Resolved Fundamental Metallicity Relation (rFMR)

• The rFMR is a spatially resolved relation that links local gas-phase metallicity with stellar mass surface density and star-formation rate surface density.
• It uses complex parameterizations and multiple diagnostic calibrations to minimize scatter and accurately capture local and global galaxy influences.
• Empirical studies reveal an inverse correlation between metallicity and star formation at fixed stellar density, highlighting the interplay of local processes and global galaxy structure.

The resolved Fundamental Metallicity Relation (rFMR) is the spatially resolved analogue of the global Fundamental Metallicity Relation, formulated on sub-galactic scales in terms of local gas-phase metallicity, usually written as $12+\log(\mathrm{O/H})$, local stellar mass surface density $\Sigma_*$, and local star-formation-rate surface density $\Sigma_{\rm SFR}$. In its strongest current formulation, the rFMR states that local metallicity depends primarily on $\Sigma_*$ and secondarily, with the opposite sign, on $\Sigma_{\rm SFR}$, while remaining sensitive to global galaxy properties such as total stellar mass $M_*$. The concept is therefore not exhausted by a purely local two-parameter law: it is a resolved chemical-scaling relation embedded in galaxy-wide regulation of metal production, retention, redistribution, and loss (Baker et al., 2022). At intermediate redshift, the empirical status is less uniform: one MUSE-Wide study found no obvious evidence for an rFMR at $z\sim0.26$, whereas MAGPI reported weak but measurable evidence at $z\sim0.3$, strongest in low-mass systems (Yao et al., 2022, Koller et al., 2024).

1. Definition and mathematical forms

The modern rFMR is usually framed as a three-parameter relation linking local metallicity, $\Sigma_*$, and $\Sigma_{\rm SFR}$. In the MaNGA-based analysis centered on local star-forming galaxies, the resolved metallicity surface is parameterized as

$\Sigma_*$0

with

$\Sigma_*$1

so that the turnover mass surface density shifts with $\Sigma_*$2. For the combined sample, the reported parameters are

$\Sigma_*$3

The same study also introduced the reduced-scatter projection

$\Sigma_*$4

finding the minimum scatter at $\Sigma_*$5, with

$\Sigma_*$6

and fitted coefficients

$\Sigma_*$7

(Baker et al., 2022).

A distinct but closely related formulation is the residual or fluctuation law. In that approach, the central relation is not written in raw $\Sigma_*$8 space but as a local anti-correlation between metallicity residuals and SFR-surface-density residuals after large-scale trends have been removed: $\Sigma_*$9 Here $\Sigma_{\rm SFR}$0 denotes the residual of $\Sigma_{\rm SFR}$1 once large-scale variations have been removed, or equivalently the difference between nearby regions within the same galaxy. This is not a full $\Sigma_{\rm SFR}$2-based rFMR, but it is a resolved metallicity–star-formation coupling from which the global FMR can be derived analytically under additional assumptions (Almeida et al., 2019).

Single-galaxy implementations also exist. In NGC 99, the resolved relation was fit directly as a plane,

$\Sigma_{\rm SFR}$3

again with metallicity increasing with $\Sigma_{\rm SFR}$4 and decreasing with $\Sigma_{\rm SFR}$5 at fixed $\Sigma_{\rm SFR}$6 (Olvera et al., 2024).

2. Observational construction and measurement

Resolved rFMR work is methodologically heterogeneous, and that heterogeneity is central to the literature. In the MaNGA analysis, the resolved measurements are at the spaxel level on explicit kpc scales. MaNGA has a PSF FWHM of about $\Sigma_{\rm SFR}$7, spaxels of $\Sigma_{\rm SFR}$8, and galaxies are sampled with roughly 3–7 radial resolution elements at $\Sigma_{\rm SFR}$9. After selection, the sample contains about $\Sigma_*$0 star-forming spaxels from 2002 galaxies, corresponding to about $\Sigma_*$1 independent regions once PSF oversampling is accounted for. The star-forming spaxels were selected using the $\Sigma_*$2 versus $\Sigma_*$3 BPT diagram with the Kauffmann et al. dividing line, with $\Sigma_*$4, $\Sigma_*$5 Å, exclusion of interacting and post-merger systems, and inclination $\Sigma_*$6 (Baker et al., 2022).

That study’s metallicity methodology is unusually elaborate. Rather than relying on a single strong-line calibrator, it jointly fits nine strong-line diagnostics tied to the $\Sigma_*$7 scale using the empirical calibrations of Curti et al. (2017, 2020), minimizing a $\Sigma_*$8 between observed and predicted line ratios with an MCMC approach. The resolved stellar mass surface density is derived from Pipe3D stellar masses per pixel divided by spaxel area. The resolved SFR surface density is derived from dust-corrected $\Sigma_*$9 luminosity using the Kroupa-IMF Kennicutt & Evans (2012) calibration,

$\Sigma_{\rm SFR}$0

and then divided by spaxel area. All surface densities are corrected for inclination through

$\Sigma_{\rm SFR}$1

For the ALMaQUEST subset, molecular gas surface density $\Sigma_{\rm SFR}$2 is taken from ALMA CO(1–0) maps and converted using a metallicity-dependent $\Sigma_{\rm SFR}$3 in the main analysis (Baker et al., 2022).

Intermediate-redshift studies adopt simpler resolved pipelines. The MUSE-Wide analysis at $\Sigma_{\rm SFR}$4 does not operate on native spaxels in the final science products but on 699 Voronoi bins across ten galaxies. Metallicity is inferred from the O3N2 index using the Marino et al. calibration,

$\Sigma_{\rm SFR}$5

with a typical calibration error of 0.08 dex. Resolved stellar mass surface density is obtained from matched HST photometry and SED fitting with FAST, while $\Sigma_{\rm SFR}$6 is derived from dust-corrected $\Sigma_{\rm SFR}$7 via

$\Sigma_{\rm SFR}$8

(Yao et al., 2022).

MAGPI, also at $\Sigma_{\rm SFR}$9, returns to a spaxel-by-spaxel resolved analysis. After line-S/N and BPT cuts, the main sample comprises 6299 star-forming spaxels from 65 galaxies. Metallicities are based on O3N2 with the Marino et al. calibration,

$M_*$0

and average metallicity uncertainty is $M_*$1 dex. Resolved stellar masses come from FADO spectral synthesis, and inclination-corrected surface densities are formed using $M_*$2 (Koller et al., 2024).

The NGC 99 study uses neither IFU spaxels nor Voronoi bins but 26 localized H II-region apertures. There, $M_*$3 and SFR are converted to $M_*$4 and $M_*$5 by dividing by a slit area of $M_*$6, and the fitted rFMR plane uses N2-based metallicities calibrated following Curti et al. (2020) (Olvera et al., 2024).

3. Empirical structure in nearby galaxies

The clearest positive detection of the rFMR in the supplied literature is the MaNGA analysis of local star-forming galaxies. In the $M_*$7–$M_*$8 plane, with bins color-coded by mean $M_*$9, the metallicity is organized along a tilted gradient rather than along purely vertical or purely horizontal bands. The interpretation is direct: at fixed $z\sim0.26$0, metallicity decreases with increasing $z\sim0.26$1, and at fixed $z\sim0.26$2, metallicity increases with $z\sim0.26$3. The authors explicitly describe this as unambiguous evidence for an rFMR. The same relation is displayed as a 3D metallicity surface and as metallicity–$z\sim0.26$4 tracks binned by $z\sim0.26$5, where higher $z\sim0.26$6 tracks lie at lower metallicity except in the high-$z\sim0.26$7, low-$z\sim0.26$8 corner where an inversion appears. Around the fitted rFMR surface, the residual metallicity distribution is reported to have a Gaussian width of $z\sim0.26$9 dex (Baker et al., 2022).

The sign and relative strength of the secondary dependence are quantified with Partial Correlation Coefficients and an arrow-angle construction in the $z\sim0.3$0–$z\sim0.3$1 plane. For the MaNGA rFMR figure, the angle is quoted as $z\sim0.3$2, and the authors interpret this as $z\sim0.3$3 contributing $z\sim0.3$4 and $z\sim0.3$5 $z\sim0.3$6 to metallicity variation in that local two-dimensional plane. The effect is mass-dependent: the optimal $z\sim0.3$7 in $z\sim0.3$8 decreases from $z\sim0.3$9 at low total stellar mass to $\Sigma_*$0 at high total stellar mass, and the trend is crudely parameterized as

$\Sigma_*$1

(Baker et al., 2022).

Single-galaxy resolved work can reproduce the same sign structure. In NGC 99, the fitted plane implies that metallicity depends proportionally on $\Sigma_*$2 and inversely on $\Sigma_*$3, with the coefficient of $\Sigma_*$4 larger in magnitude than that of $\Sigma_*$5. The metallicity scatter around the best-fit plane is $\Sigma_*$6 dex. The rFMR in that system was presented after the identification of two anomalously low-metallicity H II regions, and the paper treats the plane as evidence that the galaxy still follows the standard resolved metallicity–$\Sigma_*$7–$\Sigma_*$8 organization despite localized inflow signatures (Olvera et al., 2024).

A related nearby-galaxy result uses residuals rather than absolute resolved quantities. In 736 nearby spiral galaxies from MaNGA DR15, the local slope $\Sigma_*$9 of the relation $\Sigma_{\rm SFR}$0 is negative for low-mass galaxies, $\Sigma_{\rm SFR}$1, and becomes slightly positive near $\Sigma_{\rm SFR}$2. The paper’s central claim is that the global FMR emerges quantitatively from this local anti-correlation when local fluctuations are spatially integrated (Almeida et al., 2019).

4. Beyond a purely local law: global dependence and physical interpretation

A major development in rFMR work is the recognition that resolved metallicity is not set by local quantities alone. In the MaNGA analysis that explicitly combines local and global variables, the strongest direct dependencies of local metallicity are found to be on $\Sigma_{\rm SFR}$3 and total stellar mass $\Sigma_{\rm SFR}$4, with weaker but intrinsic inverse dependencies on both $\Sigma_{\rm SFR}$5 and total SFR. This result is visualized in the $\Sigma_{\rm SFR}$6–$\Sigma_{\rm SFR}$7–metallicity plane and formalized through a parameterization analogous to the rFMR fit,

$\Sigma_{\rm SFR}$8

with

$\Sigma_{\rm SFR}$9

for which the best-fit parameters are

$\Sigma_*$00

A compact projection is also defined through

$\Sigma_*$01

with $\Sigma_*$02 and the quadratic metallicity fit

$\Sigma_*$03

The implication is explicit: the global mass–metallicity relation does not simply emerge from the resolved rMZR, and the global FMR is not merely a coarse-grained version of the rFMR (Baker et al., 2022).

The same paper also compares $\Sigma_*$04 and $\Sigma_*$05 as secondary resolved drivers at fixed $\Sigma_*$06. In ALMaQUEST, the arrow angle for the $\Sigma_*$07–$\Sigma_*$08–metallicity relation is $\Sigma_*$09, whereas for $\Sigma_*$10–$\Sigma_*$11–metallicity it is $\Sigma_*$12. Random Forest analysis yields the same qualitative conclusion: $\Sigma_*$13 is weaker and does not replace $\Sigma_*$14 as the secondary predictor. On that basis, the paper concludes that $\Sigma_*$15 is less important than $\Sigma_*$16 for determining local metallicity, and that simple local gas accretion plus metallicity dilution is unlikely to be the primary origin of the rFMR, even if dilution contributes (Baker et al., 2022).

The preferred physical interpretations are correspondingly plural. The primary rMZR with $\Sigma_*$17 is interpreted in terms of local metal production and/or local disk potential retaining metals. The additional dependence on $\Sigma_*$18 points to global retention and redistribution, including the global gravitational potential, galaxy-scale mixing, radial migration, bars, or fountains. For the inverse dependence on SFR and $\Sigma_*$19, metal-loaded winds are emphasized: higher total SFR may launch stronger galaxy-scale outflows, and higher local $\Sigma_*$20 may reflect local SF-driven ejection of freshly enriched gas. The authors also stress timescale issues, because $\Sigma_*$21-based SFR traces $\Sigma_*$22 Myr whereas hot metal-rich gas may cool and mix on much longer timescales, leaving the physical origin of the anti-correlation open (Baker et al., 2022).

A complementary line of argument treats the resolved metallicity–SFR coupling as the local origin of the global FMR. Starting from

$\Sigma_*$23

and assuming small perturbations, a Kennicutt–Schmidt relation $\Sigma_*$24 with $\Sigma_*$25, and weak covariance between large-scale metallicity structure and local SFR fluctuations, the global relation

$\Sigma_*$26

is derived analytically. In that framework, the global FMR is the spatially integrated manifestation of a local residual anti-correlation (Almeida et al., 2019).

An even broader physical template comes from the analytic global FMR model in which gas metallicity is set by the competition between metal production, dilution by metal-poor inflow, and metal-rich outflow. That work is not a resolved model, but it argues that low-mass systems should show decreasing metallicity with increasing SFR, while high-mass systems can approach SFR-independent metallicity because enrichment is compensated by dilution and outflows become negligible. This suggests a regulator-style interpretive framework for rFMR studies, particularly for the observed weakening of the metallicity–SFR anti-correlation at high mass (Dayal et al., 2012).

5. Intermediate-redshift constraints and evolutionary status

Resolved scaling-relation work at $\Sigma_*$27–0.3 shows that the existence of an rMZR does not automatically imply the existence of an rFMR. In the MUSE-Wide study of ten emission-line galaxies at mean redshift $\Sigma_*$28, the resolved star-forming main sequence and resolved mass–metallicity relation are both detected. The preferred rSFMS slope is $\Sigma_*$29, and the rMZR has a similar shape to the local relation but is offset to lower metallicity by about $\Sigma_*$30 dex, or about $\Sigma_*$31 dex after controlling for global $\Sigma_*$32. However, the rFMR tests are negative. Metallicities and SFR surface densities are shifted relative to local galaxies in the expected sense—higher $\Sigma_*$33 and lower metallicity—but the residual test shows only weak correlation, with Pearson $\Sigma_*$34 and Spearman $\Sigma_*$35. In an FMR-style projection, the minimum occurs at $\Sigma_*$36 with $\Sigma_*$37 dex, but that is only $\Sigma_*$38 dex better than $\Sigma_*$39, effectively no SFR term. The study therefore concludes that there is no obvious evidence for an rFMR at $\Sigma_*$40, and that the data are not an extension of the local rFMR toward higher $\Sigma_*$41 (Yao et al., 2022).

MAGPI, by contrast, reports weak but non-zero evidence for an rFMR at $\Sigma_*$42. In 65 galaxies spanning $\Sigma_*$43, the rMZR exists with an average metallicity about $\Sigma_*$44 dex above the local-universe relation, though still within calibration uncertainty. The rFMR is established not through a direct plane fit but through Partial Correlation Coefficients, arrow angles, $\Sigma_*$45-binned rMZR tracks, and minimization of scatter in $\Sigma_*$46. For the full sample, the arrow angle is

$\Sigma_*$47

which the authors interpret as $\Sigma_*$48 contributing $\Sigma_*$49 and $\Sigma_*$50 $\Sigma_*$51 to local metallicity variation in the local two-dimensional plane. The best projection is obtained at $\Sigma_*$52. The mass dependence is strong: the arrow angle steepens from $\Sigma_*$53 in the highest-mass bin $\Sigma_*$54 to $\Sigma_*$55 for $\Sigma_*$56. The paper therefore argues that the rFMR at $\Sigma_*$57 is weak overall but becomes pronounced in low-mass galaxies and low-$\Sigma_*$58 regions, while global stellar mass remains a significant determinant of local metallicity (Koller et al., 2024).

Taken together, these two intermediate-redshift studies imply that evolution of the resolved mass–metallicity relation and evolution of the resolved fundamental metallicity relation are separable questions. A lower metallicity normalization at fixed $\Sigma_*$59 does not, by itself, establish a resolved SFR term. Conversely, when a weak rFMR is detected, it may be strongly conditioned by the mass distribution of the sample rather than by redshift alone (Yao et al., 2022, Koller et al., 2024).

6. Caveats, controversies, and open problems

The existence, strength, and interpretation of the rFMR remain contested because the signal is sensitive to measurement choices, selection cuts, spatial resolution, and statistical formalism. The MaNGA analysis that reports a robust rFMR also notes several caveats. Metallicity calibrators always introduce systematics, although the use of nine diagnostics and nitrogen-free subsets is intended to mitigate this. The apparent inversion in the high-$\Sigma_*$60, low-$\Sigma_*$61 corner may be physical or may reflect residual biases. The $\Sigma_*$62 cut effectively imposes a $\Sigma_*$63 threshold, so very quiescent regions are not sampled. Spatial resolution is only kpc-scale, with heavy PSF oversampling. The ALMaQUEST subset is small and biased toward massive galaxies, and the $\Sigma_*$64 result depends on the adopted CO-to-$\Sigma_*$65 conversion factor. Finally, Partial Correlation Coefficients and Random Forests establish intrinsic correlations more convincingly than simple bivariate plots, but they do not by themselves prove causality (Baker et al., 2022).

The intermediate-redshift non-detection at $\Sigma_*$66 makes the observational problem especially clear. Although the analysis uses 699 Voronoi bins, the galaxy sample contains only ten systems, the data are seeing-limited, and beam smearing can flatten gradients and blur local contrasts among $\Sigma_*$67, $\Sigma_*$68, and $\Sigma_*$69. Metallicities carry average random uncertainties of about $\Sigma_*$70 dex, plus $\Sigma_*$71 dex calibration uncertainty from O3N2, whereas the reported improvement from introducing an SFR term is only $\Sigma_*$72 dex. Under those conditions, a weak or absent rFMR is not easily distinguishable from observational washout (Yao et al., 2022).

The MAGPI result, although positive, adds further nuance rather than closure. The sample is limited to 65 emission-line galaxies, often in dense environments, and contains many high-mass systems, which likely weakens the overall rFMR signal because the metallicity–$\Sigma_*$73 anti-correlation becomes nearly absent at high mass. The analysis also depends on O3N2 metallicities with $\Sigma_*$74 dex uncertainty, BPT-selected star-forming spaxels that exclude composite and AGN-like regions, and PCCs that are meaningful only for monotonic relations. For these reasons the authors describe the $\Sigma_*$75 rFMR as tentative or weakly significant rather than definitive (Koller et al., 2024).

The central unresolved question is whether the rFMR is best regarded as a genuinely local law, a local manifestation of integrated baryon-cycle regulation, or a scale-dependent combination of both. The strongest nearby evidence indicates that local metallicity is shaped jointly by local stellar buildup, local star-forming activity, and global galaxy structure. The most direct molecular-gas comparison argues against a simple picture in which local gas accretion and dilution alone generate the relation. At the same time, analytic and single-galaxy studies continue to motivate inflow, enrichment, and outflow as the key physical ingredients. A plausible implication is that no single mechanism need dominate across all masses, spatial scales, and epochs: the rFMR may instead mark the observable intersection of local enrichment, metal retention, metal-loaded winds, mixing, and non-equilibrium timescales.