Galaxy Lookback Evolution Models -- a Comparison with Magneticum Cosmological Simulations and Observations
Rolf-Peter Kudritzki, Adelheid F. Teklu, Felix Schulze, Rhea-Silvia Remus, Klaus Dolag, Andreas Burkert, H. Jabran Zahid
DDraft version February 10, 2021
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Galaxy Lookback Evolution Models - a Comparison with Magneticum Cosmological Simulations and Observations
Rolf-Peter Kudritzki,
1, 2
Adelheid F. Teklu,
1, 3
Felix Schulze,
1, 4
Rhea-Silvia Remus, Klaus Dolag,
1, 5
Andreas Burkert,
1, 4 and H. Jabran Zahid LMU M¨unchen, Universit¨atssternwarte, Scheinerstr. 1, 81679 M¨unchen, Germany Institute for Astronomy, University of Hawaii at Manoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA Excellence Cluster Origins, Boltzmannstr. 2, 85748 Garching, Germany Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, 85748 Garching, Germany Max Planck Institute for Astrophysics, Karl-Schwarzschildstr. 1, 85748 Garching, Germany Microsoft Research, 14820 NE 36th St, Redmond, WA 98052, USA
Submitted to ApJABSTRACTWe construct empirical models of star-forming galaxy evolution assuming that individual galaxiesevolve along well-known scaling relations between stellar mass, gas mass and star formation rate follow-ing a simple description of chemical evolution. We test these models by a comparison with observationsand with detailed Magneticum high resolution hydrodynamic cosmological simulations. Galaxy starformation rates, stellar masses, gas masses, ages, interstellar medium and stellar metallicities are com-pared. It is found that these simple lookback models capture many of the crucial aspects of galaxyevolution reasonably well. Their key assumption of a redshift dependent power law relationship be-tween galaxy interstellar medium gas mass and stellar mass is in agreement with the outcome of thecomplex Magneticum simulations. Star formation rates decline towards lower redshift not becausegalaxies are running out of gas, but because the fraction of the cold ISM gas, which is capable ofproducing stars, becomes significantly smaller. Gas accretion rates in both model approaches are ofthe same order of magnitude. Metallicity in the Magneticum simulations increases with ratio of stellarmass to gas mass as predicted by the lookback models. The mass metallicity relationships agree andthe star formation rate dependence of these relationships is also reproduced. We conclude that thesesimple models provide a powerful tool for constraining and interpreting more complex models basedon cosmological simulations and for population synthesis studies analyzing integrated spectra of stellarpopulations.
Keywords: galaxies: evolution, metallicity, gas masses, accretion, star formation INTRODUCTIONObservations of galaxies through cosmic time and de-tailed hydrodynamic cosmological simulations show thatthe formation and evolution of galaxies is an extremelycomplicated process. At high redshifts, the first build-ing blocks of galaxies contract and form the first starswhile gas continues to accrete from the intergalacticmedium, providing new additional fuel for star forma-
Corresponding author: Rolf-Peter [email protected] tion. At the same time stars produce heavy elements(metals) through the nuclear fusion processes in theirinterior. Some of the newly produced metals togetherwith some hydrogen and helium are recycled to the in-terstellar medium (ISM) by a variety of complex stellarmass-loss processes. While stars continue to form andgas is continuously accreted, metals accumulate duringthe life of a galaxy, but at the same time a significantfraction of the metals appears to be expelled from theISM by large scale galactic winds. In addition, mergingprocesses with infalling other nearby galaxies influencethe evolution significantly. a r X i v : . [ a s t r o - ph . GA ] F e b Kudritzki et al.
In view of the complexity of these many processes andtheir interplay on different time scales it is surprisingthat an intriguingly simple relationship exists betweentotal galactic stellar mass and the average metallicityof galaxies, the mass-metallicity relationship (“MZR”),see for example Lequeux et al. (1979), Tremonti et al.(2004), Kudritzki et al. (2016)(herafter K16), Zahidet al. (2017) (herafter Z17). This MZR and its evo-lution with redshift appears like a true Rosetta stoneto understand the key aspects of galaxy evolution. Forinstance, Zahid et al. (2014) (hereafter Z14) show thatthe observed MZRs at different redshift can be explainedby a very simple model with galactic winds and accre-tion where the observed metallicity is a function of theratio of galactic stellar to ISM gas mass. Over their life-time star forming galaxies evolve along the (redshift de-pendent) main sequence of star formation and turn gasinto stars. During this evolution the low-mass metal-poor galaxies are gas-rich and the high-mass metal-richgalaxies are gas-poor (see Fig. 5 and 7 in Z14).Given the success of the Z14 approach in matchingand explaining the observations it appears importantto further investigate the validity of this rather simplegalaxy evolution model. An obvious way is the compar-ison with cosmological simulations, which describe thecomplicated processes during the formation and evolu-tion of galaxies in a much more comprehensive way. Thisis done in the following by using the extensive Mag-neticum simulations (see section 4).Starting from the ideas described in Z14 and Z17 wedevelop a new generation of lookback models, which de-scribe the evolution of galaxies. We then compare theproperties of these models (SFR ψ , stellar mass M ∗ ,ISM gas mass M g , luminosity weighted age t of the stel-lar population and logarithmic metallicity [Z] of the ISMand the stellar population) with the properties of galax-ies in the Magneticum Box 4 (high resolution) simula-tions. We also compare with observations. LOOKBACK MODELSOur goal is to describe the evolution of a galaxy, whichis observed at a redshift z with a stellar mass M ∗ (z )back to its origin. Because we are looking back in cosmictime, we call these models lookback models. The rela-tionship between lookback time and redshift z is givenby the standard equations t ( z ) = t ( z ) + 1 H (cid:90) zz dz (1 + z ) E ( z ) (1)with E(z) E ( z ) = (Ω Λ + Ω m (1 + z ) ) (2) and t ( z ) = 1 H (cid:90) z dz (1 + z ) E ( z ) . (3)Like the Magneticum simulations (see section 4) weadopt a flat universe with H = 70.4 km/sec/Mpc =70.4 × − yr − and densities Ω Λ = 0.728 and Ω m =1 - Ω Λ .The mass evolution is then described by M ∗ ( z ) = M ∗ ( z ) − (1 − R ) H (cid:90) zz ψ ( M ∗ , z )(1 + z ) E ( z ) dz. (4)R is the fraction of stellar mass which is returned tothe interstellar medium because of stellar winds and su-pernova explosions. Following Z14 and Z17 we adoptR = 0.45 based on the assumption of a Chabrier (2003)stellar initial mass function. ψ is the star formation rate(SFR) as a function of stellar mass and redshift. We usea modified form (see section 5 below) of the SFR lawfrom Pearson et al. (2018), Appendix C, which is basedon observations of galaxies on the star foming main se-quences out to redshift z = 6. We usually finish theintegration, when a minimum stellar mass of 10 M (cid:12) isreached.A key simplification in our approach is the assumptionthat at every redshift there is a power law correlation be-tween the total mass of the cold (molecular and atomic)gas of the ISM and the stellar mass of galaxies M g ( z ) = A ( z ) M β ∗ ( z ) (5)with A ( z ) = A (1 + z ) α . (6)This assumption is based on survey observations ofstar forming galaxies in the local Universe, which indi-cate a power law holding over several orders of magni-tude in stellar mass with an exponent of the order of β ∼ α , β and A below.With the use of equations (4) and (5) we do notaccount for potential strong variations of star forma-tion activity caused by a multitude of processes, duringwhich active galaxies become passive or passive galax-ies are rejuvenated to become active again (see, for in-stance, Trussler et al. 2020; Spitoni et al. 2020). Theconcept of our lookback models is that despite of this alaxy lookback evolution models dZdM ∗ = 1 M g ( dM Z dM ∗ − Z dM g dM ∗ ) , (7)where the change of the metallicity mass M Z of theISM is given by dM Z dM ∗ = Y N − Z (1 − R ) . (8)Y N is the effective yield, which in addition to thestellar nucleosynthesis yield Y includes the effects ofaccretion from the halo and the intergalactic medium(dM accr ) and galactic winds (dM wind ) as described byZ14 Y N = Y − ζ (9)with ζ = Z wind dM wind dM ∗ − Z accr dM accr dM ∗ . (10)Z accr is the metallicity of the accreted gas, while Z wind corresponds to the metallicity of the matter lost throughgalactic winds. Based on observational evidence pro-vided by Zahid et al. (2012) and Peeples et al. (2014),Z14 argue convincingly that ζ ∼ const. is a reasonableapproximation. We therefore use Y N ∼ const. as a freeparameter noting that it consists of two components, thenucleosynthesis yield Y and the effects of accretion andwinds described by ζ .Using eq. (5) and (6) we can express dM g dM ∗ as dM g dM ∗ = β M g M ∗ (1 − K ( M ∗ , z )) (11)with K ( M ∗ , z ) = αβ H E ( z )(1 − R ) ψ M ∗ . (12)The function K describes the influence of the evolutionwith redshift of the power law relation between stellarmass and gas mass.Eq. (11) then leads to the final form of the metallicityequation dZdM ∗ = 1 − RM g { Y N − R − Z (cid:18) β − R M g M ∗ (1 − K ) (cid:19) } . (13) We note that Z14 in their analytical approach neglectthe second term on the right hand side of eq. (7). How-ever, solving our set of lookback model equations nu-merically and calculating a large grid of models withdifferent model parameters we find that including thisterm leads to a small, but non-negligible quantitativedifference in the calculated metallicity during the evo-lution of a galaxy. We, therefore, keep this term forthe calculation of our numerical lookback models anddevelop a new analytical solution later in Appendix A. Figure 1.
Lookback model ISM metallicity [Z] as a func-tion of the ratio of stellar mass to gas mass for the evolutionof 7 galaxies with final masses of log M ∗ = 9.28, 9.65, 10.01,10.16, 10.34, 10.60, 11.13. We use different colors for thegalaxy evolution tracks with different final mass. Note thatthe tracks lie on top of each other, which means that [Z] de-pends only on M ∗ /M g . For the defintion of [Z] see equation(14). The result is discussed in detail in Appendix A and thecalculations are described in section 2. Fig. 1 shows a typical result for the lookback modelevolution of seven galaxies with different final masses (ayield of log Y N Z (cid:12) (1 − R ) = 0.3 and the star formation lawdescribed in section 5 have been used for these calcula-tions). While there is a small quantitative difference tothe Z14 results, we see that the most important prop-erty of the lookback models remains. Metallicity is toa good approximation solely a function of the ratio ofstellar mass to gas mass M ∗ /M g . As we show in Ap-pendix A, this is the consequence of the key assumptionof our lookback models, the relationship between gasmass and stellar mass as described by eq. (5) and (6).The comparison with the Magneticum models will be animportant check whether this key assumption is valid. Kudritzki et al.
In the following we will express metallicities in unitsof the solar metallicity defined as[ Z ] = log Z/Z (cid:12) . (14)For the metallicity mass fraction of the sun we willuse Z (cid:12) = 0.014 (Asplund et al. 2009).The metallicity shown in Fig. 1 is the metallicity ofthe ISM and the stars just born from this ISM. For thecomparison with stellar metallicities we will also cal-culate a V-band luminosity weighted average over thewhole galactic population of stars, which includes allstars that formed earlier at lower metallicity. Details ofthe calclulations are described in Appendix C.At the end of this section it is appropriate to brieflycompare our lookback models with most recent alterna-tive analytical models describing galaxy evolution. Spi-toni et al. (2017, 2020) present a model with an ex-ponentially decreasing gas infall rate, a star formationrate proportional to total cold gas mass and a gas out-flow rate proportional to star formation rate. They donot consider the fact that only the molecular fractionof the cold gas is involved in the star formation pro-cess, which may be important in the light of section6.5 further below. Gas accretion time scale, total ac-creted gas mass and mass loading factor are free param-eters. With assumptions about star formation time anda power law relationship between gas mass and stellarmass (both redshift independent) they use observationsof the MZR and the SFR main sequence at z=0.1 and2.2 to constrain accretion time scales, ages, mass load-ing factors and accreted gas masses of star forming andpassive galaxies and to investigate the effects of galaxydownsizing.Pantoni et al. (2019) make similar assumptions aboutthe time dependence of infall and star formation rate butthey relate the amount of infalling mass to the baryonicmass present in the host halo, which is characterized by adark matter mass distribution following an NFW-profile(Navarro et al. 1997). The accretion time scale is ob-tained from estimates of the halo cooling and dynamicaltime scales which depend on assumptions about densityand clumping. The star formation time scale is obtainedfrom a redshift independent estimate of the dynamicaltime scale in the rotating galactic disk at the radius ofcentrifugal equilibrium. Gas outflow rates are adoptedfrom theoretical work assuming energy or momentumdriven winds. For the estimate of halo masses, massgrowth by merging processes is taken into account us-ing fitting formulae obtained from the comparison withcosmological simulations. After constraining all param-eters Pantoni et al. (2019) apply their models to the starforming progenitors of early type galaxies and calculate SFR, gas mass and metallicities as a function of stellarmass at high redshifts from z = 2, 4 and 6.Lapi et al. (2020) extend the Pantoni et al. (2019)model by adding the effects of wind recycling and galac-tic fountains. They also consider the important effectthat star formation is related to the molecular fractionof interstellar gas only and they modify star formationtime accordingly. They calculate the fraction of the starforming to the total cold ISM gas assuming a linear de-pendence on the mass weighted Toomre parameter. As aresult the ratio of mass accretion time to star formationtime changes significantly compared to Pantoni et al.(2019). The agreement with observations of star forma-tion rates, gas and dust masses, metallicities and specificstellar angular momentum as a function of stellar massat redshifts z = 0 and 1 of star formation rates is re-markably good. This is a self-consistent approach, whichaims to account for the multitude of individual processesthrough a variety of parameters which are constrainedmostly by theoretical arguments and comparisons withnumerical simulations.In contrast, our lookback models do not consider themany different individual processes affecting galaxy evo-lution. They avoid explicit assumptions about mass ac-cretion and mass loss as a function of time and insteadcondense the complex interplay between accretion, out-flows and star formation into the simple redshift depen-dent relationship between gas mass and stellar mass.As a result, the understanding of chemical evolution isstraightforward and related to the effective yield andthe ratio of stellar mass to gas mass. As we will seein the following, this simple approach agrees well withobservations and detailed hydrodynamical cosmologicalsimulations. OBSERVATIONSTo compare our model calculations with observationswe use galaxy metallicities derived from quantitativestellar spectroscopy of individual blue and red super-giant stars in nearby galaxies out to 20 Mpc (see K16)and population synthesis stellar spectroscopy of stackedspectra of 250000 SDSS galaxies at a redshift of z ∼ alaxy lookback evolution models Figure 2.
Lookback model MZRs at different redshifts.The metallicity of the ISM and the young stellar populationversus the total stellar mass of the galaxy is shown in red.The blue curve represents the V-band luminosity weightedmetallicity of the whole stellar population. Observed HII-region MZRs at the same redshift are shown in green (Zahidet al. 2014), cyan (Genzel et al. 2015), yellow and violet(Sanders et al. 2020). For the lowest redshift we also showobservations of the stellar metallicities of the young stellarpopulation (pink circles: red supergiant stars; pink stars:blue supergiant stars, see Kudritzki et al. 2016). The orangecircles respresent metallicities of the young stellar populationobtained by population synthesis analysis of SDSS spectra ofthe integrated stellar population of a large sample of galaxies(Zahid et al. 2017).
Fig. 2 shows the comparison of our lookback modelswith the observations. The calculations use the param-eters α = 0.40, β = 0.60, log A = 3.73 (see eq. 5 and 6,M ∗ and M g in solar masses) and log Y N Z (cid:12) (1 − R ) = 0.225 andapply a main sequence star formation law as describedin section 5 with the correction factor c(z) set to unity.We find remarkable agreement. MAGNETICUM DATAThe Magneticum simulations are a set of fully hy-drodynamical cosmological simulations of different box-volumes and resolutions. They follow the formationand evolution of cosmological structures through cos-mic time, accounting for the complex physical processeswhich shape the first building blocks of galaxies intothe mature galaxies of today. For details on thesesimulations see Hirschmann et al. (2014) and Tekluet al. (2015). A WMAP-7 ΛCDM cosmology (Komatsuet al. 2011) is adopted with h = 0 . m = 0 . b = 0 . λ = 0 . σ = 0 . n s = 0 . with initially 2x5763 (dark matter and gas) particles.The particle masses are m DM = 3 . × M (cid:12) /h and m Gas = 7 . × M (cid:12) /h, respectively, and each gas parti-cle can spawn up to four stellar particles (i.e. the stellarparticle mass is approximately 1/4th of the gas particlemass), with a softening of (cid:15) DM = (cid:15) Gas = 1 . (cid:15) ∗ = 0 . ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ Kudritzki et al. sample. The mass bins are log M ∗ = 9.28, 9.65, 10.01,10.16, 10.34, 10.60, 11.31 and the evolution is traced upto redshift z = 4.2. The selected mass bins are an ar-birary choice to represent the range from lower to highergalaxy stellar masses. We call the first data set the red-shift “snap shot sample” and the second one the “evo-lution sample”. In the snap shot sample we distinguishbetween disk galaxies and intermediate galaxies. Forthis we use the position of a galaxy in the stellar mass–angular momentum plane, quantified by the b-value b = log( j kpc km / s ) −
23 log( M ∗ M (cid:12) ) , (15)where j is the specific angular momentum of the galaxystellar component (see especially Teklu et al. 2017, butalso Teklu et al. 2015 and Schulze et al. 2018 for moredetails). At redshift z = 0.07, galaxies with -4.35 ≥ b ≥ -4.75 are classified as intermediates, whereas galaxieswith b ≥ -4.35 are considered disks. Note that interme-diates are a transitional galaxy type between disks andspheroids.We emphasize that some of the selected galaxies in oursnap shot sample went through earlier phases of stronglyreduced star formation and were then rejuvenated againby gas rich merging processes. We keep these galaxiesin our sample. As already discussed in section 2 thisallows us to test, whether the lookback models, whichare thought to describe the averaged evolution of starforming galaxies and do not account for strong changesof star formation, describe the average properties of starforming galaxies well.For the evolution sample we select all disk galaxies atz = 0.07 with log M ∗ ≥
9. Note that at higher red-shift these galaxies could also have been intermediatesor spheroids or also passive . STAR FORMATION RATESFor the comparison with the Magneticum simulationswe need to adopt a SFR law ψ (M ∗ ,z), which providesSFRs as a function of stellar mass and redshift. Thereis a rich literature of observed SFR laws of the so-calledgalaxy main sequences (see, for instance, Pearson et al.2018, Tacconi et al. 2020 for a recent summary). Mostof the approaches describe star formation as a power law ψ ( z, M ∗ ) = ψ ( z ) M δ ( z ) ∗ , (16)where both the zero point ψ ( z ) and the slope δ ( z )increase with redshift. ψ is given in solar masses peryear and the stellar masses are in solar units. We haveselected a few published SFR laws (Elbaz et al. 2007,Behroozi et al. 2013, Speagle et al. 2014, Schreiber et al. 2015, Pearson et al. 2018, Sanders et al. 2020) and com-pare them with the Magneticum SFRs in Fig. 3. Giventhe systematic uncertainties of the observations (see theexample discussed below) the comparison is reasonableand indicates that the sub-grid physics treatment of starformation adapted from Springel & Hernquist (2003)works well. However, we note a systematic difference atlow and intermediate redshifts where the MagneticumSFRs are lower by a factor of 2 to 3. Figure 3.
Logarithm of star formation rates of the Mag-neticum snap shot samples at different redshifts as a functionof the logarithm of stellar mass (green circles: intermedi-ates, blue circles: disks). Observed star formation rates areoverplotted as solid curves: black: Elbaz et al. (2007), cyan:Behroozi et al. (2013), violet: Schreiber et al. (2015), orange:Pearson et al. (2018), red: Pearson et al. (2018), AppendixC, pink: Speagle et al. (2014), grey: Sanders et al. (2020).
In principle, we can construct lookback models withany reasonable star formation main sequence representa-tion. For instance, for the comparison with observationsin Fig. 2 we have used Pearson et al. (2018) as describedin their Appendix C. However, the purpose of this workis to compare our simple lookback model approach withthe highly complex and detailed hydrodynamic Mag-neticum simulations. Therefore, it is prudent to adopt aSFR description which matches the Magneticum SFRsreasonably well. We accomplish this by again using thePearson et al. (2018) SFR law of their Appendix C, how-ever now with several modifications. Most importantly, alaxy lookback evolution models Figure 4.
Correction factor applied to the Pearson et al.(2018) star formation rates as a function of redshift to matchthe Magneticum SFRs (see text). we apply a redshift dependent correction factor c(z) to ψ ( z ) ψ LB ( z ) = ψ ( z ) c ( z ) , (17)which is shown in Fig. 4. With this factor we obtain astar formation law ψ LB ( z, M ∗ ) which is in good agree-ment with the Magneticum SFRs at M ∗ = 10 . M (cid:12) .In addition, we introduce a broken power law with re-spect to stellar mass, which is described in more detailin Appendix B together with the complete descriptionof ψ LB ( z, M ∗ ) .Fig. 5 shows that ψ LB provides a good representationof the average SFRs of the Magneticum snap shot sam-ple. While the literature main sequence SFRs shown inFig. 3 indicate that our adopted Magneticum matchingstar formation law ψ LB may underestimate the observedmain sequence SFRs at lower and intermediate redshift,the comparison in Fig. 6 with observed SFRs obtainedby the xGASS and xCOLDGASS surveys (Saintongeet al. 2017, Catinella et al. 2018) of nearby galaxies indi-cates good agreement. We also note that the MAGMAsurvey (see Hunt et al. 2020) agrees well with Mag-neticum and ψ LB except at the very low mass end of logM ∗ ≈
9, where the MAGMA SFRs are 0.3 dex larger.While the physical reason for the introduction of thecorrection factor c(z) may, thus, be caused by system-atic uncertainties of the methods how star formationrates are determined, we also have to note a weaknessof the simulations. They adopt a local star formation ef-
Figure 5.
Same as Fig. 3 but now with the adopted look-back model star formation rates overplotted in orange (seetext). We also show isocontours enclosing 68% (red), 80%(pink) and 95% (cyan) of the Magneticum galaxies. ficiency, which is constant over time for a given amountof cold gas density above a certain threshold, with onlyminor variations due to changes of the ISM cooling func-tion caused by its dependence on metallicity. As pointedout by Miller et al. (2018) this is a more general problemof current simulations.We can also compare the time evolution of the Mag-neticum evolution sample with the ψ LB of our lookbackmodels. This is done in Fig. 7, where specific SFRs(sSFR) ψ /M ∗ are plotted for models corresponding todifferent Magneticum evolution mass bins. We concludethat the Magneticum SFRs are well represented by ourchoice of ψ LB .We note that in the complete snap shot set of Mag-neticum simulations there is a significant number of diskand intermediate galaxies with very low SFRs. Sincethis work focusses on star forming galaxies not too farfrom the main sequences, we have excluded those ob-jects in Fig. 3 and from all further comparison betweenlookback models and Magneticum. As a selection crite-rion we use the threshold of 0.8 dex below our adoptedstar formation law and include only galaxies above thisthreshold. We realize that such galaxies with low SFRsbelow our threshold may exist in the real universe, butwould be excluded from samples selected on strength ofISM emission lines. Kudritzki et al.
Figure 6.
Magneticum snap shot star formation rates atredshift z = 0.1 (green and blue circles as before) comparedwith the observed star formation rates of the xGASS andxCOLDGASS surveys (red circles, for references see text).The adopted lookback model star formation rates are over-plotted in orange.6.
COMPARISON BETWEEN LOOKBACKMODELS AND MAGNETICUM SIMULATIONS6.1.
The relationship between gas mass and stellarmass
The key simplification of the lookback models is theassumption of a redshift dependent power law betweengalaxy stellar mass and the total mass of the cold ISMgas (see eq. 5 and 6). In Fig. 8 we test whether the Mag-neticum hydrodynamic cosmological simulations sup-port the idea of such a relationship. We find good agree-ment with the Magneticum simulations. This result isvery reassuring for the lookback model approach. It is astrong confirmation of the basic concept of these mod-els. It also provides a substantial simplification for thedescription of star forming galaxy evolution.We note that the self-consistent analytical approachby Pantoni et al. (2019) and Lapi et al. (2020) discussedin section 2 also leads to a power law relationship be-tween gas mass and stellar mass similar to our lookbackmodels (see Figures 7 in both papers). The redshift de-pendence found by Pantoni et al. (2019) is in agreementwith our models and with Magneticum but it is signif-icantly steeper in the results displayed by Lapi et al.(2020).One might argue that for galaxies on the mainsquence, where star formation follows a power law with
Figure 7.
Specific star formation rate as a function oflookback time for six galaxies with different final stellar mass(at t = 0). The red curves correspond to lookback modelcalculations. The violet, green and pink squares indicateredshifts z = 1, 2 and 4.2, respectively. The blue circlescorrespond to the Magneticum evolution sample describedin the text. stellar mass, it is to be expected that ISM gas massdepends on stellar mass in a similar way, because starformation should be proportional to gas mass. However,the situation is more complicated, because not all theISM gas is involved in the star formation process andthe fraction of star forming to total ISM gas changeswith stellar mass and redshift. We will discuss thisfurther below.Nonetheless, we note that there is a weak star for-mation dependence of the relationship between ISM gasand stellar mass in the Magneticum snapshot sample.Galaxies with lower gas mass tend to have lower SFRsand higher gas mass galaxies have higher SFRs. Thishas consequences for the MZR and its SFR dependence,which will be discussed this further below.6.2.
Luminosity weighted ages of the stellar population
As the result of their different star formation histo-ries the distribution of ages of the stellar populations indifferent galaxies will vary. A crucial quantity to char-acterize the age distribution is the luminosity weightedaverage age. In Fig. 9 we show the V-band luminosityweighted ages of the Magneticum galaxies as a func-tion of stellar mass for different redshifts and compare alaxy lookback evolution models Figure 8.
Logarithm of galaxy cold ISM gas mass versuslogarithm of stellar mass at different redshifts. The Mag-neticum galaxies are shown as blue and green circles as be-fore. The lookback model relationship is plotted as the or-ange line. As in Figure 5 we also show isocontours enclosing68% (red), 80% (pink) and 95% (cyan) of the Magneticumgalaxies. with the lookback model predictions (for the calcula-tion of lookback model luminosity weighted ages see Ap-pendix C). Except for the very lowest and highest red-shifts the lookback models seem to agree with the Mag-neticum simulations. However, we note a large spreadof stellar ages for the Magneticum galaxies. At the low-est redshift we also compare with observations obtainedfrom a spectroscopic study of SDSS galaxies by Trus-sler et al. (2020), who used stellar population modellingwith FIREFLY to constrain ages of star forming galax-ies (see their section 6.1 and Figure 13). The agreementwith our lookback models is excellent.Since ages are related to star formation history, it isa crucial first test to investigate whether the age spreadis correlated with SFRs. From Fig. 5 we know thatour Magneticum snap shot sample has a wide range ofSFRs ( ≈ ± ψ Magn withrespect to the adopted lookback SFRs ψ LB in the fol-lowing way: bin1: log ψ Magn / ψ LB ≤ -0.5; bin2: -0.5 ≤ ψ Magn / ψ LB ≤ -0.25; bin 3: -0.25 ≤ ψ Magn / ψ LB ≤ +0.25; bin4: +0.25 ≤ ψ Magn / ψ LB ≤ +0.50; bin5: +0.5 ≤ ψ Magn / ψ LB . Figure 9.
V-band luminosity weighted average ages (inGyr) of the Magneticum galaxies stellar populations (greenand blue circles as before) compared with ages calculatedfrom the lookback models using the standard lookback modelSFR law ψ LB (orange) described in section 5. For the lowestredshift we also show observational results (yellow) obtainedby Trussler et al. (2020) (see text). Isocontours have thesame meaning is in the previous figures. Fig. 10 shows the ages of the Magneticum galaxiesagain but now color coded with respect to SFR bin.There is an obvious anti-correlation between age andstar formation rate, which is easy to understand. Theoldest galaxies at a given stellar mass are mostly thosewith the lowest SFRs, because it took a long time tobuild up the stellar mass, and the youngest are mostlythose with high SFRs, because the stellar mass was builtup recently.To simulate the effect of systematically higher andlower star formation rates we have also calculated look-back models with SFRs of log ψ LB ± Gas accretion
In the previous subsections we compared the look-back models with the Magneticum snap shot sample,the set of simulated star forming galaxies at differentredshifts. Now we turn to the Magneticum evolution0
Kudritzki et al.
Figure 10.
Same as Fig. 9 but now the Magneticumgalaxies are color-coded with respect to SFR. Bin1: yellow,bin2: orange, bin3: green, bin4: blue, bin5: red. Bin 1corresponds to the lowest SFRs and bin 5 to the highest (forthe extact definition see text). The green curve correspondsto ages calculated with the standard lookback model SFRlaw ψ LB . The orange and blue curves are obtained fromlookback model calculations, where log ψ LB is modified byoffsets of -0.375 dex and +0.375 dex, respectively. sample, where we combine galaxies in stellar mass binsand then follow their evolution with time calculating av-eraged properties of each mass bin as a function of time(or redshift). In a first step we study the evolution ofgalaxy gas mass and investigate the role of gas accre-tion. Both the lookback models and the Magneticumsimulations include the effect of gas accretion from thegalactic halo or the intergalactic medium. In the caseof the lookback models this is done implicitly throughthe assumption of the power law relationship betweengas and stellar mass M g = A(z)M β ∗ . In the case of theMagneticum models this is a direct result of the hydro-dynamic simulation of the galaxy formation and evolu-tion process. The change of gas mass with redshift (ortime) is given by∆ M g = ∆ M effaccr − ∆ M ∗ (18)where ∆ M ∗ is the change of stellar mass through starformation, which causes a decrease of gas mass. ∆ M effaccr is the net amount of accreted gas mass leading to anincrease of ISM gas mass and consists of two terms ∆ M effaccr = ∆ M accr − ∆ M wind , (19)where the first describes mass gain through accretionand the second term mass loss through galactic winds.The normalized effective accretion rate Λ eff is thenΛ eff = ∆ M effaccr ∆ M ∗ = ∆ M g ∆ M ∗ + 1 . (20)We note that the normalized effective accretion rateis equal to the effective mass accretion factor, which isfrequently used in chemical evolution models (see, forinstance, Kudritzki et al. 2015)Λ eff = ˙ M accr (1 − R ) ψ − ˙ M wind (1 − R ) ψ , (21)where ˙ M wind and ˙ M accr are the rates of mass-lossthrough winds and mass-gain through accretion, re-spectively. In the case of the lookback models thenormalized effective accretion rate can be calculatedanalytically (see section 2)Λ eff = 1 + β M g M ∗ (1 − K ( M ∗ , z )) . (22)Fig. 11 shows the accretion rates of the lookback mod-els (red curves) as a function of time for the galaxieswith six final masses log M ∗ . We see that accretiondominates over galactic winds (Λ eff >
0) but the massgains through accretion rates are moderate and do notexceed the SFRs by a large factor. For the lookbackmodels the value of Λ eff ≈ eff rates of the Magneticum simulationsare of the same order for the four higher galaxy masses,but are larger by a factor two to three for log M ∗ = 9.30and 9.66, respectively.While the differences between the lookback modelswith the simple assumption of a redshift dependentpower law relationship between gas mass and stellarmass and the detailed Magneticum hydrodynamic sim-ulations are obvious, we note that they are not ordersof magnitude. We take this as an additional confirma-tion of the lookback model approach. As we will see inthe next subsection, this is mostly caused by somewhathigher values of R and not so much by higher accretionrates.However, at this point we need to add a word of cau-tion. We note that our method to calculate Magneticumaccretion rates is based on the assumption that the in-crease of stellar mass is solely through star formation.If the increase of stellar mass is also partially caused by alaxy lookback evolution models Figure 11.
Normalized effective accretion rate as a func-tion of lookback time for six galaxies with different final stel-lar mass (at t = 0). The red curves correspond to lookbackmodel calculations. The violet, green and pink squares indi-cate redshifts z = 1, 2 and 4.2, respectively. The blue curvesare calculated from the Magneticum evolution sample de-scribed in the text. merging with infalling galaxies, then our approach un-derestimates the accretion rates and provides only lowerlimits. 6.4.
Stellar mass growth and merging
In order to estimate the influence of merging on thegrowth of stellar mass we determine the ratioΛ ∗ = ∆ M ∗ ψ ∆ t . (23)For the lookback models we have the constant valueΛ ∗ = 1 - R. This is the horizontal red line in Fig. 12.The blue curve shows the results obtained from the Mag-neticum evolution sample. We find good agreement withthe simple lookback model approach for the mass binslog M = 10.16 and 10.34. This implies mass return frac-tions to the ISM during the star formation process ofR ≈ ≈ (cid:46) ∗ -values clearly larger than unity.As a consequence, the Magneticum accretion rates for Figure 12.
Changes of stellar mass in units of star for-mation as a function of lookback time for six galaxies withdifferent final stellar mass. The red curves correspond tolookback model calculations. The blue curves are calculatedfrom the Magneticum evolution sample. Violet, green andpink squares indicate redshifts z = 1, 2 and 4.2, respectively.For discussion, see text. these phases of the evolution of the Magneticum highmass models are likely larger than indicated in Fig. 11.6.5.
Magneticum star forming gas masses, starformation time and the main sequence starformation law
The Magneticum simulations provide the opportunityto investigate the connection between the main sequencestar formation law ψ (z,M ∗ ) and the gas mass - stellarmass relationship M g (z,M ∗ ). We do this in three steps.We first discuss the fraction of cold ISM gas, which iscontributing to star formation. Second, we look at thetime scale of the star formation process and, third, wecombine this information with the predicted relation oftotal cold ISM gas mass (star forming and passive withrespect to the star formation process) with stellar mass.Not all the cold ISM gas in a galaxy is involved in thestar forming process. In the Magneticum simulationsonly a fraction x g x g ( z ( t ) , M ∗ ) = M SFg ( z, M ∗ ) M g ( z, M ∗ ) (24)is producing stars. M SFg is the total mass of all coldstar forming ISM gas and correponds to gas of low tem-2
Kudritzki et al. perature above a certain density threshold. Fig. 13 usesthe Magneticum snap shot sample (restricted to galax-ies with log | ψ/ψ LB | ≤ g depends on redshift and stellar mass and how it evolveswith lookback time t.The snap shot sample plots show a significant scatterat lower redshift. However, observations of galaxies inthe local universe show a similar scatter and the x g val-ues observed agree with the Magneticum observations(see the XGASS, xCOLDGASS and MAGMA surveys,Saintonge et al. 2017, Catinella et al. 2018, Hunt et al.2020).The red curve is calculated with a fit formula (seeAppendix B) to the Magneticum data, which closelymatches the mean of x g as a function of stellar massat each redshift. At low redshift x g is constant withstellar mass except at the lowest masses, where we finda strong increase. Towards higher redshift this behav-ior reverses: x g is constant at low masses, but declinestowards higher masses. Most importantly, though, themaximum of x g increases with redshift. Unfortunately,there are no direct observations of HI available in galax-ies of higher redshift to compare with the Magneticumsimulations.The evolution sample plots demonstrate very clearlythat during the course of the evolution of a galaxy x g decreases continuously. At the beginning of the life ofa Magneticum star forming galaxy practically all gas isinvolved in the star formation process. But then thefraction of gas contributing to star formation becomessignificantly smaller. The red curve is the fit taylored todescribe the snap shot sample. While it is not a perfectfit for the evolution sample, it captures the evolutionwith time (and redshift) reasonably well.We note that in observational studies - because of thelack of direct HI observations - the assumption is fre-quently made for the evolution of x g that already at z ≈ g ≈ τ SF ( z ( t ) , M ∗ ) = M SFg ( z, M ∗ ) ψ ( z, M ∗ ) . (25) Figure 13.
ISM fraction of the mass of star forming gasto total gas mass. Top: Magneticum snap shot sample atdifferent redshifts with log M g (SF)/M g as a function of stel-lar mass. Blue and green circles and isocontours as before.Bottom: M g (SF)/M g as a function of lookback time for theMagneticum evolution sample (blue circles) in six differentstellar mass bins. The red curves correspond to the fit de-scribed in the text. A formula is given in Appendix B. Fig. 14 shows τ SF for the main sequence galaxies ofthe Magneticum snap shot sample and the evolution alaxy lookback evolution models Figure 14.
Logarithm of star formation time. Top: Mag-neticum snap shot sample at different redshifts with log τ SF as a function of stellar mass. Blue and green circles andisocontours as before. Bottom: τ SF versus lookback timefor the Magneticum evolution sample in six different stellarmass bins (blue circles). The red curves are calculated ac-cording to Appendix B and are discussed in the text. Theviolet curves are given by Eq. (26). sample. The red line in the plots is a simple fit to theridge line in the snap shot sample. The corresponding fit formula is given in Appendix B. We see that in theMagneticum galaxies the star formation time decreaseswith redshift. We also find a mass dependence witha negative slope, which becomes steeper with redshift.The Magneticum simulations at z ∼ τ ( t ) = τ (1 + z ( t )) − n , (26)which we show in the plot of the Magneticum evolu-tion sample in Fig. 14 using τ = 2.3 × yrs and n=0.6(violet curve). We see that the Magneticum simulationshave a similar trend as a function of lookback time (orredshift). The fit obtained from the Magneticum snapshot sample is in agreement with eq. (26). We note thatour value for τ is a factor of two larger than the onefound by Tacconi et al. (2018). This is a consequenceof the fact that the Magneticum SFRs are lower thanthe ones used by Tacconi et al. (2018) in the range oflookback time displayed here. Figure 15.
Lookback model SFRs ψ LB as a function ofstellar mass at the ten redshifts of the Magneticum snap shotsample from z = 0.1 to 6.9 (solid curves) together with SFRscalculated from eq. (27) (dashed). Kudritzki et al.
With the fits for x g and τ SF as given in Appendix B wehave an alternative way to calculate SFR as a functionof redshift and stellar mass ψ ( z ( t ) , M ∗ ) = x fitg ( z, M ∗ ) τ fit ( z, M ∗ ) M g ( z, M ∗ ) , (27)where M g (z,M ∗ ) is given by eq. (5). Fig. 15 com-pares lookback SFRs ψ LB with SFRs calculated witheq. (27). We use the overall agreement found in Fig. 15as an argument that the SFR law along the main se-quence is a consequence of the power law relationshipbetween total ISM gas mass and stellar mass. However,it is important to stress that the complex behavior as afunction of redshift and stellar mass of x g , the ratio ofstar forming to total gas mass, and of τ SF , the star for-mation timescale are also of crucial importance. Mostimportantly, we note that the decline of specific star for-mation rates (see Fig. 7) is not caused by the relativelysmall drop of the ratio M g /M ∗ but rather by the sig-nificant continuous decrease of x g (t). In other words,galaxies form less stars in the course of their evolutionnot because they are running out of gas (or because thestar formation time changes dramatically) but ratherbecause the fraction of cold ISM gas, which is capableof producing stars, becomes smaller.6.6. Chemical evolution
One of the major goals of our lookback models hasbeen to develop a tool to understand and interprete thechemical evolution of star forming galaxies. Here the keyobservation is the MZR between stellar metallicity andtotal stellar mass. Fig. 16 shows the V-band luminosityweighted average metallicity of the stellar population ofthe Magneticum snap shot sample as a function of stellarmass at different redshifts. The stellar metallicities ofour lookback models calculated for two different effctiveyields [Z] = log Y N Z (cid:12) (1 − R ) = 0.25 (orange) and 0.40 (red)are also shown. There is a significant scatter of σ [ Z ] ≈ Figure 16.
Luminosity V-band averaged stellar metallic-ities versus total stellar mass for the galaxies of the Mag-neticum snap shot sample at ten different redshifts. Look-back model stellar metallicities calculated for two differentyields are overplotted in red and orange. At the lowest red-shift observed stellar metallicities (see Fig. 2) are also shownas small and large yellow circles and asterisks. Magneticumisocontours corresponding to the same values as in the pre-vious figures are also shown. tic metallicity is basically a function of the ratio of stellarmass to ISM gas mass. Fig. 17 confirms this conclusionby comparing lookback model stellar metallicities withMagneticum metallicities as a function of log M ∗ /M g .The metallicity of the star forming ISM gas is expectedto be slightly higher than the luminosity weighted metal-licity of the stellar population, because the ISM metal-licity represents the latest stage of the chemical evolu-tion, whereas the stellar luminosity weighted metallic-ity always contains a contribution by the older popula-tion less advanced in the formation of heavy elements.Fig. 18 confirms this expectation. Note that we repre-sent the metallicity of the star forming gas by its oxygenabundance relative to the sun [O/H] ISM , which is usuallydetermined from the observation of strong emission linesof the star forming gas. The difference ∆ = [O/H]
ISM - [Z] is somewhat larger than the lookback models formost of the Magneticum galaxies but the effect is notlarge (0.1 to 0.15 dex) and disappears towards higherredshifts. We note that a small fraction of the Mag-neticum galaxies has negative values of ∆. We inter- alaxy lookback evolution models Figure 17.
Stellar metallicities as a function of the ratioof stellar mass to total ISM gas mass. The galaxies of theMagneticum snap shot sample are compared with lookbackmodel stellar metallicities again calculated for two differentyields and overplotted in red and orange. Isocontours usethe same values as in previous figures. prete these as cases of recent accretion or merging withmetal poorer gas involved.As was discovered by Mannucci et al. (2010) from astudy of HII region emission lines, the MZR contains adependence on a third parameter, the SFR. At fixed stel-lar mass, galaxies with lower SFR tend to have highermetallicities and vice versa. Z17 in their investigationof spectra of the integrated stellar population of 250000SDSS galaxies found a similar effect for stellar metallic-ities. Most recently, Sanders et al. (2020) investigatinggalaxy gas-phase metallicities out to z ∼ Figure 18.
Difference between the oxygen abundance (rel-ative to the sun) [O/H] of the star forming ISM gas and theluminosity weighted stellar metallicity [Z] (also relative tothe sun) as a function of galaxy stellar mass. The galaxiesof the Magneticum snap shot sample are plotted as circles inthe usual way and isocontours for the same values as in pre-vious figures are also shown. The results obtained from thelookback model stellar metallicities are given by the orangeline.
In our lookback model approach metallicities do notdepend directly on SFR. As explained in Appendix A,they depend foremost on M ∗ /M g , the ratio of stellarto ISM gas mass and the evolution of the MZR withredshift is a result of the fact that the relation betweengas and stellar mass is redshift dependent. Thus at firstglance, the lookback models seem to be incapable inreproducing the observed SFR dependence of the MZRsat different redshift.However, observations of galaxies in the local Universeshow that the power law relationship between gas andstellar mass also depends on SFR. Galaxies with higherSFR have a higher ratio of gas mass to stellar mass andvice versa. Hunt et al. (2020) from the study of theirMAGMA sample of galaxies find that the shift with SFRcan be described by ∆log M g = x δ ∆log ψ with x δ = 0.37(see their eq. 10) .It is not surprising that such an additional SFR de-pendence exists. Galaxies with a higher (lower) gas massat similar stellar mass and redshift will very likely alsohave an increased mass of the star forming cold ISM gasand, in consequnce, their SFRs will be higher (lower).6 Kudritzki et al.
Figure 19.
The observed effect of SFR on the MZRs atfixed redshift. Gas phase HII region oxygen abundances (rel-ative to the sun) versus galaxy stellar mass are displayed atfour redshifts. The circles correspond to observations andare obtained from the Sanders et al. (2020) FMR relation-ship adopting their SFR main sequence law (green) and sys-tematic shifts away from the main sequence by ± For our lookback models this has an important con-sequence. We obtain a SFR dependence of the MZR byassuming that galaxies with a systematic shift away fromthe lookback model main sequence by δ = log ψ / ψ LB have also a shift in the zero point of the gas mass stellarmass relationship by ∆log A = x δ δ (see eq. 5 and 6for the meaning of A ). This is demonstrated in Fig. 19,where we compare our lookback models with the Sanderset al. (2020) MZRs calculated from their FMR formula.We adopt δ = ± δ = ± δ = | ψ/ψ LB | ≤ δ ≥ δ ≤ Figure 20.
Magneticum galaxy gas mass versus stellarmass at four redshifts. The galaxies are color coded ac-cording to their SFRs. Green circles correspond to galaxiesaround the main sequence within 0.2 dex, yellow and bluecircles are below and above the mains sequence, respectively.The pink solid line corresponds to the lookback model re-lation described by eq. 5 and 6, whereas the dashed anddashed-dotted lines use a shift of log A in eq. 6 of thisrelation by ± · We also include the standard relationship for the look-back models in Fig. 20 and for two models with ∆logA calculated with δ = ± δ = ± δ = 0, ± δ = 0, ± alaxy lookback evolution models Figure 21.
Stellar metallicities of Magneticum galaxiesversus stellar mass at four redshifts. The galaxies are colorcoded according to their SFRs as in Fig. 20. The pink solidcurve corresponds to lookback model calculations with themain sequence star formation law of eq. 17 and Appendix Badopted. The dashed and dashed-dotted curves apply shiftsin SFR and in the gas mass stellar mass relation as describedin the text.7.
SUMMARY AND DISCUSSIONThe main intention of the work presented here hasbeen to develop lookback galaxy evolution models as asimple tool to describe galaxy formation and evolution,which can then be used to interprete observational re-sults derived from spectroscopy such as the mass metal-licity relationship or to calculate model spectra usingpopulation synthesis techniques. As a crucial test ofthis new tool we compare with the Magneticum cosmo-logical simulations, which describe the process of galaxyformation and evolution in a much more comprehensiveway.An important ingredient for this comparison is theglobal galactic SFR as a function of stellar mass andredshift. A whole variety of such ’main sequence’ rela-tionships derived from observations is available in theliterature and we have compared those with the Mag-neticum SFR. We found that the Magneticum SFRs arein the right ballpark but there is a systematic differenceas a function of redshift. In order to use an SFR law forour lookback models that represents the Magneticumcalculations well we have used the Pearson et al. (2018), Appendix C, relationship but with a correction factoras a function of redshift and with a modification of thepower law, which describes the dependence of stellarmass.In summary, we find that the lookback models capturemany of the Magneticum galaxy properties reasonablywell. Most importantly, their key assumption of a red-shift dependent power law relationship between ISM gasmass and stellar mass agrees well with the Magneticumresults. While a complex interplay of star formation,gas accretion and mass-loss by galactic winds obviouslyaffects the galactic gas content, the net result, in a statis-cal sense, is still that ISM gas mass is related to the stel-lar mass at all redshifts. This has important repercus-sions for the effective galactic gas accretion rates, whichare larger than the rates of mass-loss through galacticwinds but of the same order of magnitude as star for-mation rates.The fraction of the cold star forming ISM gas to thetotal gas changes continuously with time in the Mag-neticum galaxies. At the early stage of galaxy formationall gas is involved in the star forming process but thenduring the further evolution with time the fraction ofthe gas contributing to star formation becomes signifi-cantly smaller. This is an important factor contributingto the main sequence star formation law. The specificstar formation rates of star forming galaxies strongly de-crease during the course of their evolution not becausethey are running out of gas (as described by the look-back model power law relationship between gas mass,redshift and stellar mass), but because the fraction ofthe still present cold ISM gas, which is capable of pro-ducing stars, becomes significantly smaller.The luminosity weighted ages of the stellar popula-tion in the Magneticum galaxies are also described rea-sonably well by the lookback models. At low redshiftthere is a spread in ages, which can be explained by thespread in SFR, which we encounter in the Magneticumgalaxies.With the assumption of a redshift dependent rela-tionship between ISM gas mass and stellar mass thelookback models predict that the average metallicity ofa star forming galaxy depends on the ratio of stellarmass to ISM gas mass. Metal poor galaxies are gas richand metal rich galaxies are gas poor. Indeed, the Mag-neticum galaxies follow this trend and the mass metal-licity relationships at different redshifts of Magneticumand the lookback models are in good agreement. How-ever, this requires an adjustment of the effective yield inthe lookback models, for which the yield was originallycalibrated so that observed MZRs in the local Universeobtained from spectroscopy of young stars or the inte-8
Kudritzki et al. grated stellar populations agreed with the models. Acomparison of Magneticum galaxies with these observa-tions at low redshift shows a small offset in metallicitywhich is then covered by the adjustment of the effectiveyield.The observed SFR rate dependence of the MZRs,which is also present in the Magneticum galaxies, canbe reproduced by the lookback models in a very natu-ral way. Galaxies with higher star formation rates arethose with higher gas mass content, because they alsohave more star forming gas contributing to the star for-mation process. Therefore, their ratio of gas mass tostellar mass is higher and, as metallicity in the lookbackmodels depends on the ratio of stellar to gas mass, theirmetallicity is lower.The good agreement with the Magneticum simula-tions confirms that the lookback models provide a sim-ple straightforward way to understand key aspects of theevolution of galaxies, such as the average gas accretionhistory, the star formation history and the formation ofmetals. This gives these models a great potential as toolfor population synthesis calculations of synthetic spec-tra of the integrated stellar population of galaxies, whichcan then be used to constrain stellar metallicities fromobserved spectra. A typical example is given in the workby Zahid et al. (2017).The advantage of the lookback models is that theyare simple and easy to calculate. Their chemical evo- lution can be described by a simple analytical formula(see Appendix A) and a comparison with observationsis straightforward. However, important aspects havenot yet been covered in this first investigation. For in-stance, the chemical evolution describes only metallicityas a whole and does not distinguish between α - and irongroup elements. The inspection of the Magneticum re-sults shows that this approximation does not seem tolead to large errors, because the ratio of α over ironabundances does not change by more than 0.15 to 0.2dex as a function of stellar mass or redshift, but it is asystematic and expected trend and it would be a clearimprovement, if the models described this as well. Weplan to implement this in future work in a way that stillkeeps the simplicity of these models.We thank our referee for an engaged constructive crit-ical report. This work has been supported by the Mu-nich Excellence Cluster Origins funded by the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) under Germany’s Excellence Strategy EXC-2094390783311. The Magneticum Pathfinder simulationswere performed at the Leibniz-Rechenzentrum withCPU time assigned to the Project “pr86re”. We are es-pecially grateful for the support by M. Petkova throughthe Computational Center for Particle and Astrophysics(C2PAP).APPENDIX A. ANALYTICAL SOLUTION OF THE METALLICITY EQUATIONIntroducing the variable x = 1 − R − β (1 − K ) M ∗ M g (A1)eq. (13) turns into dZdx = Y N − R − Z (1 + µx ) (A2)with µ = β (1 − K )1 − β (1 − K ) . (A3)The factor (1 - K) is roughly constant and close to unity for a large part of the evolution of a galaxy in our lookbackmodel approach except at the lowest redshifts and the largest ratios of stellar to gas mass (log M ∗ /M g > Z ≈ Y N
11 + µ x (1 −
12 + µ x ) , for x (cid:28) alaxy lookback evolution models Figure 22.
Top: Same as Fig. 1 but with the analytical solution of eq. (A6) overplotted in orange. Bottom: Enlarged plot ofthe upper right corner of the upper figure. The extended analytical solution of eq. (A8) is added in red. Z ≈ Y N M ∗ M g (1 − − R − β (1 − K ) M ∗ M g ) , for x (cid:28) . (A5)Eq. (A4) and (A5) explain - at least for small values of x - why the metallicity in our lookback models during theevolution of a galaxy depends mostly on the ratio of stellar mass to gas mass M ∗ /M g .The analytical solution of eq. (A2) for µ = 1 is Z = Y N − R x ( x − e − x ) , for µ = 1 . (A6)In Fig. 22 we overplot this analytical solution (assuming 1 - K = 0.85) to the individual numerical solutions of Fig. 1.We see that it is a good match to the numerical solutions except for large M ∗ /M g , where it saturates and is ∼ ∗ /M g values (corresponding to low redshifts) the factor (1 -K) starts to change and becomes very small andfinally negative. This is caused by the increase of stellar mass and the fact the star formation rate ψ decreases towardslower redshifts (see eq. 12 for the definition of K). As a consequence, the term corresponding to dM g /dM ∗ on theright hand side of eq. (13) becomes negative, which leads to an increase of Z rather than to a saturation. The slightdivergence of the numerical solutions for different final stellar masses is the result of slightly different (1 - K) values ineach model.The behavior of K(M ∗ , z) at low redshift and large stellar masses also means that our analytical approximationusing the variable transformation of eq. (A2) with (1 - K) = 0.85 breaks down. This contributes to the discrepancyencountered in Fig. 22. However, if we want to recover an analytical solution, we can introduce a correction term,which changes the metallicity saturation behavior for large mass ratios log M ∗ /M g (cid:61) c Z ( x ) = 16 ( µ ( x ) x − µ ( x ) x ) , for x (cid:61) x (A7)with x the x-value at log M ∗ /M g = 0.3. We limit c Z (x) to a minimum value of -0.075 at the largest x-values. Withthis correction our new analytical solution is Z = Y N − R + c Z ( x ) 1 x ( x − e − x ) , for x (cid:61) x , (A8)which is shown in the enlarged plot on the right hand side of Fig. 22. In this plot we can again see how the individualnumerical solutions for different final stellar masses diverge for log M ∗ /M g (cid:39) Kudritzki et al. the behavior of (1 - K) at large stellar masses, which is slightly different for each numerical solution. However, sincethis is a small effect, our main conclusion that in our look back models Z is a function of mainly M ∗ /M g remains valid. B. FORMULAE FOR ψ LB , X F ITG
AND τ F IT
As described in section 5 we use the Pearson et al. (2018) SFR law of their Appendix C with several modifications.For the zero point we introduce the correction factor c(z) as described by eq. (17). In addition, we introduce a brokenpower law with respect to stellar mass ψ LB = ψ LB ( z )( M ∗ / . ) δ ( z ) , M ∗ ≥ . , (B9) ψ LB = 0 . δ ( z ) ψ LB ( z )( M ∗ / . ) . , M ∗ ≤ . . (B10) δ (z) and ψ ( z ) are given in Pearson et al. (2018) in their Appendix C.For redshifts z ≥ ≤ M ∗ ≤ . . For lower masses we use ψ LB = 0 . δ ( z ) . ψ LB ( z )( M ∗ / ) δ l , M ∗ ≤ (B11)with δ l ( z ) = 1 . − . − e ( z − . . ) ) . (B12)The introduction of δ l ( z ) leads to a transition of the power law exponent in this lower mass range from 1.1 to 0.5when redshift increases. We find this trend in the low mass galaxies of the Magneticum evolution sample.Our fit of star forming to total mass of the cold ISM gas, x fitg is calculated by x fitg = 1 − a g · e − z/ (B13)with a g ( m ) = 0 .
89 + 0 . · e m min − m ) , m = log M ∗ , m min = 9 . − . − e − z ) . (B14)The factor e − z/ shifts x fitg upwards with increasing redshift z, until it saturates at unity. The second term in a g leads to a drop at low log M ∗ and m min regulates at which mass the drop sets in as function of z.For z ≥ ∗ ≥ g with stellar mass. Weaccomplish this by introducing ∆z = z - 1.5 and using x fitg = (1 − a g (9 . · e − z/ )10 − p (∆ z )( m − . , z ≥ . , m ≥ . p (∆ z ) = 0 . z (1 + 1 . z − . z ) , a g (9 .
6) = 0 .
89 + 0 . · e m min − . . (B16)The Magneticum fit for the star formation time islog τ fit = a τ ( z ) − . · z ( m − . , a τ = a ( z ) − . · z (B17)with a = 9 .