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The fundamental metallicity relation emerges from the local anti-correlation between star formation rate and gas-phase metallicity existing in disk galaxies

The fundamental metallicity relation (FMR) states that galaxies of the same stellar mass but larger star formation rate (SFR) tend to have smaller gas-phase metallicity (). It is thought to be fundamental because it naturally arises from the stochastic feeding of star-formation from external metal-poor gas accretion, a process extremely elusive to observe but essential according the cosmological simulations of galaxy formation. In this letter, we show how the FMR emerges from the local anti-correlation between SFR surface density and Zg recently observed to exist in disk galaxies. We analytically derive the global FMR from the local law, and then show that both relations agree quantitatively when considering the star-forming galaxies of the MaNGA survey. Thus, understanding the FMR becomes equivalent to understanding the origin of the anti-correlation between SFR and metallicity followed by the set of star-forming regions of any typical galaxy. The correspondence between local and global laws is not specific of the FMR, so that a number of local relations should exist associated with known global relations.

The scaling relations followed by galaxies play a central role to assess our understanding of galaxy formation. The non-linear evolution of the density perturbations created during the Big Bang gives rise to galaxies in a sequence that can be followed only through numerical simulations. Many key physical processes (e.g., the feedback produced by stars and black holes) have to be taken into account through sub-grid recipes tuned to reproduce some of the scaling properties observed in galaxies (e.g., the distribution of stellar masses). Other independent scaling relations allow us to test the consistency of the resulting model galaxies (e.g., Vogelsberger et al. 2014;Schaye et al. 2015). Among them, the fundamental metallicity relation (FMR) stands out because it relates (and so constrains) the main ingredients involved in the star-formation process, its fueling, and its self-regulation. Ellison et al. (2008), Mannucci et al. (2010), and Lara-López et al. (2010) found out that galaxies of the same stellar mass with higher SFR show lower gas-phase metallicity. As it was already put forward by Ellison et al. (2008) and Mannucci et al. (2010), the existence of this relation suggests the feeding of star-formation from external stochastic metal-poor gas accretion, a process extremely elusive to observe (e.g., Sánchez Almeida et al. 2014) but essential according the numerical simulations of galaxy formation (e.g., Dekel et al. 2009). The accretion of metal poor gas triggers star-formation, which consumes the accreted gas, which eventually leads to reducing the SFR and to increasing the gas-phase metallicity. For recent reviews on the FMR, see, e.g., Sánchez Almeida (2017), Sánchez et al. (2017), Cresci et al. (2018), or Maiolino & Mannucci (2019).
It has been recently found that star-forming disk galaxies are chemically inhomogeneous, with a decrease of metallicity spatially coinciding with enhanced SFR (Sánchez Almeida et al. 2018;Hwang et al. 2019;Sánchez-Menguiano et al. 2019). Notably, this is not only a property of some rare extremely metal poor galaxies (e.g., Sánchez Almeida et al. 2013 but it happens to be characteristic of the full population of local star-forming galaxies with stellar mass log(M /M ) 10.5, as portrayed by MaNGA (Sánchez-Menguiano et al. 2019).
In this Letter, we show that the FMR emerges from the local anti-correlation between SFR surface density and metallicity. We derive the global FMR from the local law (Sect. 2), and then show that both relations agree observationally (Sect. 3). The agreement is qualitative when considering FMRs from assorted references, but it becomes quantitative when using the same MaNGA galaxies for both the local and the global law (Sect. 3). Thus, understanding the FMR becomes equivalent to understanding the origin of the anti-correlation between SFR and metallicity followed by the set of star-forming regions of low-mass galaxies (Sect. 4). The equivalence between local and global laws is to be expected also in other known global scaling relations (Sect. 4).

FMR FROM THE LOCAL ANTI-CORRELATION
BETWEEN Σ SFR AND Z g As we discussed in Sect. 1, the galaxies forming stars have been shown to present an anti-correlation between the SFR surface density, Σ SFR , and the gas-phase metallicity, Z g , i.e., with ∆X the residual of parameter X once large scale variations have been removed or, equivalently, the difference between X in two nearby regions of the same galaxy. Equation (1) has been shown to hold by Sánchez-Menguiano et al. (2019), where the slope m, which quantifies the strength and sign of the correlation, is found to vary systematically depending on the global properties of the galaxy. Considering two galaxies with the same global properties, and thus the same m, Eq. (1) also describes the difference between two regions in the two galaxies, provided the regions are at the same galactocentric distance. Then the relation (1) also describes the excess of log Z g and log Σ SFR in one particular region of a particular galaxy with respect to the average alike galaxy 1 at the position of the region, 1 The set of alike galaxies to carry out the average includes all the galaxies with similar m in Eq. (1) whose integrated SFR and i.e., where Z g0 and Σ SFR0 stand for the large scale variation of the gas-phase metallicity and SFR in the mean alike galaxy. Defining ∆Z g = Z g − Z g0 and ∆Σ SFR = Σ SFR − Σ SFR0 , and assuming |∆Z g | Z g and |∆Σ SFR | Σ SFR , then Eq. (1) can be re-written as Note that, except for m, all variables in Eqs. (1) and (3) depend on the position on the galaxy disk, defined by the Cartesian coordinates x and y. Thus, the mean metallicity of the galaxy is defined to be with Σ g the surface density of the gas and W an arbitrary weight that goes to zero outside the galaxy so that the integral extends to all the x-y plane. The symbol M g stands for the total gas mass of the galaxy only if W equals one within the galaxy. (To ease distinguishing local from galaxy-integrated quantities, the latter are always represented as symbols within brackets.) The excess mean metallicity can be defined as with Z g0 the mean metallicity to be expected considering all the alike galaxies. Equations (3), (4), and (5) lead to dxdy. (6) The Kennicutt-Schmidt relation shows that the observed SFR surface density scales gas mass surface density through a law assumed to have the form of where A and N are numerical constants, with N 1 (e.g., Kennicutt 1998;Bigiel et al. 2008). Keeping only first order terms in both N − 1 and ∆Σ SFR , Eq. (7) becomes, metallicity will be used to derive the FMR. It is a flexible definition because the implementation can mutate depending on the actual needs. In the example given in Sect. 3, the alike galaxies are all the star-forming galaxies in MaNGA with similar stellar mass.
Introducing Eq. (8) into Eq. (6), and retaining only first order terms, with the integrated SFR defined as Equation (9) can be re-written as The second term in the right-hand side of Eq. (11) is negligible if the local excess of SFR (∆Σ SFR ) is uncorrelated with the large scale variations of metallicity (Z g0 − Z g0 ). In principle, local and large scale variations are expected to be independent from each other, allowing the condition to be met. However, for the moment, we neglect this term just as an ansatz to be justified later on in Sect. 3 using real galaxies. Thus, one can rewrite Eq. (11) as or, neglecting 2nd order terms, Equation (13) shows the global FMR, which emerges from the local anti-correlation given in Eq. (1) with exactly the same slope.  (Bundy et al. 2015). Galaxies were selected the full MaNGA parent sample to be both star-forming and large enough to allow measuring local variations of SFR and Z g . The value of m was inferred from a linear fit to the scatter plot ∆Z g versus ∆Σ SFR in each individual galaxy computed after removing the large scale trends. Z g and SFR were deduced using the strong line ratio O3N2 and the Hα flux, respectively (for details, see Sánchez-Menguiano et al. 2019). The slope m was found to depend on M , going from negative at low mass (log[M /M ] 10.5) to slightly positive at the high mass end of the mass distribution (log[M /M ] 11). The thick red line in Fig. 1 shows the resulting mean m for a given M .
As one can see from Fig. 1, the ms inferred from local and global variations agree qualitatively; the slopes are negative, and there is a tendency for |m| to decrease as M increases (see, e.g., the values from Salim et al. 2014). Qualitative differences are also evident, though. We think that these differences are mostly due to systematic errors in the various calibrations used to estimate Z g and SFR (e.g., Kewley & Ellison 2008;Kennicutt & Evans 2012), and also due to the actual set of galaxies employed to infer m (e.g., Salim et al. 2014). This impression is corroborated by the exercise described in the next paragraph, where local and global laws obtained using the same galaxies and identical Z g and Σ SFR are shown to coincide. The equivalence between the local properties and the integrated quantities is a mathematical identity, therefore, it holds provided certain conditions are met. The  , Σ SFR is a proxy for Σ g when N = 1). We employ Eq. (4) replacing Σ g with Σ SFR , and setting W to a constant = 0 only in the star-forming regions. MaNGA galaxies follow the global FMR as evidenced in Fig. 2, left panel, where the mass metallicity relation is color-coded with SFR. It is also clear from Fig. 2, right panel, showing the scatter plot Z g versus SFR for all galaxies within a narrow range of stellar masses (9.2 ≤ log[M /M ] ≤ 9.6). Z g and Σ SFR were computed with exactly the same recipes employed by Sánchez-Menguiano et al. (2019), and the actual galaxies and the star-forming regions within them 2 Zg is the fraction of metals by mass, whereas O/H in Fig. 2 is the abundance of O relative to H by number. There is no inconsistency in using any of them to infer m since O/H and Zg are proportional to each other, provided the relative abundance of the different metals is the same for all galaxies.
were also the same. A linear fit to scatter plots like the one in Fig. 2, right panel, provides m at a particular mass range. Repeating the exercise for different mass bins renders the dependence of m on M represented in Fig. 3. 3 In order to test de dependence of the result on the way the average Z g is carried out, Fig. 3 shows the result based on weighting with Σ SFR (the red symbols), with Σ 0.7 SFR (the orange symbols), without weighting (the blue symbols), and using the metallicity at the effective radius (the green symbols). The three first averages correspond to weighting with Σ g if the ) and N = 1, 1.4, and 0, respectively. N = 1.4 corresponds to the canonical value in the calibration by Kennicutt (1998).
The three main conclusions ensuing from Fig. 3 are: (1) there is a very good agreement between the slope derived from local and global laws, which reinforces the approximations leading to the analytical derivation in Sect. 2. (2) The way in which the average is computed does no seem to be very important. Four different ways of averaging yield similar results. (3) The number of galaxies used to construct Fig. 3 is large enough for the conclusions to be representative of local galaxies (as portrayed by MaNGA).

DISCUSSION AND CONCLUSIONS
3 As discussed by, e.g., Sánchez et al. (2017, the use of finite mass bins may induce a spurious correlation between Zg and SFR. However, this artificial correlation is always positive since both Zg and SFR tend to increase with increasing M . Thus, m ≤ 0 cannot result from such a bias. As a sanity check, we verified that dividing by half the mass bin does not produce a significant variation in Fig. 3. The symbols represent slopes obtained from integrated values of metallicity and SFR using exactly the same galaxies. Different colors represent different ways of computing the average metallicity (see text for details). The error bars correspond to the uncertainty associated with the determination of the slope (vertical), and the RMS variation of the stellar masses of the galaxies in the mass bin used to infer m (horizontal).
Provided the existence of a local correlation between Σ SFR and Z g , the FMR is a must -Eq. (1) ⇒ Eq. (13). Does the inverse hold true, i.e., does the existence of the FMR imply the existence a local anti-correlation? The answer is yes. If a galaxy does not have a local correlation, it cannot show a correlation in integrated quantities (i.e., if m = 0 in Eq. [1], m cannot be = 0 in Eq. [13]). Thus, understanding the FMR becomes equivalent to understanding the origin of the correlation between SFR and metallicity followed by the set of starforming regions of any typical galaxy.
A potential flaw of the above argument is the break down of the approximations used to derive Eq. (13) from Eq. (1), namely, the small variation of the physical parameters, and the lack of correlation between large-scale metallicity variations and small-scale SFR variations. The validity of these approximations was tested using a set of 736 star-forming galaxies in MaNGA, which give the same slope m from local variations and from spatially integrated quantities (Fig. 3). We note that this set is representative of the local star-forming galaxies with masses in the range 8.7 ≤ log(M /M ) ≤ 11.2.
Still, tt may break down when considering other more extreme galaxies observed at higher spatial resolution.
The exercise with MaNGA galaxies also allows us to conclude that the actual weight employed to compute the integrated quantities is not fundamental. On the other hand, the methods to estimate SFR and Z g from emission line fluxes, and the set of galaxies included in the analysis, do influence the actual value inferred for m (Fig. 1).
Our derivation in Sect. 2 is quite general, and shows that any local relation between two variables gives rise to a global relation. This is also true for the gas mass, for which an alternative FMR is known to exist. Galaxies with larger gas mass (both atomic and molecular) have smaller Z g (e.g., Bothwell et al. 2016a,b). The fact that this global law has been observed implies that there should be a local anti-correlation between Σ g and Z g . Indeed, Barrera-Ballesteros et al. (2018) find a local anti-correlation between the gas fraction inferred from dust extinction and Z g . Similarly, Lian et al. (2015) analyze a set of local analogues to Ly break galaxies, finding that objects with younger stellar populations have lower metallicity at a fixed mass. This would imply the existence of a local relation between the age of the stellar population and metallicity of the gas forming stars. These specific predictions of our equations can be tested observationally.