A New Empirical Method to Infer the Starburst History of the Universe from Local Galaxy Properties
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 14 June 2018 (MN L A TEX style file v2.2)
A New Empirical Method to Infer the Starburst History of theUniverse from Local Galaxy Properties
Philip F. Hopkins, ∗ & Lars Hernquist Department of Astronomy and Theoretical Astrophysics Center, University of California Berkeley, Berkeley, CA 94720 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
Submitted to MNRAS, September 17, 2009
ABSTRACT
The centers of ellipticals and bulges are formed dissipationally, via gas inflows over shorttimescales – the “starburst” mode of star formation. Recent work has shown that the surfacebrightness profiles, kinematics, and stellar populations of spheroids can be used to separatethe dissipational component from the dissipationless “envelope” made up of stars formed overmore extended histories in separate objects, and violently assembled in mergers. Given high-resolution, detailed observations of these “burst relic” components of ellipticals (specificallytheir stellar mass surface density profiles), together with the simple assumptions that someform of the Kennicutt-Schmidt law holds and that the burst was indeed a dissipational, gas-rich event, we show that it is possible to invert the observed profiles and obtain the time andspace-dependent star formation history of each burst. We perform this exercise using a largesample of well-studied spheroids, which have also been used to calibrate estimates of the“burst relic” populations. We show that the implied bursts scale in magnitude, mass, and peakstar formation rate with galaxy mass in a simple manner, and provide fits for these correlations.The typical burst mass M burst is ∼
10% of the total spheroid mass; the characteristic starbursttimescale implied is a nearly galaxy-mass independent t burst ∼ yr; the peak SFR of theburst is ∼ M burst / t burst ; and bursts decay subsequently in power-law fashion as ˙ M ∗ ∝ t − . .As a function of time, we obtain the spatial size of the starburst; burst sizes at peak activityscale with burst mass in a manner similar to the observed spheroid size-mass relation, butare smaller than the full galaxy size by a factor ∼
10; the size grows in time as the central,most dense regions are more quickly depleted by star formation as R burst ∝ t . . Combinedwith observational measurements of the nuclear stellar population ages of these systems –i.e. the distribution of times when these bursts occurred – it is possible to re-construct thedissipational burst contribution to the distribution of SFRs and IR luminosity functions andluminosity density of the Universe. We do so, and show that these burst luminosity functionsagree well with the observed IR LFs at the brightest luminosities, at redshifts z ∼ −
2. Atlow luminosities, however, bursts are always unimportant; the transition luminosity betweenthese regimes increases with redshift from the ULIRG threshold at z ∼ z ∼
2. At the highest redshifts z &
2, we can set strict upper limits on starburstmagnitudes, based on the maximum stellar mass remaining at high densities at z =
0, andfind tension between these and estimated number counts of sub-millimeter galaxies, implyingthat some change in bolometric corrections, the number counts themselves, or the stellar IMFmay be necessary. At all redshifts, bursts are a small fraction of the total SFR or luminositydensity, approximately ∼ − Key words: galaxies: formation — galaxies: evolution — galaxies: active — star formation:general — cosmology: theory ∗ E-mail:[email protected]
Understanding the global star-formation history of the Universe re-mains an important unresolved goal in cosmology. Of particular in- c (cid:13) Hopkins et al. terest is the role played by mergers in driving star formation and/orthe infrared luminosities of massive systems. A wide range of ob-servations support the view that violent, dissipational events (e.g.gas-rich mergers) are important to galaxy evolution, and in partic-ular that the central, dense portions of galaxy bulges and spheroidsmust be formed in such events; but less clear is their contributionto the global star formation process.In the local Universe, the population of star-forming galax-ies appears to transition from “quiescent” (undisturbed) disks –which dominate the total star formation rate/IR luminosity density– at the luminous infrared galaxy (LIRG) threshold 10 L ⊙ ( ˙ M ∗ ∼ − M ⊙ yr − ) to systems that are clearly merging and violentlydisturbed at a few times this luminosity. The most intense starburstsat z =
0, ultraluminous infrared galaxies (ULIRGs; L IR > L ⊙ ),are invariably associated with mergers (e.g. Joseph & Wright 1985;Sanders & Mirabel 1996), with dense gas in their centers provid-ing material to feed black hole growth and to boost the concentra-tion and central phase space density of merging spirals to matchthose of ellipticals (Hernquist et al. 1993; Robertson et al. 2006).Various studies have shown that the mass involved in these star-burst events is critical to explain the relations between spirals,mergers, and ellipticals, and has a dramatic impact on the proper-ties of merger remnants (e.g., Lake & Dressler 1986; Doyon et al.1994; Shier & Fischer 1998; James et al. 1999; Genzel et al. 2001;Tacconi et al. 2002; Dasyra et al. 2006, 2007; Rothberg & Joseph2004, 2006a; Hopkins et al. 2009b,e).At high redshifts, bright systems dominate more andmore of the IR luminosity function (e.g. Le Floc’h et al. 2005;Pérez-González et al. 2005; Caputi et al. 2007; Magnelli et al.2009). Merger rates increase rapidly (by a factor ∼
10 from z = −
2; see e.g. Hopkins et al. 2009l, and references therein), lead-ing to speculation that the merger rate evolution may in fact drivethe observed evolution in the cosmic SFR density, which also risessharply in this interval (e.g. Hopkins & Beacom 2006, and ref-erences therein). However, many LIRGs at z ∼
1, and possiblyULIRGs at z ∼
2, appear to be “normal” galaxies, without dra-matic morphological disturbances associated with the local star-burst population or large apparent AGN contributions (Yan et al.2007; Sajina et al. 2007; Dey et al. 2008; Melbourne et al. 2008;Dasyra et al. 2008). At the same time, even more luminous systemsappear, including large numbers of Hyper-LIRG (HyLIRG; L IR > L ⊙ ) and bright sub-millimeter galaxies (e.g. Chapman et al.2005; Younger et al. 2007; Casey et al. 2009). These systems ex-hibit many of the traits more commonly associated with merger-driven starbursts, including morphological disturbances, and maybe linked to the emergence of massive, quenched (non star-forming), compact ellipticals at times as early as z ∼ − ∼ . − inverted to obtain the full time and radius-dependent star forma-tion history of each galaxy starburst (§ 2). We present the resulting,derived burst star formation histories for samples of hundreds of lo-cal, well-observed galaxy spheroids. We show that such starburstsfollow broadly similar time-dependent behavior, with characteris-tic starburst star formation rates, durations, rise and decay rates,and spatial sizes that scale with galaxy mass and other propertiesaccording to simple scaling laws across ∼ Ω M = . Ω Λ = . h = . c (cid:13) , 000–000 tarbursts from Ellipticals As discussed in § 1, spheroid mass profiles can be decomposedinto a central relic starburst and an outer stellar envelope. The twocomponents are distinct physically in the sense that the amount anddistribution of the two are determined by dissipational and dissipa-tionless dynamics, respectively. In H08c, H09b,e, the authors com-pile large samples of ellipticals from the studies of Kormendy et al.(2009) and Lauer et al. (2007), and present decompositions forthese galaxies. We adopt these results for the present study.Briefly, Kormendy et al. (2009) present a V -band Virgo ellipti-cal survey, based on the complete sample of Virgo galaxies down toextremely faint systems in Binggeli et al. (1985) (the same samplestudied in Côté et al. 2006; Ferrarese et al. 2006). Kormendy et al.(2009) combine observations from a large number of sources (in-cluding Bender et al. 1988; Caon et al. 1990, 1994; Davis et al.1985; Kormendy et al. 2005; Lauer 1985; Lauer et al. 1995, 2005;Liu et al. 2005; Peletier et al. 1990) and new photometry from theMcDonald Observatory, the HST archive, and the SDSS for each oftheir objects which, after careful conversion to a single photometricstandard, enables accurate surface brightness measurements overa wide dynamic range (with an estimated zero-point accuracy of ± . V mag arcsec − ). Typically, the galaxies in this sample haveprofiles spanning ∼ −
15 magnitudes in surface brightness, cor-responding to a range of nearly four orders of magnitude in physicalradii from ∼
10 pc to ∼
100 kpc, permitting the best simultaneousconstraints on the shapes of both the outer and inner profiles ofany of the objects we study. Unfortunately, since this is restrictedto Virgo ellipticals, the number of galaxies is limited, especially atthe intermediate and high end of the mass function.We therefore include surface brightness profiles fromLauer et al. (2007), further supplemented by Bender et al. (1988).Lauer et al. (2007) compile V -band measurements of a large num-ber of nearby systems for which HST imaging of galactic nu-clei is available. These include the Lauer et al. (2005) WFPC2data-set, the Laine et al. (2003a) WFPC2 BCG sample (in whichthe objects are specifically selected as brightest cluster galaxiesfrom Postman & Lauer (1995)), and the Lauer et al. (1995) andFaber et al. (1997) WFPC1 compilations. Details of the treatmentof the profiles and conversion to a single standard are given inLauer et al. (2007). HST images are combined with ground-basedmeasurements (see references above) to construct profiles that typ-ically span physical radii from ∼
10 pc to ∼ −
20 kpc. The sam-ple includes ellipticals over a wide range of luminosities, down to M B ∼ −
15, but is dominated by intermediate and giant ellipticalsand S0 galaxies, with typical magnitudes M B . − ∼
2. Independent, purely empirical analyses ofthese profiles (Kormendy et al. 2009) and those of similar objects(e.g. Ferrarese et al. 2006; Balcells et al. 2007a) yield similar con-clusions. For most of the objects in our data-set, the profiles in-clude e.g. ellipticity, a / a , and g − z colors as a function of ra-dius, as well as kinematic information. In Kormendy et al. (2009)and Hopkins et al. (2009b,e), the authors show that these secondaryproperties exhibit transitions that mirror the fitted decompositions(e.g. transitions to diskier, more rotationally supported, younger in-ner components).Likewise, comparison with detailed resolved stellar popula- tion properties – i.e. independent constraints from stellar popu-lation synthesis models, abundance gradients, and galaxy colors– yields good agreement where available (McDermid et al. 2006;Sánchez-Blázquez et al. 2007; Reda et al. 2007; Foster et al. 2009;Schweizer 1996; Titus et al. 1997; Schweizer & Seitzer 1998,2007; Reichardt et al. 2001; Michard 2006). Moreover, in youngmerger remnants, where stellar populations and colors more clearlyindicate the post-starburst component, the methodology workswell (Rothberg & Joseph 2004, 2006a,b; Hopkins et al. 2008c). InH09b,e, these results are also checked against profile data used inBender et al. (1988, 1992, 1993, 1994). The latter are more limitedin dynamic range, but allow for construction of multi-color profilesin e.g. V , R , and I bands, to test whether they are sensitive to theband used and to construct e.g. stellar mass and M ∗ / L profiles.We illustrate our methodology in Figure 1. First, we show atypical stellar mass surface density profile of an ∼ L ∗ ellipticalgalaxy, from the sample of Lauer et al. (2007). We plot the de-composition into burst and extended envelope (violently relaxed)components, together with some of their integral properties (stellarmass fractions, effective radii, and fitted profile shapes in the formof the best-fit Sersic index). We compare the observed system tothe stellar mass density profile of one of the hydrodynamic simula-tions presented in H09b. The simulation is determined in that paperto be similar in total mass, profile shape, and kinematic propertiesto the observed system (see Table 1 therein). In the simulation, thetrue, physical starburst component from gas that loses angular mo-mentum and falls to the center at coalescence is known, as is thenon-starburst component from violently relaxed stars present in thedisks before the final merger. We show the separate contributionsof the two components to the stellar mass density. The agreementis good, lending confidence to our empirical de-composition pro-cedure, and allowing us to consider the simulations as a reasonablecalibration sample for our approach. Taking the observationally estimated burst component, this definesa relic surface density Σ burst ( R ) . Of course, in general, the star for-mation history that yields a given stellar surface density is non-unique. But, if we make three simple, physically and observation-ally motivated assumptions, Σ burst ( R ) can be inverted into a spaceand time-dependent burst star formation history.These assumptions are: • (1) That some form of the Kennicutt (1998) law holds for theseobjects, relating their star formation rate surface density ˙ Σ ∗ to theirgas surface density Σ gas . • (2) That the relic burst mass is dominated by of order onemost massive event, i.e. is not constituted from the later assemblyof many independent, well-separated bursts nor by an extended se-ries of independent, well-separated bursts in the same galaxy. Theremay be many small events, but most of the mass should come froma single, dominant event. • (3) That this event, at its inception, involved the central regionsbeing (however briefly) gas-dominated. In other words, the burstbegan from a gas-dominated central density and formed largely insitu, rather than from an extended, low-gas-density trickle in whichthe gas density was never nearly as large as the total/final stellarmass density.In detail, these assumptions are not exactly correct (we discussthis further in § 4), but most observational constraints indicate that c (cid:13) , 000–000 Hopkins et al. l og Σ ∗ [ M O • k p c - ] Pre-Starburst Stars(n s = 2.85)Starburst Stars(fraction = 6.1%) l og Σ ga s [ M O • k p c - ] Time (Gyr):0.00.010.10.51.02.0 Σ S F R [ M O • y r - k p c - ] -0.5 0.0 0.5 1.0t [Gyr]0.1110 S F R ( T o t a l ) [ M O • y r - ] ActualInferred (Two-Sided)
Figure 1.
Illustration of the methodology used in this paper.
Top Left:
Observed stellar mass profile of a typical ∼ L ∗ elliptical from the Lauer et al. (2007)sample (dark blue solid). We show the empirical, fitted decomposition into inner “starburst relic” (dark blue dashed) and outer violently relaxed (dark bluedotted) components, from H09b,e. We compare a simulation which yields a remnant with similar profile, kinematics, and stellar populations (total mass profilein black). For the simulation, we know the true, dissipational merger-induced starburst component (light blue, with labeled mass fraction), and dissipationlessviolently relaxed component from stars formed before the final merger (red; with labeled Sersic index). The fitting and simulation-matching procedures yieldsimilar burst decompositions. Top Right:
Implied gas surface density profile of the burst ( ignoring the violently relaxed component, formed over much moreextended periods) as a function of time. At time t = Bottom Left:
Corresponding SFR surface densityprofile. We also label the half-SFR radius at each time.
Bottom Right:
Implied total
SFR (integrating the SFR density profile over radius), at each time (blackdashed). The “two-sided” estimate assumes a light curve with a symmetric rise/fall. We compare the true simulation SFR versus time in the best-fit simulation.The two agree well – despite the simple nature of the model, it appears to be reliable in simulations with more detailed physics, and allows us to recover thestar formation history of a given spheroid starburst from its relic mass profile. deviations from them are at the factor of ∼ a couple level, compa-rable to or less than e.g. most of the systematic uncertainties in theobservations to which we will compare.Regarding assumption (1) ; observations indicate that at bothlow (e.g. Kennicutt 1998; Bigiel et al. 2008; Bothwell et al. 2009,and references therein) and high ( z ∼ −
4, e.g. Bouché et al. 2007)redshifts, a Kennicutt-Schmidt star formation law applies over alarge dynamic range in stellar mass and star formation rates (from ∼ − M ⊙ yr − ). These observations have also shown that thesame relation pertains to a diverse array of objects, from dwarfgalaxies, through local gas-poor disks, to near “quenched” red sys-tems, to clumpy, turbulent high redshift disks and irregular sys-tems, to merger-induced starbursts and ULIRGs, and high-redshiftsub-millimeter galaxies, among others. The extent of the uncer-tainty appears to be in the precise normalization/slope of the rela-tion (with the references above favoring power-law slopes varying from ∼ . − . (2) ; this is well-motivated by a com-bination of theoretical models and observations of galaxy stellarpopulations and structural properties. In most models, nuclear burstmasses are dominated by the largest, most recent event (see e.g.Khochfar & Burkert 2003; Croton et al. 2006; de Lucia & Blaizot2007; Somerville et al. 2008; Hopkins et al. 2009l). Mergers andother violent processes that can drive large quantities of gas intogalaxy centers are not so common as to yield many bursts of equalstrength, and even if systems had earlier bursts at much higher red-shifts, they would have grown by a large factor in mass since those c (cid:13) , 000–000 tarbursts from Ellipticals times, such that the most recent event would dominate (see refer-ences above and Weinzirl et al. 2009; Hopkins et al. 2009h).Observationally, evidence from nuclear kinematics and pro-file shapes supports the view that most of the mass in spheroidstarburst components, especially in the cusp ellipticals that dom-inate ( > ∼ α -enhancements of these regions require that the stars thereformed rapidly, in short-lived events or in events closely spaced intime (later series of bursts, or independent well-spaced events, be-ing ruled out by the enrichment patterns).Regarding assumption (3) ; the observations argue that thesystem must have been (at least briefly) dominated by a stronggas inflow, that then formed stars more or less in situ. The shorttimescales of star formation indicated by stellar populations do notallow for a trickle of gas over an extended period of time. If as-sumption (1) applies, the corresponding star formation timescalerequired to match the abundances is close to that obtained if allthe gas were in place at once, and formed stars according to theKennicutt (1998) relation – i.e. the shortest allowable timescale.Moreover, in hydrodynamic simulations of galaxy-galaxy mergersor dissipational collapse, this is almost always the case – gas flowsin on the local free-fall time. However, star formation is inefficienton the the free-fall timescale (the Kennicutt (1998) relation imply-ing star formation efficiencies of ∼ −
10% per free-fall time),and so catches up over a short period following the initial gas-richphase of inflow. In addition, the disky, rotational kinematics dis-tinctive of the burst relics require in situ formation from initiallygas-rich configurations (see e.g. Naab et al. 2006; Cox et al. 2006;Hopkins et al. 2009e).Note that the assumptions outlined here clearly do not applyfor stars not formed in bursts. Although observations suggest thatthe Kennicutt (1998) law applies (although there may be a surfacedensity threshold for star formation), it is established, in contrast tothe central, burst populations, that the extended, violently relaxedstars were formed through star formation histories more extended intime. Disks, of course, are fed through continuous accretion, ratherthan a single massive inflow; so both assumptions (2) and (3) breakdown. Moreover, the outer components of ellipticals are formed viaviolent relaxation – i.e. by definition from merging already-formedstars from other galaxies – and so have no reason to be a homo-geneous population. In fact, especially in massive ellipticals, it is cosmologically expected that such portions of the galaxy are madeup of debris from many small systems (Gallagher & Ostriker 1972;Hernquist et al. 1993; De Lucia et al. 2006; de Lucia & Blaizot2007; Hopkins et al. 2009f,i; Khochfar & Silk 2006; Naab et al.2009). It is likely then, that these components – ∼
90% of the stars,in typical ∼ L ∗ systems – have star formation histories that are nottrivially invertible. Together, these assumptions allow us to invert an observed Σ burst ( R ) . If the relic burst mass is dominated by of order a sin-gle major event, and that event was indeed a dissipational, gas-richand/or rapid event, then the gas density at the beginning of the burst(which we define for now, arbitrarily, as t =
0) should be given ap-proximately by Σ gas ( R , t = ) ≈ Σ burst ( R ) . (1)The Kennicutt (1998) law relates the star formation rate surfacedensity – and hence the gas depletion rate – to the gas surface den-sity through ˙ Σ ∗ = − dd t Σ gas ≈ . × − M ⊙ yr − kpc − “ Σ gas M ⊙ pc − ” n K (2)where the normalization comes from Kennicutt (1998) correctedfor our adopted Chabrier (2003) IMF, here chosen fixed for Milky-Way like systems (where the relation is best calibrated). The index n K ≈ . n K ≈ . n K .Note that since we are ultimately interested in the relic starformation rate and time-averaged gas depletion rate, stellar massloss can be absorbed into the normalization of the SFR-surfacedensity relation (in essence, adopting the instantaneous recyclingapproximation). But, in general, we find that for reasonable recy-cling fractions from e.g. the models of Bruzual & Charlot (2003),the resulting differences are less than those between the choice ofthe slope n K . Likewise, the effects of long term stellar mass loss(although, post-burst, a typical IMF will yield just ∼
20% stellarmass loss) are generally smaller than uncertainties arising from ourprimary assumptions above. Fortunately, the uncertainties in both(e.g. the singularity of the burst versus degree of stellar mass loss)have opposing signs, so experimenting with reasonable variationsin each yields little net difference in our predictions.With this relation, we can then integrate forward in time fromthe initial gas density in each annulus, computing new gas and stel-lar densities after some time interval, and then determining a new ˙ Σ ∗ in that annulus. We show the results of this procedure for our il-lustrative example in Figure 1. At the initial time, the gas density isgiven by the surface density profile of the relic starburst componentisolated from the observed profile (as seen in the Figure, it makesno difference if we take the empirical fit or starburst componentderived frmo the best-fit simulation). As discussed above, we sub-tract out the non-starburst component. This leads to a correspond-ing SFR surface density. This depletes gas, leading to a lower gassurface density. The evolution is most rapid in the central regions,where the densities are highest, because the Kennicutt-Schmidt re-lation is super-linear in surface density. At several times, we plotthe remaining gas and corresponding SFR surface densities, which c (cid:13) , 000–000 Hopkins et al. both decrease and become more extended as the dense, central re-gions become depleted.Integrating the SFR surface density over R at each time, weobtain the total SFR in the burst at t ≥ ˙ M ∗ = Z ∞ ˙ Σ ∗ π R d R . (3)Moreover, if we assume that light, in the UV, radio, or IR, traces thestar formation rate locally in some annulus, using e.g. the relation L IR ≈ . × L ⊙ “ ˙ M ∗ M ⊙ yr − ” (4)from Kennicutt (1998) (adjusted as appropriate for the Chabrier(2003) IMF adopted here), we then we obtain the light distribution,and can evaluate e.g. the half-luminosity radius at each instant intime. Figure 1 shows the resulting total SFR as a function of time.Note that the procedure above, strictly speaking, defines only a one-sided SFR for t > ˙ M ∗ ( t > ) . Real starbursts, of course, have a rise as well as a fallafter their peaks, and simulations (as well as simple dynamical con-siderations) indicate that – to lowest order – the rises and falls areroughly symmetric (di Matteo et al. 2007; Cox et al. 2008). If weassume such symmetry, it is trivial to convert our solution for theburst from t > t >
0; namely ˙ M ∗ | one sided ( t > ) ≡ f ( t > ) . Physically,our symmetry assumption means that half the time at each lumi-nosity is spent on each side of the peak. Quantitatively, for t < ˙ M ∗ ( t | t < ) → f ( − t / ) , and for t > ˙ M ∗ ( t | t > ) → f (+ t / ) . This is the symmetric or “two-sided” light curve. This choice guarantees that the total stellar massformed in the burst is preserved, and enforces the symmetry con-straint. It is, therefore, appropriate for comparison to full time-dependent light-curves. However, we stress that for all the quan-tities derived in what follows, it makes no difference. whether weassume a one-sided or symmetric two-sided lightcurve. The dura-tion of the burst is the same. (If it is defined as some t burst , we simplyhave to change our notation from referring to t = t = t burst , to t = − t burst / t = + t burst /
2; but the zero point of time is arbitraryin any case.) Moreover, for the purposes of luminosity functionsand SFR densities, the relevant quantity is the time spent in eachSFR interval – this is, by definition, identical whether or not wesplit that time across two sides of a peak.We compare the integrated SFR as a function of time from ourillustrative example in Figure 1, inferred from the relic starburstprofile, to the true simulation SFR as a function of time in the fullhydrodynamic simulation that produces such a profile. The agree-ment is excellent, verifying our procedure. This is despite the factthat, in the full simulation, many of our assumptions are not truein detail. For example, the simulations obey a local (not global)Kennicutt-Schmidt law; thus clumping and instabilities can accel-erate star formation locally. The simulations also account for feed-back from stars, stellar winds, and black hole accretion, which canremove and recycle gas. The SFR is not precisely trivial or mono-tonic in time. And the gas inflows do not occur instantaneously (so More sophisticated, luminosity and galaxy property-dependent conver-sions have been proposed, but they largely differ at low IR luminositiesin e.g. extended disks. Since the systems of interest here are massive star-bursts, we find that the alternative conversions from e.g. Buat & Burgarella(1998) or Jonsson et al. (2006) make no difference to our conclusions. that the burst would start exactly from pure gas at t =
0, as assumedhere). Nevertheless, it is clear that these details lead to almost nodeviation between the true SFR versus time and that inferred fromour simple procedure, given the observed relic starburst/inner com-ponent of the system. This is in part because some of these effectsare small. Also, several tend to offset one another, so that, on av-erage , the dominant physical considerations are the validity of anaverage Kennicutt-Schmidt law and the rapid, initially gas-rich na-ture of dissipational starbursts.
Figure 2 illustrates the results of our starburst inversion procedurefor a representative sub-sample of the Kormendy et al. (2009) Virgoellipticals. We first show ( top ) the stellar mass surface density pro-files of the inner/dissipational/burst component in each object ( not the total stellar mass density profile), as obtained using the two-component decomposition in Hopkins et al. (2009b,e), over the dy-namic range covered by the observations from Kormendy et al.(2009). In other words, we have already subtracted off the empiri-cally estimated non-burst stellar component. For each system, usingthe methodology described in § 2, we convert the inferred surfacebrightness profile to a time-dependent total star formation rate, asillustrated in Figure 1. We show the one-sided ( t >
0) SFR versustime so obtained, in a logarithmic scale (this highlights the rele-vant power-law behavior). The total burst star formation rates decayfrom some initial maximum in a power-law like fashion. We findthat good fits can be obtained to the time-dependent light curveswith the functional form ˙ M ∗ ( t ) = ˙ M ∗ ( ) [1 + ( t / t ) ] β ⇐⇒ ˙ M ∗ ( ) [1 + ( | t | / [ t / ) ] β . (5)The first is a one-sided light curve appropriate if the burst occursonly for t >
0, and the second is a two-sided curve symmetric about t = β ∼ .
5, and clearly exhibit a characteristic decaytimescale of ∼ . t / , i.e. the time for the starburst to decay to 1 / t / = ( /β − ) t . (6)Likewise, the peak SFR of the burst, ˙ M ∗ ( ) , can be trivially re-lated (knowing the burst duration t / and β ) to the total stellarmass formed in the burst, via the integral constraint that M burst = R ∞ ˙ M ∗ ( t ) d t , giving ˙ M ∗ ( ) = ( β − ) M burst t = [ ( β − ) ( /β − ) ] M burst t / . (7)For convenience, we will use t / as our timescale of interest below,as it has a straightforward interpretation independent of β . More-over, for the β values of interest, the coefficient [ ( β − ) ( /β − ) ]in Equation 7 is nearly constant at ≈ . − . c (cid:13) , 000–000 tarbursts from Ellipticals l og ( Σ ∗ ) [ M O • k p c - ] Low MassHigh Mass S F R [ M O • y r - ] R s b [ k p c ] Figure 2.
Typical results from recovering the burst star formation histo-ries of spheroids.
Top:
Surface stellar mass density profiles of the em-pirically fitted burst component alone, for each observed galaxy in theKormendy et al. (2009) Virgo elliptical sample. Colors denote the total massof each spheroid, from 10 M ⊙ (violet) to ∼ M ⊙ (red). Middle:
Im-plied burst SFR versus time since the peak of the starburst. The behavior isroughly self-similar and well-approximated by Equation 5.
Bottom:
Burstsize scale (radius enclosing 1 / starburst region; we specifically define this as the half-luminosityradius, where we assume the surface luminosity density (in e.g. theinfrared) is proportional to the gas surface density at each instant.We show this as well ( bottom ) in Figure 2. In analogy to the fittedtotal star formation rate, we can similarly describe this evolution bya power-law of the form ˙ R sb ( t ) = R sb ( ) [1 + ( t / t , r ) ] β r . (8) The radius doubling time t / , r is trivially related to t , r by the samerelation as Equation 6 above; i.e. t / , r = ( /β r − ) t , r . Figures 3 illustrates these fits and a number of global proper-ties of the bursts, using our full sample of profiles taken fromKormendy et al. (2009) and Lauer et al. (2007). We also present therelevant fits and scaling relations in Table 1. First, Figure 3 showsthe total burst mass (i.e. total mass of the inner/dissipational com-ponent) as a function of total galaxy stellar mass ( top left ). Thesevalues, and independent tests of their validity, as well as compar-ison to models of spheroid formation and observed disk gas frac-tions, are discussed extensively in Hopkins et al. (2008c, 2009b,e,2008a, 2009d). To lowest order, a fraction ∼
10% is typically in theburst/dissipational component, i.e. M burst ∼ . M ∗ . For somewhathigher accuracy, we use the fit from Hopkins et al. (2008a), moti-vated by comparison with disk gas fractions that constitute burstprogenitors, M burst ≈ + ( M ∗ / . M ⊙ ) . ] . (9)This is also shown in Figure 3, along with the simpler linear fit.Next, we show the burst half-life t / ( bottom left ), as a func-tion of burst mass (the comparison versus total stellar mass is sim-ilar, but other possible correlations are less significant). To lowestorder this is mass-independent, with a value ∼ yr – dependingon the definition, this equates to a total observable burst lifetime ofa couple 10 yr with an inherent factor ∼ − t / ≈ . × yr “ M burst M ⊙ ” . , (10)with a mass-independent scatter of σ t / ≈ . ∝ R e / V c is only weakly mass-dependent ( ∝ M ( . − . ) ∗ , if we compare R e ∝ M . ∗ and V c ∝ M . ∗ from Courteau et al. 2007) (see alsoBell & de Jong 2001; McGaugh 2005; Avila-Reese et al. 2008).Moreover, the scatter in the dynamical times expected from thesecorrelations is similar to that in Figure 3.We next show the slope of the power-law decay, β ( bottomright ). There is no obvious correlation with burst mass or anyother parameter; however, the values cluster in a reasonably nar-row range, β ∼ −
3, with a best-fit median of β ≈ .
38. Again,this matches expectations; given some Schmidt-law star formationrelation, the slope β follows from the profile shape of the relicburst component. These profile shapes, and their physical origins,are discussed extensively in Hopkins et al. (2008c, 2009b); there,the authors show that they can be reasonably well-parameterizedas Sersic functions with relatively low Sersic indices n s ∼ − β , a simple numerical cal-culation shows that β ≈ − ˙ M ∗ ( ) c (cid:13) , 000–000 Hopkins et al. ∗ ) [ M O • ]789101112 l og ( B u r s t M a ss M bu r s t ) [ M O • ] Best-Fit Trend burst ) [ M O • ]0.11101001000 P ea k B u r s t S F R [ M O • y r - ] Lauer et al. SampleKormendy et al. Sample burst ) [ M O • ]0.010.11 B u r s t H a l f - L i f e t / [ G y r ] burst ) [ M O • ]1.01.52.02.53.03.5 B u r s t D e c a y S l ope β Figure 3.
Observationally inferred properties of spheroid starbursts as a function of mass, from the relic starbursts.
Top Left:
Mass of the starburst componentof spheroids as a function of total spheroid stellar mass. Detailed determinations and tests of these component decompositions are presented in H09b,e. Weshow results for the Virgo galaxy sample of Kormendy et al. (2009, red stars) and local massive spheroid sample of Lauer et al. (2007, violet circles). Dashedblue lines show the range of best-fit scalings to the observed systems (given in Table 1).
Top Right:
Peak SFR of the starburst, as a function of total starburstmass.
Bottom Left:
Duration of the starburst (time from peak to half-peak SFR).
Bottom Right:
Best-fit power-law slope to the decay of the SFR versus time(Equation 5).
Table 1.
Starburst Scaling Relations y x a b σ log ( M burst / M ⊙ ) log ( M ∗ / M ⊙ ) − . ± .
12 0 . ± .
16 0 . ( ˙ M ∗ [0] / M ⊙ yr − ) log ( M burst / M ⊙ ) . ± .
05 0 . ± .
04 0 . ( t / / Gyr ) log ( M burst / M ⊙ ) − . ± .
04 0 . ± .
04 0 . ( ˙ M ∗ [0] ) log ( M burst / t / ) − . ± .
01 1 . ± .
01 0 . β log ( M burst / M ⊙ ) . ± .
12 0 . ± .
11 0 . ( R relic / kpc ) log ( M burst / M ⊙ ) − . ± .
07 0 . ± .
07 0 . ( R / / kpc ) log ( M burst / M ⊙ ) − . ± .
07 0 . ± .
06 0 . ( t / , r / Gyr ) log ( M burst / M ⊙ ) − . ± .
08 0 . ± .
08 0 . β r log ( M burst / M ⊙ ) . ± . − . ± .
017 0 . ( Burst Age / Gyr ) log ( M ∗ / M ⊙ ) . ± .
11 0 . ± .
04 0 . / Gyr log ( M ∗ / M ⊙ ) . ± . . ± . . x and y , the best-fit linear scaling of y ( x ) is presented, with the assumedfunctional form y = a + bx . Errors on the fitted parameters a (normalization) and b (slope)are given, along with the intrinsic scatter in y about this correlation, σ . c (cid:13) , 000–000 tarbursts from Ellipticals ( top right ). Immediately, it is clear that this is very tightly corre-lated with the total burst mass M burst . We compare the correlationexpected from Equation 7, assuming a typical β ≈ . ( β − ) ( /β − ) ≈ .
5, i.e. ˙ M ∗ ( ) ≈ . M burst / t / . We show thisboth for an assumed constant t / = × yr, and for t / givenby the weakly mass-dependent fit in Equation 10. Both cases agreevery well with the observed trend, with a small, mass-independentscatter of σ ˙ M ∗ ( ) ≈ . − . > M ⊙ yr − are obtained, but there is a largedynamic range – low mass systems can have “bursts” with impliedpeak SFRs as low as ∼ . − M ⊙ yr − . Also, since there is rel-atively little variation in t / at fixed mass, the scatter is, as men-tioned above, small. Producing a very high-SFR burst therefore re-quires an extremely high-gas-mass system; the case where a lower-mass system might compress its gas much more efficiently, leadingto a shorter-lived but arbitrarily highly-peaked burst appears to berare. Finally, we note that their is some (factor ∼
2) observationaluncertainty in the exact normalization of the correlation betweenSFR and gas surface densities. Adopting a different IMF willlikewise systematically shift the inferred relation from Kennicutt(1998). However, so long as the change in Equation 2 is purelyin normalization, it has no effect on the shape of the inferredlightcurves, and translates directly to corresponding normalizationchanges in the resulting SFR and timescales in Figures 2-4 and Ta-ble 1. Specifically, if the normalization in Equation 2 is multipliedby a factor η , then the implied peak SFR ˙ M ∗ [0] increases by thesame factor η , and the burst timescale t / (and t / , r ) decrease bya factor η . All other parameters in Table 1 are unchanged. Figure 4 continues our analysis, plotting parameters as in Figure 3but for the burst spatial sizes. First, we consider the effective (pro-jected half-stellar mass) radius of the burst remnant ( top left ), i.e.the radius of all stars after the burst has completed, namely of therelic profile as shown in Figure 2 ( top left ). We consider this as afunction of burst mass, and find a correlation of the form R relic ≈ .
37 kpc “ M burst M ⊙ ” . (11)with scatter σ R relic ≈ . total mass distri-bution, it is studied in more detail in Hopkins et al. (2008c, 2009b).There, and in Hopkins et al. (2009j), it is shown that such sizes arisenaturally from dissipational inflows, as a consequence of where theinflows stall and become self-gravitating. This is the reason that thepower-law scaling in Equation 11 reflects the observed scaling ofspheroid sizes with mass (see e.g. Shen et al. 2003), and is a conse-quence of simple dynamics, independent of the star formation law.Next, we consider R burst ( top right ), the effective (half- light ,assuming light traces local SFR) radius of the burst itself duringthe starburst episode (as shown in Figure 2; bottom ). For conve-nience, and as a reference point, we define the plotted quantity as R burst ( t / ) , i.e. the size at time t = t / . This is a convenient param-eterization and also has the advantage of being more representativeof what might be observed, since the duty cycle at peak is vanish-ingly small. In any case, there is a similar correlation, with R / ≡ R burst ( t / ) ≈ .
27 kpc “ M burst M ⊙ ” . (12)with σ R relic ≈ . R burst near the peak of star formation ismuch smaller (by a typical factor ∼ a few) than the size scale ofthe relic stars. This owes to two factors. First, near-peak, it is thehighest-density material just at the galaxy center that dominates theSFR, while almost none of the more extended material is contribut-ing at a significant level – hence the size of the burst itself will stillgrow by a large factor with time (see Figure 2). Second, this sizeis weighted by SFR, i.e. by gas density to some super-linear poweraccording to the Kennicutt (1998) relation. If Σ gas follows a Sersiclaw with some index n s and effective radius R e , and Σ SF ∝ Σ n K gas ,then it is trivial to show that the half-SFR radius will simply be R e × n − n s K . For n s = n K = .
5, this gives a factor of almost5 smaller R e for the SFR distribution (for n s =
2, more typical ofthe inner components of the systems here, this is a more moderatefactor ∼ same systems that reach > M ⊙ yr − peakSFRs, the effective radii are also large, ∼ −
10 kpc. Moreover, werefer here only to the strict half-SFR radius; estimating an observedsize in detail requires radiative transfer modeling. Analysis of simu-lations in Wuyts et al. (2009b,a), and similar analysis of some well-studied local starburst galaxies (Laine et al. 2003b), suggests that –especially at the peak of activity – the IR or radio size can be biasedto factors of ∼ t / , r .Unsurprisingly, the values are very similar to the SFR decay times t / , with similar weak dependence on mass. We find a best-fit me-dian t / , r t / , r ≈ . × yr “ M burst M ⊙ ” . , (13)with a mass-independent scatter of σ t / , r ≈ . β r , of the starburst size versus time.Here, there may be a weak correlation with mass at the high-massend, but formally this is only marginally significant ( ∼ σ ) andonly if we restrict ourselves to M burst ∼ − M ⊙ . We there-fore hesitate to quote a correlation; but do note the relatively ro-bust median value of β r ∼ . − .
5, with scatter σ β r ≈ .
1. Again,these naturally follow from the combination of profile shapes andthe Kennicutt (1998) relation.
Thus far, we have adopted the Kennicutt-Schmidt relation as cal-ibrated at low redshifts (Equation 2), with an index n K = . ˙ Σ ∗ ∝ Σ n K gas ). However, some recent observations of high-redshift,massively star-forming systems have suggested that the SFRs ofsuch systems exceed the predictions with this index and favor asomewhat steeper slope of n K = . ± . ˙ Σ ∗ ≈ . × − M ⊙ yr − kpc − “ Σ gas M ⊙ pc − ” . . (14)We caution that the high-redshift observations – in particular the in-ferences of both gas masses and total bolometric luminosities andhence SFRs – remain uncertain. Nevertheless, this is an importantsource of error, and generally larger than many of the other sys-tematic uncertainties (those being at the factor of ∼ c (cid:13) , 000–000 Hopkins et al. burst ) [ M O • ]0.010.1110 B u r s t R e li c S i z e R r e li c [ k p c ] Best-Fit Trend burst ) [ M O • ]0.010.1110 B u r s t S i z e R / [ k p c ] Lauer et al. SampleKormendy et al. Sample burst ) [ M O • ]0.010.11 B u r s t S i z e - G r o w t h T i m e t / , r [ G y r ] burst ) [ M O • ]0.00.20.40.60.81.0 B u r s t S i z e - G r o w t h S l ope β r Figure 4.
As Figure 3 – observationally inferred properties of spheroid starbursts as a function of mass, from the relic starburst stars – but here giving thespatial size distribution of starbursts.
Top Left:
Size (projected half-stellar-mass radius) of the burst relics (generally a factor ∼ . R e ),directly from the observed galaxy properties. Top Right:
Size of the starburst at time t = t / (when the SFR is half-peak). Size here is defined as half-SFR orhalf-light (assuming light traces SFR) radius; hence is more compact than the relic size. Bottom Left:
Size-doubling time of the starburst (time for e.g. relativegas exhaustion in central, high density regions to leave only star formation at larger radii, giving a larger apparent radius). This is simply related to the SFRdecay timescale in Figure 3.
Bottom Right:
Best-fit power-law slope to the increase of starburst scale radius versus time (Equation 8). gas densities. We therefore re-consider our previous comparisons,but instead adopt this modified (steeper) index.Table 2 shows the resulting revised fits, for quantities that areaffected by this (e.g. the SFR and size versus time). The resultsare, in all cases, qualitatively very similar to those in Figures 2-4.We therefore show only the results of fitting the correlations, andpresent a simple illustration of how the inferred burst propertieschange with n K in Figure 5. Specifically, we repeat our analysisof the observed Virgo systems from Figure 2, but include both theprevious results (with n K = .
4) and the steep-index results (with n K = . peak SFR of bursts is systematically higher. The differ-ence is large – a factor of ≈ h ˙ M ∗ ( ) i ≈ M ⊙ yr − ( M burst / M ⊙ ) . ; in otherwords, this implies that most systems with M burst > × M ⊙ will exceed 1000 M ⊙ yr − SFRs at the peak of their starburst phase.However, with a higher SFR comes faster gas exhaustion, so thehalf-life of this peak, t / , is systematically shorter, by a factor of ≈ t / ≈ yr. The decay slope β is also systematically different, now β ≈ . . t ≪ t / is more peaked with n K = .
7, and then at t ≫ t / (when the gas density is lower) more extended (because the steeperindex yields lower SFRs at low surface densities).In practice, if one defines a starburst “duration” by time spentabove some fixed, high SFR, this change in the Kennicutt-Schmidtslope does not actually have as dramatic an effect as implied bythe change in t / . At t > . t < . Having quantified how burst properties scale with mass, and howthese are connected to overall properties of their host galaxies, werequire only a simple extension of our analysis to infer the cosmichistory of starbursts.Figure 6 shows the observed ages, at z =
0, of the domi- c (cid:13) , 000–000 tarbursts from Ellipticals Table 2.
Starburst Scaling Relations with Alternative Kennicutt-Schmidt Relation y x a b σ log ( ˙ M ∗ [0] / M ⊙ yr − ) log ( M burst / M ⊙ ) . ± .
12 0 . ± .
11 0 . ( t / / Gyr ) log ( M burst / M ⊙ ) − . ± .
10 0 . ± .
09 0 . ( ˙ M ∗ [0] ) log ( M burst / t / ) − . ± .
01 1 . ± .
01 0 . β log ( M burst / M ⊙ ) . ± .
05 0 . ± .
04 0 . ( R / / kpc ) log ( M burst / M ⊙ ) − . ± .
07 0 . ± .
06 0 . ( t / , r / Gyr ) log ( M burst / M ⊙ ) − . ± .
23 0 . ± .
21 0 . β r log ( M burst / M ⊙ ) . ± . − . ± .
01 0 . ˙ Σ ∗ ∝ Σ . ), as suggested by observations of high-redshift,high-SFR systems. S F R [ M O • y r - ] n K = 1.4n K = 1.7 Figure 5.
As Figure 2 ( middle ), but comparing a subset of the inferredstar formation histories from the observed systems with different assumedKennicutt-Schmidt law indices. Solid lines, as Figure 2, assume the z = n K ≈ .
4. Dashed lines assume a steeper relation suggested by somehigh-redshift observations, n K ≈ .
7. The latter leads to more sharplypeaked bursts with larger peak SFRs in early ( t < yr) stages. The starformation histories after ∼ . n K = . nant starburst in z = t lookback , burst ≈ . “ M ∗ M ⊙ ” . . (15)We also plot the 1 σ scatter in ages at each mass determined fromeach sample; these too agree well. We find that this is reasonablyparameterized as a mass-independent, Gaussian ± ∗ ) [ M O • ]02468101214 B u r s t/ N u c l ea r SSP A ge a t z = [ G y r ] Best-Fit Trend Trager et al. 2000Caldwell et al. 2003Nelan et al. 2005Thomas et al. 2005Bernardi et al. 2006Gallazzi et al. 2006
Figure 6.
Observed average single stellar population age (at z =
0) ofspheroid nuclear/most recent burst populations, as a function of totalspheroid stellar mass. Points show the median age at each mass from variousdeterminations; error bars show the scatter (not the much smaller error inthe mean). Observations are compiled from various sources (labeled). Solidblack lines show the best-fit log-log and log-linear trends for the medianvalues (Equation 15 and Table 1).
Various analyses, for example observationally analyzing mockSEDs constructed from hydrodynamic galaxy formation simu-lations or semi-analytic models of galaxy formation (see e.g.Wuyts et al. 2009b,a; Trager & Somerville 2009, and referencestherein) have shown that, for typical spheroids that combine starsformed over an extended (potentially still short in absolute terms)star formation history in pre-merger disks (assembled dissipation-lessly in mergers) with stars formed in bursts via gas dissipation inmergers, these observed ages do reflect the cosmic time at whichthose bursts occurred. Since a number of constraints and theoret-ical models show that the time of last major gas-rich merger andtime when star formation in “quenched” systems must shut downare tightly coupled (even though there may not necessarily be acausal relationship between the two), this can also be thought ofas dating the time of the last burst of star formation that con-sumed the residual gas in the systems (see Bundy et al. 2006, 2008;Haiman et al. 2007; Hopkins et al. 2007, 2006a; Cattaneo et al.2006; Hopkins et al. 2008b; Shankar et al. 2009). Moreover, mostof these ages are derived specifically from the central stellar popu-lations, those that are clearly formed dissipationally; thus they rep-resent a clear constraint on when this burst occurred independent ofmore complex assembly and star formation histories for the more c (cid:13) , 000–000 Hopkins et al. extended components. Finally, where available, detailed, resolvedmeasurements of stellar population gradients show that these cen-tral populations do represent a distinct component, in agreementwith dissipational models, and that their properties favor a singleburst, rather than an extended star formation history or dissipation-less assembly of many distinct sub-components (Schweizer 1996;Titus et al. 1997; Schweizer & Seitzer 1998, 2007; Reichardt et al.2001; Michard 2006).Given these comparisons, it seems reasonable to take the agedistributions in Figure 6 as a first approximation to the times whenthese starbursts occurred. Together with the analytic fits to the dis-tributions of burst properties versus mass, this enables us to recon-struct the burst history of the Universe. Specifically, we begin withthe mass function of bulges/spheroids at z =
0, here adopting thedetermination in Bell et al. (2003). We ignore disks/late-type galax-ies and very low-mass bulges, as these contribute negligibly to thequantities of interest (they have very low burst masses, hence lowpeak SFRs, where we will show the population is dominated bynon-burst star formation). At a given mass, we know the num-ber density of systems today, and the distribution of burst masses M burst ( M ∗ , tot ) (from e.g. Equation 9, with appropriate scatter); i.e.we know the (co-moving) total number density of bursts with var-ious masses that must have occurred before z = t / (Equation 10) and power law decay slope (median ≈ .
4; again with scatter). Together these define the correspond-ing peak SFR ˙ M ∗ ( ) . The time t = ≈ t / , but by this pointthe luminosity has decayed sufficiently that the burst is negligible.We then make mock observation of the Monte Carlo population atany redshift z , and construct the luminosity function or luminositydensity of bursts. Of course, we are really predicting a distribu-tion of star formation rates; for comparison with observations, weconvert these to total (8 − µ m) infrared luminosities with theempirically-calibrated conversion factor in Equation 4. We have experimented with alternative determinations of the bulge-dominated galaxy mass function at z =
0, including those presented inBernardi et al. (2009) and Vika et al. (2009). We have also attempted toconstruct the mass function of bulges specifically, following Driver et al.(2007) or by adopting the morphology-separated mass functions fromKochanek et al. (2001) with a type-dependent B / T from Balcells et al.(2007b). We find these make little difference to any of our conclusions, asthe resulting differences lie primarily in identification of low mass bulges orbulges in late-type galaxies which are negligible in the total IR luminositydensity. Note that, in comparing to observed starburst light curves or luminosityfunctions, the average M ∗ / L will depend systematically on the IMF. How-ever, the inferred stellar surface mass density on which we base our analysis We can also trivially repeat this procedure for a different bolo-metric correction from ˙ M ∗ to L IR (which will systematically shiftthe predicted luminosities by some factor ∼ n K ≈ . n K ≈ .
4, given in Table 2.
Figure 7 shows the resulting burst luminosity functions at a varietyof redshifts from z = −
4. We compare with observations com-piled from a number of sources, available from z ∼ −
3. Note thatall of these are corrected to a total IR luminosity from observationsin some band; we adopt the corrections compiled in Valiante et al.(2009), but emphasize that some caution, and at least a systematicfactor ∼ L IR , should be considered in estimatesfrom most if not all observed wavelengths.At low luminosities, the predicted LFs are well belowthose observed, but they grow rapidly in importance, and areroughly consistent with the observations, at the high-luminosityend. This is expected – it is well-established that star forma-tion at relatively low rates is dominated by quiescent star for-mation in normal (e.g. non-merging) galaxies; i.e. distributedstar formation in disks, rather than dissipational starbursts (seee.g. Sanders & Mirabel 1996; Tacconi et al. 2008; Noeske et al.2007b,a; Bell et al. 2005; Jogee et al. 2009; Robaina et al. 2009;Veilleux et al. 2009; Sajina et al. 2007). At low redshifts, for exam-ple, nuclear starbursts (typically merger-induced) are negligible atluminosities . L ⊙ ; but at the highest luminosities, > L ⊙ ,they become dominant (Sanders & Mirabel 1996). At higher red-shifts, the luminosity function of bursts increases rapidly, as doesthe global luminosity function. This again is expected, as increas-ing gas fractions and specific star formation rates lead all sys-tems (mergers and quiescent disks) to higher SFRs at fixed mass.Thus, at all redshifts the (relatively) low-luminosity population re-mains non-burst dominated, and the threshold (in terms of L IR ) forburst domination moves up to higher luminosities ( L IR & L ⊙ at z ∼
1, and L IR & L ⊙ at z ∼ ∼ − Mpc − log − L IR ,with much weaker redshift dependence. This is similar to the be-havior predicted in models of merger-induced star formation bursts(Hopkins et al. 2009k).We show predictions for both the cases of the low-redshiftKennicutt-Schmidt law slope of n K ≈ .
4, and for the suggestedsteeper value of n K ≈ . ∼ . is determined from observed luminosity profiles, with the same systematicdependence on M ∗ / L . Thus, so long as the IMF does not evolve with red-shift, variations between typical choices make little difference.c (cid:13) , 000–000 tarbursts from Ellipticals
10 11 12 13 14 -10-8-6-4-2 l og ( Φ ) [ M p c - l og - ( L I R ) ] z = 0.1 Bursts (n K =1.4)Bursts (n K =1.7) (No Scatter)
10 11 12 13 14 -10-8-6-4-2 z = 0.5
Merger ModelPredictions
10 11 12 13 14 -10-8-6-4-2 z = 1.0
10 11 12 13 14 -10-8-6-4-2 z = 1.5
10 11 12 13 14 -10-8-6-4-2 log(L IR ) [ L O • ] z = 2.0
10 11 12 13 14 -10-8-6-4-2 z = 4.0
Figure 7.
Total (8 − µ m) IR luminosity functions as a function of redshift. We show the contribution from bursts, using the methodology presented here(black lines), assuming a Kennicutt-Schmidt index of n K = . . ∗ ’s) Babbedge et al. (2006, red pentagons), Chapman et al.(2005, dark green + ’s), and Pérez-González et al. (2005, blue open circles). Shaded green range shows the prediction (with systematic uncertainty) for the IRLF of merger-induced bursts from the semi-empirical models and hydrodynamic simulations in Hopkins et al. (2009k)4. The burst contribution dominates thebright end of the IR LF, agreeing well with predicted merger-induced bursts and other constraints, but is a small fraction of the typical ∼ L ∗ activity at anyredshift. The luminosity threshold above which bursts are important increases with redshift, along with the entire LF. Fits to the burst LF inferred here areprovided in Table 3 of bursts, as a function of redshift, for both the shallow and steepKennicutt-Schmidt slopes. Following standard convention, we fitthe luminosity function with a double power law form, Φ ≡ d n d log L = φ ∗ ( L / L ∗ ) α + ( L / L ∗ ) γ (16)where the parameters φ ∗ (normalization), L ∗ (break luminosity), α (faint-end slope, i.e. Φ ∝ L − α for L ≪ L ∗ ), and γ (bright-end slope,i.e. Φ ∝ L − γ for L ≫ L ∗ ) depend on redshift, with that dependenceconveniently approximated aslog L ∗ = L + L ′ ξ + L ′′ ξ log φ ∗ = φ + φ ′ ξ + φ ′′ ξ α = α + α ′ ξ + α ′′ ξ γ = γ + γ ′ ξ + γ ′′ ξ ξ ≡ log ( + z ) (17)(Note that log here and throughout refers to log .) We perform thisfit using our results only up to redshift z =
4, as the uncertaintiesgrow rapidly at higher redshift. The fits should be considered withcaution at higher redshifts. We provide the best-fit parameters inTable 3, along with their uncertainties.We note briefly that, especially because bursts are importantat the high-luminosity end of the total LF, incorporating the scat-ter in the relevant relationships in § 3.2 is critical to the predicted abundance of IR-bright systems. In Figure 7, we compare the pre-diction if we were to ignore all scatter – i.e. simply construct ourMonte Carlo population using just the median values of all param-eters and their correlations. As expected, this cuts off much morequickly at the high-luminosity end, reflecting the rapid exponentialcutoff in the observed number of high-mass galaxies.Recently, Hopkins et al. (2009k) presented predictions fromgalaxy formation models for the contribution to the SFR and IR lu-minosity distributions from normal, quiescent star-forming galax-ies, from merger-induced bursts of star formation, and from AGN.The models used a halo-occupation based approach to populategalaxies at each redshift (i.e. simply beginning with observedgalaxy stellar mass functions and gas masses, with merger ratesdetermined from evolving this forward in agreement with observedmerger fractions), then mapping each population to suites of high-resolution simulations, to predict the distribution of SFRs in vari-ous systems. We compare their predictions for the merger-inducedstarburst population to our inferred burst SFR and IR luminositydistributions in Figure 7. Given the systematic uncertainties theyquote (shown as the shaded range in the figure), and ours here, the The merger rates determined in this model are presented inHopkins et al. (2009l). A “merger rate calculator” script to give themerger rate as a function of galaxy mass, mass ratio, and gas frac-c (cid:13) , 000–000 Hopkins et al. agreement is reasonable. The two diverge at z ≫
3, but this is wherethe uncertainties in both the modeling and our empirical inferencesbecome large. The faint-end extrapolations are also different, butin neither case are these well-constrained (in a systematic sense, atvery low SFR/late times, the designation as “burst” is somewhat ar-bitrary). In general, the agreement seen supports our interpretationof the burst components of galaxies, and favors a possible mergerorigin for these bursts. More important, the burst history inferredhere should not correspond to the normal or quiescent star formingpopulation.
We also note that, at z ∼ −
3, our reconstruction appears to some-what under-predict the abundance of the most luminous starburstsystems relative to observations. However, at this redshift, the onlyconstraints at the high luminosities of interest come from the sub-millimeter populations observed in Chapman et al. (2005). This isa relatively small sample, with a number of difficult completenesscorrections involved in estimating the number density, and non-trivial cosmic variance given the sample selection. Recently, analy-ses of the number counts of similarly bright sources in much largerIR surveys have suggested that the average counts may actually bemuch lower – a factor ∼ L IR , a small error in the bolomet-ric correction translates to order-of-magnitude differences in num-ber density at fixed L . And indeed, such a conversion from thesub-millimeter to total IR flux is highly uncertain, depending onquantities that are not well-known for these populations, such asthe dust temperature, and subject to large object-to-object varia-tion in the local Universe. It should also be stressed that the ob-served high-redshift points in Figure 7 have not been corrected forpossible AGN contributions to the IR luminosity, although (fromindirect constraints) it has been argued that AGN are unlikelyto contribute more than ∼
30% of the bolometric luminosity inthese systems (Menéndez-Delmestre et al. 2009; Casey et al. 2009;Bussmann et al. 2009). For these reasons, we consider the compar-ison at these redshifts and luminosities to be largely qualitative.A more detailed comparison would require full modeling from e.g.high-resolution simulations that include spatially dependent, multi-phase star formation, metallicity, gas, and dust distributions thatcan forward-model the full SED of such bursts (for some prelim-inary such results, we refer to Li et al. 2008; Jonsson et al. 2009;Narayanan et al. 2009b,a; Younger et al. 2009).But the comparison in Figure 7 emphasizes an importantpoint: if the number density and bolometric corrections of suchhigh-luminosity systems are correct, then either the Kennicutt(1998) law must break down severely at high redshifts (givingmuch higher star formation rates than implied by e.g. observationsof high-density systems in Bouché et al. 2007, at fixed surface den-sity), or something must be fundamentally wrong in our conver-sion between mass and light, either at low redshifts (i.e. some dra-matic errors in measurements of the surface brightness and stellarmasses of local ellipticals, which appears unlikely) or at high red-shifts (i.e. the results of a strongly time-dependent stellar IMF, or tion, which determines these luminosity functions, is publicly available at . large heavily-obscured AGN contributions to the high- z luminosi-ties).Taken at face value, the number density of high- z bursts in-ferred, coupled with their implied star formation rates, would implyfar too much high-density material in massive systems today. Allof our analyses of burst star formation rates effectively set an upperlimit on the burst number density at high- L . Consider, for example,the consequences of relaxing our assumptions. If starbursts weresplit into several separate events, with lower initial burst rates, thenthe duty cycle would go up, but the absolute L would go down (andin this portion of the luminosity function, the net result would be asevere decrease in the high- L IR prediction). If systems were not ini-tially gas-dominated or 100% gas, but maintained some lower gasfraction in quasi steady-state for the time needed to build up theirdensities, then again the peak luminosities decrease severely. If thepresent-day densities are the result of assembling many systems,this is the same as breaking up bursts, and leads to a lower pre-diction. The bursts cannot be more concentrated in time (yieldinghigher peak SFR) without violating the Kennicutt (1998) law; evendoing so, they become more luminous but shorter, and it requiresan order-of-magnitude change to the star formation relation beforethis yields a match.Ultimately, a number of other obvious comparisons make thisclear. Yielding the implied SFRs of the observed bright SMG sys-tems (taken at face value) without breaking the Kennicutt (1998)relation requires ∼ M ⊙ kpc − gas (and ultimately stellar) sur-face densities, over spatial radii of ∼ − .
100 pc scales; not at kpc scales (where the typical surface den-sity in a massive elliptical is ≈ M ⊙ kpc − ). Also, given thecharacteristic SMG lifetime of ∼ yr, from observational con-straints, it is straightforward to estimate the total amount of stellarmass that would be formed at or above these surface densities, fromSMGs over the redshift range z ∼ −
4, and the number comes to ∼ M ⊙ Mpc − [ Σ ∗ > M ⊙ kpc − ]. Hopkins et al. (2009a)calculate the actual observed stellar mass density in all galaxiesabove such a threshold, and find that it is ≈ . × M ⊙ Mpc − ,well below the range of uncertainties owing to e.g. stellar mass lossand the somewhat uncertain values above. Figure 8 integrates the luminosity functions shown in Figure 7 togive the total IR luminosity density, and corresponding total SFRdensity, owing to bursts, as a function of redshift. Note that manyof the uncertainties that might affect our other results are integratedout here (e.g. the exact lightcurve shape, or Kennicutt-Schmidtslope). We compare with the total SFR density from a numberof observations in IR, radio, and UV wavelengths as compiled inHopkins & Beacom (2006), as well as that inferred at high redshiftsfrom observations of the Lyman- α forest in quasar spectra andimplied ionizing background, compiled in Faucher-Giguère et al.(2008). As expected from Figure 7, the predicted SFR density ow-ing to bursts is well below the total SFR density observed. Thisclearly demonstrates that bursts do not, and cannot, dominate thetotal SFR density.We compare with a number of observations that attempt to es-timate the luminosity or SFR density induced by mergers. First, westress that this is different from the total SFR density in objects identified as mergers. A merger can take ∼ c (cid:13) , 000–000 tarbursts from Ellipticals Observed:
TotalMerger-Induced:
Brinchmann et al. 1998Bell et al. 2005Jogee et al. 2009Robaina et al. 2009Sanders & Mirabel 1996
Bursts l og ( ρ I R ) [ L O • M p c - ] -4.0-3.5-3.0-2.5-2.0-1.5-1.0 l og ( ρ S F ) [ M O • y r - M p c - ] Figure 8.
Total IR luminosity density (and corresponding SFR density) as a function of redshift. We show the inferred contribution owing to bursts (solidblack line) from the observed z = total IR luminositydensity/SFR density (green diamonds), and at high redshifts the SFR density inferred from Lyman- α forest measurements in Faucher-Giguère et al. (2008,green circles). We also compare observed estimates of the SFR density induced in mergers (blue points), from Jogee et al. (2009, triangles), Sanders & Mirabel(1996, inverted triangle), Robaina et al. (2009, circle), Brinchmann et al. (1998, squares), and Bell et al. (2005, star). The burst component of star formationis a small fraction ∼ −
10% of the total IR luminosity density (most stars are formed quiescently in disks and are then violently relaxed into spheroids).This agrees well with observational estimates of the fraction of star formation owing specifically to mergers, as predicted if these mergers induce the angularmomentum loss that drives gas dissipation in forming spheroids. escent disks, with at most a modest enhancement. The burst itself,where gas dissipation drives inflows into the center of the mergerremnant, enhancing the SFR significantly, has a duration of only ∼ . Typically, in these cases, the SFR density of some merger sam-ple (identified in a similar manner) is considered, but only aftersubtracting away the expected contribution from quiescent star for-mation. In general, this is accomplished via comparison to somecontrol sample of star-forming galaxies with similar stellar massesand at comparable redshifts. Robaina et al. (2009) attempt this froma pair-selected sample at z ∼ . − .
8; Jogee et al. (2009) per-form such an estimate from morphologically selected galaxies at Note that most of these authors actually measure the fraction of theSFR density in or induced by mergers, not the absolute value. We con-vert this to an absolute density by rescaling with the observed total SFRdensity at the same redshift from the best-fit observed trend presented inHopkins & Beacom (2006). z ∼ . −
1. We also compare with somewhat less well-defined,but similar samples that estimate the total amount of star forma-tion in observationally identified ongoing mergers or recent (mor-phologically disturbed) merger remnants. We compile observationsfrom Brinchmann et al. (1998) and Bell et al. (2005), who estimatethis quantity in morphologically-selected objects at z ∼ − .
5. Weperform a similar exercise at low redshift using the fraction of late-stage major merger systems as a function of IR luminosity fromSanders & Mirabel (1996), together with the IR luminosity func-tions from Saunders et al. (1990) and Yun et al. (2001), to estimatethe fraction of the luminosity density in mergers. Because the selec-tion of ongoing mergers is somewhat strict in these samples (isolat-ing near-peak times), they are not very different from the estimatesof merger-induced star formation, but formally should still be con-sidered upper limits.Our prediction agrees well with these observations, over therange z = −
1. In fact, given the (probable factor ∼
2) system-atic uncertainties involved in both, the level of agreement is sur-prising, and may be somewhat coincidental. But, in general, bothdirect estimates and our indirect constraint imply that bursts consti-tute ∼ −
10% of the SFR density, and that this is not dramatically c (cid:13) , 000–000 Hopkins et al. larger at high redshifts. The agreement here also provides furtherevidence that most of the bright, massive bursts of interest here arereally merger-induced, since that is specifically what is observed. Ifthere were a large non-merger population, the observations shouldbe much lower than our prediction.We have also compared both to the predictions of the mostrecent generation of cosmological models. Somerville et al. (2008)use semi-analytic models, and Hopkins et al. (2008d, 2009g) em-ploy semi-empirical (halo-occupation based models), both com-bined with the predictions from high-resolution galaxy mergersimulations (Cox et al. 2008; Hopkins et al. 2009c) to predict themerger-induced burst SFR and luminosity density as a function ofredshift. Most recently, Hopkins et al. (2009k) present a revisedversion of these semi-empirical models based on larger suites ofsimulations and improved halo occupation constraints and model-ing, and specifically present fits to the merger-induced burst pop-ulation. These are compared to our predicted luminosity functionsin Figure 7. Comparing all of these results to our predictions inFigure 8, we find good agreement (unsurprising, given that thepredicted luminosity functions in Figure 7 also agree well). Theimportant point is that most recent models also predict that only ∼ −
10% of the SFR density comes from mergers, at all redshifts.Finally, we hesitate to extrapolate our results beyond z ∼ − ∼ . − z & It has long been believed that the centers of galaxy spheroidsmust be formed in dissipational starburst events, such that gas inthe outer parts of a galaxy must lose angular momentum rapidly,and fall in on roughly a dynamical time to the galaxy center. Re-cently, observations have shown that it is possible to robustly sep-arate the “burst” component of galaxy profiles from the outer, vi-olently relaxed component owing to the scattering of progenitorgalaxy stars formed over earlier, more extended periods. As de-tailed, high-resolution observations of e.g. spheroid stellar massprofiles, shapes, kinematics, and stellar population properties im-prove, such decompositions become increasingly robust and appli-cable to a wide range of systems.In this paper, we use these observations as a novel, inde-pendent constraint on the nature of galactic starbursts. We stressthat by “starburst” here, we do not mean simply any high-SFRsystem; rather, we refer specifically to star formation in central,usually sub-kpc scale bursts owing to large central gas concentra-tions driven by angular momentum loss as described above. Theseare commonly associated with galaxy-galaxy mergers, which havelong been known to efficiently drive starbursts and violent relax-ation (e.g. Lynden-Bell 1967; Toomre 1977; Barnes & Hernquist1991, 1996; Barnes 1998; Mihos & Hernquist 1994, 1996). Butthey could also owe to any other sufficiently violent process; e.g.gas inflows owing to disk bars or during dissipational collapse. Regardless of origin, we can robustly identify the starburstrelics and stellar mass surface density profiles in well-observedspheroids at z =
0. This methodology and tests of its accuracyhave been discussed in Hopkins et al. (2008c, 2009b,e, 2008a).This alone is a powerful integral constraint on the star formationhistories of these systems. But we show that they can in fact beused to provide more detailed information, by allowing for the in-version and recovery of the star formation history of each burst. Wecan thus use local observations to recover the full time-dependentand scale-dependent star formation history of each burst, if we cou-ple these observed constraints to simple, well-motivated assump-tions. Namely, that some form of the Kennicutt-Schmidt relation(between star formation rate and gas surface density) applied in theformation of these stars, and that they formed in dissipational (i.e.rapid and initially gas-rich, on these scales) events. Both assump-tions are directly motivated by observations, but we also considerhow uncertainties in them translate into uncertainties in the result-ing constraints.Performing this exercise, we recover a large number of empir-ically determined parameters of starbursts, and quantify how theyscale as a function of mass and other properties. This includes thetotal mass of the starburst and the time-dependent SFR. In gen-eral, we find that the constrained starbursts can be reasonably well-approximated by a simple power-law behavior (Equation 5), ris-ing and decaying from a characteristic maximum SFR with char-acteristic half-life t burst and a typical late-time power-law slope of ˙ M ∗ ∝ ( t / t burst ) − β where β ∼ . − . M burst is ∼
10% of the to-tal spheroid mass, but it scales weakly with galaxy mass ina manner similar to how disk galaxy gas fractions scale withtheir stellar masses (expected if disks are the pre-merger pro-genitors of spheroids), M burst ≈ / ( + [ M ∗ / . M ⊙ ] . ) . De-tailed discussion of this trend, and its consequences for theglobal structure and kinematics of spheroids, are presented inHopkins et al. (2008a). However, it already makes it clear thatbursts should not dominate the SFR density. They are a smallfraction ∼
10% of all stars in spheroids (let alone all stars). Thatdoes not mean that bursts are unimportant, however. It is clearthat they control many of the properties of galaxies (Cox et al.2006; Robertson et al. 2006; Naab et al. 2006; Oñorbe et al. 2006;Ciotti et al. 2007; Jesseit et al. 2007, 2009; Covington et al. 2008;Hopkins et al. 2009b,e), and they can account for the short-lived,highest-SFR systems in the Universe. Moreover, the scatter in burstmass is significant, ∼ . − . t burst ∼ yr. Both the value ofthis timescale, and the weak scaling with galaxy and/or burst mass,agree well with the dynamical times in the central ∼ kpc of galaxies.As above, though, there is considerable scatter of order ∼ . ∼ t burst / t Hubble , or ∼ −
5% from z = −
3. Indeed,observations have shown that most galaxies at these redshifts lie ona normal star-forming sequence, without a large ∼ σ scatter frome.g. merger-induced bursts (Noeske et al. 2007b,a; Papovich et al. c (cid:13) , 000–000 tarbursts from Ellipticals ∼ − σ level in the wings of theSFR distribution at a given mass).These conclusions are supported by independent evidencefrom observational stellar population synthesis studies. Specificcomparisons to the objects considered here, where available, arepresented in detail in Hopkins et al. (2009b,e) and (Foster et al.2009). It is well-established that constraints from abundances re-quire the central portions of spheroids be formed in a similar,short timescale. And detailed decomposition of stellar populationsinto burst plus older stellar populations have yielded consistentresults for the typical burst fractions and sizes (Titus et al. 1997;Schweizer & Seitzer 1998; Reichardt et al. 2001; Michard 2006).As should be expected from the generic behavior above, burstspeak at SFRs of ∼ M burst / t burst , which follows M burst (and hence totalspheroid mass) in a close-to-linear relation. The most massive localellipticals – especially those with total stellar masses of & M ⊙ – had extreme peak SFRs of > M ⊙ yr − . Thus, it is at leastpossible that some local systems reached the highest SFRs inferredfor massive, high-redshift starburst galaxies (Papovich et al. 2005;Chapman et al. 2005; Walter et al. 2009). More moderate, but stillmassive ellipticals with M ∗ ∼ − × M ⊙ , reached a range ofpeak SFRs from ∼ − M ⊙ yr − , corresponding to their form-ing fractions from ∼ −
20% of their total masses in starbursts.These match well with the observed SFRs in more typical, localand z ∼ ∼ L ∗ galaxies), which have been specif-ically associated with mergers driving gas to galaxy centers, andforming the appropriate nuclear mass concentrations to explain e.g.elliptical kinematics, sizes, phase space densities, and fundamentalplane scalings (Cox et al. 2006; Naab et al. 2006; Robertson et al.2006; Jesseit et al. 2009; Hopkins et al. 2008c,a, 2009d).We similarly quantify starburst spatial sizes as a function oftheir mass, peak SFR, and time. Starburst sizes scale with starburstmass in a similar fashion as the spheroid mass-size relation, butare smaller than their host spheroids by a fraction similar to theirmass fraction. The most massive starbursts reach half-SFR (i.e.half-light, in IR or mm wavelengths) size scales of ∼ − ∼
10 kpc. These size scales also agree wellwith observations of the most massive high-redshift starburstingsystems (Younger et al. 2008; Tacconi et al. 2006; Schinnerer et al.2008), and of massive, compact ellipticals formed at high redshift,believed to be the relics of such starbursts (with, at that time, lit-tle envelope of dissipationless, low-density material yet accreted;van Dokkum et al. 2008; Cimatti et al. 2008; Trujillo et al. 2006;Bezanson et al. 2009; Hopkins et al. 2009i).For more typical, ∼ L ∗ starbursts, sizes range from ∼ . − shapes .Some claims have been made that the large sizes of high-redshiftstarbursts could imply that they are not scaled up analogues of lo-cal extreme starbursts; but we find here that they correspond nat-urally. Scaling up a starbursting system in the starburst mass frac-tion will not preserve spatial size, but rather will scale along thestarburst size-mass relation here, which appears to be smooth andcontinuous from the smallest starbursts with masses ∼ M ⊙ tothe largest with masses > M ⊙ . The origin of the size-mass scaling is of considerable physical interest. It has been proposedthat in such starbursts the Eddington limit from radiation pressureon dusty gas sets a universal maximum central surface density, overall mass scales, from which this follows (Hopkins et al. 2009f). Theimportant constraint, from our analysis, is that there is no discon-tinuity, and we provide the scaling relations that any such modelmust satisfy.Combining these constraints with observational measure-ments of the nuclear stellar population ages of these systems –i.e. the distribution of times when these bursts occurred – we showthat it is possible to re-construct the dissipational burst contributionto the distribution of SFRs and IR luminosity functions and lumi-nosity density of the Universe. We show that the burst luminosityfunctions agree well with the observed IR LFs at the brightest lu-minosities, at redshifts z ∼ −
2. At low luminosities, however,bursts are always unimportant, as expected from their short dutycycles, noted above. Although the burst luminosity functions risewith redshift, they always represent low space densities, and theoverall LF evolves rapidly. As such, the transition luminosity abovewhich bursts dominate the IR LFs and SFR distributions increaseswith redshift from the ULIRG threshold at z ∼ z ∼
2. This appears to agree well with recent esti-mates of the transition between normal star formation and merg-ers, along the observed luminosity functions. Systematic morpho-logical studies at low redshifts (Sanders & Mirabel 1996) yield theconventional wisdom that – locally – the brightest LIRGs and es-sentially all ULIRGs are merging systems (see also references in§ 1). At high redshifts, similar studies have now been performed(see e.g. Tacconi et al. 2008, and references therein). They too findthat the brightest sources are almost exclusively mergers, but with atransition point an order-of-magnitude higher in luminosity. Othermorphological studies at intermediate redshifts z ∼ . − . ∼ −
5% at z ∼ ∼ −
10% at z >
1. This agrees wellwith recent attempts to estimate the contribution to the SFR den-sity at z = − Given the completely independent nature ofthe constraints, and significant uncertainties involved in both, theagreement is good. The small value is what is expected, given ourprevious determination that the typical burst mass is just ∼ ∼ L ∗ spheroids (the galaxies that dominate the stellar mass den-sity). But it clearly rules out merger-induced bursts driving the SFRdensity evolution of the Universe.At the highest redshifts z &
2, we can put strict upper lim-its on starburst intensities, based on the maximum stellar mass re-maining at high densities at z =
0, and find some tension betweenthese and estimated number counts of sub-millimeter galaxies fromChapman et al. (2005). This implies that some change may be nec-essary in either the number counts themselves, the bolometric cor-rections used to convert these observations to total IR luminosities, Note that it is important here to distinguish estimates of the SFR induced by mergers, i.e. that above what some control population would exhibit,from that simply in ongoing/identifiable mergers (since many criteria iden-tify mergers for a timescale ∼ Gyr, much longer than the burst timescale).For example, a merger fraction of 10% would imply at least 10% of starformation in ongoing mergers, even if there were no starbursts and thosesystems were only forming stars at the same rate as they would in isolation.c (cid:13) , 000–000 Hopkins et al. or the stellar IMF used to convert between SFR and IR luminosity.However, the observations remain considerably uncertain, with thebolometric corrections relying sensitively on assumed dust temper-atures, and the number counts subject to significant cosmic vari-ance (see e.g. Austermann et al. 2009). More observations, at newwavelengths and in larger, independent fields, are needed to resolvethese discrepancies.We compare our constraints on these histories with recent pre-dictions from galaxy formation models and simulations, and findreasonably good agreement. Both exhibit similar tension with es-timates of the high-redshift, extremely luminous number densities.However, the models are able to match the inferred number den-sities presented here without any change to the stellar IMF or re-quiring other exotic physics. The systematic uncertainties in themodels, especially at high redshift, are large, however, so the em-pirical constraints presented here provide a powerful new means ofconstraining the models and their input parameters.For example, if models are constrained via e.g. a halo occupa-tion approach or otherwise, so as to match the observed mergerfractions of galaxies as a function of redshift, then these num-bers become relatively large ( > z & ∼ −
50% of the SFR density at z > − ∼ yrfrom the beginning of a burst, the predicted behavior at very earlytimes in the bursts is quite different. A larger Kennicutt (1998) re-lation index implies higher peak SFRs and more sharply peakedstarbursts, increasing the predicted number counts of the most lu-minous sources and making it relatively more easy for models toaccount for the most extreme star-forming systems. It therefore re-mains of considerable importance for observations to probe both gas density measurements and full SFR indicators in extreme sys-tems, at low and high redshifts.Finally, we note that our analysis is only possible because,as indicated by basic dynamics, simulations, and observations ofstellar populations, starburst components of spheroids were formedin dissipational (i.e. gas-rich, at their centers), rapid star formingevents. The dissipationless “envelopes” surrounding the central,dense components in spheroids were not formed in such a manner(again indicated by both their structural and kinematic properties,simulations of their formation, and stellar population observations).Rather, they represent the debris of stars from disks which wereformed pre-merger, and assembled (and violently relaxed) dissipa-tionlessly. These stars were formed over extended periods of time,with new gas accretion onto the disk fueling new or continuousstar formation (e.g. Kereš & Hernquist 2009), as opposed to a sin-gle massive inflow. Especially in the most massive systems, theyare also assembled from multiple systems, via e.g. minor mergerscontributing tidal material to the extended “wings” well-known inmassive galaxies. As such, there is not a straightforward means toinvert their surface stellar mass density profiles to obtain their starformation history. Such constraints will depend on other methods,such as e.g. direct stellar population analysis. It should be borne inmind that this represents ∼
90% of the mass in most spheroids, andso understanding star formation in disks remains critical to under-standing the origin of stellar populations in spheroids.
ACKNOWLEDGMENTS
We thank Chris Hayward and Josh Younger for helpful dis-cussions throughout the development of this manuscript. Supportfor PFH was provided by the Miller Institute for Basic Research inScience, University of California Berkeley.
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