A fundamental relation between mass, SFR and metallicity in local and high redshift galaxies
F. Mannucci, G. Cresci, R. Maiolino, A. Marconi, A. Gnerucci
MMon. Not. R. Astron. Soc. , ?? – ?? (2010) Printed 22 October 2018 (MN L A TEX style file v2.2)
A fundamental relation between mass, SFR and metallicityin local and high redshift galaxies
F. Mannucci (cid:63) , G. Cresci , , R. Maiolino , A. Marconi , A. Gnerucci INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125, Firenze, Italy Max-Planck-Institut f¨ur extraterrestrische Physik (MPE), Giessenbachstr.1, D-85748 Garching, Germany INAF - Osservatorio Astronomico di Roma, via di Frascati 33, I-00040 Monte Porzio Catone, Italy Dip. di Fisica e Astronomia, Universit`a di Firenze, Largo E. Fermi 2, I-50125, Firenze, Italy
Submitted 2010 March
ABSTRACT
We show that the mass-metallicity relation observed in the local universe is due to amore general relation between stellar mass M (cid:63) , gas-phase metallicity and star forma-tion rate (SFR). Local galaxies define a tight surface in this 3D space, the FundamentalMetallicity Relation (FMR), with a small residual dispersion of ∼ ∼ ∼ µ α = log(M (cid:63) ) − α log(SFR), with α =0.32, wedefine a projection of the FMR that minimizes the metallicity scatter of local galaxies.The same quantity also cancels out any redshift evolution up to z ∼ µ . and metallicity and have the same range ofvalues of µ . . At z > µ . . The small metallicity scatter around theFMR supports the smooth infall scenario of gas accretion in the local universe. Key words:
Galaxies: abundances; Galaxies: formation; Galaxies: high-redshift;Galaxies: starburst
Gas metallicity is regulated by a complex interplay betweenstar formation, infall of metal-poor gas and outflow of en-riched material. A relation between magnitude and metal-licity was discovered in the ’70 (McClure & van den Bergh1968; Lequeux et al. 1979), in which more luminous galaxiesalso have higher metallicities. Later on, it was understoodthat this luminosity-metallicity relation is a manifestation of (cid:63)
E-mail:fi[email protected] a more fundamental stellar mass-metallicity relation wheregalaxies with larger stellar mass M (cid:63) have higher metallicities(Garnett 2002; P´erez-Gonz´alez et al. 2003; Pilyugin et al.2004; Tremonti et al. 2004; Savaglio et al. 2005; Lee et al.2006; Cowie & Barger 2008; Panter et al. 2008; Kewley &Ellison 2008; Hayashi et al. 2009; Michel-Dansac et al. 2008;Liu et al. 2008; Rodrigues et al. 2008; P´erez-Montero et al.2009).The origin of this relation is debated, and many differ-ent explanations have been proposed, including ejection ofmetal-enriched gas (e.g., Edmunds 1990; Lehnert & Heck- c (cid:13) a r X i v : . [ a s t r o - ph . C O ] J u l F. Mannucci et al. man 1996; Garnett 2002; Tremonti et al. 2004; Kobayashiet al. 2007; Scannapieco et al. 2008; Spitoni et al. 2010),“downsizing”, i.e., a systematic dependence of the efficiencyof star formation with galaxy mass (e.g., Brooks et al. 2007;Mouchine et al. 2008; Calura et al. 2009), variation of theIMF with galaxy mass (K¨oppen et al. 2007), and infall ofmetal-poor gas (Finlator & Dav´e 2008; Dav´e et al. 2010).Recently, evidence has been found that at high redshiftthe infall of pristine gas can have a dominant role (Bour-naud & Elmegreen 2009; Dekel et al. 2009; Brooks et al.2009; Agertz et al. 2009). The mass-metallicity relation hasbeen studied by Erb et al. (2006) at z ∼ (cid:54) M (cid:63) . Recently, also Lopez-Sanchez (2010)presented evidence for a link between SFR and metallicity,while Peeples et al. (2009) reported high values of SFR ina sample of outliers, toward low metallicities, of the mass-metallicity relation.To test the hypothesis of a correlation between SFRand metallicity in the present universe and at high redshift,we have studied several samples of galaxies at different red-shifts whose metallicity, M (cid:63) , and SFR have been measured.In the next section we present the data samples we areusing for this study. In sec. 3 we study the mass-metallicityrelation as a function of SFR, and in the following sectionwe introduce the Fundamental Metallicity Relation. Insec. 7 we discuss the physical origin of this relation. Weadopt a ΛCDM cosmology with H =70 km/sec, Ω m =0.3and Ω Λ =0.7. Stellar mass M (cid:63) and SFR are expressed inM (cid:12) and in M (cid:12) /yr, respectively. λ (cid:48)(cid:48) aperture ofthe spectroscopic fiber samples a significant fraction of thegalaxies (3 (cid:48)(cid:48) correspond to ∼ α of SNR >
25 was used to have reliable values of metallicity,while no SNR threshold is used for the other lines. Such ahigh SNR on H α is needed to ensure that all the main opticallines are generally detected with enough SNR without intro-ducing metallicity biases. For example, the [NII] λ α flux at high metallicities, and about1/10 at the lowest metallicities sampled by SDSS. Noise andintrinsic dispersion of the [NII] λ α ratio could pro-duce very low fluxes of the [NII] λ λ α means that,on average, the faintest [NII] λ > A V < .
5, in order not todeal with very large extinction corrections, and galaxies withBalmer decrements below 2.5 were removed. These selectionsremove 0.2% of the galaxies. Finally, AGN-like galaxies (22%of the sample) were excluded by using the BPT classificationby Kauffmann et al. (2003a).Total stellar masses M (cid:63) from Kauffmann et al. (2003b)were used, as listed in the same MPA/JHU catalog, witha correction factor of 1.06 to scale the masses down froma Kroupa (2001) to a Chabrier (2003) initial mass function(IMF).SFRs inside the spectroscopic aperture were measuredfrom the H α emission line flux corrected for dust extinctionas estimated from the Balmer decrement. The conversionfactor between H α luminosity and SFR in Kennicutt (1998)was used, corrected to a Chabrier (2003) IMF.Oxygen gas-phase abundances were measured from theemission line ratios as described in Nagao et al. (2006) andMaiolino et al. (2008). Two independent measurements ofmetallicity are available for these galaxies, based either onthe [NII] λ α ratio or on the R23 quantity, definedas R23=([OII] λ λ β . When bothquantities are inside the useful range for metallicity cali-bration, i.e., log([NII] λ α ) < –0.35 and log(R23) < ∼ c (cid:13) , ?? ––
5, in order not todeal with very large extinction corrections, and galaxies withBalmer decrements below 2.5 were removed. These selectionsremove 0.2% of the galaxies. Finally, AGN-like galaxies (22%of the sample) were excluded by using the BPT classificationby Kauffmann et al. (2003a).Total stellar masses M (cid:63) from Kauffmann et al. (2003b)were used, as listed in the same MPA/JHU catalog, witha correction factor of 1.06 to scale the masses down froma Kroupa (2001) to a Chabrier (2003) initial mass function(IMF).SFRs inside the spectroscopic aperture were measuredfrom the H α emission line flux corrected for dust extinctionas estimated from the Balmer decrement. The conversionfactor between H α luminosity and SFR in Kennicutt (1998)was used, corrected to a Chabrier (2003) IMF.Oxygen gas-phase abundances were measured from theemission line ratios as described in Nagao et al. (2006) andMaiolino et al. (2008). Two independent measurements ofmetallicity are available for these galaxies, based either onthe [NII] λ α ratio or on the R23 quantity, definedas R23=([OII] λ λ β . When bothquantities are inside the useful range for metallicity cali-bration, i.e., log([NII] λ α ) < –0.35 and log(R23) < ∼ c (cid:13) , ?? –– ?? he Fundamental Metallicity Relation Figure 1.
Left panel:
The mass-metallicity relation of local SDSS galaxies. The grey-shaded areas contain 64% and 90% of all SDSSgalaxies, with the thick central line showing the median relation. The colored lines show the median metallicities, as a function of M (cid:63) ,of SDSS galaxies with different values of SFR. Right panel: median metallicity as a function of SFR for galaxies of different M (cid:63) . At all M (cid:63) with log( M (cid:63) ) < albeit small systematic difference of 0.05 dex ( ∼ λ α . This small difference is likely to bedue to the different sample used here and in Maiolino et al.(2008), which use a SNR threshold of 10 on the flux of eachline. This may introduce a small bias in the calibrations rel-ative to our sample.The final galaxy sample contains 141825 galaxies. Many galaxies has been observed at high redshift and thesedata can be used to study the evolution of metallicitywith respect to the other properties of galaxies. We ex-tracted from the literature three samples of galaxies atintermediate redshifts, for a total of 182 objects, havingpublished values of emission line fluxes, M (cid:63) , and dust ex-tinction: 0.5 < z < < z < < z < λ α , depending on which lines are avail-able. AGN are removed using the BPT diagram (Kauffmannet al. 2003a) or, when [OIII] λ β are not available,by imposing log([NII] λ α ) < –0.3. The [NII] λ α , is not detected inseveral galaxies, but removing these galaxies from the sam-ple would bias it towards high metallicities. For these ob-jects we have assumed a value of the intrinsic [NII] λ M (cid:63) have been converted to a Chabrier (2003) IMF. For galaxies without observations of both H α and H β , dust extinction is estimated from SED fitting, andwe assume that continuum and the emission lines suffer thesame extinction. In local starburst lines often suffer of higherextinctions ( A V (lines) ∼ A V (SED) according to Calzettiet al. 2000). We have checked that the inclusion of this ef-fect would have little effect on the final relations and on theconclusions of this paper.Erb et al. (2006) have observed a large sample of 91galaxies at z ∼ M (cid:63) , which has the re-sults of mixing galaxies of different SFRs. Despite this prob-lem, no systematic differences in metallicity are detectedwith respect to the other galaxies measured individually,and the Erb et al. (2006) galaxies are included in the high-redshift sample, although without binning them with therest of the galaxies. A significant sample of 16 galaxies at redshift between 3 and4 was observed by Maiolino et al. (2008) and Mannucci et al.(2009) for the LSD and AMAZE projects. Published valuesof stellar masses, line fluxes and metallicities are available forthese galaxies, which can be compared with lower redshiftdata. The same procedure as at lower redshift was used, withthe exception that SFR is estimated from H β after correc-tion for dust extinction, and metallicities are measured by asimultaneous fitting of the line ratios involving [OII] λ β and [OIII] λ c (cid:13) , ?? – ?? F. Mannucci et al.
Figure 2.
Three projections of the Fundamental Metallicity Relation among M (cid:63) , SFR and gas-phase metallicity. Circles without errorbars are the median values of metallicity of local SDSS galaxies in bin of M (cid:63) and SFR, color-coded with SFR as shown in the colorbaron the right. These galaxies define a tight surface in the 3D space, with dispersion of single galaxies around this surface of ∼ The grey-shaded area in the left panel of Fig. 1 shows themass-metallicity relation for our sample of SDSS galaxies.Despite the differences in the selection of the sample andin the measure of metallicity, our results are very similarto what has been found by Tremonti et al. (2004). The metallicity dispersion of our sample, ∼ ∼ log ( O/H ) = 8 .
96 + 0 . m − . m − . m + 0 . m (1) c (cid:13) , ?? – ?? he Fundamental Metallicity Relation where m =log( M (cid:63) )–10 in solar units.We have computed the median metallicity of SDSSgalaxies for different values of SFR. Median have been com-puted in bins of mass and SFR of 0.15 dex width in bothquantities. On average, each bin contains 760 galaxies, andonly bins containing more than 50 galaxies are considered.The left panel of Fig. 1 also shows these median metallicitiesas a function of M (cid:63) . It is evident that a systematic segrega-tion in SFR is present in the data. While galaxies with high M (cid:63) (log( M (cid:63) ) > M (cid:63) more active galaxies also show lowermetallicity. The same systematic dependence of metallicityon SFR can be seen in the right panel of Fig. 1, where metal-licity is plotted as a function of SFR for different values ofmass. Galaxies with high SFRs show a sharp dependenceof metallicity on SFR, while less active galaxies show a lesspronounced dependence.A hint of this effect was already noted by Ellison et al.(2008), but the different sample selection and the large binsin SFR reduced the observed dependence on SFR to a smallcorrection of ∼ The dependence of metallicity on M (cid:63) and SFR can bebetter visualized in a 3D space with these three coordinates,as shown in Figure 2. SDSS galaxies appear to define atight surface in the space, the Fundamental MetallicityRelation, with metallicity well defined by the values of M (cid:63) and SFR. All the data on this FMR are shown in table 1.The introduction of the FMR results in a significant re-duction of residual metallicity scatter with respect to thesimple mass-metallicity relation. The dispersion of individ-ual SDSS galaxies around the FMR, shown in Fig. 3, com-puted in bins of 0.05 dex in M (cid:63) and SFR, is ∼ ∼ ± σ ) is contained inside 0.8 dex.Galaxies with very low SFRs are not selected due to thehigh SNR threshold on H α , while high SFR galaxies arerare in the local universe. As a consequence the scatter onthe full sample is dominated by galaxies having a small cor-rection due to SFR. In contrast, considering only galaxieswith high SFRs, the scatter is reduced by a large factor. Forexample, for log(SFR) > ∼ ∼ Figure 3.
Metallicity dispersion of single SDSS galaxies aroundthe FMR. This histogram shows the differences from the mediancomputed in bins of 0.05 dex in M (cid:63) and SFR. The red line is agaussian with σ =0.053 dex. Positive differences mean metallicitiesof single galaxies larger than the median. calibration, to be added to the uncertainties in the line ra-tios), on mass (estimated to be 0.09 dex by Tremonti et al.2004), and on the SFR, which are dominated by the uncer-tainties on dust extinction. Nevertheless, the scatter aroundFMR tends to reduce when the minimum redshift z min ofthe galaxy sample is increased. This means that part of theresidual scatter is probably due to the different aperturesused to measure mass, based on a total magnitude, andmetallicity and SFR, derived for the central 3 (cid:48)(cid:48) . In partic-ular, the effect of metallicity gradients are expected to be-come less important at larger redshifts, and for this reasonthe increase of z min is able to reduce the scatter.A close inspection of fig. 3 reveals the presence of anextended wing toward lower metallicities in the distributionof scatter. This extension contains ∼
3% of the objects.Most of these galaxies have low M (cid:63) and high SSFR, andcould be objects in special conditions. For example, theycould be interacting galaxies, which will be discussed insec.7.We have fit the median values of metallicity of the SDSSgalaxies in table 1 with a second-order polynomial in M (cid:63) andSFR, obtaining:12 + log ( O/H ) = 8 .
90 + 0 . m − . s − . m +0 . ms − . s (2)where m =log( M (cid:63) )–10 and s =log(SFR) in solar units. Theresidual scatter of median metallicities around this fit is0.001 dex. Such a fit, shown in fig. 2, provides a clearrepresentation of the dependence of metallicity both on M (cid:63) and SFR, and allows to compare the local FMR with highredshift galaxies.The shape of the FMR surface depends on a numberof factors, such as the selection of the galaxy sample and c (cid:13) , ?? – ?? F. Mannucci et al.
Table 1.
The Metallicity Fundamental Relation for SDSS galaxies selected as in sec. 2.1. For each value of M (cid:63) (reported in the firstline) and SFR (first column) we list the median value of metallicity, the 1 σ dispersion around this value, and the number of galaxies ineach bin. log(SFR) log( M (cid:63) )9.10 9.25 9.40 9.55 9.70 9.85 10.00 10.15 10.30 10.45 10.60 10.75 10.90 11.05 11.20 11.35 − − − − − − − − − − the way metallicity, M (cid:63) and SFR are measured. We havedone a number of checks to test whether the result dependscritically on either of our assumptions. First, we changedthe thresholds in SNR and redshift used to select thegalaxy sample, and checked that the results do not changesystematically with these thresholds. Second, we havestudied the effect of considering only metallicities derived either from R23 or from [NII] λ α . As discussed insec. 2.1, systematic differences are found but are limited tothe level of 0.05 dex. Apart from this metallicity offset, theshape of the FMR does not change by more than 0.05 dexat any point. In particular, there is no large, monotonicdependence of the difference with SFR or SSFR. This isinteresting because there could be systematic effects related c (cid:13) , ?? – ?? he Fundamental Metallicity Relation either to the density of the HII regions or to the ionizationparameter U, which could depend on the SSFR (e.g.,Shapley et al. 2005; Hainline et al. 2009) and introducespurious behavior. R23 and [NII] λ α show oppositedependencies with U (see, for example, Liu et al. 2008 andNagao et al. 2006). For this reason the absence of systematicdifferences between these two line ratios with SSFR is anindication that the dependence of measured metallicity ondensity or U, if present, is small, in agreement with thefindings of Liu et al. (2008) and Brinchmann et al. (2008).There are two points that could affect the shape of theFMR. First, SFR is estimated from H α luminosity correctedfor extinction using the Balmer decrement. Several authors(Kennicutt 1998; Moustakas et al. 2006) have shown thatthis is a reliable SFR indicators over a large range of galaxyproperties. Others (Charlot & Longhetti 2001; Brinchmannet al. 2004) have discussed that systematic effects with massand metallicity could be present. As we have split galaxies inbins of mass and SFR, and inside each bin metallicity spansa small range, any systematic effect on SFR does not hamperthe existence of the FMR but could change its shape. Sec-ond, we are using SFRs and metallicities that apply only tothe central 3 (cid:48)(cid:48) of the galaxies, corresponding to 4–11 kpc pro-jected angular size given our redshift range. These quantitiesare compared with total mass derived from integrated pho-tometry (Kauffmann et al. 2003b; Salim et al. 2007). In theSDSS sample, the fraction of mass contained inside the pro-jected fiber aperture (as listed in the MPE/JHU catalog) isabout 1/3 of the total, with log(total mass) – log(fiber mass)= 0.50 ± α scales radially as luminosityor mass. If SFR scales with radius as mass, we expected thattotal SFR are 3 times larger than the fiber ones consideredhere. The use of total SFR would produce a FMR shifted to-wards higher SFRs, and its shape could have some changes.Nevertheless the small scatter observed in the FMR meansthat the fiber SFR must be well correlated with the totalone. Summarizing, even if the overall shape of the FMR canchange in different samples of galaxies and depends on sev-eral details, the main properties of the FMR are very robustand passed all our tests. Using the samples described in Sect. 2.2 and 2.3 we comparethe dependence of metallicity on M (cid:63) and SFR in galaxies atz ∼ ∼ ∼ ∼ ∼ M (cid:63) objects. Median val-ues of M (cid:63) , SFR and metallicities are computed for each ofthese samples.Galaxies at intermediate and high redshifts show, on av-erage, larger SFR with respect to local SDSS galaxies. Thisis easily explained by selection, because only galaxies withsignificant SFRs are selected and observed spectroscopically.Galaxies at z ∼ M (cid:63) and SFR which overlapwith the SDSS sample, and therefore the two samples canbe directly compared. These galaxies are found to be com-pletely consistent with the FMR defined by SDSS galaxies,with no evidence for evolution. This is shown in Figs. 2 and4. At redshift above 1, some extrapolation towards higherSFRs of the fit in eq. 2 in required. All galaxies are within0.6 dex from the most active SDSS galaxies, while most mas-sive galaxies at z=2.2 require an extrapolation of 1 dex. Forcomparison, SDSS galaxies span two orders-of-magnitude inSFR (see Fig.4). The necessity of an extrapolation intro-duces some uncertainty, but we have checked the the resultdoes not depend critically on the characteristic of the fit,such as the degree of the used polynomial.When taking into account the uncertainties, data up toz ∼ M (cid:63) ( ∼ ∼ c (cid:13) , ?? – ?? F. Mannucci et al.
Figure 4.
Left:
Metallicity as a function of SFR for galaxies in the three bins of M (cid:63) containing high-redshift galaxies. The values oflog( M (cid:63) ) are shown by the labels on the left. Empty square dots are the median values of metallicity of local SDSS galaxies, with errorbars showing 1 σ dispersions. Lines are the fits to these data. Solid dots are median values for high-redshift galaxies with z < Right: metallicity difference from the FMR for galaxies at different redshifts, color-codedin mass as in the left panel. The SDSS galaxies defining the relation are showing at z ∼ ∼ ∼ enrichment of SDSS galaxies. This is related to the fact thatgalaxies with high SFRs, the objects showing the strongestdependence of metallicity on SFR (see the right panel offig. 1), are quite rare in the local universe. At high redshifts,mainly active galaxies are selected and the dependence ofmetallicity on SFR becomes dominant. > Galaxies at z ∼ < z < > λ α . As discussed in sec. 2.3, in the local universeboth indicators give consistent results, with systematic dif-ferences limited to 0.05 dex. Also, galaxies at z ∼ ∼ α and [NII] λ ∼ µ m and are not observed in any galaxy of this group.For this reason we cannot use this line ratio to remove AGNs.X-ray and mid-IR data on these targets have been analyzedin order to exclude dominant AGN (Maiolino et al. 2008),but it is possible that some faint AGN is still present amongthese galaxies (Wright et al. 2010). The presence of suchobjects would tend to reduce the measured metallicity. Asalmost all galaxies at z ∼ ∼ ∼ λ λ λ α observed at z <
2, and because the cosmo-logical dimming of the surface brightness, proportional to(1 + z ) , is a much more severe problem at higher redshift.In our data at z=3.3, metallicity does not seem to increase c (cid:13) , ?? ––
2, and because the cosmo-logical dimming of the surface brightness, proportional to(1 + z ) , is a much more severe problem at higher redshift.In our data at z=3.3, metallicity does not seem to increase c (cid:13) , ?? –– ?? he Fundamental Metallicity Relation Figure 5.
Left:
Residual dispersion of the median values of metallicity of SDSS galaxies as a function of α as defined in eq. 3. The valuescorresponding to the minimum dispersion ( α =0.32), to α =0 ( µ α =log( M (cid:63) )) and and α =1 ( µ α =–log(SSFR)) are shown. Right: metallicityas a function of µ . , which minimize the residual scatter. Colored lines are local SDSS galaxies, with colors as in the left panel of Fig. 1.The black line shows the polynomial best fit. Black dots are high-redshift galaxies, labelled with redshifts. These galaxies, except thoseat z ∼ with increasing photometric aperture, although we are lim-ited by the low SNR in the external regions of the galaxies.This effect could be present but is not likely to produce theobserved difference of a factor of 3.Finally, there could be selection effects resulting in areduction of the average metallicity of observed sample.For example, if more metal-rich galaxies also have largeramounts of dust, it is possible that our UV-selected galaxiesat z ∼ At a given mass, galaxies with higher SFR have lower metal-licities and, therefore, have the metallicity properties oflower mass galaxies. As a consequence, we expect that acombination of M (cid:63) and SFR could be better correlated withmetallicity. This can be seen in the central panel of fig. 2,showing a projection of the FMR which considerably re-duces the metallicity scatter. To investigate this point weintroduce a new quantity µ α obtained as linear combinationof SFR and M (cid:63) as: µ α = log(M ∗ ) − α log(SFR) (3)where α is a free parameter. For α =0, µ corresponds tolog( M (cid:63) ), while for α =1, µ =–log(SSFR).The value of α that minimizes the scatter of medianmetallicities of SDSS galaxies around the relation corre-sponds to the quantity µ α that is more directly correlatedwith metallicity. Fig. 5 shows the scatter of data in table 1 asa function of α . These results show that neither M (cid:63) ( α =0),nor, SSFR ( α =1) are the quantities producing the smallestscatter. In fact, α ∼ .
32 produces a minimum in the disper-sion. The resulting diagram is shown in the right panel ofFig. 5, where metallicity is plotted against µ . . The medianof the resulting distribution cab be fitted by:12+ log ( O/H ) = 8 . . x − . x − . x +0 . x (4)where x = µ . − µ . and metallicity as inthe local universe, and also have the same range of values of µ . . This is the same effect noted in the previous section,where high-redshift galaxies have been found to follow theextrapolation of the FMR, but with two important changes:first, no extrapolation from the SDSS galaxies is now needed,because both samples have similar values of µ . ; second,it is possible to search for simple physical interpretation of µ . in terms of the physical processes in place.In practice, metallicity of star-forming galaxies of anymass, any SFR and at any redshift up to z=2.5 follow thefollowing relation:12 + log ( O/H ) = 8 .
90 + 0 . x if µ . < . .
07 if µ . > . c (cid:13) , ?? – ?? F. Mannucci et al. with x = µ . − It is interesting to plot metallicity as a function of SSFR,as in fig. 6, because several properties of galaxies depend onthis quantity. In this plot it can be seen that SDSS galax-ies of any mass have the same dependence of metallicityon SSFR. A threshold SSFR exists, about 10 − yr − whichdiscriminates the abundance effect of the SFR. Above thislimit, metallicity decreases rapidly with SFR in galaxies onany mass. Below this limit, metallicity is constant in massivegalaxies (log( M (cid:63) ) > M (cid:63) . Most of the low-mass galax-ies, and only a small fraction of high-mass galaxies in ourSDSS sample are in the “high SSFR” regime, and this is thewell-known ”downsizing” effect (Gavazzi & Scodeggio 1996;Cowie et al. 1996). As M (cid:63) and SFR are independent vari-ables of the FMR, downsizing does not shape the relationbut defines how it is populated, i.e, how many galaxies of agiven M (cid:63) have a certain level of SFR and, as a consequence,metallicity. In sec. 4 we have shown that in the local universe a tightrelation exists between metallicity, stellar mass, and SFR,in which metallicity increases with M (cid:63) and decreases withSFR in a systematic way. In sec. 5 we have shown that thesame relation, without any evolution, holds up to z=2.5, andthat the observed evolution of the mass-metallicity relationis simply due to the sampling of different parts of this re-lation at different redshifts. In sec. 6 we have seen that thesystematic dependence of metallicity on M (cid:63) and SFR at allredshifts can be expressed in an easy form by introducingthe quantity µ . , linear combination of M (cid:63) and SFR.The interpretation of these results must take into ac-count several effects. In principle, metallicity is a simplequantity as it is dominated by three processes: star forma-tion, infall, outflow. If the scaling laws of each of these threeprocesses are known, the dependence of metallicity on SFRand M (cid:63) can be predicted. In practice, these three processeshave a very complex dependence of the properties of thegalaxies, and can introduce scaling relations in many differ-ent ways.First, it is not known how outflows depend on the prop-erties of the galaxies. In many models, star formation pro-duces SNe which inject energy, radiation and momentuminto the interstellar medium, with the result of ejecting partof the enriched gas (Veilleux et al. 2005; Spitoni et al. 2010).A central Active Galactic Nucleus (AGN) can also providefeedback (Somerville et al. 2008). The properties of thisgalactic winds are debated. Dekel & Woo (2003) reproducedthe mass-metallicity relation with a wind related to the en-ergy of SNe, which is proportional to stellar mass. In largegalaxies with deep potential wells, such a wind is not effec-tive in producing an outflow (Tremonti et al. 2004). Murrayet al. (2005), Dav´e et al. (2007) and Oppenheimer & Dav´e Figure 6.
Metallicity of SDSS galaxies as a function of SSFR.The grey-shaded areas contain 64% and 90% of all SDSS galaxies,with the thick central line showing the median relation. The col-ored lines show the median metallicities for different values of M (cid:63) .The lines with black dots show fits to these metallicity distribu-tion for four values of log( M (cid:63) ), 9.4, 9.7, 10, and 10.9. The modelis described in sec. 7.1 and includes only dilution for infalling gasand the SK law. (2008) discuss a different scheme, the ”momentum-drivenwind”, in which wind speed increases with galaxy mass whileits efficiency decreases.Second, infalls are expected to influence metallicity intwo ways. On the one hand, infall of metal poor gas directlyreduces the observed metallicity by diluting the metal-richgas, as discussed, for example, by Finlator & Dav´e (2008)in the context of their wind scheme. On the other hand,infall is expected to produce star-formation activity follow-ing the Schmidt-Kennicutt law, and the metals produced bynew stars are expected to increase metallicity. If mergingbetween galaxies, rather than smooth infall from the IGM,is the main channel to drive gas into galaxies, the fuel ofstar formation can be significantly enriched, and the dilu-tion effect could be absent.Third, in some semianalytical models of galaxy forma-tion (de Rossi et al. 2007; Mouchine et al. 2008) the be-havior of metallicity is dominated by the dependence of star-formation efficiency of galaxy mass: less massive galax-ies are less evolved and, therefore, show lower metallicities.Some other models (Brooks et al. 2007; Dayal et al. 2009;Salvaterra et al. 2009) put together downsizing and feed-back: metallicity is mainly related to different star formationefficiencies in different galaxies, but the efficiency is regu-lated by SN feedback. Tassis et al. (2008) agree that lowstar-formation efficiency in low-mass galaxies is the maindriver of the mass-metallicity relation, but also pointed outthat mixing of heavy elements in the outer regions of galax-ies could help hiding a significant fraction of metals. AlsoK¨oppen et al. (2007) attribute the different abundances to c (cid:13) , ?? – ?? he Fundamental Metallicity Relation different levels of productions of metals, but their model in-clude systematic variation of IMF, which is proposed to bemore top-heavy in galaxies with higher SFRs.A full exploitation of our results requires a full discus-sion of these models, that is beyond the scope of this paperand will be subject of a future work. Here we discuss somesimple implications of our results on the role and the prop-erties of infalls and outflows. In the next two subsections wewill assume two limiting cases for the timescale of chemicalenrichment as compared to the other relevant timescales. The dependence of metallicity on SFR can be explained bythe dilution effect of the infalling gas. In a very simple model,we assume that galaxies with low SFR have a given gas frac-tion f g . When a certain amount M inf of metal-poor gas isaccreted, galaxies start forming stars at a given SFR de-fined by the Schmidt-Kennicutt (SK) law on the infallinggas. This is an empirical relation between surface densitiesof star formation and surface density of gas:Σ SFR ∼ Σ ngas (6)where n has values around 1.4–1.5 both at low and highredshifts (Kennicutt 1998; Bouch´e et al. 2007; Kennicutt2008; Gnedin & Kravtsov 2010; Verley et al. 2010). For SDSSgalaxies, the spectroscopic aperture is the always same andthe SK law becomes a relation between masses, obtaining:SFR ∼ M ninf (7)In this model, galaxies are observed during the phase whenthe dilution effect of the infall is predominant over the metal-licity enrichment due to new stars, i.e., the observed metal-licity [12+log(O/H)] obs is related to the initial metallicity[12+log(O/H)] in by the ratio of pre-existing and infallinggas:[12+log(O / H)] obs = [12+log(O / H)] in − log (cid:18) inf f g M (cid:63) (cid:19) (8)The results of this simple model are presented in fig. 6. Forsimplicity, only 4 values of mass are shown, using the best-fitting values of f g =0.3% and M inf between 10 . and 10 . M (cid:12) . This range of values of M inf is the same for all massesand is determined by the range of observed SSFR. For eachmass, the metallicity level at low SSFR, [12+log(O/H] in , is afree parameter, whose value is possibly determined by othereffects, such as outflow. Our very simple model is capable ofreproducing all the main properties of the FMR: metallicityreduces with increasing SFR, a threshold of SSFR exists,larger metallicity effects are produced in smaller galaxies,both above and below the threshold, the slope of the relationat high SSFR is exactly as observed.For this scenario to work, the timescales of chemicalenrichment must be longer than the dynamical scales of thegalaxies, over which the SFR is expected to evolve. In otherwords, galaxies on the FMR are in a transient phase : afteran infall, galaxies first evolve towards higher SFR and lowermetallicities. Later, while gas is converted into stars andnew metals are produced, either galaxies drop out of thesample because they have faint H α , or evolve toward highervalues of µ . and higher metallicities along the FMR. A detail modelling of this evolution and a comparison of thedifferent timescales involved is needed to test if this is aviable explanation.In this scenario, the dependence of metallicity on SFR isdue to infall and dominates at high redshifts, where galaxieswith massive infalls and large SFRs are found. In contrast, inthe local universe such galaxies are rare, most of the galaxieshave low level of accretion, and abundances are dominatedby the dependence on mass, possibly due to outflow. In many local galaxies, timescales of chemical enrichmentcan be shorter than the other relevant timescales (e.g., Silk1993), and galaxies can be in a quasi steady-state situation ,in which gas infall, star formation and metal ejection occursimultaneously (Bouche et al. 2009). This is the oppositesituation of what is discussed in the previous section.Assuming this quasi steady-state situation, in whichinfall and SFR are slowly evolving with respect to thetimescale of chemical enrichment, our results can be usedto derive information on the mechanisms of infall and out-flow. Fig. 6 implies that a process exists that depends onlyon SSFR which is effective in reducing metallicity from alevel that depends on mass. In a steady-state situation, in-fall cannot be the only dominant effect. The exponent n ofthe SK relation is larger than 1, and this means that theefficiency of star formation increases with gas density, i.e.,more active galaxies should also be more efficient in con-verting metal-poor gas into stars. As a consequence, the SKlaw alone, applied to the infalling gas, would predict theopposite to what is actually observed, i.e., metallicities in-creasing with SFR at constant mass. Some other effect mustbe present.One obvious candidate is the presence of outflows, asusually observed in starburst galaxies and discussed bymany authors. As discussed above, there are several pos-sible types of galactic winds, which follow different scalingrelations with mass and SFR. When the dilution by the in-falling gas M inf is considered together with enrichment dueto the SFR, we can reproduce the dependence of metal-licity on µ . by introducing an outflow proportional toSFR s M (cid:63) − m , with s and m free parameters. In this case,at the first order we obtain:12 + log ( O/H ) ∼ log (cid:18) SFRM inf
SFR s M − m (cid:63) (cid:19) (9)and, using the SK law:12 + log ( O/H ) ∼ m log(M (cid:63) ) + (1 − − s) log(SFR) (10)where n = 1 . µ α , and comparing this equation withthe best-fitting value of α = 0 .
32 we obtain m = 1 and s = 0 .
65. In other words, in a steady-state situation theoutflow must to inversely proportional to mass and mustincrease with SFR . .We note that, using an index n of the SK relation of 1.5,the best-fitting value of α =0.32 corresponds almost exactly c (cid:13) , ?? – ?? F. Mannucci et al. to 1 − /n = 0 .
33. Using this, the simplest way to combinethe relevant physical parameters to produce µ . is:log (cid:16) M inf M (cid:63) SFR (cid:17) ∼ log(M (cid:63) ) − (1 − / n)log(SFR) ∼ µ . (11)In this form, metallicity increases with M (cid:63) , an effect easilyattributable either to downsizing or to outflow. In contrast,the dependence of metallicity on M inf /SF R is not obviousand require complete modelling. The small scatter of SDSS galaxies around the FMR derivedin sec. 4 can be used to constrain the characteristics of gasaccretion. For this infall/outflow scenario to work and pro-duce a very small scatter round the FMR, two conditionsare simultaneously required: (1) star formation is alwaysassociated to the same level of metallicity dilution due toinfall of metal-poor gas; (2) there is a relation between theamount of infalling and outflowing gas and the level of starformation. These conditions for the existence of the FMRfits into the smooth accretion models proposed by severalgroups (Bournaud & Elmegreen 2009; Dekel et al. 2009),where continuos infall of pristine gas is the main driver ofthe grow of galaxies. In this case, metal-poor gas is contin-uously accreted by galaxies and converted in stars, and along-lasting equilibrium between gas accretion, star forma-tion, and metal ejection is expected to established.In contrast, larger scatter around the FMR are expectedin case of merging, for two reasons. First, in interacting sys-tems, part of the gas producing the starburst could be metal-poor material due to interaction-induced infall (Rupke et al.2008, 2010), but part is expected to be metal-rich materialalready present inside the interacting galaxies. In this casethe initial dilution is not present, and higher metallicities areexpected. This is true, in particular, for the smaller mem-ber of a merger between galaxies with very different masses.The secondary, smaller mass, galaxy is expected to showhigher metallicity because its star formation activity is ex-pected to be fuelled by gas coming from the other, larger,more metal-rich, galaxy. This is exactly what is observed byMichel-Dansac et al. (2008): higher metallicities, above themass-metallicity relation, are present in the secondary mem-ber of minor mergers. Second, the level of SFR is related tothe properties of the merging galaxies, and is expected tovary significantly during the different stages of the interac-tions. As a consequence, a large range on SFR is expectedfor a given level of metallicity during the merging historyof the systems. Both effects are expected to produce largerspreads in merging galaxies, and this is what is actually ob-served (Kewley et al. 2006; Rupke et al. 2008; Michel-Dansacet al. 2008; Peeples et al. 2009). As discussed in sec 4, thepresence of interacting galaxies in the SDSS could be at theorigin of the extended wing in the distribution of differencewith the FMR, and this point will be investigated in a futurepaper.The situation at intermediate redshift, up to z=2.5, isless clear. Galaxies show larger dispersion in metallicity, butthis can be explained by larger uncertainties in the measuredvalues of metallicity, mass and SFR. With the present datasample it is not possible, therefore, to study whether smoothaccretion is dominant up to z=2.5 as in the local universe or if merging has a larger impact on dispersion. Nevertheless,the absence of any evolution of the FMR up to these redshiftsupport a single physical process of accretion in all thesegalaxies. At even higher redshift, z ∼ and different average metallicities, it is likelythat new physical effects become important.The uncertainties on SFR can be a critical point. Underthis respect, the Herschel satellite is expected to improvesignificantly the accuracy of the estimate of total SFRs ina short time. With its contribution it will be possible toreduce the scatter, especially at high redshifts, study thepresence of an intrinsic dispersion of properties among thegalaxies, and obtain a much better characterization of theFMR relation at high redshifts. > The interplay between gas accretion, star formation, and gasoutflow seems to be the same at any redshift up to z=2.5, asall these galaxies follow the same FMR. What is importantin this context is the ratio between the different rates andthe various timescale involved: gas infall, dynamical times,star formation, stellar evolution, supernova explosion, chem-ical mixing, outflow. Apparently the relative importance ofthese processes does not evolve up to z=2.5. It is possiblethat at higher redshifts this constant balance does not applyany more and lower metallicities are observed while galaxiesevolve towards the FMR. This could be related to an increas-ing importance of merging as a way to drive cold gas intothe galaxies, at least in the most luminous objects that arepreferentially selected as LBG. This point can be addressedby studying the morphology and the mass-SFR relation inthese objects.It should be noted that the metallicity of galaxies atz=3.3 can also be reproduced by the same model in sec. 7.1,in which the infall dilution is dominant. Local galaxies arereproduced by M inf of the order of 10 − M (cid:12) , while gasmasses of the order of 10 − M (cid:12) are needed for galaxiesat z=3.3. This is in very good agreement with what obtainedby Mannucci et al. (2009) and Cresci (2010). Also, galaxiesat these high redshifts could be preferentially detected dur-ing the first stages of the starburst, when the dilution effectis maximum. This is possible if the starburst has a peakon short timescales, shorter than the timescales of metalproduction and chemical mixing, and declines afterwards.Galaxies could also become more dust-rich during the laterphases of the starburst, and drop-out from the UV-selectedsamples. We have studied the dependence of gas-phase metallicity12+log(O/H) on stellar mass M (cid:63) and SFR on a few sam-ples of galaxies from z=0 to z=3.3. In the local universe,we find that metallicity is tightly related to both M (cid:63) andSFR (fig. 1), and this defines the Fundamental Metallic-ity Relation (fig. 2). The residual metallicity dispersion oflocal SDSS galaxies around this FMR is about 0.05 dex(fig. 3), i.e, about 12%. The well-known mass-metallicity re-lation, together with the luminosity-metallicity and velocity-metallicity relations, is one particular projection of this rela- c (cid:13) , ?? ––
33. Using this, the simplest way to combinethe relevant physical parameters to produce µ . is:log (cid:16) M inf M (cid:63) SFR (cid:17) ∼ log(M (cid:63) ) − (1 − / n)log(SFR) ∼ µ . (11)In this form, metallicity increases with M (cid:63) , an effect easilyattributable either to downsizing or to outflow. In contrast,the dependence of metallicity on M inf /SF R is not obviousand require complete modelling. The small scatter of SDSS galaxies around the FMR derivedin sec. 4 can be used to constrain the characteristics of gasaccretion. For this infall/outflow scenario to work and pro-duce a very small scatter round the FMR, two conditionsare simultaneously required: (1) star formation is alwaysassociated to the same level of metallicity dilution due toinfall of metal-poor gas; (2) there is a relation between theamount of infalling and outflowing gas and the level of starformation. These conditions for the existence of the FMRfits into the smooth accretion models proposed by severalgroups (Bournaud & Elmegreen 2009; Dekel et al. 2009),where continuos infall of pristine gas is the main driver ofthe grow of galaxies. In this case, metal-poor gas is contin-uously accreted by galaxies and converted in stars, and along-lasting equilibrium between gas accretion, star forma-tion, and metal ejection is expected to established.In contrast, larger scatter around the FMR are expectedin case of merging, for two reasons. First, in interacting sys-tems, part of the gas producing the starburst could be metal-poor material due to interaction-induced infall (Rupke et al.2008, 2010), but part is expected to be metal-rich materialalready present inside the interacting galaxies. In this casethe initial dilution is not present, and higher metallicities areexpected. This is true, in particular, for the smaller mem-ber of a merger between galaxies with very different masses.The secondary, smaller mass, galaxy is expected to showhigher metallicity because its star formation activity is ex-pected to be fuelled by gas coming from the other, larger,more metal-rich, galaxy. This is exactly what is observed byMichel-Dansac et al. (2008): higher metallicities, above themass-metallicity relation, are present in the secondary mem-ber of minor mergers. Second, the level of SFR is related tothe properties of the merging galaxies, and is expected tovary significantly during the different stages of the interac-tions. As a consequence, a large range on SFR is expectedfor a given level of metallicity during the merging historyof the systems. Both effects are expected to produce largerspreads in merging galaxies, and this is what is actually ob-served (Kewley et al. 2006; Rupke et al. 2008; Michel-Dansacet al. 2008; Peeples et al. 2009). As discussed in sec 4, thepresence of interacting galaxies in the SDSS could be at theorigin of the extended wing in the distribution of differencewith the FMR, and this point will be investigated in a futurepaper.The situation at intermediate redshift, up to z=2.5, isless clear. Galaxies show larger dispersion in metallicity, butthis can be explained by larger uncertainties in the measuredvalues of metallicity, mass and SFR. With the present datasample it is not possible, therefore, to study whether smoothaccretion is dominant up to z=2.5 as in the local universe or if merging has a larger impact on dispersion. Nevertheless,the absence of any evolution of the FMR up to these redshiftsupport a single physical process of accretion in all thesegalaxies. At even higher redshift, z ∼ and different average metallicities, it is likelythat new physical effects become important.The uncertainties on SFR can be a critical point. Underthis respect, the Herschel satellite is expected to improvesignificantly the accuracy of the estimate of total SFRs ina short time. With its contribution it will be possible toreduce the scatter, especially at high redshifts, study thepresence of an intrinsic dispersion of properties among thegalaxies, and obtain a much better characterization of theFMR relation at high redshifts. > The interplay between gas accretion, star formation, and gasoutflow seems to be the same at any redshift up to z=2.5, asall these galaxies follow the same FMR. What is importantin this context is the ratio between the different rates andthe various timescale involved: gas infall, dynamical times,star formation, stellar evolution, supernova explosion, chem-ical mixing, outflow. Apparently the relative importance ofthese processes does not evolve up to z=2.5. It is possiblethat at higher redshifts this constant balance does not applyany more and lower metallicities are observed while galaxiesevolve towards the FMR. This could be related to an increas-ing importance of merging as a way to drive cold gas intothe galaxies, at least in the most luminous objects that arepreferentially selected as LBG. This point can be addressedby studying the morphology and the mass-SFR relation inthese objects.It should be noted that the metallicity of galaxies atz=3.3 can also be reproduced by the same model in sec. 7.1,in which the infall dilution is dominant. Local galaxies arereproduced by M inf of the order of 10 − M (cid:12) , while gasmasses of the order of 10 − M (cid:12) are needed for galaxiesat z=3.3. This is in very good agreement with what obtainedby Mannucci et al. (2009) and Cresci (2010). Also, galaxiesat these high redshifts could be preferentially detected dur-ing the first stages of the starburst, when the dilution effectis maximum. This is possible if the starburst has a peakon short timescales, shorter than the timescales of metalproduction and chemical mixing, and declines afterwards.Galaxies could also become more dust-rich during the laterphases of the starburst, and drop-out from the UV-selectedsamples. We have studied the dependence of gas-phase metallicity12+log(O/H) on stellar mass M (cid:63) and SFR on a few sam-ples of galaxies from z=0 to z=3.3. In the local universe,we find that metallicity is tightly related to both M (cid:63) andSFR (fig. 1), and this defines the Fundamental Metallic-ity Relation (fig. 2). The residual metallicity dispersion oflocal SDSS galaxies around this FMR is about 0.05 dex(fig. 3), i.e, about 12%. The well-known mass-metallicity re-lation, together with the luminosity-metallicity and velocity-metallicity relations, is one particular projection of this rela- c (cid:13) , ?? –– ?? he Fundamental Metallicity Relation tion into one plane, and neglecting the dependence of metal-licity on SFR results in doubling the observed dispersion.When high redshift galaxies are compared to the FMRdefined locally, we find no evolution up to z=2.5, i.e., high-redshift galaxies follow the same FMR defined by SDSSgalaxies even if they have higher SFRs (fig 4). This meansthat the same physical processes are in place in the local uni-verse and at high redshifts. The observed evolution of themass-metallicity relation is due to the increase of the aver-age SFR with redshift, which results in sampling differentparts of the same FMR at different redshifts.At even higher redshift, z ∼ ∼ M (cid:63) ,SFR and 12+log(O/H), metallicity is found to be tightlycorrelated with µ . = log(M (cid:63) ) − .
32 log(SFR). Galaxiesat any redshifts up to z=2.5 follow the same µ . -metallicityrelation and have the same range of values of µ . (fig.5).Metallicity in galaxies of any mass is found to have thesame dependence of SSFR, with galaxies above the thresholdof SSFR=10 − yr − showing a rapidly decreasing metallic-ities with increasing SSFR (fig. 6).The interpretation of the existence of the FMR, itsdependence of SSFR, and the role of µ . depend onthe relevant timescales. If dynamical times are shorterthan timescales for chemical enrichment, the dependence ofmetallicity on SFR can be easily explained by dilution byinfall. In this case this effect dominates the metallicity evolu-tion of galaxies at high redshift, when galaxies grow becauseof massive accretions of metal poor gas and produce largeSFR. Also galaxies at z=3.3 can fit into this scheme, withlarge masses of infalling gas. This is in agreement with otherrecent independent results (Mannucci et al. 2009; Cresci2010). In the local universe, galaxies with large SFR arerare and often associated to merging events, and other ef-fects becomes dominant which relate metallicity mainly to M (cid:63) . Outflow is an possibly, although downsizing could alsowork. If, in contrast, infall and SFR evolve on timescalesmuch longer than the chemical enrichment timescale, a sortof steady-state situation is created: continuos infall of metal-poor gas, which both sustains SFR and dilutes metallicity,and outflow of metal-rich gas in galactic winds. In this casethe outflows must depend on both mass and SFR.The small residual scatter around the FMR in thelocal universe supports the smooth accretion scenario,where galaxy grow is dominated by continuos accretion ofcold gas. Interacting and merging galaxies are expected toshow larger spread around the FMR, in agreement to whatis actually observed. Galaxies at intermediate and highredshifts show larger metallicity dispersions, which couldbe due either to uncertainties in the measurements, or tointrinsic dispersion, or both. This effect prevents us to studythe evolution of residual scatter with redshift. Nevertheless,the absence of significant biasses in metallicity or in SFR upto z=2.5 points toward the existence of the same physicaleffects and the dominance of smooth accretion even atintermediate redshift. Acknowledgements
We are grateful to the MPA/JHU teams which madetheir measured quantities on SDSS galaxies publicly avail-able, and to the members of the italian Virtual Observatoryfor support on data visualization. We also acknowledge stim-ulating discussions with T. Nagao, R. Dav´e, S. Ellison, andthe Arcetri MEGA group.
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