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Mon.Not.R.Astron.Soc.000,1–13(2011) Printed10January2012 (MNLATEXstylefilev2.2) Cosmic star formation rate: a theoretical approach L. Vincoletto1⋆ F. Matteucci1,2 F. Calura3 L. Silva2 G. Granato2 1Dipartimento di Fisica - Sezione di Astronomia - Universita`degli Studi di Trieste- Piazzale Europa, 1, I-34127 Trieste,Italy 2 2INAF - Osservatorio Astronomico di Trieste, ViaG.B. Tiepolo 11, I-34131 Trieste,Italy 1 3INAF - Osservatorio Astronomico di Bologna, ViaRanzani 1, I-40127 Bologna, Italy 0 2 n a J 9 ABSTRACT ] The cosmic star formation rate (CSFR), namely the star formation rate in a unitary O comoving volume of the Universe, is an important clue to investigate the history of C theassemblyandevolutionofgalaxies.Here,wedevelopamethodtostudytheCSFR from a purely theoretical point of view. Starting from detailed models of chemical . h evolution, which best fit the properties of local galaxies, we obtain the histories of p star formation of galaxies of different morphological types (ellipticals, spirals, irregu- - o lars).These histories are then usedto determine the luminosity functions of the same r galaxiesbymeansofaspectro-photometriccode.WeobtaintheCSFRunderdifferent st hypothesesaboutgalaxyformationscenarios.First,westudythehypothesisofapure a luminosity evolution scenario, in which all galaxies are supposed to form at the same [ redshift and then evolve only in luminosity without any merging or interaction.Then weconsiderscenariosinwhichthenumberdensityortheslopeoftheluminosityfunc- 1 v tionsareassumedto varywithredshift.After comparisonofourresultswiththe data 1 availableinliterature,weconcludethatapureluminosityevolutiondoesnotprovidea 5 goodfit to the data, especially at very high redshift, although many uncertainties are 7 still present in the data because of the unknown dust corrections and assumed initial 1 massfunction. Onthe otherhand,a variationinthe number density ofellipticalsand 1. spirals as a function of redshift can provide a better fit to the observed CSFR. We 0 criticallydiscussthe possiblescenariosforgalaxyformationderivedfromthis finding. 2 We also explore cases of variable slope of the luminosity functions with redshift as 1 well as variations of number density and slope at the same time. We cannot find any : of those cases which can fit the data as well as the solely number density variation. v Finally, we compute the evolution of the average cosmic metallicity in galaxies with i X redshift and show that in the pure luminosity evolution case the ellipticals dominate r themetalproductionintheUniverse,whereasinthecaseofnumberdensityevolution a are the spirals the main producers of metals. Key words: galaxies: evolution - galaxies: fundamental parameters - galaxies: pho- tometry - galaxies: luminosity function - ISM: general - ISM: evolution. 1 INTRODUCTION by Lilly et al. (1996) and thanks to deeper and deeper sur- veysithasbeenpossibletoextendthedeterminationofthe Therateatwhichgalaxieshaveformedstarsthroughoutthe cosmicstarformationrateuptoz∼10(Madau et al.1998; whole cosmic time is a fundamental clue to investigate the Hopkins 2004; Bouwens et al. 2008; Li 2008; Kistler et al. history of the assembling and evolution of structures in the 2009; Bouwens et al. 2011a; Ishida et al. 2011). Constrain- Universe. The cosmic star formation rate (CSFR), defined ingtheCSFRathighredshiftwouldbeofparamountimpor- as the comoving space density of the global star formation tancetoputconstraintsinthegalaxyformationscenario.In rate in a unitary volume of the Universe, is not a directly particular,todecidebetweentheMonolithicCollapse (MC) measurable quantity.It can be inferred only from the mea- andtheHierarchicalClustering (HC),becausetheCSFRde- surement of the luminosity density in different wavebands, pendsonboththestarformationrate(SFR)ingalaxiesand whicharethentransformedintostarformationratebysuit- on the evolution of the galaxy luminosity function (LF). In ablecalibrations.Startingfromitsfirstdeterminationmade the MC scenario, spheroids and bulges formed at high redshift (e.g. z >2−3) as the result of a violent burst of star ⋆ E-mail:[email protected] formation, following the collapse of a gas cloud. After the 2 L. Vincoletto & al. development of a galactic wind quenching star formation, formation,theproductionrateofchemicalelementsandthe galaxies evolved passively until present time (Larson 1974; chemical abundances in the stars and in the gas, starting Matteucci & Tornambé 1987). Moreover, Matteucci (1994) from the matter reprocessed by the stars and restored into introduced in this scenario the assumption that more mas- the ISM through stellar winds and supernova explosions. sivespheroidshadhigherstarformationefficienciesthanless Starting from observational constraints such as the present massiveones,andstoppedtoformstarsearlier.Itiscommon daychemicalabundances,itispossibletoperformadetailed nowtorefertothisbehaviouras“downsizinginstarforma- backward evolution of galaxies of different morphological tion”.Theseassumptionsallowustoreproducethemajority type. We consider three galactic types namely irregulars, of chemical and photometric properties of local ellipticals. spiraldisksandspheroidswhichincludebothbulgesandel- Ontheotherhand,intheHCscenarioforgalaxyformation, lipticalgalaxies.Thisismotivatedbythefactthatellipticals baryon assembly basically mirrors thehierarchical build up andbulgesarelikely tohaveacommon origin, i.e.both are of dark matter structures. Therefore, spheroids and bulges likelytohaveformedtheirstarsonveryshorttimescalesand formed as a series of subsequential mergers among gas-rich alongtimeago(Jablonka et al.1996).Detaileddescriptions galaxiesorwithgalaxiesthathavealreadystoppedtheirstar oftheadoptedmodelscanbefoundinMatteucci & Francois formation.Inthisscenario,galaxiesformstarsatlowerrates (1989) for spirals, Pipino & Matteucci (2004) for ellipticals thanintheMCscenario,withmoremassivespheroidsreach- and Bradamante et al. (1998) for irregulars. ing their final mass at later times (z .1.5) (White& Rees Inourpicture,spheroidsform astheresultofasudden 1978;Baugh et al.1998;Cole et al.2000;Menci et al.2004). collapse of a gas cloud of primordial chemical composition. In this paper, we calculate the evolution of the CSFR and The gas is supposed to be accreted over a finite timescale, the mean metallicity of the interstellar medium (ISM) in so in this respect we do not assume pure MC models. A galaxies. This is done by means of detailed chemical evolu- galaxy can be modelled either as a one-zone object or a tionmodelsforgalaxiesofdifferentmorphologicaltypes,i.e., multi-zoneobjectinwhichthegalaxyisdividedintoseveral ellipticals, spirals, and irregulars, which successfully repro- concentric shells independent of each-other. The spheroids ducethelocalpropertiesofsuchgalaxies,andfromwhichwe are modelled as a one zone objects and they are supposed derive their star formation histories (SFH). Star formation to suffer a strong initial burst of star formation, that stops histories obtained with chemical evolution models are then as soon as the galactic wind develops; in this sense their usedtodeterminethephotometricevolutionofthegalaxies evolutioncruciallydependsonthetimeatwhichthegalactic (spectra and magnitudes), through a spectro-photometric wind occurs. From that moment on the galaxy is supposed code.ThisallowsustocomputetheLF,thatgivesthenum- to evolvepassively. berofgalaxiesinaunitaryvolumeoftheUniverseinalumi- The chemical evolution of spiral disks is studied by us- nosity bin. In this work, we normalize our LFs to the local ingtheone-infallmodelofMatteucci & Francois(1989),de- LFs derived by Marzke et al. (1998), which are based on veloped to reproduce the observational constraints of the the“Second Southern Sky Redshift Survey” (SSRS2) using Milky Way disk. This model belongs to the class of the so- data from 5404 galaxies at z 6 0.05. These LFs are deter- called“infallmodels”,wherethediskisthoughtasgradually mined in the B band for galaxies of different morphological formingbyinfallofprimordialgas.Instantaneousmixingof type. Then we compute the evolution in luminosity of the gasis assumed,buttheinstantaneousrecyclingapproxima- galaxyatthebreakoftheLFforeachgalactictype.Thisis tionisrelaxed.Thegalacticdiskissupposedtohaveformed done under different scenarios: first, we study the hypoth- on timescales increasing with the galactocentric distance, esis of a pure luminosity evolution scenario (PLE). In this being larger at larger radii, in agreement with the dissipa- case, galaxies are supposed to form all at the same red- tive models of galaxy formation of Larson (1976). The gas shift and then evolve only in luminosity without any merg- accumulates faster in the inner than in the outer region, ing or interaction. In this case, we follow the approach of according to the so-called “inside-out” scenario. The pro- Calura & Matteucci(2003).Thenwedefinetwonewscenar- cessof disk formation is muchlongerthan thehalo and the ios: one in which the number density of the LF is assumed bulge formation (they should form on timescales no longer to vary with redshift, and one in which the slope of the LF than 1 Gyr) with typical timescales varying from ∼ 2 Gyr is assumed to vary with redshift. All the results have been in the inner disk, ∼ 7 Gyr in the solar region and up to compared with the available data found in literature. Fi- 15−20 Gyr in the outer disk. This mechanism is impor- nally, we calculate the mean metallicity of the ISM for the tant to reproduce the observed abundance gradients (see PLEandthenumberdensityevolution(NDE)scenario.The Matteucci & Francois1989,Chiappini et al.2001).Thespi- paperisorganizedasfollows:in§2wedescribethechemical raldiskisapproximatedbyseveralindependentrings,2kpc and photometric models, in § 3 we present our results and wide, without exchange of matter between them. In this in §4 we draw some conclusions. work thedisk has been set toextend from 2 to16 kpc. Throughout this paper thecosmology adopted will fol- Irregulargalaxiesaremodelledasonezoneobjectsand low the ΛCDM paradigm with Ω0 = 0.3, ΩΛ = 0.7 and areassumed toassemble all theirgas by meansof a contin- h=0.65. uous infall of primordial gas and to form stars at a lower rate than the other morphological types. Their stellar pop- ulations appear to bemostly young,their metallicity is low and their gas content is large. All these features indicate 2 CHEMICAL AND PHOTOMETRIC thatthesegalaxiesarepoorlyevolvedobjects.Furthermore, EVOLUTION OF GALAXIES irregular galaxies are allowed to develop moderate galac- Chemical evolution models allow us to compute the evolu- tic winds which do not halt the star formation as in the tion in timeand space ofseveral quantitiessuch asthestar spheroids (Bradamante et al. 1998). CSFR: a theoretical approach 3 2.1 The chemical evolution models Matteucci & Greggio (1986). The parameter A represents thefraction of binary stars giving rise to Type Ia SNe, and Theequationthatdescribestheevolutionofthei-thelement although itsreal valueis unknownit is fixed in order tore- has thefollowing form: producetheobserved present time SNIa rate. This param- eterobviouslydependsontheadoptedIMFand,ingeneral, dGi(t) =−ψ(t)X (t) values between 0.05 and 0.09 are adopted. This ensures to dt i reproducetheactualSNIaratebothintheMilkyWayand MBm in other galaxies. It is worth noting that, in this term, the + ψ(t−τ )Q (t−τ )φ(m)dm ZML m mi m quantitiesψ andQmi refertothetime(t−τm2),whereτm2 indicatesthelifetimeofthesecondarystarofthebinarysys- MBM +A φ(m) tem, which regulates theexplosion timescale of the system. Z MBm The coefficient µ= M2/MB, is the ratio between the mass 0.5 ofthesecondarycomponentandthetotalmassofthebinary · f(µ)ψ(t−τm2)Qmi(t−τm2)φ(m)dµdm system,while f(µ) is thedistribution function of thisratio. Z µmin It is calculated on a statistical basis indicating that values   MBM close to 0.5 are preferred. Its analytical expression can be +(1−A) ψ(t−τm)Qmi(t−τm)φ(m)dm written as: Z MBm MU f(µ)=2(1+γ)(1+γ)µγ, (5) + ψ(t−τ )Q (t−τ )φ(m)dm m mi m ZMBM with γ = 2 as a parameter (Greggio & Renzini 1983) and +XAiA(t)−XiW(t). (1) µministheminimummassfractioncontributingtotheSNIa rate at thetime t, and is given by: Where G (t) is the normalized fractional mass of gas i within a galaxy in theform of theelement i: M (t) M −0.5M G (t) = Mg(t)Xi(t), (2) µmin =max M2B , 2 MB B. (6) i M   tot The fourth term of eq. 1 represents the enrichment due to where M and M (t) are the total galaxy mass and the tot g starsintherange[M ,M ]whicharesingleor,ifbina- mass of gas at timet, respectively. Bm BM ries,donotproduceaSNIaevent.Allthestarsinthisrange X (t) = Gi(t) (3) with mass larger than 8M⊙ are assumed to explodeas core i G(t) collapse supernovae. Thefifthtermrepresentsthecontributiontothechem- representstheabundanceinmassoftheelementi,thesum- icalenrichmentcomingfromstarsmoremassivethanM mation over all elements in the gas mixture being equal to BM (supposed to explode like Type II and I SNe). In all the unity.Thus, the quantity: b/c models the upper mass limit contributing to chemical en- M (t) G(t)= g , (4) richment is assumed to be 100 M⊙. Mtot Finally, the last two terms are the rate of accretion of is the total fractional mass of gas present in the galaxy at matter with primordial abundances XA, and the outflow timet.InthecaseofspiraldisksM (t)andM arereplaced rate for the element i, respectively. g tot by thecorresponding surface densities σ(r,t) and σ(r,t ) G Thefirsttermontherighthandsideofeq.1,including 2.1.1 The Star Formation Rate the star formation rate ψ(t), represents the rate at which each element disappears from the ISM owing to the star The main feature characterizing a particular morphological formation. galactic typeis represented by the prescription adopted for The second term is the rate at which each element its star formation history. In the case of elliptical galaxies is restored into the ISM by single stars with masses in theSFR ψ has the simple form given by: the range [M , M ], where M is the minimum stellar L Bm L mass contributing, at a given time t, to chemical enrich- ψ = νG(t), (7) ment (≃ 0.8M⊙) and MBm is the minimum total binary where ν is the star formation efficiency, namely the inverse massof systemswhichcan giverise toTypeIaSNe(3M⊙, ofthetypicaltimescale forstar formation, and assumed to Matteucci & Greggio1986).ThequantityQmi(t−τm)indi- be15 Gyr−1. catesthefractionofmassrestoredbythestarsinformofan In the case of irregular galaxies, a continuous star for- elementi thatcanbeproducedordestroyedinstarsorboth. mation rate is assumed as in eq. 7, but characterized by This is the so-called “production matrix” (Talbot & Arnett a lower efficiency than the one adopted for ellipticals (0.1 1973). Gyr−1 in this case). The third term represents the enrichment due to bi- In the case of spiral galaxies theformulation is thefol- naries which become Type Ia SNe, i.e. all the binary sys- lowing(seeMatteucci & Francois1989 andChiappini et al. temswithtotalmassintherangebetweenM andM , Bm BM 1997): with M being the maximum total mass of a binary sys- BM tem which can give rise to Type Ia SNe and is equal to 2(k−1) (k−1) σ(r,t) σ(r,t ) 16 M⊙ (8+8 M⊙). The model adopted for the progeni- ψ(r,t)=ν·  · G  ·Gk(r,t), (8) tors of the Type Ia SNe is the single degenerate one as in σ(r⊙,t) σ(r,t)     4 L. Vincoletto & al. Morphological Mtot τinf ν Reff Type (M⊙) (Gyr) (Gyr−1) (Kpc) Ellipticals 1011 0.3 15 3.0 100 Spirals 4×1010 7.0 1.0 3.5 Irregulars 1010 0.5 0.1 1.0 Table 1. Model parameters for the galaxies reproduced in this 10 work.Mtotisthebaryonicmass,Reff istheeffectiveradius,τinf is the infall timescale and ν is the star formation efficiency. For spiralsweindicatetheaverageinfalltimescaleforthedisk. 1 where ν is set in this case equal to 1 Gyr−1, σ(r,t) is the total surface mass density at a given radius and at a given time,σ(r⊙,t)isthetotalsurfacemassdensityattheposition of the Sun (set equal to 8.0 Kpc). Finally σ(r,tG) is the 0.1 surface mass density at present time. The exponent k is assumed equal to 1.4, as suggested by Kennicutt (1998). All these models are tuned to reproduce at best the main observed properties of local galaxies of different mor- 0.01 0.1 1 10 phologicaltype.InTable1weshowthemaincharacteristics of the galaxies of different morphological type that we use inthiswork.Itisimportanttostressthatinourmodels,we consider only one galaxy per morphological type and take Figure1.Starformationratesasafunctionoftime,foratypical it as representative of the whole population. For the spiral ellipticalof1011 M⊙ (green solidline), spiralof ∼4×1010 M⊙ galaxy we take theaverage SF and metallicity of the disk. (red dotted line) and irregular of 1010 M⊙ (blue dashed line) In Figure 1 we show the histories of star formation of galaxy. thegalaxiesofdifferentmorphologicaltypedescribedabove, for a Salpeter (1955) IMF. It is worth noting that for a spiralliketheMilkyWayanIMFsteeperthantheoneused equaltothesolarpartition).Atthispoint,thespectralsyn- is usually adopted. Here we assume Salpeter (1955) IMF thesis is performed summing up the spectra of each stellar forallgalaxiesbecausetheCSFRderivedobservationally is generation provided by the SSP of the appropriate age and obtainedbymeansofthisIMF.AsitisclearfromtheFigure metallicity,weightedbytheSFR(ψ)atthetimeofthestar a typical elliptical (1011 M⊙ of luminous mass) experiences birth.The flux at a given wavelength is defined as: ahighburstofstarformationlastingforabout0.6Gyrwith tG amaximumpeakofmorethan∼100M⊙yr−1.Aftersucha Fλ(tG)=Z ψ(t)×SSPλ(tG−t,Z(t))dt, (9) period,thestarformationceasesabruptlyowingtotheonset 0 of the galactic wind. The spiral disk is characterized by a where Fλ(tG) is the integrated monochromatic flux at the continuousSFR with a large peak around 1∼2 Gyrof less present time tG and SSPλ(tG −t,Z(t)) is the integrated than∼10M⊙yr−1andapresenttimevalueof∼2M⊙yr−1 monochromaticfluxofanSSPataget andmetallicity Z.If (fortheMilkyWay).Finally,theirregulargalaxyformsstars theeffectsofdustareneglected(asinourcase)thespectral at a rate smaller than the previous morphological types, energy distribution (SED) is simply given summing up the withanincreasingSFRthatreachesamaximumoflessthen spectra of all stars. 1M⊙yr−1,withasmallerpresentvalue,duetotheonsetof thegalactic wind. 2.3 The luminosity evolution of galaxies To derive the CSFR, it is of fundamental importance the 2.2 The photometric model knowledge of the relative distribution of galaxies, of any To study the evolution of the photometric properties morphological type, as a function of redshift. This can be of galaxies, we use the spectro-photometric code grasil donethrough theLF,which gives thedistribution of galax- (Silvaet al. 1998). Its starting point is the output of a ies per unit volume in the luminosity interval [L, L+dL]. chemical evolution code that follows the time evolution of TheLFiswellreproducedbytheSchechter(1976)function: SFR, metallicity (Z) and gas fraction. The building blocks −α oafndgatlhaexystpelhlaortoamtmetorsicphmeroidcemlsoadreelst.hgerlaibsrialriysobfasiseodchornonthees Φ(L)dLL∗ =φ∗LL∗ e(−L/L∗)dLL∗, (10) Padova stellar models (Bertelli et al. 1994), with a major   difference consisting in the computation of the effects of where φ∗ is the number of galaxies per unit volume; L∗ is dusty envelops around AGB stars, and on the stellar at- thecharacteristic luminositywhichseparatesbrightsources mosphericmodelsbyKurucz(1992).Thesinglestellarpop- from faint sources and α is the slope of the luminosity ulations(SSPs)spanarangefrom1Myrto18Gyrforwhat function.Differentdeterminationsofthelocal LFexist (see concernsages,andZ =0.004, 0.008, 0.02, 0.05, 0.1forthe deLapparent 2003 for a review) and they show several dif- metallicities (keeping the relative proportion of the metals ferences, in the sense that the parameters of the Schechter CSFR: a theoretical approach 5 can calculate the CSFR according to the method adopted Morphological M∗ α φ∗ byCalura & Matteucci (2003): Type Ellipticals −19.37 −1.00 4.4 M Spirals −19.43 −1.11 8.0 ρ˙∗(z)= ρBi(z)L  (z)ψi(z), (12) Irregulars −19.78 −1.81 0.2 Xi  Bi Table2.ParametersoftheLFasdefinedinMarzkeetal.(1998). whereρ˙∗(z)istheSFRdensity,ρBiistheB-Bandluminosity HereM∗ refersto the magnitude of the galaxy at the break, φ∗ density, M is the B-band mass-to-light ratio and ψ refersto the normalizationof the LF(expressed inMpc−3) and (cid:18)L(cid:19)Bi i αistheslopeoftheLF. (expressedinyr−1)isthestarformation rateperunitmass of thei-th morphological type. In this work, we make use of the B-band luminosity, function are not fully constrained by observations. The lu- becauseit is agood starformation tracersince hotmassive minosity function can be measured in several bands, obvi- stars emit in thisband. ously dependingon theredshift regime underinvestigation. In principle, the CSFR, could be computed simply by In this work, we make use of the determination of the lo- summingtheSFRofeachgalaxytypemultipliedbythecor- calB-bandluminosityfunctionofMarzke et al.(1998),that responding number density. However, this simple approach hasthecharacteristic ofconsideringseparately thedifferent wouldnotallowustoexploreindetailthecaseinwhichthe galactic morphological types. For each morphological type slope of the LF varies with redshift (see section 3.1.3). We theparameters of the LFare summarized in table 2. tested that thetwo approaches lead to the same result. Once we have the LF, we can compute the luminosity density (LD), which is what is really observed. The luminosity density, in a given band, is defined as the integrated 3.1.1 The Pure Luminosity Evolution Scenario lightradiated perunitvolumefromtheentiregalaxypopu- lation.Itstemsfromtheintegraloverallluminositiesofthe Here we describe the determination of the CSFR in the observed luminosity function: framework of a PLE scenario. With pure luminosity evolution,wemeanthatgalaxiesofallmorphological typesare L supposed to form at high redshift and then to evolve only ρL=Z Φ(L)L∗dL. (11) in luminosity and not in number. We also assume that the   slope of the LF is constant with redshift. In this context, At z = 0, the LDs for the single galaxy types are given by galaxies are assumed tobe isolated, i.e. theeffects of merg- theaboveintegralofthepresenttimeluminosityfunctionsof ers or interactions are irrelevant at all redshifts, and it is Marzke et al. (1998). At redshifts other than zero, for each assumed that theystart forming stars all at thesame time. morphologicaltypeweconsidertheluminosityevolutionob- Inthissimpleapproach,thedifferencesbetweenthevar- tained with thespectrophotometric code. ious morphological types should be ascribed only to their different star formation histories. Early-type galaxies (i.e. ellipticals, bulgesandS0)aredominatedbyold stellarpop- 3 RESULTS ulations so they must have formed their stars with high efficiency and in a short period, while late-type galaxies (i.e. Asalready pointedout,theCSFRisnotadirectlymeasur- spiralsandirregulars)shouldhaveformedstarsforthewhole ablequantity.Itcanbeinferredonlyfromthemeasurement Hubbletime, as seen in Fig. 1. of the luminosity density in different wavebands and then In Figure 2 we can see the predicted evolution of the convertedintoSFbymeansofsuitablecalibrations.Itsfirst luminosity density for ellipticals, spirals and irregulars, in determination was made by Lilly et al. (1996) (at 2800 ˚A, theU (centeredat 3650 ˚A),B(centered at 4450 ˚A),I (cen- 4400 ˚A, and 1 µm), followed in the subsequent years by tered at 8060 ˚A) and K (centered at 21900 ˚A) band. These Hogg et al. (1998) (using the [OII] line) and Wilson et al. evolutions are computed by adopting the SFRs of Fig. 1 (2002) (in the UV). These studies constrained the CSFR and constant slope and number density in the luminosity from z =0 to z=1, establishing that it is steadily increas- function. inginthatrange.TheCSFRseemstopeakaroundz∼2−3 From the Figure it is clear that, at early times in ev- and then decrease at higher redshifts, where there is not a ery band, the luminosity density is dominated by the light well defined trend. For example Bouwens et al. (2008) on emitted by elliptical galaxies. This is obviously due to the the basis of optical and UV data and Hopkins (2004) on fact that in the monolithic scenario they all suffer a strong the basis of a compilation of data taken in different bands, initial burst of star formation which lasts for < 1 Gyr and foundthattheCSFRfromroughlyz∼3toz∼9issteadily thenevolve passively.At this timea galactic wind develops decreasing. On theother hand,Kistler et al. (2009), on the andstarformationstops,sotheirluminosityintheUandB basisofthegammarayburst(GRB)datafromSwift found bands, where young, newborn stars emit, begins rapidly to a CSFR that is roughly constant from z∼4 to z∼8. decrease.Ontheotherhand,spiralsformstarscontinuously and thedecrease in luminosity in theU and Bband should beascribedmostlytotheconsumptionoftheirgasreservoir. 3.1 The cosmic star formation rate Finally,also irregulars form starsfor theentirecosmic time Bymeansofthestarformationhistoriesofgalaxiesofdiffer- but their luminosity density is lower than that of the other ent morphological type and their photometric evolution we morphological types due to their low star formation. Their 6 L. Vincoletto & al. 9 9 8 8 -0.5 7 7 6 6 0 5 10 0 5 10 -1 9 9 8 8 -1.5 7 7 6 6 -2 0 5 10 0 5 10 0 2 4 6 8 10 Figure 3. Cosmicstar formationhistory inthe case of pure lu- Figure 2. Prediction of the luminosity density evolution for minosityevolution.Bluesolidline,modeldescribedinthiswork; galaxiesofdifferenttypesinU(topleft),B(topright),I(bottom greendashedlinemodelfromCalura&Matteucci(2003).Points left)andK(bottom right)bandrespectively,inthecaseofpure aretakenfrom:Hopkins(2004)redopencircles,Lyetal.(2007) luminosityevolution.Herethex-axisrepresentstheelapsedtime brown squares, Takahashi etal. (2007) purple open squares, sincez=10.Solidgreenlinereferstoellipticals,reddottedlineto Li (2008) cyan open hexagons, Seymouretal. (2008) yellow spiralsandlongdashedbluelinetoirregulars. pentagons, Kistleretal. (2009) orange crosses, Lyetal. (2011) magenta triangles, Bouwensetal. (2011a) green open stars, Cucciatietal.(2011)lightgreencrosses. luminosity densityisalso decreased bytheonset ofgalactic winds at late times. we underpredict the observed values by a factor of ∼ 3. The I andK bandsare dominated bythelight emitted This is the redshift interval in which only spirals and irreg- by less massive, long living stars. Also in this case, ellipti- ulars contribute to the cosmic star formation history. Also calsarethemainsourcesatearlytimes,whilespiralsbecome the Calura & Matteucci CSFR suffers of the same problem dominant for cosmic time >2 Gyr.However,thedifference when compared with data. On the other side, at high red- in the values reached by ellipticals and spirals is less pro- shift, our model seems to overpredict the data, as well as nounced than in U and B bands. The luminosity density that of Calura & Matteucci. The high observed peak is to of spirals, after an initial phase of constant increase lasting ascribe to ellipticals which, in order to reproduce the local ∼3Gyr isobservedtodecreaseslowly intheIbandandto observationalfeatures,(liketheincreaseofthe[Mg/Fe]ra- bequiteconstantintheKband.Irregulars,alsointhiscase tio with galactic mass) have to form stars in a short time have values lower than ellipticals and spirals which tend to interval(lasting less than1Gyr,Matteucci 1994).Thisim- decrease veryslowly through the whole cosmic time. plies that the SFR of these objects must lie in the range ThePLEmodelbelongstothecategoryoftheso-called ∼ 100−1000 M⊙yr−1. At z ∼ 6 ellipticals stop to form “backward evolution” models(Tinsley & Danly1976);these stars and the CSFR decreases abruptly. The discrepancies models start from a well known and well constrained de- with the model of Calura & Matteucci (2003) can be ex- scription of the local Universe to reconstruct the evolution plained through the differences in the parameters adopted of galaxies back to their formation. in the chemical evolution models and in the different pho- In our model, all the galaxies are supposed to form at tometric code used. In fact, in Calura & Matteucci (2003), z = 10. This choice is motivated by the fact that more thephotometric code ofJimenez et al. (1998) was adopted. and more objects are revealed at z > 5 (Mobasher et al. Moreover, thechemical modelfor theellipticals adoptedby 2005,Vanzella et al.2008),withevidencesofaz∼10detec- Calura & Matteucci(2003)isaclosedboxone,whereasour tion (Bouwens et al. 2011b). The predicted CSFR is shown model includes gas infall. in Figure 3; for comparison weplot in thesame Figure also the CSFR as obtained by Calura & Matteucci (2003) who adopted verysimilar assumptions and star formation histo- 3.1.2 The Number Density Evolution Scenario (NDE) ries. Asitcanbeseen,bothmodelsshowsomediscrepancies Inthissection,wedescribeadifferentapproachforthecalcu- whencomparedwiththeobservationaldata.Inourcase,the lationoftheCSFR.Inwhatfollows weintroduceascenario discrepancy begins at z ∼0.5 and it is particularly evident inwhichweletthenumberdensityofgalaxiestoevolvewith in the redshift interval running from ∼ 0.5 to ∼ 5. Here redshift while the SFHsof galaxies are thesame of Fig. 1. CSFR: a theoretical approach 7 The main consequence of relaxing the hypothesis of modelfortheCSFRinthecaseofnumberdensityevolution. the constancy of the number density of galaxies is that FromtheplotwecanseethatthepeakobservedinthePLE we consider galaxy interactions. It has been claimed that scenario completely disappears. The model seems to better giant ellipticals could form, as the result of the occur- reproducethehighredshifttrendiftheerrorbarsaretaken rence of 1 to 3 major dry-mergers during galactic lifetime into account. Also in the redshift interval between 0 and (Pipino & Matteucci 2008 and reference therein) and this ∼ 5 the agreement is improved although the curve is still view is in agreement with the observations. Moreover it slightly lower than thedata, but this is a problem common hasbeenobservedthatthefractionofdry-mergersincreases to all the other models of CSFR, as we will see later. Also with redshift from z ∼1.2 to z =0 (Lin et al. 2008). in this case we underpredict the decrease of the CSFR be- Toincludethenumberdensityvariationwemodify the tweenz ∼2andz =0.This,inprinciple,canbeduetothe expression of the luminosity function as defined in eq. 10, lack in our models of strong starburst galaxies at low and letting the parameter φ∗ (i.e. the number density of galax- intermediate redshift (see e.g. Elbaz et al. 2011), even if as ies) to vary with the redshift. The new formulation of the claimed by Rodighiero et al. (2011) starbursts seem to ac- luminosity function is: countonlyforthe10% oftheSFRdensityatz∼2.Onthe otherhand,ourpredictedslopeoftheCSFRinthisredshift −α Φ(L)dLL∗ =φ0·(1+z)βLL∗ e(−L/L∗)dLL∗, (13) r(a2n01g1e).goTohdesaesawutehllorassufosirngthceosmmoodloelgiocfalvsamnodoethVeodorptaretticalle.   hydrodynamicssimulations,explainthisdropwithadecline where φ =φ∗(z =0), i.e. the present time numberdensity inthecold-accretionratedensityontohaloesandwithAGN 0 ofgalaxies. Thismodelisconstrained inordertoreproduce feedbackthatdecreasesthecontributiontotheCSFRofthe the number densities observed at z = 0 by Marzke et al. gas accreted in thehot mode. (1998). In Figure 5 we show the predicted evolution of the lu- We let the parameter β to run from −6 to +6 in steps minosity density, for our best model, in the case of num- of0.2. Thechoiceofthisintervalhasbeenmadeinorderto berdensityevolution.Comparingthesepredictionswiththe exploreasetofvaluesaswideaspossible withoutgivingus onesinFigure2,itispossibletoseethatinthiscasespirals meaninglessresultsand,atthesametime,lettingustotest dominate the luminosity density in every band and at any cases of extreme number density variation. For each value epoch, at variance with thePLE case. ofβ,wecalculatetheluminositydensityusingthemodified Itmustbesaidthatthevariationofthenumberdensity version of the Schechter function. Then, using the star for- ofgalaxies ofdifferentmorphological typesisstillmatterof mation histories depictedin section 2.1.1,wedeterminethe debate, since there is, in general, no concordance among star formation rate density,using theequation 12. variousauthors.Forexample,Totani & Yoshii(1998)found We perform a careful check of the contribution to the that, parametrizing the comoving number density of E/S0 total CSFR from the different morphological types, and on asproportionalto(1+z)γ,thefavouredvalueforγis−0.8± itsbasiswedecidetoletonlyspiralsandellipticalstoevolve. 1.7, in perfect agreement with our value. This analysis is Therefore,weconsiderthenumberdensityofirregularscon- valid in the redshift range 0.3 < z < 0.8, and the large stant through the cosmic time. The reason resides in their error is due to the limited redshift range of their sample. contributiontothetotalCSFR,thatis,atanyredshift,two deLapparent et al. (2004), using data from theESS (ESO- orders of magnitude lower than the one of ellipticals and SculptorSurvey),foundanumberdensityevolutionoflate- spirals and therefore negligible. Our best values of the β typegalaxies following thesame parametrization, with γ = parameter for each morphological typeare: 2±1. On the other hand, for example Lilly et al. (1998) found at z < 1 an evolution of large disk structures (with • Ellipticals: β=−0.8; a bulge-over-total ratio B/T > 0.5) following a (1+ z)γ • spirals: β=1; relation, with γ = ±0.5, lower than the one found in our • irregulars: β =0. model. Thismeansthatthenumberdensityofellipticalsissup- Even morechallenging isto asses theevolution at high posedtoincreasefromz=10toz =0(with∼15%ofellip- redshift, since very deep surveys are required and the cor- ticals already in place at z = 10). The opposite behaviour respondencebetween actualgalaxies andtheirhighz coun- ispredictedforspirals,whicharesupposedtodecreasewith terpartsisnotfirmlyestablished.Thiscausesalossofinfor- decreasingredshift.Undertheseassumptions,thereisapos- mation about the morphology of the various objects giving sibleinterpretationintermsofgalaxyformation.Forexam- controversial results. Moreover, the uncertain dust correc- ple,thepredictednumberdensityevolutioncouldmeanthat tions and the adopted IMF (Salpeter 1955) used to derive alarge fraction ofelliptical galaxies can form thankstodry the CSFR, could underestimate the actual CSFR. There- mergersofspirals,asproposedbyToomre & Toomre(1972) fore, wedonot completely excludethattheCSFR couldbe and Naab & Burkert (2003). Since we are considering only constant or increasing at high z. dry-mergersitmeansthatthemergingprocessquenchesthe star formation in the progenitors and that the residual gas 3.1.3 The Variation of α islost.From ourmodelwecan seethatthenumberdensity of spirals decreases of roughly 60% from z ∼ 2 to z ∼ 0, In this section we describe a modification of the Schechter and of 50% from z ∼1 to z =0. This is in good agreement functioninwhich weconsidertheevolutionwith redshiftof with Boissier et al. (2010) who found a decrease of 50% of the slope of the LF for galaxies of different morphological Milky Way siblings from z ∼ 1 to z = 0, indicating this type.Westartagain from theLFasdefinedinsec.2.3,and percentage as a lower limit. In Figure 4 we show our best assume that α = α (1+z)β, where α = α(z = 0) is the 0 0 8 L. Vincoletto & al. 0 -0.5 -0.5 -1 -1.5 -2 -1 0 2 4 6 8 10 -0.5 -1.5 -1 -1.5 -2 -2 0 2 4 6 8 10 0 2 4 6 8 10 Figure4.Thecosmicstarformationhistoryinthenumberden- Figure 6.TheCSFHinthecaseofvariationoftheLFslopeα. sity evolution scenario. Blue line refers to model results. Data Hereweshowthecasescorrespondingtoβ=0.4(toppanel)and pointsarethesamedescribedinFigure3. β = −0.2 (bottom panel). Blue lines indicate the results of the model.DatapointsarethesamedescribedinFigure3.Notethe differenty-axisscaleadopted forclarity. 9 9 slope of the LF at present time and β is a free parameter. Therefore, theLF assumes theform: 8 8 [−α0(1+z)β] 7 7 Φ(L)dL =φ∗· L  e(−L/L∗)dL. (14) L∗ L∗ L∗   6 6 This meansthatat any redshift wechangetheslopeof 0 5 10 0 5 10 the LF. As in the case of the number density evolution, we let β to run from −6 to +6 in steps of 0.2, and in no case wehavefoundanimprovementinfittingthedatarelativeto 9 9 the pure luminosity or number density variation scenarios. From the observational point of view, there are some sug- 8 8 gestions of variations of the slope of the LF but referred to thewholepopulationofgalaxies, withoutdifferentiatingfor 7 7 morphological types. For example, Reddy& Steidel (2009) found that α=−1.1 locally and increases up to ∼−1.7 at 6 6 high redshift (from z=3 to 6). From our numerical exper- iments this case corresponds to β ∼ 0.3. We can exclude 0 5 10 0 5 10 all the cases with β > 0.4 and those with β < −0.2 since theypredictvaluesofαwhichareverymuchoutsidetheob- served range, and also a CSFR very much at variance with Figure 5. Prediction of the luminosity density evolution for observations. galaxiesofdifferenttypesinU(topleft),B(topright),I(bottom In Figure 6 we show the case β = 0.4 (top panel) left)andK(bottom right)bandrespectivelyinthecaseofnum- which is similar to the suggested observational variation berdensityevolution(bestmodel).Herethex-axisrepresentsthe (Reddy& Steidel 2009), and the case β = −0.2 (lower elapsedtimesincez=10. Solidgreenlinereferstoellipticals,red panel). Therefore, although we do not exclude a variation dottedlinetospiralsandlongdashedbluelinetoirregulars. oftheslopeoftheLFsofgalaxies,wecanconcludethatthe casesofαvariationsareprovidingafittotheCSFRwhichis worsethan thecasein whichthereisnumberdensityvaria- tion,asshowninFigure4.Finally,werunseveralmodelsby varyingbothφ∗ andαandwedidnotfindanycombination ofparameterswhichcan fittheobservedCSFRbetterthan our best case with φ∗ only variation. In Figure 7 we show CSFR: a theoretical approach 9 -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 0 2 4 6 8 10 0 2 4 6 8 10 Figure 7.Comparisonbetween ourbestmodelwithNDE(blue Figure 8. A compilation of determinations of the CSFH, com- solid line) and a model with both NDE (the same as the best pared to the results of this work. Models are from:Steideletal. model)andαevolution(green dashedline). Datapoints arethe (1999)magenta longdash-dotted line,Porciani&Madau(2001) samedescribedinFigure3. greenshort-dashedline,Coleetal.(2001)darkgreendottedline, Mencietal. (2004) blue solid line, Strolgeretal. (2004) cyan dot short-dashed line, this work(PLE) black short-dashed long- a comparison between a model with the same variation of dashed line, this work (NDE) turquoise long dashed line. Data φ∗ as our best model (see section 3.1.2) and a model with pointarethesameasFigure3. the same φ∗ variation plus a variation of α, obtained with β = 0.3 for ellipticals and β = 0.2 for spirals and irregu- thehypothesis that the rate of GRBs traces theglobal star lars.Asonecansee,theagreementbetweenthemodelwith formation history of theUniverse. variable α and φ∗ does not fit the data as well as our best • Strolger et al. (2004): model. Itisamodelbasedonamodifiedversionoftheparametric form suggested byMadau et al. (1998),takingintoaccount dustextinction.Modelparametersaredeterminedfittingthe 3.1.4 Comparison with other determinations collection of measurementsof theCSFRof Giavalisco et al. In Figure 8 we show the results of several determinations (2004). of the CSFR that we compare to our model predictions, in • Menci et al. (2004): particularwiththePLEandwithourbestmodelwithNDE. Prediction from a semi-analytic model for galaxy forma- ThedeterminationsoftheCSFRweuseforthiscomparison tionintheframeworkofthehierarchicalclusteringscenario. are: For further information we address the reader to the • Steidelet al. (1999): abovementioned papers. Itisbasedonasampleofstar-forminggalaxies atz& 4. From Figure 8 it is possible to see that our model ThecorrespondingCSFRbasedontheirdatahasbeenthen with number density evolution shows a complete agree- parametrized by Porciani & Madau (2001) (their model ment with the semi analytical model of Menci et al. (2004) SF2). and, in general, shows a better agreement with the data, • Cole et al. (2001): if compared to the determinations of Steidelet al. (1999) Combining the data of the Two Micron All Sky Survey and Porciani & Madau (2001). For what concerns the PLE (2MASS) Extended Source Catalog and the 2dF Galaxy model, this presents lower values than the other CSFRs, in Redshift Survey, they produce an infrared selected galaxy the redshift range [0−6]. On the other hand, for z > 6 it catalogue. They use it to estimate the galaxy LF and to exceedsalltheotherpredictionsconsideredhere,aswehave infer the total mass fraction in stars. Then they use their discussed already. resultstogetherwiththedataofSteidel et al. (1999)toob- tain a parametric fit of the CSFR. Their parametric form 3.1.5 The cosmic ISM mean metallicity has also been used byHopkins& Beacom (2006). • Porciani & Madau (2001): Here we compute the evolution of the cosmic, luminosity HerewerefertotheirmodelSF3.Itisaparameterization weighted, mean interstellar metallicity of the galaxies of of the CSFR based on the data collected by the Burst and different morphological type. With this definition, we in- TransientSourceExperiment(BATSE).Itisadoptedtotest dicate the mean metallicity of the gas from which stars are 10 L. Vincoletto & al. 10 10 1 1 0.1 0.1 0.01 0.01 0.001 0.001 0.0001 0.0001 0 2 4 6 8 10 0 2 4 6 8 10 Figure9.Thecosmicluminosityweightedmeanmetallicityrela- Figure10.Thecosmicluminosityweightedmeanmetallicityrel- tivetotheSuninthecaseofpureluminosityevolution.Magenta ativetotheSuninthecaseofnumberdensityevolution.Magenta solid line: total; red dotted line: spirals; green long-dashed line: solid line: total; red dotted line: spirals; green long dashed line: ellipticals;bluedashedline:irregulars.Allthevaluesarenormal- ellipticals;bluedashedline:irregulars.Allthevaluesarenormal- ized to the solar metallicty Z⊙ = 0.0134, from Asplundetal. ized to the solar metallicty Z⊙ = 0.0134, from Asplundetal. (2009). (2009). born, in a unitary volume of the Universe. Assuggested by Kulkarni& Fall(2002),wecomputeitthroughthefollowing expression: formation lasts for the whole cosmic time. We note that in this case two distinct effects are involved: the metallicity Z (L )L Φ (L )dL Z¯= i i i i i i, (15) which is increasing and the luminosity which is decreasing R L Φ (L )dL i i i i i with decreasing z. P R whereZ (L)istheaverageinterstellarmetallicityinagalaxy InFigure10weshowthesameasinFigure9butunder i of luminosity L , at any given cosmic time, and of the i-th the hypothesis of number density evolution. In particular, i morphological typeandΦ (L )istheluminosityfunctionof we show the results of our best model, as described in sec- i i thei-th morphological type. tion3.1.2.Themost evidentfact inthiscaseisthatspirals, Wecomputethisquantitybothinthecase ofPLEand instead of ellipticals, are the main contributors to the to- NDE, using the metallicity evolutions obtained with the tal cosmic luminosity weighted mean metallicity. However, chemical evolution models, together with the local B-band weneedfurtherinvestigation tobetterinterpretthisresult. luminosities obtained with the spectrophotometric code. It has to be considered that this quantity cannot be com- The parameters of the local luminosity functions, are from pared directly with observations, since it is a cosmic aver- Marzke et al. (1998), as in the previousparagraphs. age, while measurements are performed in single objects. From Figure 9 it is clear that in the PLE scenario el- Another important thing we note is that, in this case the lipticals dominate themean galactic metallicity throughout cosmic luminosity weighted mean metallicity of ellipticals the whole cosmic time. This is due to their higher average is increasing throughout the whole cosmic time. This can metallicity compared to spirals and irregulars and to the be explained by the fact that the decreasing of the B-band constant number density of all galaxies. The spiral galax- luminosity is compensated by the increase of their number ies are the second most important contributors to the cos- densityatlowerredshift.Wealsofindthatourpresenttime mic chemical enrichment, whereas irregulars give a negligi- valueofthecosmicluminosityweightedmeanmetallicityof blecontribution.Wecanseethatourmodelisinagreement theISMinallgalaxiesandineachgalaxyformationscenario with Calura & Matteucci (2004) who found that for z & 6 is ∼ 2.4 Z⊙, a value that is slightly higher than 1.3 Z⊙ as the main metal producers are ellipticals whereas for z . 2 found by Calura & Matteucci (2004). This is partly due to themainsourcesofmetalsproductionarespirals.Itisworth thefactthatweadoptthesolarmetallicityofAsplund et al. noting that, thedecrease of theluminosity weighted metal- (2009), that is lower than the Z⊙ adopted in the previous licity of ellipticals is due to the fact that, at z ∼ 6 in our paper. model, ellipticals stop forming stars and from that moment Thisaveragemetallicityisnottheglobalaveragemetal- on their B-bandluminosity abruptlydecreases. Also spirals licityoftheUniversesincewedidnotconsiderthemetallic- decreasetheirB-bandluminositybutmildlysincetheirstar ity of the intracluster and intergalactic medium.