their surrounding subhalo are assigned a surviv-ing merging timescale proportional to the actual Chan-drasekhar dynamical friction timescale via a fudge factor(see De Lucia et al. 2010). Thus relative velocities and dis-tances would still need to be modelled according to thistimescale (see also Neistein et al. 2011).Thus, given the inevitable inclusion of some parame-ters in the modelling of bulge sizes, we decided for this workto stick with Eq. (1) that requires only one truly free extraparameter, f orb /c , and at the same time allows a closer com-parison with previous semi-analytic and numerical works inthe Literature.As anticipated in the previous Section, another route toform bulges in the model is via secular evolution (Guo et al.2011). The adopted criterion for instability is expressed as V max < p GM disc / R disc , with V max the maximum circularvelocity of the host subhalo, and M disc and R disc the massand exponential scalelength of the disc, respectively. In theevent of instability, a fraction δM star of the disc stellar massis transferred to a central bulge to restore equilibrium, andthe size R b of the newly formed bulge is computed assum-ing an exponential profile. If a bulge is already present, thesize of the bulge is computed via a “merger-type” relationbetween the old and new bulge stellar mass as the one inEq. (1), with M and R the mass and half-mass radius ofthe pre-existing bulge, and M and R equal to δM star and R b , respectively, and f orb = 2 to take into account that theinteraction in concentric shells is stronger than in a merger(see Guo et al. 2011 for further details).After each merger or disc-instability event, we also com-pute the velocity dispersion associated to each galaxy follow-ing the analytical fit given by Covington et al. (2011) c (cid:13) , 1– 21 ize Evolution σ = k GM star R H (cid:18) − M H ( < R H ) M H ( < R H ) + f ( M + M ) (cid:19) − , (5)where we set k = 0 .
15 and f = 0 .
1. Here M H ( < R H ) isthe fraction of the subhalo mass associated to the remnantwithin the final half-mass radius. Throughout the paper we will compare model predictionswith a sample of ∼ ,
000 galaxies from the Sloan DigitalSky Survey (SDSS). This is a random subset of the well de-fined and complete sample defined in Bernardi et al. (2010).This sample ranges between 3 × M ⊙ to ∼ M ⊙ , themass range of interest here. In addition to the photometricparameters (e.g., cmodel magnitudes and sizes) presentedin Bernardi et al. (2010), this subset of ∼ ,
000 galaxiesalso provides bulge-to-total
B/T light from a de Vaucouleurs(e.g., de Vaucouleurs 1948) plus exponential decomposition(see Meert et al. and Vikram et al. in preparation), whichwe will extensively use in this work to select the most bulgedominated galaxies to compare with the model. The sam-ple is characterized by stellar masses with a Chabrier InitialMass Function (Chabrier 2003), consistent with the one usedin the model.
In Figure 1 we show a first comparison between model pre-dictions and data from SDSS. The left panel shows the me-dian 3D half-mass radius R H versus stellar mass for dif-ferent cuts in B/T , as labelled. From here onwards, unlessotherwise stated, we will compute R H as the mass-weightedaverage of the half-mass radiuses of the bulge and the disccomponents. We have checked that, especially for the bulge-dominated systems of interest here, this is equivalent (atthe percent level) to computing a full mass profile assuming,e.g., an Hernquist (Hernquist 1990) plus an exponential pro-file for the bulge and the disc, respectively. Moreover, simplyneglecting the disc component in systems with B/T & . R e as a function of stellar mass for the same cuts in B/T . Themodel predicts an increasing size with stellar mass, how-ever, it is evident that below a “characteristic mass scale”of M star ∼ M ⊙ the predicted sizes flatten out, at vari-ance with the data that continue to show a steep declinedown to much lower masses .It is quite unlikely that such a strong discrepancy can be We note that the behaviour of increasing half-mass radius with
B/T at fixed mass is induced by the fact that the model predictslarger bulge and disc sizes with increasing
B/T . We verified thatjust the opposite is true for disc-dominated galaxies with
B/T . . − .
5, in agreement with observations. The apparent flattening of the measured size-mass relation atvery low masses M star . M ⊙ is most probably induced bycontamination of later-type galaxies, and thus not relevant for thepresent discussion (see details in Bernardi et al. 2011b). simply explained by some mass/luminosity-dependent con-version factor between R H and R e , as we will also discuss inmore detail later. We checked that the predicted R H - M star relation shows a very similar flat behaviour at low massesalready at z ∼
2. The latter implies that the wrong shape ofthe size-mass relation yielded by the model is a consequenceof some wrong “initial conditions” and not necessarily linkedwith any later galaxy assembly.
If the merger is gas-rich, a significant fraction of the initialenergy E diss of the system will be dissipated away, induc-ing a more compact remnant. Several groups have studiedthis issue in some detail using high-resolution hydrodynamicsimulations and semi-numerical models (e.g., Naab et al.2006; Ciotti et al. 2007; Hopkins et al. 2009, and referencestherein).Some of these groups have provided basic analytic for-mulations that can be included in SAMs to study the impactof dissipation on cosmological structure formation of galax-ies. One method is based on the conservation of energy, asproposed by Covington et al. (2008) E fin = E int , + E int , + E orb + E diss , (6)with the dissipation energy parameterized byCovington et al. (2011) in terms of the final energy ofthe remnant as E diss = 2 . f gas E fin , (7)with f gas the ratio between the total mass of cold gas andthe total cold plus stellar mass (inclusive of the mass formedduring the burst) of the progenitors.Hopkins et al. (2009) parameterize the decrease in sizedue to dissipation by the simple relation R H [ final ] = R H [ dissipationless ]1 + f gas /f , (8)with f = 0 .
25, and R H [ dissipationless ] computed fromEq. (1).It is clear that Eqs. (6) and (8) can significantly reducethe sizes predicted in a dissipationless merger. We assumethat Eqs. (6) and (8) only hold in major mergers, whenthe gas actually gets into the bulge (see Section 2.2). Thedecrease is proportional to the gas fractions in the progen-itors. As shown in the right panel of Figure 2, the modelpredicts increasing gas fractions at lower stellar masses and,at fixed mass, increasing with redshift, in broad agree-ment with observations (e.g., Kannappan 2004; Erb et al.2006; Catinella et al. 2010; Peeples & Shankar 2010, andreferences therein). The results shown here refer to galax-ies with B/T > . B/T . Low-mass galaxies caneasily have most of their baryonic mass still in gaseousform at z &
2. However, at any epoch, galaxies above M c & × M ⊙ tend to have progressively lower gas frac-tions down to . − M c is an interestingmass scale several times reported in the Literature to be in-dicative of some basic physical process in galaxy evolution(feedback from Active Galactic Nuclei?) as several spectraland structural properties change when galaxies transition c (cid:13) , 1– 21 F. Shankar et al.
Figure 2.
Left panel : The dotted , long-dashed , and solid lines show the predicted median size-mass relation for galaxies with B/T > . vertical dotted line marks the transition above which (dry) mergers are believed to dominate galaxyassembly (see text). Right panel : median cold gas fractions predicted in the model as a function of stellar mass at redshifts z = 0, z = 1,and z = 2, as labelled. above it (e.g., Kauffmann et al. 2004; Shankar et al. 2006;Khochfar & Silk 2009; Bernardi et al. 2011a,b).This mass- and time-dependent behaviour of f gas canthen easily explain the decrease in size shown in the leftpanel of Figure 2. The dotted, long-dashed, and solid linesshow the predicted median size-mass relation for galaxieswith B/T > . M star =10 M ⊙ marks the transition above which dissipation doesnot play a significant role in shaping the sizes of galaxiesbecause mergers become progressively gas-poorer. It is re-markable that dissipation tends to erase the flattening belowthe characteristic mass producing a nearly single power-lawin agreement with the data. In the following we will useEq. 8 as our reference model with gas dissipation, thoughcomparable results are obtained when switching to Eq. 6. We now attempt a closer comparison between model predic-tions and data measurements by converting 3D half-massradiuses R H into 2D projected half-light radiuses R e . As-suming that light traces mass we convert R H to R e using thetabulated factors from Prugniel & Simien (1997). The lattercomputed for a system of total mass M , the scaling factors S ( n ), dependent on the S´ersic index n (S´ersic 1963), con-necting the gravitational energy W to their effective radius,i.e., | W | = S ( n ) GM /R e = GM /R g , with G the gravita-tional constant and R g the gravitational radius. Assumingthe systems are virialized we can approximate R g ≈ R H ,thus having R e ≈ S ( n ) R H . (9) Figure 3.
Effective radius R e as a function of stellar massat z = 0 from the SDSS sample of early-type galaxies with B/T > .
7. The contours mark the region of plane containing68%, 95%, and 99.7% fraction of the total sample. For complete-ness, the solid squares represent the median size-mass relation forthe early-type galaxy sample discussed by Bernardi et al. (2011b).The dotted , long-dashed , solid , and triple dot-dashed lines repre-sent, respectively, the predicted size-mass relations without dissi-pation, with dissipation, with dissipation plus f orb = 0, and withdissipation plus a fraction of dark matter in the merger (see textfor details). By setting n = 4 in Eq. (9) (i.e., S (4) = 0 .
34 from Table 4 ofPrugniel & Simien 1997), we can convert the predicted 3Dhalf-mass radiuses into 2D projected half-light radiuses .Figure 3 shows the z = 0 SDSS effective radius R e (assuming de Vaucouleurs plus exponential profiles) as a We have checked that our conclusions do not significantly de-pend on the exact profile chosen for the bulges. For example,we find broadly similar results, although somewhat steeper cor-relations at the highest stellar masses, when assigning to eachspheroid a S´ersic index according to their luminosity following,e.g., the empirical relation by Terzi´c & Graham (2005), and thenconverting from R H to R e using the appropriate S ( n ).c (cid:13) , 1– 21 ize Evolution Figure 4.
Relative size distribution of galaxies of a given mass and minimum
B/T as labelled. The dotted and long-dashed , and solid lines are, respectively, for models with no dissipation, with dissipation including and excluding pseudobulges. More specifically, the solid lines refer to models with dissipation but include only the subsample of galaxies grown mainly through mergers. Each distribution isnormalized in a way that the sum of the galaxy fractions in each bin of size equals unity. function of stellar mass for galaxies with
B/T > .
7, withthe contours marking the regions containing 68%, 95%, and99.7% of the total sample (the results discussed below donot depend on the exact choice of
B/T threshold). For com-pleteness, we report in the same Figure with solid squaresthe median size-mass relation for the early-type galaxy sam-ple discussed by Bernardi et al. (2011b).The lines in Figure 3 show the predicted median size-mass relations for galaxies selected to have the same mini-mum
B/T threshold as in the data. The dotted line refers toa model with no dissipation. As discussed above, the lattermodel better lines up with the data for galaxies above thecharacteristic mass, but below it the relation inevitably flat-tens out to larger sizes. The long-dashed line is the size-massrelation for a model that includes dissipation in major merg-ers. As expected, the sizes get progressively smaller towardslower mass spheroids that have formed out of gas-richer pro-genitors. However, spheroids of all masses get shrunk. Thus,including dissipation not only steepens the size-mass rela-tion, but it also lowers its overall normalization.In order to improve the fit to the data with a modelwith dissipation we therefore need to increase the normal-ization of the predicted sizes at fixed stellar mass. Fol-lowing Eq. (2), we have that the size of the remnant is R H ∝ ( M + M ) / ( E + E + E orb + E diss ), so in orderto increase the size at each merger event, we need to eitherdecrease the denominator, and/or increase the numerator.The solid line in Figure 3 is the predicted size-mass rela-tion assuming that most of the merger events happen onparabolic orbits with null orbital energy, i.e., f orb = 0, a farfrom uncommon condition in numerical simulations. Actu-ally, Khochfar & Burkert (2006) studied the orbital param-eters of major mergers of cold dark matter halos using ahigh-resolution cosmological simulation finding that almosthalf of all encounters are nearly parabolic. This simple vari-ation to the basic model significantly improves the match tothe data.We can however also increase the sizes by assuming thatthe total mass actually taking part in the merger is the sum of the baryonic plus a fraction of the dark matter host halomass, i.e., M i = M star , i + M cold , i + α × M H ( < R i ) (10)where R i is the half-mass radius of the progenitor, and α a constant parameterizing the still uncertain effect of adia-batic contraction. For each progenitor we take its mass atinfall and compute the fraction within the half-mass radiusassuming a Navarro et al. (1997) profile, and assigning aconcentration following the mean relation by Bullock et al.(2001). The result is shown with a triple dot-dashed linein Figure 3, where we set α = 1. The predicted sizes arelarger, as expected, though the inclusion of a constant frac-tion of dark matter also produces somewhat larger sizes atlow masses than actually observed (a similar behaviour wasdiscussed by Gonz´alez et al. 2009).We conclude that dissipation inevitably shrinks galax-ies and thus some additional ingredient must be includedin the model to reestablish the normalization of the size-mass relation. In the following we will use the model with f orb = 0 and α = 0 as the reference one, unless otherwisestated. We note, however, that including some amount ofdark matter participating in the merger could still representa viable model if we somehow tune α to properly increasewith halo/stellar mass . So far we discussed the median shape of the size-mass re-lation. We now turn to the discussion of the dispersion insizes at fixed stellar mass. Figure 4 shows the relative sizedistribution of galaxies in the mass range 10 < log M star < . B/T as labelled, with each distribution P ( R e | M star ) normalized in a way that the sum of the galaxyfractions in each size bin equals unity. We stress here that a model with α = 0 in the merger cannot re-produce the tilt of the fundamental plane discussed in Section C,that requires a halo mass-dependent velocity dispersion.c (cid:13) , 1– 21 F. Shankar et al.
Figure 5.
Predicted logarithmic 1 − σ scatter in sizes at fixedstellar at different redshifts, as labelled. Red and cyan lines referto model predictions with and without dissipation, respectively.For comparison, also shown the scatter measured in our SDSSsubsample ( thick , dot-dashed line), by Shen et al. (2003, filledsquares ), and by Nair et al. (2011, yellow stripe ) for early-typegalaxies. The dotted lines show the P ( R e | M star ) distributionscompeting to the no-dissipation model. While galaxies with B/T > . P ( R e | M star )distribution, galaxies with lower B/T > . P ( R e | M star ) distribution, proving that the latter is not pro-duced by mergers but rather by disc instabilities. We findthat most of the remaining galaxies after the filtering have B/T & . & × M ⊙ we always findsingle-Gaussian distributions irrespective of the chosen B/T threshold. We thus conclude that galaxies of any mass with
B/T > . M s tend on average to havea significant population of pseudobulges with B/T < . B/T & . B/T yield similarresults, as long as only classical bulges are considered). Wehomogeneously compute the scatter in the data and in themodel by binning galaxies in stellar mass, build the distri-bution in sizes and compute the 68% percentile as repre-sentative of its 1 − σ dispersion. Predictions for the modelswithout and with dissipation are shown with cyan and redlines, respectively, at redshifts z = 0, z = 1, and z = 2, aslabelled. The thick dot-dashed line is the scatter derived for ourSDSS galaxy subsample with the same cut in B/T . We findit to be at the constant level of ∼ . / ln(10) ∼ .
22 dexbelow M c , steeply decreasing to ∼ . M star & × M ⊙ , the scale M c above which galaxiesbecome progressively gas-poorer and their structural evolu-tion becomes controlled by dry mergers. The disagreementbetween model predictions and data is mass dependent, andcontained to be . B/T thesignificant contamination from pseudobulges tends to fur-ther increase the disagreement with observations. We furthernote that the level of predicted scatter is only marginallydependent on dissipation. In fact, galaxies with stellar mass M star & × M ⊙ have comparable levels of scatter inthe two models, as expected given that the role of (dry)mergers in the size evolution becomes progressively moreimportant at higher masses (see Section 4.1). The latterfeature is in broad agreement with some previous studies,though the role of dissipation in determining the final scatterof sizes was more emphasized (e.g., Khochfar & Silk 2006b;Covington et al. 2011). Also, the scatter does not stronglydepend on redshift, especially for more massive galaxies,again those with M star & × M ⊙ . As discussed in Section 1 massive and passive spheroids athigh redshifts appear more compact with respect to theirlocal counterparts in SDSS. This section is dedicated to un-derstand the degree of size evolution at fixed stellar masspredicted by the model and discuss it in light of the avail-able data.Figure 7 shows the predicted size-mass relation fromour reference model, in terms of the projected radius R e , atdifferent redshifts, as labelled, and compared to SDSS localvalues. Here at all redshifts we select only bulge-dominatedgalaxies with B/T > .
7. We find a progressive decreaseof sizes at all masses. Galaxies with stellar masses above c (cid:13) , 1– 21 ize Evolution Figure 6.
Predicted median size-mass relation for bulge-dominated galaxies with
B/T > .
7, in terms of the projectedradius R e , at different redshifts, as labelled, and compared toSDSS local data (coloured regions). There is a progressive de-crease in sizes of up to a factor of ∼ z = 3, similar at allstellar masses. M star & M c = 3 × M ⊙ seem to experience a similar de-gree of evolution, i.e., a progressive average decrease in sizeup to . z .
3. This type of nearly mass-independentevolution implies that the slope of the size-mass relation ispredicted to be almost constant at all redshifts, at least forbulge-dominated systems with
B/T > . M star & M c .Figure 7 presents a more specific study of the redshiftevolution in the median half-mass radius R H , normalized tothe median local value, for two different intervals of bulge-to-total ratios, i.e., B/T < . B/T > .
7, limits chosento select statistically significant disc- and bulge-dominatedgalaxy samples, respectively. Each point in the Figure repre-sents normalized median values with their associated errorbars. The upper panel shows the evolution competing tolower mass galaxies with 10 < M star /M ⊙ < × , themiddle panel for galaxies with 3 × < M star /M ⊙ < ,while the lower panel refers to higher mass galaxies with10 < M star /M ⊙ < × . Globally, for all galaxies, evenfor the bulge-dominated ones (solid lines), we do not findstrong dependence of size evolution on gas dissipation (redand cyan lines refer to model outputs with and without dis-sipation, respectively). We have also checked that the degreeof evolution in bulge sizes does not significantly depend onthe amount of orbital energy included in the model (Eq.1).This is not unexpected given that the degree of bulge sizeevolution in hierarchical models is mainly governed by thenumber and type of mergers (Section 4.1). In other words,dissipation mainly acts in deciding how compact spheroidsappear after the initial gas-rich major merger event, leavingthe degree of evolution, controlled by other processes, notsignificantly perturbed.From Figure 7 it is evident that the model, as antici-pated in Section 2.1, can manage to reproduce the moder-ate redshift evolution in the half-mass stellar radius of disk-dominated galaxies (long-dashed lines), which are observedto decrease in size by a factor of ∼ z = 2 (see,e.g., Somerville et al. 2008, and references therein). On av-erage, however, we do not find the empirical trend for which Figure 7.
Predicted median redshift evolution (normalized to themedian value at z = 0) for the total half-mass radius of galaxies ofdifferent stellar mass as labelled and B/T as labelled. The upper , middle and lower panels refer to, respectively, the size evolutionof low, intermediate, and high stellar mass galaxies divided intotwo intervals of B/T , as labelled. For bulge-dominated galaxies,the redshift evolution does not depend much on stellar mass or thedegree of gas dissipation. Massive galaxies tend to evolve slowerthan what the data suggest ( black , solid and long-dashed lines). bulge-dominated galaxies tend to grow faster than disc-dominated ones (e.g., Somerville et al. 2008; Ryan et al.2012; Huertas-Company et al. 2012). If anything, we startseeing a flatter size evolution in disc-dominated galaxies onlyfor galaxies above & M ⊙ and at z & . c (cid:13) , 1– 21 F. Shankar et al.
Figure 8.
Left panel : Predicted median size-mass relation for all central ( dot-dashed line) and Type 2 satellite galaxies ( long-dashed line;with no restriction in host halo mass), compared to SDSS data. A systematic difference of ∼
30% with centrals being larger than satellites,is apparent at all masses. The dotted and solid lines refer to the size-mass relation of centrals in haloes with mass 10 < M H /M ⊙ < (Groups) and M H > M ⊙ (BCGs), respectively. Central galaxies residing in more massive haloes tend be larger. Right panel : medianfractional size evolution for BCGs, centrals in groups, and Type 2 satellites. BCGs tend to have a much faster evolution than all othergalaxies of similar mass. H imaging and data from the Literature. Themodel predictions are in good agreement with the empiricalfit for masses below ∼ × M ⊙ , where the inferred sizeevolution is weaker, but tend to progressively depart fromthe data at higher masses (we stress that this comparison isstill at a qualitative level as the fits to observations have notbeen derived for a homogeneous sample of galaxies selectedto have the same B/T as in the model).Similar results on the possible inefficiency of merg-ers in puffing up massive galaxies have now beenclaimed also by several independent works Cimatti et al.(2012); Huertas-Company et al. (2012); Nipoti et al. (2012).Huertas-Company et al. (2012) more recently claimed evi-dence for a weaker size evolution at intermediate masses3 × < M star /M ⊙ < , in better agreement withmodel predictions, but tend to confirm the strong dropin sizes for the highest stellar mass bins. For complete-ness, we also report in the bottom panel of Figure 7 thefit recently inferred by Newman et al. (2012) (long-dashedline) that better lines up with the model predictions, atleast at z .
1. We caution, however, that due to theirbroader selections, their sample may not be restricted toonly early-type, bulge-dominated galaxies (see discussion inHuertas-Company et al. 2012, and references therein).In our study of size evolution at fixed stellar mass wealso tried to separate spheroidal galaxies that have mainlygrown their bulges via mergers from those that mainly grewtheir bulges via disc instabilities. We found tentative evi-dence for pseudobulges to evolve slower in sizes with respectto classical bulges of similar stellar mass, but the statistics insome bins is poor and the systematic difference is confined atthe .
20% level. SAMs that adopt stronger disc instabilitiescould provide different conclusions in this respect.
Not all early-type galaxies may follow the same size-mass re-lation. Environment, or simply the special location of galax-ies at their formation epoch, might induce different evolu-tion at later times in galaxies of similar mass. For exam-ple, if mergers dominates the structural growth of galax-ies, at least at lower redshifts, then galaxies in denser en-vironments where mergers are more efficient might appearlarger at fixed stellar mass. From the observational pointof view this is still debated. At high redshifts, while somegroups find clear evidence for larger galaxies in denser en-vironments, at fixed stellar mass (e.g., Cooper et al. 2012;Papovich et al. 2012), other don’t or claim some stellar massdependence (Huertas-Company et al. 2012; Raichoor et al.2012). In the local Universe, when selecting galaxies of agiven stellar mass in the field and in overdense regions suchas Clusters, two main issues have emerged recently. Galaxiesin Clusters less massive than ∼ × M ⊙ tend to appearsmaller at fixed stellar mass than their local counterparts inthe field (Valentinuzzi et al. 2010a), and non-central clustergalaxies might have had a slower or even negligible evolu-tion down to z = 0 (Valentinuzzi et al. 2010b; Saracco et al.2010). The observational evidence though is still sparse ornot secure. Weinmann et al. (2009) investigated size distri-butions, among other properties in the SDSS Data Release4 group catalogue of Yang et al. (2007), finding no cleardifference in the sizes of early-type centrals and satellites.Even the degree of evolution for the brightest cluster galaxies(BCGs), is not yet well understood. Bernardi (2009) foundthat BCGs have evolved by 50% in size in the last few Gyrs,while Stott et al. (2011) claim a milder evolution of & z = 1. Ascaso et al. (2011) also claim evidence for sig-nificant size evolution though not in the light profile (themeasured S´ersic index is nearly invariant with time).Given the non-trivial impact that environment mightinduce on galaxy size evolution, it is thus mandatory tostudy what the model predictions are with respect to thisimportant issue. We recall that the Guo et al. (2011) modelfollows in great detail the fate of the gas and stellar compo- c (cid:13) , 1– 21 ize Evolution nents of galaxies becoming satellites in larger dark matterhaloes. Satellites suffer tidal and ram pressure stripping thatcan remove a large part of their gas reservoir. In particular,the model further assumes that when the host subhalo iscompletely disrupted during its journey within the largesthalo, the galaxy’s stellar component starts also being dis-rupted (see Guo et al. 2011 for details). The stars whichare stripped away can then later become part of the centralgalaxy of the parent dark matter halo hosting the satellitegalaxy. Given all of these physical prescriptions, it is nat-ural to expect some structural differences between centraland satellite galaxies in the code.The left panel of Figure 8 shows the predicted size-massrelation for central and satellite galaxies (dot-dashed andlong-dashed lines, respectively). Here satellites are only theones defined to be “Type 2” in the code, i.e., the ones thathave completely lost their associated subhalo due to disrup-tion. We find a relatively small, although systematic, differ-ence with satellites being smaller by .
30% with respect tocentrals of the same stellar mass, somewhat in between thefindings of Weinmann et al. (2009) and Valentinuzzi et al.(2010a). The dotted and solid lines refer to the size-massrelation of centrals in haloes with mass 10 < M H /M ⊙ < and M H > M ⊙ , respectively. Central galaxies re-siding in more massive haloes tend to be larger mainly be-cause they have undergone a larger number of mergers overcosmic time.We have also analyzed size evolution for galaxies liv-ing in different environments. The right panel of Figure 8shows the median fractional size evolution for galaxies atfixed stellar mass and environment. We consider galaxieshaving stellar masses above & M ⊙ and B/T > . M H > M ⊙ (solid line), centrals in galaxy groups with10 < M H /M ⊙ < (dotted line), and Type 2 satellitesgalaxies with no restriction in host virial mass (long-dashedline). The model predicts that, at fixed stellar mass, galaxiesresiding in progressively more massive haloes have a propor-tionally stronger size evolution evolution, mainly induced bythe larger number of mergers. In particular, BCGs are pre-dicted to increase in size by &
50% at z <
1, a degree of evo-lution in between the one calibrated by Stott et al. (2011)and Bernardi (2009).
Several additional clues on the hierarchical evolution ofspheroidal galaxies can be obtained when considering ve-locity dispersions and the virial mass of galaxies. In thissection we will discuss the physical implications that can bederived from the comparison of model predictions with a va-riety of key observables that include velocity dispersion. Tothis purpose, unless otherwise stated, we will only considerhere the subsample of pure elliptical galaxies, i.e., those with
B/T > . B/T does not alter theconclusions below).All 3D velocity dispersions are computed followingEq. (5) and then converted, consistently with what we dis-
Figure 10.
Predicted median σ - V vir relation at different red-shifts, as labelled, for a model with a dark matter mass-dependent σ and galaxies with B/T > .
9. The grey stripe indicates the ve-locity dispersion-circular velocity correlation by Baes et al. (2003)for early-type galaxies with circular velocity converted to veloc-ity at the virial radius using the velocity-dependent correction ofDutton et al. (2010). cussed for sizes (Section 3.3), to line-of-sight 1D σ (1 D ) us-ing the Prugniel & Simien (1997) S K ( n ) coefficients, i.e., σ (1 D ) = [3 S K ( n )] / σ (3 D ), where we set n = 4 (the ef-fect of the latter correction is however relatively small anddoes not minimally alter our conclusions).The very first correlation with velocity dispersion thatis usually studied is the one between luminosity/stellarmass and velocity dispersion, the Faber-Jackson relation(Faber & Jackson 1976). Our results are presented in Fig-ure 9 where the predicted M star - σ relation is plotted at theredshifts z = 0 , . , , , M star - σ relation from our sub-sample of SDSS galaxies with B/T > .
9. The left panel ofFigure 9 shows predictions for a model with σ computed viaEq. (5), that includes dependence on the host dark matterhalo, while in the right panel σ simply scales inversely withhalf-mass radius, ∝ M star /R H . It is clear that the modelwith σ computed via Eq. (5) provides a much better matchto the data, with stellar mass properly increasing with in-creasing σ . Neglecting any mass dependence in σ inevitablyproduces a flattening at high masses where the model pre-dicts a quasi-linear dependence between half-mass radiusand stellar mass (cfr., e.g., Figure 3). It is also interest-ing to note that neglecting any halo mass dependence in σ produces a much stronger evolution in velocity dispersion atfixed stellar mass, while a model with halo mass dependencecontains the evolution in σ to .
30% in good agreement withdirect observations (e.g., Bernardi 2009; Cenarro & Trujillo2009; Cappellari et al. 2009; van de Sande et al. 2011).Several groups have proven that a clear correlationexists between velocity dispersion and circular velocity atthe outer optical radius (e.g., Ferrarese 2002; Baes et al.2003; Pizzella et al. 2005; Chae 2011). This is expectedfrom basic dark matter theory (e.g., Loeb & Peebles 2003;Cirasuolo et al. 2005; Shankar et al. 2006; Lapi & Cavaliere2009) as during the early fast-collapse phase of a halo, c (cid:13) , 1– 21 F. Shankar et al.
Figure 9.
Left : Predicted M star - σ relation at different redshifts, as labelled, for a model a dark matter-dependent σ as suggested bynumerical simulations ( left panel) and a model with no dependence on dark matter ( right panel). Only galaxies with B/T & . B/T cut ( grey areas ). its potential well is established and a dynamical link be-tween baryon velocity at the center and halo circular ve-locity should be established. Several high-resolution simula-tions have confirmed this behaviour (e.g., Zhao et al. 2003;Diemand et al. 2007; Wang et al. 2011). We plots in Fig-ure 10 the predicted median correlation between 1D velocitydispersion and virial velocity V vir of the halo at in fall (seeGuo et al. 2011), at different redshifts, as labelled, for themodel with a dark matter mass-dependent σ and galaxieswith B/T > .
9. The grey stripes in Figure 10 indicate thecorrelation between velocity dispersion and circular velocityinferred by Baes et al. (2003) for early-type galaxies withcircular velocity converted to velocity at the virial radiususing the empirically-derived velocity-dependent correctionof Dutton et al. (2010) for early-type galaxies (their Eq. 3).A good agreement is found in the local Universe. The modelthen predicts some evolution in the zero point of this rela-tion with galaxies at fixed velocity dispersion being mappedinto haloes with higher virial velocity but the correlation ispreserved despite mergers (e.g., Boylan-Kolchin et al. 2005;Robertson et al. 2006; Ciotti et al. 2007). We also find (notshown in the Figure) that the scatter in velocity dispersionat fixed V vir increases with increasing redshift, in line withsome observations (Courteau et al. 2007; Ho 2007). Having described here and in the previous sections globalstructural properties of spheroids of different masses and
B/T , we now attempt to sketch a more comprehensive pic-ture of spheroid evolution. To this purpose, we select sub-samples of 100 galaxies of a given stellar mass and
B/T at z = 0 and trace back in time the most massive progenitorof each galaxy and record its properties.The result is given in Figure 11 which shows the ex-pected median redshift evolution of several properties char-acterizing spheroid progenitors. We show results for threesubset of galaxies classified at z = 0 by having 10 < log M star < . < log M star < . M star > . B/T > . z = 0. The model predicts thatthe progenitors’ half-mass radiuses shrink when moving tohigher redshifts (top panel; note that we are here plottingprogenitors identified from the merger trees, while beforewe always considered different galaxies of the same stellarmass at different epochs). Noticeably, all galaxies that endup being large spheroids in the local Universe are predictedto share, on average, quite similar size evolutions, at leastat z . . −
2, though different morphologies at higher red-shifts.We find that most of the progenitors of bulge-dominatedgalaxies at high redshifts are disc-dominated (middle panel).More specifically, galaxies that today lie below the character-istic mass of 10 M ⊙ , are found to turn into disc-dominatedsystems at z ∼ . −
2, with
B/T . .
2, and extremelygas-rich (Figure 2). Even more massive galaxies with stellarmass at z = 0 in the range 10 < M star /M ⊙ . × M ⊙ rapidly turn into discs with a median B/T ∼ .
3. In fact,we found that only 20% of the galaxies in this mass regimeremain bulge-dominated with
B/T & . B/T ∼ . z & . − z . M star ( z = 0) . M ⊙ , while the compact and red galax-ies should end up being the most massive ellipticals we ob-serve in the local Universe. Full exploration of high and lownumber densities of galaxies of a given property will help tofurther constrain the model (see Section 4.3).We also note that beyond z & z = 2 (bottom panel). Solving the discrepancy betweenmodel and data is beyond the scope of the present paper.We note however that it is a common feature of many galaxy c (cid:13) , 1– 21 ize Evolution Figure 12.
Fractional average mass ( left ) and size ( right ) growth of the subsets of galaxies discussed in Figure 11.
Long-dashed , solid ,and dotted lines refer, respectively, to the fractional cumulative growth down to z = 0 as a function of final stellar mass, experienced bygalaxies via major mergers, minor mergers, or any other mechanism (disc instability, star formation, etc...). formation models, and it has been recently pointed out anddiscussed by Khochfar & Silk (2011) and Weinmann et al.(2011).Figure 12 shows the fractional average mass (left) andsize (right) growth of the subsets of galaxies discussed inFigure 11. Long-dashed, solid, and dotted lines refer, respec-tively, to the fractional cumulative growth down to z = 0as a function of final stellar mass, experienced by galax-ies via major mergers, minor mergers, or any other mecha-nism (such as disc instability and/or in loco star formation).We find that the massive spheroids considered here mainlygrow through mergers in the model. Above M star & M ⊙ these galaxies grow more than &
50% of their final stel-lar mass and size via minor mergers, while major mergersdominate the growth at lower masses and become progres-sively less important at higher masses (e.g., Khochfar & Silk2009). This is expected given that the median accretion his-tory of their typical host dark matter haloes, in the range10 − h − M ⊙ , are found in high-resolution numericalsimulations to be dominated by mergers with satellites thatare ∼
10% of the final halo mass (e.g., Stewart et al. 2008).We also find that most of the minor mergers, especiallyin the most massive galaxies, are dry, i.e., have a (cold) gasmass fraction in the progenitors that is lower than 0.15. Mi-nor dry mergers can roughly preserve the projections of thefundamental plane (e.g., Ciotti et al. 2009, and referencestherein), and galactic central densities (e.g., Cimatti et al.2008; Bezanson et al. 2009, and references therein), thoughthey may also increase the scatter in the scaling relations(e.g., Nipoti et al. 2009, and references therein). The find-ings of Figure 12 is consistent with an inside-out evolution-ary scenario, where stellar matter is continuously added tothe outskirts of the compact high-redshift galaxies as timegoes on (e.g., Naab et al. 2009; Oser et al. 2010). More de-tailed comparisons with metallicity, age and colour gradientsare needed to set this model on firmer foots (see discussionin Section 4.3).
While all galaxy formation models have discussed predic-tions for stellar mass distributions, just a handful have takena step further to also consider structural properties, espe-cially for early-type galaxies.Khochfar & Silk (2006b) within the context of a fullsemi-analytic model, emphasized the role of gas dissipationin mergers in forming compact massive spheroids at highredshifts, deriving progenitors’ gas fractions in good agree-ment with the ones discussed here.Almeida et al. (2007, 2008), following Cole et al.(2000), studied the galaxy size-mass and global fundamentalplane relations predicted by two significantly different rendi-tions of the GALFORM semi-analytic model by Baugh et al.(2005) and Bower et al. (2006). After varying most of theparameters relevant for determining bulge sizes, they con-cluded that both models fail to reproduce the sizes of brightearly-type galaxies, though they noted that a better matchto the data was achieved with no adiabatic contraction (e.g.,Tissera et al. 2010; Covington et al. 2011), as assumed inthis work. On a similar note, Gonz´alez et al. (2009), fol-lowing on Almeida et al. (2007), further studied the galaxysize-mass relations in the GALFORM models varying otherparameters, such as orbital energy in the merger, but stillfinding significant disagreement with the data.Shankar et al. (2010b) and Shankar et al. (2010a)showed that the size-mass relation at z = 0 predicted bythe Bower et al. model is much flatter than the observedone due to too large low-mass galaxies with stellar mass M star . M ⊙ , similarly to what found here. They sug-gested that the latter problem may be linked to the ini-tial conditions, given that large and low-mass galaxies arepresent at all epochs in the model, in line with what dis-cussed here.We have also adapted the recipes for bulge size growthvia mergers and disc instability discussed in Section 2.2 to aprevious version of the Munich code by De Lucia & Blaizot(2007). The latter differs from the Guo et al. (2011) versionin several respects, from physical recipes to values of the c (cid:13) , 1– 21 F. Shankar et al.
Figure 11.
Evolution with redshift of the average propertiesof a subset of 100 galaxies derived by following the most mas-sive progenitor back in time along its merger tree. We show re-sults for three subsets of galaxies classified by having at z = 010 < log M star < . long-dashed lines), 11 < log M star < . solid lines), and log M star > . dotted lines). All galaxies have B/T > . z = 0. The top panel shows the median fractionalsize evolution, the middle panel the average bulge-to-total ratios,the bottom panel the specific star formation rate. best-fit parameters. We have checked, however, that most ofthe results discussed here remain globally similar.Monaco et al. (2007) implemented the recipes fromCole et al. (2000) for size growth in their MORGANA model(with no dissipation) finding acceptable agreement with thelocal size-mass relation for early-type galaxies, though witha scatter larger than the observed one (see their Figure 16).Understanding the success of the latter model with respectto the previous mentioned ones relies on pinning down thedifferences in the physical inputs of the MORGANA modeland the impact they have on size evolution.Other works aimed at studying the global structuralproperties of local early-type galaxies through semi-analytictechniques was pursued by Cirasuolo et al. (2005), withinthe framework of the Granato et al. (2004) model for the co-evolution of super-massive black holes and their host mas-sive spheroids. Cirasuolo et al. showed that by tighteningvelocity dispersion of spheroids to the virial velocity at theepoch of their formation (see also Loeb & Peebles 2003),the local early-type velocity dispersion function (Sheth et al.2003; Shankar et al. 2004; Bernardi et al. 2010) and Gaus-sian dispersion in sizes was fully recovered. Their study isquite intriguing as the match to the photometric and dy-namical properties of local ellipticals only relies on galaxy properties at virialization epoch and minimizes the role oflater merger events.A more refined theory of structural evolution ofspheroids besides mergers, has been presented by (Fan et al.2008, 2010). As anticipated in Section 1, this classof models explains size evolution of spheroids viaexpansion consequent to the blow-out of substantialamounts of mass via quasar and/or stellar feedback(Fan et al. 2008, 2010; Damjanov et al. 2009). Numericalsupport to the latter models was recently provided byRagone-Figueroa & Granato (2011) who showed that evenin the presence of dark matter, baryons can indeed expandby a factor of a few consequent to substantial mass losses.They also pointed out that the puffing up via expansion maybe too rapid with respect to the old ages measured for thecompact high-z early-type galaxies. Understanding the ac-tual role played by expansion versus mergers is beyond thescope of the present work. However, we discussed in Sec-tion 3.5 that the strong size evolution for the most massiveellipticals claimed by some observational groups (Ryan et al.2012; Huertas-Company et al. 2012; Newman et al. 2012)can hardly be reproduced in our only-merger model, andpossibly some extra-expansion (a factor of ∼
2) in the firstphases of evolution might help.Relevant numerical work has been pursued in the lastyears to explore the size evolution of spheroids in a full cos-mological context. We recall here the work by Naab et al.(2009) and Oser et al. (2010, see also Scannapieco et al.2011), who developed high resolution hydrodynamical cos-mological simulations of massive spheroidal galaxies. Theyparticularly emphasized that galaxies above the character-istic mass of M star & M ⊙ can accrete via minor mergerabout 80% of their final stellar mass, in agreement with whatfound here (Figure 12). In this work we emphasized the role of gas dissipation as aviable mechanism to shrink bulge sizes, especially in lowermass galaxies, thus improving the match to a variety of dif-ferent observables. Gas dissipation is indeed a natural out-come of gas-rich mergers and thus should to be properlyincluded in complete models of galaxy formation. Neverthe-less, we cannot rule out that part of the discrepancy betweenmodel predictions and data, especially regarding the matchwith the size mass relation (Section 3.2), could also be in-duced by some other wrong model inputs. For example, ithas been recognized that this SAM, like several others, over-produces the stellar mass function at high redshifts and lowstellar masses (e.g., Henriques et al. 2012). This in turn im-plies more massive and larger galaxies in lower mass haloeswith respect to what expected from, e.g., cumulative abun-dance matching arguments (Moster et al. 2012), thus possi-bly contributing to the flattening at low masses in the sizemass relation (Section 3.1). Besides the actual performanceof the model in properly predicting the size-mass relation,we nevertheless stress that gas dissipation can improve thematch to several other observables, as discussed above andfurther in the Appendices (e.g., size-age relation, correla-tions with velocity dispersion, etc...).In this work we have focused our attention on the mostrelevant median scaling relations among structural proper- c (cid:13) , 1– 21 ize Evolution ties of bulged galaxies. It is clear, however, that several otherproperties are equally important to pin down and better con-strain the viable models.First of all, it is fundamental for a model not only toproduce the correct structure but also the correct numberof galaxies of a given type. Galaxy formation models havemost seriously investigated the match to the stellar massfunction, finding good agreement adopting different physi-cal prescriptions (e.g., Cole et al. 2000; Benson et al. 2003;Granato et al. 2004; Croton et al. 2006), though significantuncertainties still affect high and low redshift measurements(Bernardi et al. 2010; Marchesini et al. 2009, 2010).The study of number densities in other properties suchas size, is instead still at its infancy. The first study wascarried out by Cole et al. (2000). A more recent attempt hasbeen carried out by Shankar et al. (2010a) who compareddetailed predictions from hierarchical models with the latestreleases of the Φ( R e ) galaxy size function, finding that theBower et al. (2006) model overpredicts the number of verycompact and very large galaxies (see also Trujillo et al. 2009;Taylor et al. 2010).Higher redshift measurements of portions of the sizefunction of spheroids are also now becoming available (e.g.,Mancini et al. 2009; Saracco et al. 2010; Valentinuzzi et al.2010a; van Dokkum et al. 2010). At higher redshifts thescatter in sizes at fixed stellar mass apparently seems to in-crease as large and compact galaxies seem to co-exist at thesame epoch (e.g., Mancini et al. 2010; Saracco et al. 2011).This increase in scatter may suggest a faster evolution insizes at fixed stellar mass (Fan et al. 2010), although trendsbetween size and age/star formation rate, may bias this re-sult (e.g., Mosleh et al. 2011; F¨orster Schreiber et al. 2011;Saracco et al. 2011).In order to use number density measurements of sizeand velocity functions (Cirasuolo et al. 2005) at differentredshifts, a homogeneous and well studied spectroscopic andphotometric sample of galaxies is needed, now not yet re-leased. Larger statistical samples will become available inthe near future (such as multi-wavelength optical surveysfrom the Canada French Hawaii Telescope and the NextGeneration Virgo Cluster Survey).Ages, colours, metallicities, and other dynamical prop-erties (e.g., Cappellari et al. 2009) can be key observables toprobe galaxy evolution. For example, Bernardi et al. (2011a)and Bernardi et al. (2011b), making use of the latest SDSSdata releases, showed that all correlations with stellar masssteepen above M star & × M ⊙ (see also Fasano et al.2010; van der Wel et al. 2011), while relations with ve-locity dispersion don’t. Bernardi et al. claimed, in linewith other observational works (e.g., Kauffmann et al. 2004;van der Wel et al. 2009) and with the results presented inFigure 12, the presence of two mass scales, one at M star ∼ × M ⊙ , below which gas dissipation controls galaxyformation, and a higher mass scale M star & × M ⊙ above which mergers dominate the evolution. The empiri-cal mass scales noticed by Bernardi et al. are in line withthe characteristic masses M c and M s emphasized in Sec-tion 3.1. Metallicity gradients have been recently calibratedby a number of groups (e.g., Foster et al. 2009; Spolaor et al.2010; Forbes et al. 2011) and can provide invaluable insightsinto the evolutionary patterns of early-type galaxies (e.g.,Di Matteo et al. 2009; La Barbera & de Carvalho 2009). Understanding structural evolution of spheroids has becomeone of the hottest topics in cosmology in the last years asit can provide invaluable insights into the true physical pro-cesses that regulated galaxy evolution. While angular mo-mentum conservation may explain many properties of discs,the origin of bulges is still largely debated. The situation iseven more puzzling given that at higher redshifts galaxiespresent further disparate structural and physical properties,from clumpy star-forming discs, to very compact, red andmassive galaxies. Such a complicated zoology is difficult toreconcile within a coherent framework of galaxy formation,and in fact we discussed in Section 1 that models sometimespropose conflicting scenarios.Our aim in this work was to study the predictions ofa state-of-the-art hierarchical model of galaxy formation,which evolves the sizes of spheroids via mergers and discinstabilities, against the most recent local and high redshiftdata. To this purpose we updated the source code of thelatest release of the Munich semi-analytic galaxy formationmodel by Guo et al. (2011), by modifying the computationof bulge radii exploring a variety of possibilities, and addedthe calculation of the coupled velocity dispersions. In orderto properly compare model predictions with available datawe made use of a large sample of early-type galaxies fromSDSS for which bulge-to-disc decompositions have been per-formed both via the SDSS automated “on-the-fly” analysisand by applying a detailed fitting code.Our main results can be summarized as follows. • Sizes are computed in the model via energy conserva-tion in dissipationless mergers. Taking the model at facevalue at masses below M s . M ⊙ the model predictsa flattening of the size-mass relation at variance with thedata, already at z ∼ • Following the results of hydro-simulations, we have in-cluded the energy dissipated in gas-rich major mergers inthe energy budget. This modification produces progressivelymore compact remnants with decreasing stellar mass, pro-portionally to the fraction of cold gas in the progenitors,improving the match with a variety of observables. • We confirm and discuss evidence for two characteris-tic masses. One is at M c ∼ × M ⊙ , below which theinitial bulge sizes are controlled by dissipation (higher gasfractions) and then evolve under mergers and disc instabil-ities. Galaxies above M s & M ⊙ , instead mainly growthrough minor dry mergers, especially at z < • We find that the global scatter (1- σ uncertainty) in sizesat fixed stellar mass for galaxies with B/T > . .
40% than theobserved one. The predicted amount of scatter for galax-ies with stellar mass M star & × M ⊙ does not dependmuch on dissipation and/or amount of orbital energy in themerger. • Spheroids are predicted to be, on average, more com-pact at higher redshifts at fixed stellar mass. More specif-ically, at fixed
B/T a nearly mass-independent relativelymild decrease in sizes is predicted, in possible disagreementwith some observations. • The model predicts that environment plays a significantrole in defining the structural properties of bulged galaxies.(Central) galaxies residing in denser environments are pre- c (cid:13) , 1– 21 F. Shankar et al. dicted to undergo more mergers, thus evolve faster and endup having larger sizes at fixed stellar mass. • The progenitors of the massive spheroids with M star ∼ (1 − × M ⊙ today with B/T > . z ∼ B/T ∼ .
3, with only ∼
20% remaining bulge-dominated.The progenitors of lower-mass spheroids with
B/T > . • Finally, we also discuss a number of related issues (inthe text and in the Appendices), ranging from the correla-tions with galaxy age, with central black hole mass, withvelocity dispersion, to the scaling relations of pseudobulges.
ACKNOWLEDGMENTS
FS warmly thanks Enzo Gattorno for his kind and helpfulsupport. We thank Bruno Henriquez and Gerard Lemson formany useful inputs. We thank Guinevere Kauffmann and Si-mon White for making the source code of the Munich modelavailable to us to properly carry out this work and for gen-eral comments. We also thank Qi Guo, Luca Ciotti, LucaGraziani, Sadegh Khochfar, Juan Gonz´alez, Frederic Bour-naud, Simone Weinmann, Ravi Sheth, Luigi Danese, An-drea Lapi, Alister Graham, Eyal Neistein, Fabio Fontanot,Silvia Bonoli, Thorsten Naab, Avishai Dekel, and Matt Cov-ington for several interesting discussions. We thank DimitriGadotti and David Fisher for providing us their data onpseudobulges. FS acknowledges support from the Alexan-der von Humboldt Foundation and partial support from aMarie Curie grant. MB is grateful for support provided byNASA grant ADP/NNX09AD02G. We thank the referee fora useful report that significantly improved the presentationof the paper.
REFERENCES
Almeida C., Baugh C. M., Lacey C. G., 2007, MNRAS,376, 1711Almeida C., Baugh C. M., Wake D. A., Lacey C. G., BensonA. J., Bower R. G., Pimbblet K., 2008, MNRAS, 386, 2145Ascaso B., Aguerri J. A. L., Varela J., Cava A., Bettoni D.,Moles M., D’Onofrio M., 2011, ApJ, 726, 69Auger M. W., Treu T., Brewer B. J., Marshall P. J., 2011,MNRAS, 411, L6Baes M., Buyle P., Hau G. K. T., Dejonghe H., 2003, MN-RAS, 341, L44Barnes J. E., 1992, ApJ, 393, 484Baugh C. M., Lacey C. G., Frenk C. S., Granato G. L., SilvaL., Bressan A., Benson A. J., Cole S., 2005, MNRAS, 356,1191Bennert V. N., Auger M. W., Treu T., Woo J., MalkanM. A., 2011, ApJ, 708, 1507Benson A. J., Frenk C. S., Baugh C. M., Cole S., LaceyC. G., 2003, MNRAS, 343, 679Bernardi M., 2009, MNRAS, 395, 1491Bernardi M., Shankar F., Hyde J. B., Mei S., Marulli F.,Sheth R. K., 2010, MNRAS, 404, 2087Bernardi M., Roche N., Shankar F., Sheth R. K., 2011a,MNRAS, 412, 684 Bernardi M., Roche N., Shankar F., Sheth R. K., 2011b,MNRAS, 412, L6Bezanson R., van Dokkum P. G., Tal T., Marchesini D.,Kriek M., Franx M., Coppi P., 2009, ApJ, 697, 1290Blain A. W., Smail I., Ivison R. J., Kneib J.-P., FrayerD. T., 2002, Phys. Rep., 369, 111Bonoli S., Marulli F., Springel V., White S. D. M., Bran-chini E., Moscardini L., 2009, MNRAS, 396, 423Bonoli S., Shankar F., White S. D. M., Springel V., WyitheJ. S. B., 2010, MNRAS, 404, 399Bournaud F., Chapon D., Teyssier R., Powell L. C.,Elmegreen B. G., Elmegreen D. M., Duc P., Contini T.,Epinat B., Shapiro K. L., 2011a, ApJ, 730, 4Bournaud F., Dekel A., Teyssier R., Cacciato M., DaddiE., Juneau S., Shankar F., 2011b, ApJ, 741, L33Bower R. G., Benson A. J., Malbon R., Helly J. C., FrenkC. S., Baugh C. M., Cole S., Lacey C. G., 2006, MNRAS,370, 645Boylan-Kolchin M., Ma C.-P., Quataert E., 2005, MNRAS,362, 184Boylan-Kolchin M., Springel V., White S. D. M., JenkinsA., Lemson G., 2009, MNRAS, 398, 1150Buitrago F., Trujillo I., Conselice C. J., Bouwens R. J.,Dickinson M., Yan H., 2008, ApJ, 687, L61Bullock J. S., Kolatt T. S., Sigad Y., Somerville R. S.,Kravtsov A. V., Klypin A. A., Primack J. R., Dekel A.,2001, MNRAS, 321, 559Cappellari M. et al., 2009, ApJ, 704, L34Catinella B. et al., 2010, MNRAS, 403, 683Cenarro A. J., Trujillo I., 2009, ApJ, 696, L43Chabrier G., 2003, PASP, 115, 763Chae K.-H., 2011, MNRAS, 413, 887Chapman S. C. et al., 2008, ApJ, 689, 889Cimatti A. et al., 2008, A&A, 482, 21Cimatti A., Nipoti C., Cassata P., 2012, MNRAS, 422, L62Ciotti L., van Albada T. S., 2001, ApJ, 552, L13Ciotti L., Lanzoni B., Volonteri M., 2007, ApJ, 658, 65Ciotti L., 2009, Nuovo Cimento Rivista Serie, 32, 1Ciotti L., Ostriker J. P., Proga D., 2009, ArXiv e-printsCirasuolo M., Shankar F., Granato G. L., De Zotti G.,Danese L., 2005, ApJ, 629, 816Cole S., Lacey C. G., Baugh C. M., Frenk C. S., 2000,MNRAS, 319, 168Cooper M. C. et al., 2012, MNRAS, 419, 3018Courteau S., McDonald M., Widrow L. M., Holtzman J.,2007, ApJ, 655, L21Covington M., Dekel A., Cox T. J., Jonsson P., PrimackJ. R., 2008, MNRAS, 384, 94Covington M. D., Primack J. R., Porter L. A., Croton D. J.,Somerville R. S., Dekel A., 2011, MNRAS, 415, 3135Croton D. J. et al., 2006, MNRAS, 365, 11Croton D. J., 2006, MNRAS, 369, 1808Damjanov I. et al., 2009, ApJ, 695, 101De Lucia G., Blaizot J., 2007, MNRAS, 375, 2De Lucia G., Boylan-Kolchin M., Benson A. J., FontanotF., Monaco P., 2010, MNRAS, 406, 1533De Lucia G., Springel V., White S. D. M., Croton D., Kauff-mann G., 2006, MNRAS, 366, 499de Vaucouleurs G., 1948, Annales d’Astrophysique, 11, 247Decarli R., Falomo R., Treves A., Labita M., KotilainenJ. K., Scarpa R., 2010, MNRAS, 402, 2453Dekel A. et al., 2009a, Nature, 457, 451 c (cid:13) , 1– 21 ize Evolution Dekel A., Sari R., Ceverino D., 2009b, ApJ, 703, 785Di Matteo P., Jog C. J., Lehnert M. D., Combes F., SemelinB., 2009, A&A, 501, L9Diemand J., Kuhlen M., Madau P., 2007, ApJ, 667, 859Djorgovski S., Davis M., 1987, ApJ, 313, 59Dutton A. A., Conroy C., van den Bosch F. C., Prada F.,More S., 2010, MNRAS, 407, 2Eggen O. J., Lynden-Bell D., Sandage A. R., 1962, ApJ,136, 748Erb D. K., Steidel C. C., Shapley A. E., Pettini M., ReddyN. A., Adelberger K. L., 2006, ApJ, 646, 107Faber S. M., Jackson R. E., 1976, ApJ, 204, 668Fan L., Lapi A., De Zotti G., Danese L., 2008, ApJ, 689,L101Fan L., Lapi A., Bressan A., Bernardi M., De Zotti G.,Danese L., 2010, ApJ, 718, 1460Fasano G. et al., 2010, MNRAS, 404, 1490Ferrarese L., 2002, ApJ, 578, 90Ferrarese L., Ford H., 2005, Space Science Reviews, 116,523Fisher D. B., Drory N., 2008, AJ, 136, 773Fisher D. B., Drory N., 2010, ApJ, 716, 942Fisher D. B., Drory N., 2011, ApJ, 733, L47Forbes D. A., Spitler L. R., Strader J., Romanowsky A. J.,Brodie J. P., Foster C., 2011, MNRAS, 413, 2943F¨orster Schreiber N. M. et al., 2011, ApJ, 739, 45Foster C., Proctor R. N., Forbes D. A., Spolaor M., HopkinsP. F., Brodie J. P., 2009, MNRAS, 400, 2135Franx M., van Dokkum P. G., Schreiber N. M. F., WuytsS., Labb´e I., Toft S., 2008, ApJ, 688, 770Fu J., Guo Q., Kauffmann G., Krumholz M. R., 2010, MN-RAS, 409, 515Gadotti D. A., 2009, MNRAS, 393, 1531Gaskell C. M., 2009, ArXiv:0908.0328Genzel R., Newman S., Jones T. et al., 2011, ApJ, 733, 101Gonz´alez J. E., Lacey C. G., Baugh C. M., Frenk C. S.,Benson A. J., 2009, MNRAS, 397, 1254Gonz´alez J. E., Lacey C. G., Baugh C. M., Frenk C. S.,2011, MNRAS, 413, 749Governato F., Willman B., Mayer L., Brooks A., StinsonG., Valenzuela O., Wadsley J., Quinn T., 2007, MNRAS,374, 1479Granato G. L., De Zotti G., Silva L., Bressan A., DaneseL., 2004, ApJ, 600, 580Granato G. L., Silva L., Lapi A., Shankar F., De Zotti G.,Danese L., 2006, MNRAS, 368, L72Graves G. J., Faber S. M., 2010, ApJ, 717, 803Guo Q., White S., Boylan-Kolchin M. et al., 2011, MNRAS,413, 101H¨aring N., Rix H.-W., 2004, ApJ, 604, L89Henriques B. M. B. et al., 2012, MNRAS, 421, 2904Hernquist L., 1990, ApJ, 356, 359Hernquist L., 1992, ApJ, 400, 460Ho L. C., 2007, ApJ, 668, 94Hopkins P. F., Hernquist L., Cox T. J., Keres D., WuytsS., 2009, ApJ, 691, 1424Hopkins P. F. et al., 2010, ApJ, 724, 915Hyde J. B., Bernardi M., 2009, MNRAS, 394, 1978Huertas-Company M. et al., 2012, arXiv:1207.5793Kannappan S. J., 2004, ApJ, 611, L89Kauffmann G. et al., 2004, MNRAS, 353, 713Kauffmann G. et al., 2012, MNRAS, 422, 997 Khochfar S., Burkert A., 2006, A&A, 445, 403Khochfar S., Silk J., 2006a, MNRAS, 370, 902Khochfar S., Silk J., 2006b, ApJ, 648, L21Khochfar S., Silk J., 2009, MNRAS, 397, 506Khochfar S., Silk J., 2011, MNRAS, 410, L42Kormendy J., Kennicutt Jr. R. C., 2004, ARA&A, 42, 603La Barbera F., Busarello G., Merluzzi P., de la Rosa I. G.,Coppola G., Haines C. P., 2008, ApJ, 689, 913La Barbera F., de Carvalho R. R., 2009, ApJ, 699, L76La Barbera F., Lopes P. A. A., de Carvalho R. R., de LaRosa I. G., Berlind A. A., 2010, MNRAS, 408, 1361Lapi A., Cavaliere A., 2009, ApJ, 692, 174Loeb A., Peebles P. J. E., 2003, ApJ, 589, 29Magdis G. E., Elbaz D., Hwang H. S., Pep Team, & Her-mes Team 2011, Galaxy Evolution: Infrared to MillimeterWavelength Perspective, 446, 221Mancini C., Matute I., Cimatti A., Daddi E., Dickinson M.,Rodighiero G., Bolzonella M., Pozzetti L., 2009, A&A,500, 705Mancini C. et al., 2010, MNRAS, 401, 933Marchesini D., van Dokkum P. G., F¨orster Schreiber N. M.,Franx M., Labb´e I., Wuyts S., 2009, ApJ, 701, 1765Marchesini D. et al., 2010, ApJ, 725, 1277Marconi A., Hunt L. K., 2003, ApJ, 589, L21Marulli F., Bonoli S., Branchini E., Moscardini L., SpringelV., 2008, MNRAS, 385, 1846Marulli F., Bonoli S., Branchini E., Gilli R., Moscardini L.,Springel V., 2009, MNRAS, 396, 1404Menci N., Cavaliere A., Fontana A., Giallongo E., Poli F.,Vittorini V., 2004, ApJ, 604, 12Menci N., Fontana A., Giallongo E., Grazian A., SalimbeniS., 2006, ApJ, 647, 753Merlin E. et al., 2012, arXiv:1204.5118Monaco P., Fontanot F., Taffoni G., 2007, MNRAS, 375,1189Mosleh M., Williams R. J., Franx M., Kriek M., 2011, ApJ,727, 5Moster B. P., Naab T., White S. D. M., 2012,arXiv:1205.5807Naab T., Jesseit R., Burkert A., 2006, MNRAS, 372, 839Naab T., Johansson P. H., Ostriker J. P., Efstathiou G.,2007, ApJ, 658, 710Naab T., Johansson P. H., Ostriker J. P., 2009, ApJ, 699,L178Naab T., Ostriker J. P., 2009, ApJ, 690, 1452Nair P. B., van den Bergh S., Abraham R. G., 2010, ApJ,715, 606Nair P., van den Bergh S., Abraham R. G., 2011, ApJ, 734,L31Napolitano N. R., Romanowsky A. J., Tortora C., 2010,MNRAS, 405, 2351Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490,493Neistein E., Weinmann S. M., 2010, MNRAS, 405, 2717Neistein E., Li C., Khochfar S., Weinmann S. M., ShankarF., Boylan-Kolchin M., 2011, MNRAS, 416, 1486Newman A. B., Ellis R. S., Bundy K., Treu T., 2012, ApJ,746, 162Newton E. R., Marshall P. J., Treu T. et al., 2011, ApJ,734, 104Nipoti C., Treu T., Bolton A. S., 2009, ApJ, 706, 86Nipoti C., Treu T., Leauthaud A. et al., 2012, MNRAS, c (cid:13) , 1– 21 F. Shankar et al.
APPENDIX A: RELATION WITH AGE
Although a large number of massive early-type galaxiesat high redshifts have been observed to be more compactand with higher velocity dispersion than their local coun-terparts, still a significant fraction shows to be alreadyevolved. Mancini et al. (2009) suggested a downsizing sce-nario in sizes, with the most massive galaxies approachingthe local size-mass relation earlier than less massive ones.Cappellari et al. (2009) also discussed that a large fractionof their galaxies at 1 . . z . z ∼ . ∼ − assembly c (cid:13) , 1– 21 ize Evolution Figure A1.
Predicted median size of galaxies of a given stellar mass as a function of mass-weighted age for different
B/T thresholds,and with and without dissipation included, as labelled. This relation for bulge-dominated galaxies is flat at fixed stellar mass, as observedin the local Universe (e.g., Shankar et al. 2010b). bias , i.e., the youngest galaxies at any epoch may be largerthan their older counterparts of similar stellar mass (but seeWhitaker et al. 2012).In the local Universe the assembly bias seems insteadto be erased, at least above M star ∼ M ⊙ and large B/T . Shankar & Bernardi (2009), Shankar et al. (2010b),Bernardi et al. (2010) showed from a large sample of early-type galaxies extracted from SDSS that old and young galax-ies of similar stellar mass share similar sizes, i.e., the size-agerelation is rather flat. Trujillo et al. (2011), more recently,confirmed these results showing that the size-age relationis flat already at z ∼
1. When moving to lenticular galax-ies with lower
B/T the relation gets tilted, with the morecompact bulges being the oldest (e.g., van der Wel et al.2009; Bernardi et al. 2010). Whatever mechanism puffs upspheroids must be fine-tuned to allow all galaxies of a givenstellar mass to end up on the same size-mass and size-agerelations in the local Universe.Preliminary theoretical studies to interpret the size-agerelation were performed by Shankar et al. (2010b) withinthe context of the Bower et al. (2006) hierarchical model.Their study showed that because older galaxies usually un-dergo more mergers, they naturally grow more and are ableto “catch up” with the younger ones producing a rather flatsize-age relation. Here we want to extend the preliminarystudy by Shankar et al. (2010b) to the case of the present hi-erarchical model that, at variance with the Bower et al. one,includes dissipation and better matches the local size-massrelation. We will focus here only on the predicted size-agerelation in the local Universe. At redshifts z & − . B/T thresholds. For lower masses (left panel)the model predicts that in the absence of dissipation (dot-ted lines) more compact galaxies are also the oldest ones.This is mainly induced by redshift evolution in the sizes ofthe progenitors that shrinking at higher redshifts (Figure 7)produce more compact early-type remnants. Including dis- sipation (solid lines) flattens the size-age relation out. Asdiscussed above, dissipation acts in a way to reduce sizesproportionally to the gas fractions in the progenitors. Withina given mass bin, the lower mass galaxies have, on average,higher f gas thus suffering more dissipation than the galax-ies lying closer to the more massive end of the bin. Thus, inthis model both mergers and dissipation act in a coordinatedway to flatten out the relation.At higher masses (right panel) the inverse relation ofsizes with age disappears, even in the absence of dissipation.In fact, including dissipation slightly steepens the relation.These results are all in broad agreement with what observedin SDSS. However, there are also some discrepancies. As an-ticipated above, data on lenticulars, or in general galaxieswith lower B/T ratios, seem to follow a different trend, witholder galaxies being more compact (Bernardi et al. 2010). Asseen in Figure A1, however, the model seems not to show anysignificant change in the shape of the size-age relation whenlowering the
B/T threshold (long-dashed lines; only galaxieswith bulges mainly grown through mergers are consideredhere). This inconsistent behaviour needs to be further un-derstood both observationally and theoretically.
APPENDIX B: PROPERTIES OFPSEUDOBULGES
There is increasing empirical evidence that not allbulges in the local Universe can be formed via mergers(e.g., Kormendy & Kennicutt 2004; Fisher & Drory 2008;Gadotti 2009). Recent works suggest in fact that a largefraction, possibly the majority of local bulges are pseudob-ulges (Fisher & Drory 2011), with well defined propertiesdifferent from those of classical bulges of similar mass. Pseu-dobulges are usually characterized by younger stellar popu-lations, they are usually rotation rather than pressure sup-ported, have less concentrated surface brightness profiles,and tend to follow well distinct scaling relations in theirglobal structural properties with respect to classical bulges.In particular, pseudobulges are found to be much morecompact by a factor of a few at fixed stellar mass with re- c (cid:13) , 1– 21 F. Shankar et al.
Figure B1.
Predicted median effective radius as a function ofstellar mass for bulges with sizes grown mainly by mergers ( long-dashed line with their 1 − σ dispersion), and bulges mainly grownvia disc instability ( solid lines). The classical bulges are shownfor B/T > .
5, while the upper and lower solid lines refer topseudobulges with
B/T > . B/T > .
5, respectively. The colored contours are the SDSS subsample with corresponding
B/T > .
5, while the triangles and circles are pseudobulges mea-surements from Gadotti (2009) and Fisher & Drory (2010), re-spectively, with no cut on
B/T . spect to classical bulges. The open triangles and circles inFigure B1 are the measured sizes of pseudobulges from thesamples of Gadotti (2009) and Fisher & Drory (2010), re-spectively. Both their stellar mass measurements have beenconverted to a common initial mass function using the tab-ulated values in Bernardi et al. (2010). It is interesting tonote that despite the completely independent and differentmethods utilized, these groups agree in finding that pseu-dobulges are more compact by a factor of a few.The model predicts a size-mass relation for pseudob-ulges (solid lines and filled squares) - that we here label asthose bulges that have grown more than 50% of their finalsize via disc instability - with a slope and scatter similar tothe one predicted for classical bulges (long-dashed line andopen squares) but lower in normalization by a factor of a few.The upper and lower solid lines referring to pseudobulgeswith B/T > . B/T > .
5, respectively, are comparedwith data by Gadotti (2009, triangles) and Fisher & Drory(2010, circles), with no cut on
B/T . The predictions are ingood agreement with the data, and even more so given thatno fine-tuning was imposed in the model parameters. Themodel also predicts that no pseudobulges are found above M star & × M ⊙ , again in line with the data. As discussedabove, we stress that most of the pseudobulges produced inthe model have B/T < . M star ∼ × M ⊙ , and sharply decreases at highermasses, where the contribution of classical bulges domi-nates. This behaviour is in good agreement with the recentstudy by Fisher & Drory (2011, see also Kormendy et al.2010), who presented an inventory of galaxy bulge types in Figure B2.
Fraction of pseudo ( cyan lines) and classical ( red lines) bulges as a function of stellar mass for redshift z = 0 ( solid lines) and z = 1 ( dotted lines). There is a tendency to have an in-creasing fraction of pseudobulges at lower stellar masses in broadagreement with the data. a volume-limited sample within the local 11 Mpc volume us-ing Spitzer and HST data (see their Figure 3). Theoretically,this is expected because mergers dominate the size evolutionabove the characteristic mass of 10 M ⊙ , as anticipated inSection 3.1 and further developed in Section 4.1. On theother hand, Fisher & Drory (2011) also claim that pseu-dobulges tend to be the dominant class of bulges at lowermasses, being close to 60% around M star ∼ M ⊙ . Themodel instead predicts a lowering of the fraction of pseudob-ulges towards lower masses. More detailed observational andtheoretical studies of pseudobulges in the low-mass regime,beyond the scope of the present paper, are needed to fullyunderstand this discrepancy. APPENDIX C: THE TILT OF THEFUNDAMENTAL PLANE
Early-type galaxies obey a tight scaling relation linking σ , M star , and R e , the so-called fundamental plane (FP), as ex-pected from basic virial arguments (e.g., Djorgovski & Davis1987). If the mass-to-light ratio is constant at all masses,i.e., L ∝ M dyn , from the virial relation one would thensimply expect L ∝ σ R e . However, the observed FP re-lation has small but significant departures from the theo-retical predictions, suggesting that M dyn /L has a non triv-ial dependence on mass. This “tilt” of the FP is possiblya consequence of stellar effects and/or progressive differ-ence in the inner dark matter content of the most mas-sive galaxies (e.g., La Barbera et al. 2008; Tortora et al.2009; La Barbera et al. 2010; Napolitano et al. 2010;Graves & Faber 2010; Tortora et al. 2012).Shankar & Bernardi (2009) showed from a large sampleof SDSS early-type galaxies that the ratio M dyn /M star ∝ M γ star , with γ ≈ .
13, i.e., it increases with increasing M star ,in a way approximately independent of the age of the galax-ies. The left panel of Figure B3 shows the predicted ratio ofdynamical mass to stellar mass as a function stellar mass forgalaxies of different (mass-weighted) age at z = 0 and with c (cid:13) , 1– 21 ize Evolution Figure B3.
Predicted ratio of dynamical mass to stellar mass as a function of Age at z = 0 ( left ) and at different redshifts ( right ).The red long-dashed line is the tilt measured by Shankar & Bernardi (2009). The tilt of the fundamental plane is predicted to be ratherindependent of galaxy age at z = 0, and to flatten out at z > B/T > .
9. In agreement with the SDSS data, the modelcorrectly predicts a positive tilt in the FP relation very sim-ilar to the observed one (red long-dashed line), and withweak dependence on age. The parametrization adopted inEq. (5) naturally induces a tilt because lower mass galaxieshave proportionally smaller sizes, therefore lower dark mat-ter fractions within R H , thus proportionally lower velocitydispersions. The right panel of Figure B3 shows that the tiltis predicted to decrease at higher redshifts because galax-ies at higher redshifts are associated to less massive darkmatter hosts and also more compact galaxies, thus, again,proportionally lower dark matter fractions within R H . Thescatter of FP is measured to be small, thus being an impor-tant quantity to compare models with. However, a detailedstudy of the FP scatter is beyond the scope of this briefAppendix, and we will postpone it to future work. APPENDIX D: THE CONNECTION WITH THECENTRAL BLACK HOLE: EVOLUTION INSCALING RELATIONS
Dynamical observations have revealed that super-massiveBlack Holes (BHs) are ubiquitous at the centres of most,if not all, local massive, bulge-dominated galaxies, withtheir mass M bh ∼ − M ⊙ , tightly correlated withthe mass and velocity dispersion of the host bulge (seeFerrarese & Ford 2005; Shankar 2009, for recent reviews). Itis therefore natural that constraining the evolution of mas-sive spheroidal systems contemporarily implies understand-ing the origin and evolution of BHs and of such tight scalingrelations.The Munich SAM self-consistently follows the evolu-tion of BHs during the hierarchical evolution of galaxies.The model assumes that a fixed fraction of the cold gasis destabilized during merger events and feeds the centralBH (see, e.g., Marulli et al. 2008 for details). A parametercontrols how much gas mass is funnelled onto the centralBH and it is fine-tuned to reproduce the local M bh - M star relation. The model has been found to be consistent withAGN luminosity functions and quasar clustering at differ- ent redshifts and luminosities (see details in Marulli et al.2008, 2009; Bonoli et al. 2009, 2010).The left panel of Figure D1 shows the M bh - σ relationat different redshifts, as labelled, for a model with σ com-puted with Eq. (5). For reference, the grey stripe indicatesthe fit by Tundo et al. (2007) with its intrinsic scatter. Themodel produces a reasonable match to the data and wehave checked that neglecting any halo mass dependence in σ would have produced a flattening similar to the one observedfor the M star - σ relation in Figure 9.Interestingly, at variance with the evolution found forthe M star - σ relation, the model predicts a positive evolutionfor the M bh - σ relation, i.e., comparable or higher BH massesat higher redshifts at fixed velocity dispersion. This is inline with direct and indirect measurements of the M bh - σ at higher redshift in quasar host galaxies (e.g., Shields et al.2006; Woo et al. 2006, 2008; Treu et al. 2007; Shankar et al.2009; Gaskell 2009; Bennert et al. 2011). Though BH hostgalaxies get more compact and thus possess higher velocitydispersions at higher redshifts, the redshift evolution in the M bh - σ relation shows the opposite trend.The reason behind the opposite time behaviour of thetwo relations can be understood by looking at the right panelof Figure D1, which shows the predicted M bh - M star relationfor the same early-type galaxy subsample. The grey stripeis a linear relation of the type M bh = 2 × − M bulge withsome scatter, indicative of what suggested by a variety of lo-cal data (e.g., Marconi & Hunt 2003; H¨aring & Rix 2004).The overall agreement with the data at z = 0 in the normal-ization and slope of the M bh - M star relation is mostly a sim-ple consequence of the underlying model that assumes bothBH accretion and star formation rate to be proportional tocold gas reservoir. A genuine prediction of the model is in-stead that the M bh /M star ratio evolves at fixed stellar mass,possibly in a mass dependent way, increasing by a factorof a few at higher redshifts (see also Croton 2006), consis-tently with what derived by many groups (e.g., Decarli et al.2010). Thus the model predicts higher velocity dispersionsbut also more massive BHs at fixed stellar mass at higherredshifts, in a way to erase or even reverse the predicted evo-lution in the M bh - σ relation, in good agreement with the c (cid:13) , 1– 21 F. Shankar et al. data. In other words, while most of the BH mass is accretedduring the high- z gas-rich merger phase, the growth of thestellar mass and velocity dispersion of the host spheroid isprolonged to later times. c (cid:13) , 1– 21 ize Evolution Figure D1.
Left panel : Predicted M bh - σ relation at different redshifts, as labelled, for a model with a dark matter mass-dependent σ for galaxies with B/T > .
9. The grey stripe indicates the fit by Tundo et al. (2007) with its intrinsic scatter.
Right panel : Predicted M bh - M star relation at the same different redshifts for galaxies with B/T > .
9. The grey stripe shows a linear relation of the type M bh = 2 × − M bulge with some scatter, indicative of what suggested by a variety of local data.c (cid:13)000