Discriminating Between the Physical Processes that Drive Spheroid Size Evolution
Philip F. Hopkins, Kevin Bundy, Lars Hernquist, Stijn Wuyts, Thomas J. Cox
aa r X i v : . [ a s t r o - ph . C O ] O c t Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 21 August 2018 (MN L A TEX style file v2.2)
Discriminating Between the Physical Processes that Drive SpheroidSize Evolution
Philip F. Hopkins ∗ , Kevin Bundy , Lars Hernquist , Stijn Wuyts , , & Thomas J.Cox , , Department of Astronomy, University of California Berkeley, Berkeley, CA 94720 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA W. M. Keck Postdoctoral Fellow
Submitted to MNRAS, August 10, 2009
ABSTRACT
Massive galaxies at high- z have smaller effective radii than those today, but similar cen-tral densities. Their size growth therefore relates primarily to the evolving abundance oflow-density material. Various models have been proposed to explain this evolution, whichhave different implications for galaxy, star, and BH formation. We compile observations ofspheroid properties as a function of redshift and use them to test proposed models. Evolutionin progenitor gas-richness with redshift gives rise to initial formation of smaller spheroids athigh- z . These systems can then evolve in apparent or physical size via several channels: (1)equal-density ‘dry’ mergers, (2) later major or minor ‘dry’ mergers with less-dense galaxies,(3) adiabatic expansion, (4) evolution in stellar populations & mass-to-light-ratio gradients,(5) age-dependent bias in stellar mass estimators, (6) observational fitting/selection effects.If any one of these is tuned to explain observed size evolution, they make distinct predic-tions for evolution in other galaxy properties. Only model (2) is consistent with observationsas a dominant effect. It is the only model which allows for an increase in M BH / M bulge withredshift. Still, the amount of merging needed is larger than that observed or predicted. Wetherefore compare cosmologically motivated simulations, in which all these effects occur, &show they are consistent with all the observational constraints. Effect (2), which builds up anextended low-density envelope, does dominate the evolution, but effects 1, 3, 4, & 6 each con-tribute ∼
20% to the size evolution (a net factor ∼ M BH − σ similar to that observed. Key words: galaxies: formation — galaxies: evolution — galaxies: active — quasars: general— cosmology: theory
Observations have suggested that high-redshift spheroids havesignificantly smaller effective radii than low-redshift analoguesof the same mass (e.g. Daddi et al. 2005; van Dokkum et al.2008; Zirm et al. 2007; Trujillo et al. 2006b; Franx et al. 2008;Damjanov et al. 2009; van der Wel et al. 2008; Cimatti et al. 2008;Trujillo et al. 2007, 2006a; Toft et al. 2007; Buitrago et al. 2008).Whatever process explains this apparent evolution must be particu-lar to this class of galaxies: disk galaxies do not become similarlycompact at high redshift (Somerville et al. 2008; Buitrago et al.2008, and references therein). In addition, these high-redshift pop- ∗ E-mail:[email protected] There are of course different ways of defining galaxy “density” or“compactness,” for which the disk/spheroid difference is not the same (see ulations have been linked to observed sub-millimeter galaxies, themost rapidly star-forming objects in the Universe, and bright, high-redshift quasar hosts (Younger et al. 2008b; Tacconi et al. 2008;Hopkins et al. 2008d,b; Alexander et al. 2008). As such, these ob- e.g. Buitrago et al. 2008). In this paper, we will generally use these termsto refer to the coarse-grained phase-space density, which can be approx-imated by f ∼ M / ( R V ) ∼ / ( GR V ) , where R is the characteristicmajor-axis radius (proportional to e.g. effective radius of spheroids or diskscale-length), and V the characteristic velocity (dispersion or circular ve-locity). This is the quantity of theoretical interest as, in dissipationless pro-cesses, it is conserved or decreased ( f final ≤ f initial ; see e.g. the discussionin Hernquist et al. 1993). Quantities such as the orbital streaming motionand disk scale heights enter in the fine-grained phase-space density, whichalthough formally conserved is not observable. Similarly, as discussed in§ 2, we take “size” to refer to the semi-major axis half-light size.c (cid:13) Hopkins et al. servations represent a strong constraint on models of galaxy andbulge formation.More recently, Hopkins et al. (2009a) showed that the cen-tral densities of these high-redshift systems are similar to those ofmassive ellipticals at low redshifts; the primary difference between“small” (high-redshift) and “large” (low-redshift) systems relatesto the amount of observed low-density material at large radii, ab-sent in the high-redshift systems (see also Bezanson et al. 2009).There are therefore two important, related questions. First, how dohigh-redshift massive spheroids form, apparently without low den-sity material, but with their dense cores more or less in place rela-tive to their z = at the time of its formation primarily reflects the degree of dissipation involved – i.e. the lossof angular momentum by disk gas and its participation in a dense,central starburst (Cox et al. 2006b; Oñorbe et al. 2006; Ciotti et al.2007; Jesseit et al. 2007; Hopkins et al. 2009b,e). If all the massof a spheroid were formed in such a starburst, then one would ex-pect an extremely small size . kpc, comparable to the sizes of e.g.ULIRG starburst regions. If, on the other hand, none of the masswere formed in this way, the size of the remnant would simply re-flect the (large, ∼ −
10 kpc) extents of disk/star-forming progeni-tors. This leads to a natural expectation for size evolution.Disks at z = ∼ L ∗ disks at z > ∼ M ⊙ ellipticals with R e ∼ R e , allowing them to “catch up” to later forming sys-tems. Thus not only does such a general scenario anticipate evolu-tion in the median size-mass evolution, but also the nature of suchevolution: early buildup of dense regions via dissipation, followedby later growth in the extended, low-density wings as the progeni-tor population becomes less gas-rich with cosmic time.That being said, cosmological galaxy formation models havedifficulty explaining how systems could catch up from the mostextreme size evolution seen in the observations: a factor of ∼ effective radii R e at fixed stellar mass in the most massivegalaxies at z = ∼ largest radii for their mass – if so, these systems have evolvedto “overshoot” the median of the local size-mass relations, evolv-ing by more like a factor ∼
10 in effective radius (Gallazzi et al.2006; Bernardi et al. 2007; Graves et al. 2009). (Although this de-pends on whether “mass” is defined by dynamical or stellar mass,and we note van der Wel et al. 2009; Valentinuzzi et al. 2009, whoreach different conclusions.) As a consequence, other interpreta-tions of the observations and theoretical models for size evolutionhave been debated.In this paper, we show how dissipation drives the formationof smaller ellipticals at high redshift (§ 2) and consider the differ-ent explanations that have been proposed for how these massive,high-redshift ellipticals increase their apparent sizes at lower red-shifts, and construct the predictions made by each individual modelfor other observable quantities, including their velocity dispersions,central densities, masses, and profile shapes (§ 3). We compile ob-servations of these quantities and other constraints to break the de-generacies between the different models (§ 4). We then compare amodel motivated by cosmological simulations, in which many ofthese effects occur at different points in the galaxy’s evolution. Weshow how such a mixed model tracks through the predicted space,and how relatively small contributions from each of the proposedexplanations combine to yield order-of-magnitude cumulative sizeevolution (§ 5). We summarize and discuss our conclusions in § 7.Throughout, we assume a WMAP5 (Komatsu et al. 2009) cos-mology, and a Chabrier (2003) stellar IMF, but the exact choicesmake no significant difference to our conclusions.
First, we must consider how small, high-mass ellipticals are formedinitially. This is discussed in detail in e.g. Khochfar & Silk (2006);Naab et al. (2009); Feldmann et al. (2009) and Hopkins et al.(2009d); but we briefly review these results here.Figure 1 summarizes the important physics. First, considerthe mass density in passive ellipticals as a function of redshift( top ; from Bell et al. 2003; Bundy et al. 2005, 2006; Abraham et al.2007; Daddi et al. 2005; Labbé et al. 2005; van Dokkum et al.2006; Grazian et al. 2007). It is well-established that this declinesrapidly with redshift; we fit the observations shown in Figure 1 withthe simple functional form ρ ell ∝ exp ( − A z ) and find A ≈ . − . z =
2, only ∼
5% of the z = ∼ L ∗ ellip-tical mass density is in place. It is possible to consider this ingreater detail in terms of number counts or as a function of galaxymass, but the qualitative results are similar in each case: the pop-ulation “in place”, represented by the compact, high-redshift mas- Specifically, we plot the mass density in bulge-dominated galaxies, whichis not the same as the absolute mass density in all bulges. At high redshifts z > . z < (cid:13) , 000–000 esting Models for Spheroid Size Evolution l og ( ρ bu l ge ) [ M O • M p c - ] Bell et al.Bundy et al.Abraham et al.Daddi et al.Grazian et al.Labbe et al. ∝ exp{-Az} (A = 1.2 - 1.6) f ga s ( M ba r ∼ . × M O • ) Bell & de JongMcGaughKannappanPeuch et al.Shapley et al.Erb et al.Manucci et al. 〈 f gas 〉 ∝ (1+z) R e ( M s ph > M O • ) [ k p c ] Observed (van der Wel compilation)Sims with 〈 f gas (z) 〉 Fit to Simulations + 1Gyr delaySims with 〈 f gas (z) 〉 & R disk (z) Fit to Simulations Figure 1.
Top:
Evolution in the mass density in spheroid-dominated (com-pact) galaxies. Points are observations; dashed lines a fit. Evolution is steep;at all z , most spheroids are recently-formed and have not experienced e.g.dry mergers. The size-mass relation must primarily reflect how spheroidsform in situ . Middle:
Evolution in typical gas fractions of star-forming(spheroid progenitor) galaxies of total (baryonic) mass such that their majormerger will yield a & M ⊙ spheroid. Points observed; error bars showthe scatter in f gas at fixed mass, not the (smaller) uncertainty in the mean.Line shows a fit. Bottom:
Predicted sizes of ∼ M ⊙ ellipticals formed in situ at each redshift from gas-rich mergers with the expected h f gas ( z ) i .Circles show observed (mean) sizes compiled in van der Wel et al. (2008).Black diamonds show actual hydrodynamic simulation remnants with thesame progenitor disk size/structure but with different gas fractions appro-priate for each h f gas ( z ) i . Black solid line shows a fit, given the median scal-ings of h f gas ( z ) i and R e ( f gas ) . Dotted line is the same, allowing for a 1Gyrdelay after each merger before it is observed as a “passive” remnant. Reddiamonds are similar simulations, but also include viewing biases and stel-lar population effects (e.g. mock images at the appropriate times) and allowfor the maximum observationally inferred (weak) progenitor disk size evo-lution ( R disk ∝ ( + z ) . − . ). Red dashed line shows the appropriate fit in-cluding this additional scaling. Evolving gas-richness and rapid new buildupof ellipticals drives the evolution in the size-mass relation. Dry mergers andother effects do not dominate the relation, but explain how early-formingsystems “catch up to” (or exceed) the relation at low redshifts. sive elliptical population, is only a small fraction of the popula-tion that will be present at any significantly lower redshift (e.g.van Dokkum et al. 2008; Kriek et al. 2008a; Pérez-González et al.2008; Marchesini et al. 2008; van der Wel et al. 2009; Ilbert et al.2009). As such, evolution in the median size-mass relation – i.e.the average size of galaxies at a given stellar mass – must reflectevolution in the sizes at the time of formation .This is very important: at any redshift, most of the spheroidpopulation is recently formed , and has not had to evolve from someearlier redshift via e.g. dry mergers or any other channel. The evo-lution in the size-mass relation cannot, therefore, be the result ofall ellipticals forming early (with some size) and then dry mergingor experiencing other processes that increase their size to z = slope of the disk/rotationally supported galaxy size-massrelation is distinct from that of the spheroid size-mass relation at all z = − R e ( M ∗ | z ) ∝ ( + z ) − β , the observations constrain the maximum β for disk galaxies to be β < . β < . β ≈ β ≈ . − .
7. It is clear, there-fore, that the sizes of ellipticals at formation, and the evolution intheir size-mass relation, cannot simply reflect the sizes of their pro-genitors.Simple phase-space considerations, however, make it impos-sible to increase densities in dissipationless (purely stellar) merg-ers (Hernquist et al. 1993). But in sufficiently gas-rich mergers,the remnant size can be much smaller than that of the progen-itor. Gas dissipation makes this possible; disk gas loses angularmomentum via internal torques in the merger (Barnes & Hernquist1991, 1992), and can then dissipate energy, fall to the center,and build stars in a compact central starburst on scales ≪ kpc(Mihos & Hernquist 1994b, 1996). High-resolution hydrodynamicsimulations, and basic physical arguments, have shown that this Although there is some debate regarding the degree of evolution innumber density of the very most massive galaxies at z < .
8, the ∼ L ∗ , M ∗ ∼ M ⊙ population on which we focus here (and which de-fines the observed samples to which we compare) dominates the ef-fects shown in Figure 1 and clearly shows a rapid decrease with red-shift even over this range (see e.g. the references above and Bundy et al.2005; Pannella et al. 2006; Franceschini et al. 2006; Borch et al. 2006;Brown et al. 2007; Pozzetti et al. 2009). Moreover, the rapid decline in pas-sive spheroid number density with redshift is clear at all masses at z & . (cid:13) , 000–000 Hopkins et al. degree of dissipation is the primary determinant of the remnantspheroid size (Cox et al. 2006b; Robertson et al. 2006a; Naab et al.2006a; Oñorbe et al. 2006; Ciotti et al. 2007; Jesseit et al. 2007,2008; Covington et al. 2008; Hopkins et al. 2009b,e). To lowestorder, this degree of dissipation – i.e. the mass fraction formeddissipationally – simply reflects the cold gas fractions availablein the progenitor disks at the time of the merger (Hopkins et al.2009c). To rough approximation, one can fit the results of high-resolution simulations of gas-rich mergers (references above) andestimate how the remnant spheroid size scales with this gas frac-tion: R e ( M ∗ | f gas ) ≈ R e ( M ∗ | f gas = ) exp ( − f gas / . ) . (Based onthe arguments above, R e ( M ∗ | f gas = ) is equivalent, modulo a ge-ometric prefactor, to the pre-merger disk effective radii.) If the gasmass is sufficiently large ( ∼ / middle )compiles observations from Bell & de Jong (2001), Kannappan(2004), and McGaugh (2005) at low redshift, and Shapley et al.(2005); Erb et al. (2006); Puech et al. (2008); Mannucci et al.(2009) at redshifts z ∼ − Specifically, we plot thegas fractions observationally inferred for disk/rotationally domi-nated, star-forming galaxies of baryonic masses ∼ . × M ⊙ (the stellar masses may be a factor of a couple lower, correspond-ing to the observed f gas ) such that after a major merger (whichwill increase the total mass and turn the gas into stars), the sys-tem will be comparable to the ∼ M ⊙ observed massive, com-pact ellipticals. Figure 1 shows that f gas grows from ∼ . − . z = ∼ . − . z = . − h f gas i = . ( + z ) . ).This leads to an expected evolution in the sizes of spheroidsat their time of formation. A detailed set of predictions forthis size evolution as a result of evolving gas fractions is pre-sented in Hopkins et al. (2007a, 2009d) and a similar model inKhochfar & Silk (2006); in Figure 1 we simply summarize thekey result. Figure 1 ( bottom ) plots the expected size of major gas-rich ∼ M ⊙ merger remnants, considering the mean gas frac-tions as a function of redshift (shown above). We compare thesewith the observed average sizes of spheroids of the given massat each redshift. We show the results of simulated remnants fromHopkins et al. (2009b), with the same initial disk sizes appropri-ate for this mass, but with the relevant f gas for the median at eachredshift. We also show the corresponding median trend estimatedby simply combining the fitted R e ( M ∗ | f gas ) scaling above with the At z =
0, the gas fractions shown are based on measured atomic HI gasfractions; Bell & de Jong (2001) correct this to include both He and molec-ular H ; McGaugh (2005) correct for He but not H ; Kannappan (2004)gives just the atomic HI gas fractions (this leads to slightly lower esti-mates, but still within the range of uncertainty plotted; H may account for ∼ −
30% of the dynamical mass, per the measurements in Jogee et al.2005). We emphasize that these gas fractions are lower limits (based onobserved HI flux in some annulus). At z =
2, direct measurements are notalways available; the gas masses from Erb et al. (2006) are estimated indi-rectly based on the observed surface densities of star formation and assum-ing that the z = h f gas ( z ) i scaling. In these cases the disks are all just as extendedas low-redshift disks (i.e. yield R e ( M ⊙ | f gas = ) ≈ − ∼ Gyr ago; and so had correspondingly somewhat higher f gas reflecting the expectation at that earlier time); this gives similarbut slightly stronger evolution. We can also allow for some mod-erate disk size evolution. If we scale the sizes of progenitor disks(and, as a consequence, the remnant R e in the absence of gas) bythe maximum allowed by the observations, ∝ ( + z ) − . , we againfind similar but slightly stronger evolution. These simple size pre-dictions agree very well with the observed spheroid sizes at all in-termediate and high redshifts.In other words, it is straightforward to form ∼ kpc-sized ∼ M ⊙ ellipticals at z = , and in fact such sizes are thenatural expectation given the observed/expected gas-richness ofspheroid-forming mergers at these redshifts. The difficulty is not “how to form” such ellipticals, nor is it even to explain the av-erage evolution in the size-mass relation. Rather, the difficulty isthat, as discussed in § 1, such systems clearly do not passivelyevolve to z = z =
0. It may even bethe case that early forming systems would to be the largest fortheir mass (i.e. have lower effective densities) at z = above the Shen et al. (2003) size-mass relation for more recently-assembled field galaxies (see e.g.Batcheldor et al. 2007; von der Linden et al. 2007; Kormendy et al.2009; Lauer et al. 2007c; Bernardi et al. 2007).Some process, therefore, not only increases the sizes of thesehigh-redshift early-forming systems so as to “keep pace” with themean evolution in the size-mass relation, but may even need to“overshoot” the relation. Some hint of this can be seen even in Fig-ure 1; at the lowest redshifts, assuming all ellipticals are formedin situ from gas-rich mergers actually under -predicts the medianlow-redshift sizes. By this late time (unlike at high redshifts) a non-trivial fraction of the population has formed earlier and undergonesome subsequent evolution, bringing up the average size at thesemasses. In what follows, therefore, we consider how such systemsmight grow in size at a pace equal to or greater than the rate ofchange in the size of newly-forming systems shown in Figure 1.Note that the sizes shown in Figure 1, and the simulation sizesto which we refer throughout this paper (and, where possible, theobserved sizes) refer to the semi-major axis lengths, for ellipticalsystems. This is almost identical to the (projected) circular radius R that encloses 1 / not identical to the “circularized”radius R circ ≡ √ a b , where a and b are the major and minor axislengths ( a ≈ R e ), respectively ( ǫ ≡ − b / a being the standard def-inition of ellipticity). The reason for our choice is that the majoraxis length or projected half-light circular radius is more physi-cally robust, and relevant to the constraints from phase space den-sities and merger histories. A thin disk, for example, viewed edge-on, has a vanishingly small circularized radius (arbitrarily small b ), c (cid:13) , 000–000 esting Models for Spheroid Size Evolution even though the parameter of physical interest, the scale length,is the same. Examination of the suite of simulations shown here,for example, shows that while at low ellipticity systems can be ei-ther large or small, there are no systems with very large ellipticityand large R circ , even though such systems are merely flattened rel-ative to their counterparts, and have similar scale lengths, velocitydispersions, energetics, and physical phase space densities. Follow-ing Dehnen (1993), one can show analytically that dissipationlesslyrandomizing a system which is flattened owing to some rotation,conserving total energy and phase space density, leads to very littlechange in the projected major-axis radius (this is the basic reasonwhy the projected R e of dissipationless disk-disk merger remnantsis similar to the in-plane R e = . h of their progenitor disks, dis-cussed above), but will obviously increase R circ by an arbitrarilylarge factor depending on the thickness of the original system. Be-cause more gas-rich mergers lead to remnants with more rotation(on average), the ellipticity of the systems in Figure 1 can be some-what higher at high redshift, and therefore their circularized radiievolve even more steeply (averaging over all inclinations, though,the effect is weak, adding a power ∼ ( + z ) − . to the redshiftevolution). We consider the following sources of size evolution for systemsinitially formed at some high redshift ( z ∼
2) as compact, massivegalaxies ( R e ∼ M ∗ ∼ M ⊙ ), illustrated in Figure 2: “Identical” Dry Mergers: Often, when the term “drymergers” is used in the context of models for size evolution,what is actually assumed is not general gas-poor merging butspecifically spheroid-spheroid re-mergers, between spheroids withotherwise identical properties (or at least identical profile shapesand effective densities, in the case of non-equal mass mergers).In a 1:1 such merger, energetic arguments as well as simulations(Hernquist et al. 1993; Hopkins et al. 2009e) imply profile shapeand velocity dispersion are conserved, while mass and size double(more generally, R e ∝ M ). Minor/Late Accretion:
In fact, the scenario above is not expected to be a primary growth channel for massive ellipticals,nor is it expected to be the most common form of “dry merger.”Rather, in models, at later times the typical secondary (even inmajor mergers) is a later-forming galaxy – a gas-poor disk ormore “puffy” spheroid that was itself formed from more gas-poor mergers and therefore less compact (Hopkins et al. 2009d;Naab et al. 2009; Feldmann et al. 2009). At the highest masses,growth preferentially becomes dominated by more and more minormergers with low-effective density galaxies (Maller et al. 2006;Hopkins et al. 2009g). The secondaries, being lower-density, buildup extended “wings” around the high central density peak inthe elliptical. The central density and velocity dispersion remainnearly constant, while the Sersic index of the galaxy increaseswith the buildup of these wings (see e.g. Naab & Trujillo 2006;Hopkins et al. 2009e,c). The effective radius grows much faster perunit mass added in these mergers – it is possible to increase R e of agiven elliptical by a factor ∼ R e ∝ M − . ). Adiabatic Expansion:
If a system loses mass from its centralregions in an adiabatic manner, the generic response of stars anddark matter will be to “puff up,” as the central potential is less deep. For a spherically symmetric, homologous contraction of shells withcircular orbits, this reduces to the criterion that M ( r ) r = constant.More general scenarios behave in a similar manner (modulo smallcorrections; see Zhao 2002; Gnedin et al. 2004). If a galaxy couldlose a large fraction of its central (baryon-dominated) mass, eitherby efficiently expelling the material from stellar mass loss or byblowing out a large fraction of the baryonic mass in an initiallyvery gas-rich system (from e.g. quasar feedback; Fan et al. 2008),the radius will grow correspondingly ( R e ∝ M − ). The profileshape will be conserved (to lowest order, although this dependsat second order on the stellar age distribution versus radius) butuniformly inflated, the central density will decrease sharply, andthe velocity dispersion will decrease ∝ R − e . M ∗ / L Gradients:
Massive, old ellipticals at z = R e . However, the same is not necessarily true for young ellipticalsrecently formed in mergers; these can have blue cores at theircenters with young stars just formed in the merger-driven starburst(e.g. Rothberg & Joseph 2004). Simulations and resolved stellarpopulation analysis suggests that the resulting gradient in M ∗ / L (brighter towards the center) can lead to smaller R e by up to a factor ∼ B ), relative to the stellar mass R e (Hopkins et al. 2008c). As the system ages, these gradients willvanish and R e in optical bands will appear to increase; moreover,depending on the exact band observed, the late-time R e , light mayactually over-estimate the stellar mass R e (as e.g. age gradientsfade and long-lived metallicity gradients remain, yielding aredder center and hence apparently less concentrated optical lightdistribution). Thus at both early and late times, M ∗ / L gradientscan, in principle, yield evolution in the size at fixed wavelength,while conserving the stellar mass R e . Obviously, the central massdensity will remain constant, and the velocity dispersion will onlyweakly shift (with the appropriate luminosity-weighting). Seeing/Observational Effects:
The large effective radii ofmassive, low-redshift ellipticals are driven by material in low sur-face density “envelopes” at large radii. This is difficult to recover athigh redshifts. Moreover, galaxy surface brightness profiles are notperfect Sersic (1968) profiles, so the best-fit Sersic profile and cor-responding R e will depend on the dynamic range observed (see e.g.Boylan-Kolchin et al. 2005; Hopkins et al. 2009a). Together, theseeffects can, in principle, lead to a smaller fitted R e (from samplingonly the central regions, inferring a smaller n s and less low-densitymaterial) at high redshifts (although it is by no means clear that thebiases must go in this direction). At lower redshifts, where surfacebrightness limits are less severe, more such material would be re-covered, leading to apparent size-mass and profile shape evolution( R e changes at fixed M ∗ ), without central surface density or velocitydispersion evolution.Another effect that might occur is that high-redshift ellipticalscould be more flattened than those at low redshift. If the defini-tion of effective radius used is that of the “circularized” radius R circ = √ a b , then a flattened high-redshift system could have small R circ from edge-on sightlines (and smaller median R circ , by a lesserfactor), per the discussion above in § 2. Indeed, many such compactsystems are observed to be relatively elliptical (Valentinuzzi et al.2009; van Dokkum et al. 2008). This could in fact be a realphysical effect – high redshift systems might have more rotationor larger anisotropy (see e.g. van der Wel & van der Marel 2008). c (cid:13) , 000–000 Hopkins et al. l og ( Σ [ P h ys i c a l ] ) [ M O • k p c - ] Identical Dry Mergers
ProgenitorDescendant M ∗ (i) = 2.0e10 M O • R e (i) = 1.1 kpc σ c (i) = 210 km s -1 n s (i) = 5M ∗ (f) = 4.0e11 M O • R e (f) = 10.0 kpc σ c (f) = 270 km s -1 n s (f) = 7 Minor/Late Accretion M ∗ (i) = 2.0e11 M O • R e (i) = 2.2 kpc σ c (i) = 250 km s -1 n s (i) = 3 Adiabatic Expansion M ∗ (i) = 8.0e11 M O • R e (i) = 5.0 kpc σ c (i) = 550 km s -1 n s (i) = 7 l og ( Σ [ A ppa r en t] ) [ M O • k p c - ] M ∗ /L Gradients InferredTrue M ∗ (i) = 4.0e11 M O • R e (i) = 3.7 kpc σ c (i) = 280 km s -1 n s (i) = 10 Observational Effects M ∗ (i) = 3.3e11 M O • R e (i) = 3.5 kpc σ c (i) = 270 km s -1 n s (i) = 3 M ∗ Uncertainties M ∗ (i) = 2.0e12 M O • R e (i) = 10.0 kpc σ c (i) = 270 km s -1 n s (i) = 7 Figure 2.
Evolution of physical ( top ) or observationally inferred ( bottom ) surface stellar mass density profiles, according to different models. We consider sixmodels, described in the text: the physical stellar mass profile can change owing to identical dry mergers (doubling M ∗ and R e ; top left ), minor/late accretion(building up an extended envelope; top center ), or adiabatic expansion (mass loss leading to uniform inflation; top right ). The inferred mass profile (fitted fromobservations, assuming standard stellar populations and constant M ∗ / L ) can change owing to the presence of stellar mass-to-light ratio gradients (from youngcentral stellar populations; bottom left ), seeing effects and surface brightness dimming (points show a simulated z = r / -law profile given the observed range; bottom center ), or discrepancies between the true and best-fit stellar mass(owing to e.g. contribution of AGB stars; bottom right ). In each case, the plotted initial profile is a factor of ∼ − z = Some process, e.g. dynamical heating from bars, or minor mergers,or clumpy star formation, could then vertically heat the systemsby scattering stars and make them more round, while contributinglittle net mass or energy. Major axes would be little affected,while circular radii would increase. This effect could also occurowing to selection effects; if high-redshift samples (selected via acombination of Sersic indicex estimates and/or stellar populationproperties) include diskier systems or more early-type disks (e.g.Sa galaxies). In either case, we include it as part of this categorybecause the systems would appear to evolve along similar tracks,as both the apparent and real evolution in R circ would involve nosignificant change in σ , M ∗ , or Σ c . Stellar Mass Uncertainties:
If the (uncertain) contributionof AGB stars to near-infrared light is large in young ellipticals(ages . z ∼
2; Kriek et al. 2006),then the stellar mass M ∗ as derived from commonly used stellarpopulation models lacking a proper treatment of the TP-AGBphase (e.g. Bruzual & Charlot 2003) may be over-estimated byfactors ∼ a few (Maraston 2005). The difference will vanish as thepopulations age. This change in inferred M ∗ will lead to apparent R e − M ∗ evolution ( M ∗ changes as fixed R e ); systems will appear less massive, but conserve R e , σ , and profile shape. Likewise,redshift evolution in the stellar initial mass function, suggested(indirectly) by some observations (Hopkins & Beacom 2006;van Dokkum 2008; Davé 2008), could yield a similar effect.Figure 3 plots how individual galaxies (or, since the historyof an individual galaxy will be noisy, the median of a populationof similar galaxies) evolve forward in time, according to these dif-ferent models. We assume that at z =
2, all systems “begin” onan observed R e − M ∗ relation similar to that inferred for observedsystems at high mass – specifically at an observed 10 M ⊙ with R e = z =
0, they will lie on theobserved size-mass relation from Shen et al. (2003). In each case,we assume that one and only one of the effects above operates, andwe consider the strength of the effect to be arbitrary – we make itas strong as necessary to evolve the systems onto the z = z = R e − M ∗ relation along these tracks.This is not directly comparable to what is done observation- c (cid:13) , 000–000 esting Models for Spheroid Size Evolution gal / M O • )110 R e [ k p c ] z = z = Tracks of Individual Galaxies R e ff ( z ) / R e ff ( z i ) Identical Dry MergersMinor/Late AccretionAdiabatic Expansion M ( z ) / M ( z i ) M ∗ /L GradientsObservational EffectsM ∗ Uncertainties σ ( z ) / σ ( z i ) Σ c ( z ) / Σ c ( z i ) n s ( z ) - n s ( z i ) Figure 3.
Evolution of a fixed population of galaxies, from fixed initial conditions, according to different models.
Top Left:
Tracks made by galaxies in thesize-mass plane. Solid line is the adopted “initial” z = M ⊙ with R e ∼ z = Top Center:
Corresponding evolution in effective radius (relativeto the initial z = Top Right:
Corresponding evolution in stellar mass, required to produce the given size evolution.
Bottom Left:
Correspondingevolution in the velocity dispersion σ . Bottom Center:
Evolution in the central/maximum surface stellar mass (not luminosity) density (i.e. some M ∗ kpc − averaged inside e.g. ∼ ∼
1% of the light; not the effective surface brightness).
Bottom Right:
Evolution in apparent galaxy profile shapein e.g. fixed rest-frame B -band, parameterized with the best-fit Sersic index n s . The different models clearly map out different average tracks in this space. ally, as the systems “end up” at different stellar masses. However,knowing how a galaxy population evolves forward in each model,it is straightforward to calculate how properties should evolve atfixed stellar mass, looking back on populations with the same ob-served M ∗ at different redshifts. We show this in Figure 4. In detail,we force the z = R e ( M ∗ ) for ∼ M ⊙ galaxies, where the mostdramatic size evolution has been seen. Integrating all populations Specifically, we adopt the observed z = R e = . ( M ∗ / M ⊙ ) . (Shen et al. 2003), velocity dispersion σ = − ( M ∗ / M ⊙ ) . (Hyde & Bernardi 2009), central stel-lar mass surface density Σ c = . × M ⊙ kpc − ( M ∗ / M ⊙ ) − . (fitted from the compilation in Hopkins et al. 2009f), and Sersic index (fit-ting the entire galaxy profile to a single Sersic index; as noted above thisshould be treated with some caution as the results depend on the fitteddynamic range) n s = . ( M ∗ / M ⊙ ) . (Ferrarese et al. 2006). Thecentral surface density Σ c must be defined: one can adopt either the ex-trapolation of the best-fit Sersic profile to r =
0, or the average surfacedensity within some small annulus (e.g. fixed physical ∼ ∼ R e / Σ isonly a very weak function of R in massive systems at these radii). For con-venience we adopt the fixed 100pc mean definition. We assume the specificform for the evolution of the size-mass relation R e ( M ∗ | z ) = R e ( M ∗ | z = ) × ( + z ) − . . This is convenient and yields a good fit to the observationsshown in Figure 4, appropriate for massive galaxies, but the evolution maybe weaker at lower masses (which has no effect on our conclusions). backwards in time, we then reconstruct the properties at fixed stel-lar mass ( ∼ M ⊙ ) according to each model.Clearly, these correlated properties can break degeneracies be-tween different models tuned to reproduce the size distribution. Forexample, if adiabatic expansion were the explanation for the ob-served evolution, the velocity dispersions of ∼ M ⊙ galaxiesat z ∼ − ∼ − σ ∼
600 km s − at these masses. Onthe other hand, if minor/late accretion is responsible, the primarychange has been the buildup of low-surface brightness “wings” inthe profile, which contribute negligibly to σ (leading to dispersions ∼ −
250 km s − at z ∼ − M ∗ ∼ M ⊙ )galaxies are compiled in van der Wel et al. (2008); we adopttheir compilation (see Trujillo et al. 2006a; Longhetti et al.2007; Zirm et al. 2007; Toft et al. 2007; Cimatti et al. 2008;van Dokkum et al. 2008; Franx et al. 2008; Rettura et al.2008; Buitrago et al. 2008). The Sersic indices of the high-redshift systems are presented in van Dokkum et al. (2008) andvan der Wel et al. (2008); we compare these to the distributionof Sersic indices as a function of stellar mass at z = z = c (cid:13) , 000–000 Hopkins et al. R e ( z ) / R e ( ) Population at Fixed M ∗ Identical Dry MergersMinor/Late AccretionAdiabatic Expansion σ ( z ) / σ ( ) Σ c ( z ) / Σ c ( ) M ∗ /L GradientsObservational EffectsM ∗ Uncertainties n s ( z ) - n s ( ) Figure 4.
Evolution in properties of galaxies at fixed stellar mass, given the different models from Figure 3.
Top Left:
Size evolution (median size observed atredshift z , for galaxies of the same fixed observed stellar mass, relative to the median size observed at z = & M ⊙ ) galaxies – i.e. attribute the observed size-massrelation evolution in each case entirely to just the one model. We compare to the observed size evolution (points; from the compilation in van der Wel et al.2008). In the following panels, we see the predicted consequences of this for other quantities measured at fixed stellar mass. Top Right:
Velocity dispersion.Observations are compiled in Cenarro & Trujillo (2009) (squares; the z > . Bottom Left:
Cen-tral/peak stellar mass surface density ( M ∗ kpc − ). Observations are compiled in Hopkins et al. (2009a). Bottom Right:
Profile shape/Sersic index. Observationscompiled from Hopkins et al. (2009b, z = z ∼ z ∼ vations, is also compatible with their sample). The high-redshiftobservations do not resolve the small radii needed to directlymeasure the central stellar mass density Σ c ; we adopt the estimatesfrom Hopkins et al. (2009a) based on the best-fit profiles pre-sented in van Dokkum et al. (2008) and van der Wel et al. (2008),extrapolating the fits from ∼ − ∼ − z = . − z = . −
2, along with a couple of individual object measurementsof σ ; we show their results as well (from the stacked spectra; theindividual detections are on the low- σ end of the allowed rangefrom the stack). We also show the recent detection presented invan Dokkum et al. (2009), of a very large σ ∼
500 km s − (albeitfor a more massive M ∗ = × M ⊙ system); however, we notethat this is a single object (one of the brightest in the field), not a statistical sample, and the uncertainties in the measurement of σ are large. None of the models is ideal. Identical dry mergers can easily ex-plain core creation in massive ellipticals (owing to the “scouring”action of binary supermassive BHs), but move systems relativelyinefficiently with respect to the R e − M ∗ relation. It requires a verylarge (order-of-magnitude) mass growth to get systems onto the z = z =
2; this would yield too many ∼ M ⊙ systems today (relative to e.g. the local mass functionfrom Bell et al. 2003), given the number density of compact z = Σ c is much too strong, and that in σ somewhat so.Minor/late mergers are a more efficient way to increase effec-tive radii relative to the size-mass relation. This model fares bestin Figure 4 – in fact, it is the only model that appears at least c (cid:13) , 000–000 esting Models for Spheroid Size Evolution marginally consistent with all the observational constraints. How-ever, the implied number of such mergers, to yield the full factor ∼ R e ( M ∗ ) , is large – larger than that predicted by cos-mological models (Khochfar & Silk 2006; Hopkins et al. 2009d;Naab et al. 2009) or permitted by observational constraints imply-ing ∼ − ∼ z = z ∼ ∼ . − z = z = R e ). This is ultimately reflected in the factthat models including just this effect tend to predict more moderatefactor ∼ − loss , so there is no issuewith the mass function. However, the mass loss required to yielda large change in R e is correspondingly large, >
50% of the z = z > ∼ . − ∼ σ and Σ c is much larger than that observed.Mass-to-light ratio gradients should be present at z =
2, if theobserved stellar population gradients at low redshift are extrapo-lated back in time. However, using these to obtain more than afactor ∼ M ∗ / L in the central regions (since thereis effectively an upper limit to M ∗ / L at large radii given by stel-lar populations with an age equal to the Hubble time). Such pop-ulations would require near-zero age, not the ∼ . − n s at high- z (since high- n s profiles have moreconcentrated central light), in conflict with the observations in Fig-ure 4.Attributing the entire evolution to incorrect stellar mass es-timates appears similarly unlikely. It requires invoking some-thing more than just the known differences between e.g. theMaraston et al. (2006) and Bruzual & Charlot (2003) stellar pop-ulation models – applied to the observed z = ∼ . σ clearly disagrees with the observations.Observational biases from e.g. profile fitting appearmarginally consistent with the constraints in Figure 4. However, attempts to calibrate such effects typically find they lead to bias inhigh-redshift sizes at the factor ∼ ∼ −
10 desired. Stacking the high-redshift dataalso appears to yield similar sizes, so it is unlikely that a verylarge fraction of the galaxy mass lies at radii not sampled by theobservations (van der Wel et al. 2008). In fact, experiments withhydrodynamic simulations suggest that, if anything, biases in fit-ting Sersic profiles may lead to over -estimates of the high-redshiftsizes (S. Wuyts et al., in preparation), and the higher dissipationalfractions involved in forming compact ellipticals can yield sharptwo-component features that bias the fits to higher Sersic indicesand corresponding effective radii. Moreover, it is difficult toinvoke these effects to explain the observed R e ( M ∗ ) evolution atlower stellar masses, where z = z = − .
3, and use this to compare the evolution in ma-jor and minor axis lengths. Restricting our analysis to major axislengths alone, we find that the evolution is slightly weaker thanthat using circularized radii (in other words, there is some increasein the median ellipticity of the samples with redshift); however,the effect appears to be relatively small, accounting for a factor ∼ ( + z ) − ( . − . ) in evolution (i.e. ∼
20% of the total size evolu-tion). This is comparable to the effects anticipated from simulations(see § 2).As observations improve, it appears unlikely that these effectscan account for the full evolution, although it may be important(altogether) at the factor ∼ . − In principle, all of these effects can occur. We therefore considera cosmological model for galaxy growth in which they are all in-cluded. We follow a “typical” massive spheroid formed at z ∼ z = ∼ L ∗ disks, with typical gas fractions for their mass and red-shift. We specifically choose as representative one of the simula-tions described in detail in Hopkins et al. (2008a) (typical of merg-ers with such gas fractions, with common orbital parameters andhalo properties). The simulation “begins” at z = c (cid:13) , 000–000 Hopkins et al. -20 -10 0 10 20x [kpc]-20-1001020 y [ k p c ] l og ( Σ ) [ M O • k p c - ] z = 2M ∗ = 10 M O • R e (M ∗ ) = 1.3 kpc σ c = 263 km s -1 〈 Age 〉 = 0.5 GyrTotalDissipational (25%)Dissipationless (75%) µ B [ m ag a r cs e c - ] R e (L B ) = 1.1 kpcR e (fit) = 0.8 kpcn s (fit) ∼ -20 -10 0 10 20x [kpc]-20-1001020 y [ k p c ] l og ( Σ ) [ M O • k p c - ] Original Profile + Adiabatic Expansion z = 0M ∗ = 8 × M O • R e (M ∗ ) = 1.5 kpc σ c = 200 km s -1 〈 Age 〉 = 11 Gyr µ B [ m ag a r cs e c - ] R e (L B ) = 1.6 kpcR e (fit) = 1.6 kpcn s (fit) ∼ Aged & Observed at z=0 -20 -10 0 10 20x [kpc]-20-1001020 y [ k p c ] l og ( Σ ) [ M O • k p c - ] + Identical (1:3) Dry Merger z = 0M ∗ = 1.2 × M O • R e (M ∗ ) = 1.9 kpc σ c = 215 km s -1 〈 Age 〉 = 11 Gyr µ B [ m ag a r cs e c - ] R e (L B ) = 2.1 kpcR e (fit) = 1.9 kpcn s (fit) ∼ -20 -10 0 10 20x [kpc]-20-1001020 y [ k p c ] l og ( Σ ) [ M O • k p c - ] + Minor/Late Mergers z = 0M ∗ = 2.6 × M O • R e (M ∗ ) = 6.7 kpc σ c = 240 km s -1 〈 Age 〉 = 10 GyrTotalDissipational (17%)Dissipationless (83%) µ B [ m ag a r cs e c - ] R e (L B ) = 7.3 kpcR e (fit) = 9.3 kpcn s (fit) ∼ Figure 5.
Illustration of a typical history from cosmological simulations, realized in high-resolution hydrodynamic simulations, for a massive, early-formingspheroid.
Top:
Image of stellar surface density ( left ), and axially-averaged projected stellar surface mass density profile ( center ; solid) after the spheroid firstforms in a z ∼ f gas ∼ − z = ∼ B -band profile ( right ) with half-lightradius R e ( L B ) , and a mock z = R e ( fit ) and index n s . We compare to the observed z > Second from Top:
Same, after adiabatic expansion and stellar evolution are allowed to operate. Stars lose appropriate mass foreach stars age/metallicity evolved forward to z =
0, which then virializes; the resulting mass profile is compared to the original ( z =
2) profile ( center ). We alsore-construct the B -band profile, with the stellar populations aged in this way (allowing for M ∗ / L evolution per the observed/simulated properties), and re-fit itassuming image depth comparable to local observations. Second from Bottom:
Same, including a single (mass ratio 1:3) “identical” dry merger (high-redshiftdry merger with similarly gas-rich disk or compact elliptical) in the history.
Bottom:
Further adding a small series of later major and minor mergers of less-dense systems (typical lower-redshift material in disks and spheroids). Since this material has lower dissipational content (forming from lower-redshift, moregas-poor disks), the net dissipational fraction goes down, and extended envelopes preferentially build up. We compare the B -band profile to a few observedmassive spheroids in Virgo (Kormendy et al. 2009). c (cid:13) , 000–000 esting Models for Spheroid Size Evolution R e ( z ) / R e ( ) Σ c ( z ) / Σ c ( ) Net Evolution (all effects)Contributions from : Identical Dry MergersMinor/Late AccretionAdiabatic ExpansionM ∗ /L GradientsObservational EffectsM ∗ Uncertainties σ ( z ) / σ ( ) n s ( z ) - n s ( ) Figure 6.
Predictions of a cosmological model where all the effects in the text contribute to size evolution (as in Figure 5). In each, we consider the predictionfor the net evolution (black solid line) as in Figure 4. We show the contribution to evolution in each track from each of the individual effects described in thetext. The “late/minor merger” channel dominates growth, yielding a factor ∼ − ∼
2) mass growth, but each of the othereffects here contributes a small ∼
20% additional effect. Together, this gives an additional factor ∼ − at about z ∼ . − ∼ −
40% is entirely chan-neled into the central starburst, the effective radius – 50% mass ra-dius – lies just outside the compact central starburst region). Sincethe halo mass has crossed the “quenching threshold” where coolingbecomes inefficient ( ∼ M ⊙ ; see Kereš et al. 2005) with thismerger, and the observations indicate that these high-redshift sys-tems do not form many stars from their formation redshifts to today(likewise massive ellipticals today have not formed significant starssince z ∼ − z = top ) shows the resulting stellarmass surface density profile, together with some salient parameters.The system has a true stellar mass of M ∗ = . × M ⊙ , a mass-weighted central velocity dispersion of σ =
260 km s − , a cen-tral/peak stellar surface mass density of Σ c = × M ⊙ kpc − ,and a stellar effective radius R e = . σ , Σ c , and R e are projected quantities; here, we sample the system at each time byprojecting it along ∼
100 lines of sight, uniformly sampling the unit sphere, and quote the median, but the sightline-to-sightline variancein these quantities is small ( . . z =
2. Thestellar population properties (ages and metallicities) are determinedself-consistently from the star formation and enrichment model inthe simulation; dust properties are computed self-consistently fromthe simulation gas and sub-resolution ISM model, following themethodology in Hopkins et al. (2005b,a) (but at this time, the gasis depleted or in the hot halo, so this is a relatively small correc-tion). The B -band light weighted stellar population age at this timeis ∼
500 Myr, very similar to that inferred from the observed high-redshift systems (Kriek et al. 2006). Figure 5 shows the true rest-frame B -band light profile, and corresponding parameters. There isa gradient in the stellar mass-to-light ratio, owing to the recently-forming starburst populations; as a result, the profile shape and ef-fective radius are slightly different in B -band than in stellar mass.The B -band effective radius is 1 . M ∗ / L gradients in this sys-tem are not negligible, but contribute only a ∼ −
30% effect inthe size.We compare this directly to the observed best-fit (de-convolved) profiles of z > c (cid:13) , 000–000 Hopkins et al. sampled and not severely affected by the PSF (roughly ∼ − B -band, with instru-ment quality comparable to HST imaging, and a simple representa-tive PSF, and (simple noise) sky background, and fit it in a mannerdesigned to mimic the observations (here a one-dimensional fit tothe circularized profile allowing for varying ellipticity and isopho-tal twists with radius). The mock “observed” best-fit Sersic param-eters are shown in Figure 5. The best-fit observed R e is 0 . n s = = . n s = not the same as in the outer re-gions (thus leading to a fit that reflects that central portion), lead toa slightly smaller fitted size than the true B -band size. But the ef-fect is small, only ∼ − ǫ = − b / a ≈ . − .
3. Thus, if we were to adopt the circularized radius insteadof the circular half-light radius or major axis radius, we wouldobtain a smaller radius by a factor of p b / a ≈ . − .
89. Al-though a small correction, this is comparable to the other effectshere and is not negligible – even in a relatively round system viewedfrom a random viewing angle, the circularized radius yields another ∼ −
20% smaller apparent size. In more flattened systems, orfrom the more extreme viewing angles for this particular simula-tion, the effect can be as large as ∼ − M ∗ = . × M ⊙ . Inother words, the maximum of this effect is a ∼
20% bias in stellarmass, which given the observed correlation R e ∝ M . ∗ , would leadto just a 5% effect in the apparent size-mass relation.As the system evolves in time, the stellar mass-to-light ra-tio gradients become progressively weaker (the system becomesuniformly old, and moreover metallicity gradients increasingly off-set age gradients; see Hopkins et al. 2009b), as does the differencebetween the stellar mass inferred from different stellar populationmodels, and observed at z = z = For the sake of generality, we adopt a simple Gaussian PSF with 1 σ width = . = . z & . B -band R e here is about ∼
10% larger than the stellar mass R e atlate times. If this is all that would happen, of course, the observed B -band size will simply converge or slightly over-estimate the truestellar-mass size; giving only a factor 1 . − ∼
50% of the stellar mass is recycledby stellar mass loss, much of this occurs when the stellar popula-tions are very young (to rough approximation, mass loss rates atlate times decline as ∼ t − ). Given the Bruzual & Charlot (2003)models and the distribution of stellar population ages already inplace at z =
2, the stars here will lose only ∼
20% of their massover the remainder of a Hubble time. At z = . × M ⊙ , inflate the ef-fective radius to 1 . σ =
200 km s − (note that σ has contributions from dark mat-ter and large radii, so is not quite as strongly affected as might beexpected).The system experiences a significant merger history, involvingseveral different mergers. Here, we select a representative mergerhistory from the cosmological models in Hopkins et al. (2009g) ,where these are discussed in detail, for a galaxy that is already amassive ∼ M ⊙ at z >
2. We simulate that merger history athigh-resolution, and show the results in Figure 5.For simplicity, and to correspond to the model classificationscheme used throughout this paper, we divide the mergers in thecosmological model into “identical dry mergers” and “minor/latemergers”, and consider each in turn (note that, for the final rem-nant, the exact time-ordering of the mergers does not make a signif-icant difference). First, consider the “identical dry merger”: around z ∼
2, the system experiences a (marginally) major merger (massratio ≈ ∼ −
40% gas merger (again, since this is still at high redshift,these numbers are typical; more gas-poor systems are not typi-cal), and should therefore be similarly dense. We model this byconsidering a 1:3 merger with a similar spheroid formed, itself, inan equally gas-rich merger. The effective radius expands R e ∝ M ∗ , The merger rates from this model can be obtained asa function of galaxy mass, redshift, and merger mass ratiofrom the “merger rate calculator” script publicly available at .c (cid:13) , 000–000 esting Models for Spheroid Size Evolution as expected, and the velocity dispersion increases very slightly (itis not perfectly constant because there is some preferential trans-fer of energy to the least-bound outer material instead of the cen-tral regions). The net effect is a relatively small increase in size( R e = . ∼
5% to the final mass). In general,Hopkins et al. (2009g) show that > L ∗ systems have enhancedmerger rates, but the mergers become progressively more minor(on average) as the system grows in mass. This simply reflects thefact that most of the mass in the Universe is in ∼ L ∗ systems, sothe growth will be dominated by mergers of those systems (whichare minor when the system is ≫ L ∗ ). In the particular Monte Carlohistory modeled here, this is manifest in the series of “minor/latemergers.”The system, around z ∼ . −
1, experiences its last majormerger, with a more gas-poor disk of mass ∼ . L ∗ (a roughly1:3 merger). The disk has a gas fraction of ∼
20% at the timeof merger – still not negligible, but significantly lower than the ∼
40% that made the original spheroid. Therefore it contributesproportionally more dissipationless (stellar disk) material, whichis low-density relative to the dissipational (gas/starburst) material,and therefore preferentially builds up the “wings” of the profile.Similarly, this is accompanied and followed by a sequence of a ∼ ∼ ∼ z ∼ − .
5. Whether those secondary galaxies are modeledas disks or spheroids makes little difference – the key is that theyhave the appropriate gas or dissipational fractions for their massand redshift ( ∼ − z = ∼ . z > M ∗ & M ⊙ ) passive galaxy grew by such a factor, itwould account for only ∼ −
50% of the z = M ∗ & − × M ⊙ spheroid population (see e.g. Pérez-González et al. 2008;Marchesini et al. 2008; van der Wel et al. 2009). The velocity dis-persion grows by a small amount for the same reasons as in the“identical dry merger”, but is still moderate for the total stellarmass, σ ≈
240 km s − . But the effective radius has grown by a fac-tor ∼ −
10. The stellar mass and B -band effective radii are now ∼ R e is slightly larger, owing to the large best-fit Sersic index n s ∼ ∼ ∼
10 most massive Virgo ellipticals (from Kormendy et al. 2009), spanning a mass range ∼ . − . M ⊙ .The profile appears quite typical. Note that although the centraldensity inside ∼
100 pc may be slightly high, by a factor ∼ . − ∼ −
100 pc, for galaxies with the masses of our finalremnant) and convert initially “cuspy” nuclear profiles (formed ingas-rich starbursts as those here) into “cored” profiles. Including atoy model for such a process following Milosavljevi´c et al. (2002),we find that the predicted profile is fairly typical.Therefore, allowing for the combination of all the effects con-sidered here, we find it is possible to account for even order-of-magnitude size evolution in the most massive, early-forming com-pact spheroids. The most important contribution to that evolution isthe “minor/late merger” channel, as expected based on our com-parisons in § 3. But as noted in § 4, given cosmologically re-alistic merger histories and observational constraints on galaxymerger/growth rates, this channel is only expected to contribute afactor ∼ − ∼ −
3, wefind, comes from a combination of the other effects consideredhere: M ∗ / L gradients, observational effects, adiabatic expansion,and identical dry mergers. Each of these effects contributes onlya relatively small ∼ −
30% effect to the size evolution – but to-gether, this yields the net factor ∼ − z > z = Given these models, we briefly consider their implications for theevolution in the correlations between black hole (BH) mass andvarious host galaxy properties. c (cid:13) , 000–000 Hopkins et al. M B H ( z ) / M B H ( ) Minor/Late Accretion: ExpectedMinor/Late Accretion: Upper Limit M B H ( z ) / M B H ( ) Net Evolution (all effects)Net Evolution (upper limit)Contributions from : Identical Dry MergersMinor/Late AccretionAdiabatic ExpansionM ∗ /L GradientsObservational EffectsM ∗ Uncertainties
Figure 7.
Predictions for the evolution in BH mass at fixed host galaxy stel-lar mass, in the style of Figure 4 ( top ) and Figure 6 ( bottom ), for the samemodels. Observational effects, M ∗ / L gradients, and identical dry mergersall give no evolution; adiabatic expansion and M ∗ uncertainties lead tohigher (real or inferred) bulge masses at high- z , hence lower M BH / M bulge .Accounting for subsequent BH accretion only makes the predicted evolu-tion more negative. Minor/late mergers allow for evolution towards larger M BH / M bulge at high-redshift; the bulge “catches up” via later mergers withe.g. gas-poor disks. We show both the upper limit of such evolution (as-suming that the late/minor mergers contribute no BH mass) and the cosmo-logically expected evolution (given BH masses and merger histories fromhydrodynamic simulations).
Bottom:
Same, from the full model of § 5, withcontributions from each effect. Observational constraints are compared,from Häring & Rix (2004, black star), Peng et al. (2006, blue squares),Woo et al. (2006); Treu et al. (2007, green triangle), and Salviander et al.(2007, orange inverted triangles). Upper limits derived from observations ofthe spheroid mass density and the fact that BH mass cannot decrease withtime are also shown, from Hopkins et al. (2006, red circles). The modelagrees reasonably well, but the observations are systematically uncertain,and various biases probably affect the determinations at high redshifts (seetext).
In the models which do not involve mergers, consideredhere, the expected evolution of BH-host correlations is particu-larly simple. We perform the same exercise as in § 3, and re-quire that all the models reproduce the observed z = z = M BH = . M ∗ ; Häring & Rix2004), although adopt instead the z = M BH − σ relation ( M BH = . ( σ/
200 km s − ) . ; Tremaine et al. 2002) makes no differ-ence. Since the BH cannot decrease in mass with time, this sets atleast a limit on the evolution. We evolve the systems backwards intime, and predict the observed BH mass at fixed galaxy properties.In the case of stellar mass-to-light ( M ∗ / L ) ratio gradients andobservational/seeing effects, the prediction is trivial. Neither ofthese effects has any affect on either the galaxy stellar mass or theBH mass – systems simply evolve passively and appear to changein radius.The case of stellar mass uncertainties is only slightly morecomplex. Here, there is no effect that can change the BH mass; butat high redshifts, the system appears to be higher-stellar mass, byan amount tuned to match the apparent evolution in the size-massrelation. Therefore, for an (apparent) mass-selected sample at highredshift, since the “true” stellar mass is lower, the BH mass mustbe lower, and there will appear to be strong negative evolution in M BH .Adiabatic expansion requires significant host bulge mass lossto have occurred since the time of BH and bulge formation, so itpredicts that BH masses at high redshifts should be substantiallysmaller than those today (at fixed stellar mass; such that after thishost mass loss they will lie on the z = top ), we assume that the BH does notgrow in mass over this time (i.e. the entire z = M BH at fixed bulge mass would be even more negative (towardslower M BH ( z ) / M BH ( ) ). Likewise for the other models above, iffurther accretion occurs subsequent to the initial bulge formation.The case of identical dry mergers is also straightforward. Un-like the cases above, the BH will grow in time – with each drymerger, the BH should grow via the merger of the two progenitorBHs. But since the systems are, by definition, structurally identical,they should obey the same proportionality between BH mass andhost galaxy stellar mass. So BH mass and stellar mass simply addlinearly, and the BH-host mass relation (being linear itself) is con-served by these mergers. As a consequence, there is no predictedevolution in M BH at fixed stellar mass (i.e. M BH / M bulge ), although M BH and M bulge can both grow significantly for an individual sys-tem. The minor/late merger case is more interesting. As before,BH mass will grow via mergers. However, unlike the identical drymerger case, it is not necessarily true that the BH masses of bothmerging systems obey the same BH-host galaxy correlations (andtherefore that BH and host mass will add linearly and conserve the M BH − M bulge relation). The previous scale-invariance can be brokenin two ways. First, imagine the case of a merger with a late-forming,gas-poor disk-dominated galaxy (for the case of simplicity, takethe extreme limit of a gas-free, nearly pure disk secondary). Sincethe secondary has little or no gas, there will be no new accretion;since it has little or no bulge, it has corresponding little or no BH.As a consequence, the secondary merger contributes negligible BHmass. But the entire secondary disk stellar mass will be violentlyrelaxed, adding substantially to the total bulge mass. Similarly,spheroids and disks that merge later will have formed later, frommore gas-poor mergers. Such mergers build up a less-deep centralpotential and fuel less material to the BH, and so will form lessmassive BHs, relative to their bulge masses (see e.g. Hopkins et al.2007a). The result in either case is that M BH at high redshifts shouldbe larger at fixed stellar mass. c (cid:13) , 000–000 esting Models for Spheroid Size Evolution An upper limit to the magnitude of this effect is easy to de-termine. In this case, assume that no BH mass is contributed fromthe subsequent late/minor mergers. This could be the case if e.g.the depletion of gas fractions occurs sufficiently rapidly with cos-mic time, or if the late/minor mergers involve preferentially verydisk-dominated galaxies, or if BH-BH mergers are inefficient andthe systems are relatively gas-poor. Because the BH mass still can-not decrease, and must lie on the observed z = z = ∼ M BH / M bulge from z = − σ , M BH ∝ M . ∗ σ . , at the timeof bulge formation in gas rich mergers. Direct analysis of the ob-served BH-host correlations appear to support such a driving cor-relation (see e.g. Hopkins et al. 2007b; Aller & Richstone 2007).Moreover, Younger et al. (2008a) showed that such a model pre-dicts a unique difference between e.g. the M BH − σ relation of“classical” bulges (believed to be formed in mergers) and “pseu-dobulges” (formed in secular events, from e.g. disk bars), becausethe observed velocity dispersion relates differently to the centralbinding energy in structurally distinct objects. Since then, vari-ous observations have found that such a difference exists, withthe sense and magnitude predicted (Hu 2008; Greene et al. 2008;Gadotti & Kauffmann 2008). In any case given such a model forthe BH mass as a function of bulge properties (at e.g. fixed massor σ ), it is straightforward to calculate the contribution from sub-sequent minor/late mergers (which we do following, in detail, theequations and approach in Hopkins et al. (2009b)) and (given the z = z = − z = − z = .
36 from narrow-redshift Seyfert samples (Woo et al.2006; Treu et al. 2007). We caution, however, that these estimates(indirect BH mass estimates based on the virial mass indica-tors) are systematically uncertain, and various biases are presentin AGN-selected samples that will tend to systematically over-estimate the amount of evolution at high redshift (see Lauer et al.2007a) (although the Salviander et al. 2007, estimates do attemptto correct for this bias). Alternatively, Hopkins et al. (2006) de-termine a non-parametric upper limit to the degree of evolutionby comparing the observed spheroid mass density at each red-shift to the BH mass density at z =
0, given the constraint thatBHs cannot decrease in mass; we show this as well. Other in-direct constraints, from e.g. clustering and integral constraints,give qualitatively similar results, but with a large scatter (seee.g. Merloni et al. 2004; McLure & Dunlop 2004; Alexander et al.2005, 2008; Adelberger & Steidel 2005; Fine et al. 2006). Al-though the various biases and uncertainties involved are large, itappears robust that there is probably some evolution towards higher M BH at fixed galaxy stellar mass. We hesitate to use this as a quan-titative constraint on the models here, but note that this appears tostrongly conflict with the adiabatic expansion and stellar mass errormodels, and moreover stress that only the late/minor merger chan-nel provides a mean to evolve in the observed sense, at any level. We have illustrated how different physical effects combine to ex-plain the observationally inferred size evolution of massive ellipti-cals with redshift, and shown how various observations can distin-guish between these models.Observations have shown that, at each time, most of thespheroid population is recently assembled. As a consequence, theevolution in the size-mass relation as a whole must reflect a red-shift dependent scaling in the in-situ sizes of ellipticals at the timeof formation . As is discussed in detail in Khochfar & Silk (2006)and Hopkins et al. (2009d), this occurs naturally in merger mod-els: at the same mass, higher-redshift disks have larger observedgas fractions. The gas fraction at the time of merger, which yieldsthe compact, remnant starburst stellar populations required to ex-plain elliptical stellar population gradients, kinematics, mass pro-file shapes, and densities, is the dominant determinant of the sizeof any merger remnant (see e.g. Hopkins et al. 2008a, and refer-ences therein). As a consequence, mergers of these gas-rich disksat high redshifts will yield spheroids with smaller effective radii.We show that simulated merger remnants with the appropriate gasfractions as a function of redshift naturally reproduce the observedevolution in the mean trends.That being said, there does not appear to be a “relic” popula-tion of high-mass, very compact spheroids (see e.g. Trujillo et al.2009). It may even be the case that the most highly clustered (andtherefore earliest-to-assemble) spheroids (i.e. BCGs) may havesomewhat larger sizes for their mass. Therefore, some other pro-cess must increase the apparent sizes of these systems after theirformation, at least keeping pace with the evolution in the medianpopulation owing to redshift-dependent gas fractions. c (cid:13) , 000–000 Hopkins et al.
We discuss six effects that can contribute to such evolution,that cover the range of possibilities in the literature. These include:“identical” dry mergers (mergers of identically compact spheroids),“late/minor accretion (major or minor mergers at later times, withless dense material – for example, less gas-rich disks or otherspheroids formed in lower-redshift, less gas-rich mergers), adia-batic expansion (owing to mass expulsion from feedback or grad-ual stellar mass loss from stellar population evolution), the pres-ence and subsequent evolution of stellar mass-to-light ratio gradi-ents (changing e.g. the observed optical or near-IR R e relative tothe stellar mass R e ), age or redshift-dependent uncertainties in stel-lar masses (owing to e.g. unaccounted-for contributions of AGNstar light in young stellar populations), and seeing/observationaleffects (e.g. possible redshift-dependent fitting biases from limiteddynamic range and resolution effects, or issues arising from differ-ent choices of definition of effective radii and different distributionsof galaxy shapes in different samples).In each case, we show how these different mechanisms shouldmove systems through the size-mass space with redshift, andhow this relates to other observable properties, including spheroidvelocity dispersions, central stellar mass densities, light profileshapes, and black hole-host galaxy correlations.Given this, we construct the predictions from each model, ifit were solely responsible for the observed evolution. In particular,we demand that each model obey the observational constraints asboundary conditions: newly-formed ellipticals must appear to havethe appropriate observed size at each redshift, and then, evolvedforward allowing only the given affect to alter the size, must lie onthe appropriate parameter correlations of z = z = only model that is consistent with all ofthese constraints is the “late/minor merger” model. Hopkins et al.(2009a) have shown, for example, that the observed profiles ofthe high-redshift systems are actually very similar to the observedcores of massive ellipticals today – they are not more dense, byeven a factor ∼ a few (see also Bezanson et al. 2009, who reachsimilar conclusions). It appears that the dominant effect has beenthe buildup of extended, lower-density wings, from high redshiftsto today, yielding higher Sersic indices in low-redshift systems andmore extended envelopes that drive the effective radii to larger val-ues, while leaving the central properties relatively intact. Support-ing evidence for this comes form Cenarro & Trujillo (2009), whosee only weak growth in the velocity dispersions of galaxies atfixed mass with redshift, consistent with evolution via buildup oflow-density material that does not strongly affect σ . In contrast,an adiabatic contraction or identical dry merger model would pre-dict dramatically higher central densities at high redshifts, with cor-respondingly larger velocity dispersions. Invoking stellar mass bi-ases would predict dramatic inverse evolution in velocity disper-sions with redshift. And invoking mass-to-light ratio gradients pre-dicts weaker velocity dispersion evolution, and opposite light pro-file shape evolution, relative to that observed.However, although it is in principle possible to explain the en-tire necessary size evolution via “late/minor mergers” without vio-lating these observational constraints, both theoretical models andindependent constraints on e.g. merger rates and the stellar mass function suggest that the required number of mergers would be toolarge, at least in the extreme case of factor ∼ −
10 evolution insizes of the most massive, early-forming spheroids as required. Apriori theoretical models and semi-empirical models based on ob-served clustering and stellar mass function evolution predict closerto a factor ∼ − ∼ − ∼ . −
2) stellar mass growth.However, the other effects described here also all contribute someapparent or real size evolution. Each of the them contributes a rel-atively small effect, ∼ − B -band R e at early times and larger R e at late times, but each at the ∼
10% level. Observational fit-ting/resolution effects, and/or small shifts in the typical flatteningof systems in observed samples coupled to the adoption of a “circu-larized” effective radius definition, can contribute a relatively small ∼
20% to size estimates, for the observed and model profile shapes.Given the stellar ages, star formation rates, and masses already inplace at high redshift, and low-redshift constraints on star formationhistories of massive ellipticals, there could not be a very large gassupply awaiting removal in the compact passive systems, but theywill inevitably lose ∼
20% of their mass to later stellar evolution,and will adiabatically expand as a result. Also, given the observedphotometry, stellar mass estimates appear uncertain at less than the ∼
20% level. And the earliest (highest redshift) mergers after thesystem first forms are likely to be similarly gas-rich and/or com-pact, contributing an average of ∼
20% in mass via the “identical”dry (or mixed) mergers channel.Each of these effects is small; together, however, they con-tribute a net additional factor ∼ − ∼ − R e . The combination of secondary effects – all similarly impor-tant – explains the remaining factor needed to reconcile models ofe.g. mergers alone and the observations. Moreover, because eachof these effects is, individually, small, the success of theoreticalmodels does not critically depend on any one of the effects operat-ing (other than, of course, late/minor mergers). The size evolutionof even the most massive, early forming spheroids, therefore, ap-pears to be a natural consequence of hierarchical, merger-drivenspheroid formation; but understanding this evolution in detail re-quires accounting for a number of effects related to e.g. stellar pop-ulations, small deviations in profile shape, multi-component stellarmass profiles, and different mergers that can only be followed inhigh-resolution simulations.Interestingly, it also appears that only the “late/minor merger”channel provides a means for evolution in the BH-host galaxy cor-relations, in the sense of more massive BHs at high redshift rel-ative to their host galaxies. The other mechanisms predict eitherno or inverse evolution in these correlations, which both appear c (cid:13) , 000–000 esting Models for Spheroid Size Evolution to conflict with the observations and would make reconciling theBH population and high-redshift quasar population (which requiresa large fraction of the present-day massive BH population be ex-tant and active at these times (Hopkins et al. 2007c)) difficult. Thefull model above appears to be consistent with recent observationsof such evolution, but observational uncertainties remain large. Inthis scenario, early-forming BHs are relatively massive, owing tothe deep potential wells of their compact hosts; the spheroid masscan then “catch up” to the z = original , gas-rich merger history (allowing direct tests athigh redshift of the hypothesis, which appears successful at lowredshift, that gas-richness dominates the determination of the rem-nant size) and on the subsequent merger history, potentially map-ping between gas-rich or equal density mergers and dry, lower-density mergers (Mihos & Hernquist 1994a; Barnes & Hernquist1996; Naab et al. 2006a; Cox et al. 2006b; Robertson et al. 2006a;Oñorbe et al. 2006; Jesseit et al. 2007; Hopkins et al. 2009b,e).We wish to emphasize that the various observations dis-cussed here, and the magnitude or nature of size evolution, do not themselves strongly constrain whether or not the “late/minoraccretion” channel is actually dominated by a few “major” or alarger number of “minor” mergers. Provided that the same to-tal amount of low-density material (stellar mass) is added to asystem, in a dissipationless/collisionless fashion, it makes no dif-ference whether or not it is “brought in” by a major or minormerger. In other words, so long as the major companions are suf-ficiently low-density (unlike in the “identical dry mergers” case,where they are, by definition, high-density), the characteristic stel-lar densities and radii of that material in the post-merger rem-nant will be the same as if the material was merged via a se-ries of much more minor mergers. This follows necessarily frombasic phase-space considerations (e.g. Gallagher & Ostriker 1972;Hernquist et al. 1993), and has been seen as well in various nu-merical studies (Bournaud et al. 2005; Boylan-Kolchin et al. 2006;Hopkins et al. 2009e; Naab et al. 2009; Feldmann et al. 2009). Re-alistic cosmological models predict that both major and minormergers contribute comparably to this aggregation of low-densitymaterial (see e.g. Hopkins et al. 2009g, and references therein),with major mergers dominating at masses near and somewhatabove ∼ L ∗ , and minor mergers becoming progressively more im-portant at higher masses (essentially, most of the mass is near ∼ L ∗ ; so a “typical” z = ∼ L ∗ system will reflect buildup from ∼ ∼
10% of systemsmay escape much subsequent merging (Hopkins et al. 2009d), sug-gestively similar to the recently-observed abundance of popula-tions of present-day compact systems in galaxy clusters, their ex-pected environments (Valentinuzzi et al. 2009). Distinguishing ob-servationally between e.g. a series of minor mergers and a coupleof major mergers will require additional, independent checks. Ob-viously, direct constraints on merger rates/fractions as a functionof mass, redshift, and mass ratio are important. Also, higher-orderkinematics, such as e.g. galaxy isotropies and the presence/absenceof certain kinematic subsystems can distinguish between differentmerger histories (see references above and e.g. Burkert et al. 2008;Hoffman et al. 2009). These constraints will be particularly valu-able in informing theoretical models that attempt to follow the de-tailed formation histories of such systems.However, it should also be emphasized, as above, that the vastmajority of ellipticals/spheroids do not form via these high redshift,compact channels. The observed mass density of bulge-dominatedgalaxies at z ∼ ∼
5% of its z = z = ∼ −
35% of its z = > M ⊙ , the regimewhere “core” ellipticals (believed to have survived dry mergers) be-come important, in good agreement with the models for evolutionconsidered here. ACKNOWLEDGMENTS
We thank Eliot Quataert, Norm Murray, Pieter van Dokkum,and Ignacio Trujillo for a number of very helpful conversationsand suggestions throughout the development of this manuscript.Support for PFH was provided by the Miller Institute for BasicResearch in Science, University of California Berkeley. SWand TJC gratefully acknowledge support from the W. M. KeckFoundation.
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