Physical origin of the secondary star-formation-rate dependence of the galaxy mass–metallicity relation (the Fundamental Metallicity Relation) on the Te abundance scale

Curti et al. (2020) re-derived the local mass–metallicity relation (MZR) and the fundamental metallicity relation (FMR) on a self-consistent abundance scale tied to the direct electron-temperature (Te) method, explicitly parametrizing the FMR through the SFR-dependence of the turnover mass, and left open the physical origin of the secondary dependence of gas-phase metallicity on star formation rate (SFR) at fixed stellar mass: dilution by pristine inflows, metal-enriched outflows, or gas-fraction/evolutionary-state variations, and which mechanism dominates in each mass regime. The 2021–2026 literature shows intense activity but no consensual resolution: direct gas-flow observations, non-equilibrium chemical-evolution fits, resolved integral-field analyses, and a new mass-continuity theoretical framework give partially conflicting mechanism attributions, while new Te-based work finds the canonical FMR absent below $10^9M_\odot$ and simulations find it inverted at high mass. The problem — a physically explained, observationally adjudicated account of the origin and regime-dependence of the $M_\star$–SFR–metallicity relation, together with its robust structure in the previously unsampled low-mass high-SFR region — remains open.

Background

Star-forming galaxies in the nearby Universe obey the mass–metallicity relation (MZR): more massive galaxies contain more metals in their ionized gas, usually quantified as the oxygen abundance $12+\log(\mathrm{O/H})$, with the relation flattening above a turnover stellar mass of roughly $10^{10}M_\odot$. The scatter of the MZR correlates with star formation rate (SFR): at fixed stellar mass, galaxies currently forming stars faster tend to be more metal-poor. Mannucci et al. (2010) showed that local and high-redshift ($z\lesssim2.5$) galaxies appear to populate a common, nearly redshift-invariant surface in the three-dimensional space defined by stellar mass $M_\star$, SFR and gas-phase metallicity — the Fundamental Metallicity Relation (FMR) — which has been widely interpreted as evidence that galaxy evolution proceeds through a near-equilibrium interplay between gas accretion, star formation, and gas ejection. Gas-phase metallicities are usually inferred from 'strong-line' diagnostics (ratios of bright forbidden emission lines), calibrated either against photoionization models or against the 'direct' Te method, in which the electron temperature $T_e$ measured from faint auroral lines such as [OIII]λ4363 yields an oxygen abundance nearly free of model assumptions; different calibrations can differ by several tenths of a dex (a dex being a factor of ten in the measured quantity), so statements about the FMR are meaningful only relative to a stated abundance scale. Curti, Mannucci, Cresci & Maiolino (2020) anchored metallicities from the Sloan Digital Sky Survey (SDSS), a large local spectroscopic galaxy survey, to the Te scale using Te-calibrated strong-line diagnostics from earlier work of the same collaboration, refit the MZR with a functional form whose parameters are the saturation metallicity $Z_0$, the turnover mass $M_0$, the low-mass slope $\gamma$ and a knee-width parameter $\beta$, and built a new FMR parametrization by letting $M_0$ increase with SFR, reducing the residual metallicity scatter to 0.054 dex for individual galaxies. The standard physical frames for the secondary SFR dependence are 'gas-regulator'-type open-box models: pristine gas inflow both dilutes the interstellar medium and fuels star formation, while galactic winds expel metal-enriched gas with a mass-loading factor $\eta$ (the ratio of outflow rate to star-formation rate); in such models the metallicity–SFR anticorrelation can also arise as a projection of a more fundamental relation between metallicity and gas fraction, because at fixed mass the more strongly star-forming galaxies are more gas-rich. Which effect dominates, and in which mass regime, is not a matter of bookkeeping: it determines whether observed metallicities can be used to measure inflow and outflow rates, and whether the local FMR can serve as a redshift-invariant benchmark against which high-redshift galaxies (now observed with the James Webb Space Telescope, JWST) are compared. Within its own analysis, Curti et al. (2020) found the SFR dependence strongest at low and intermediate masses, no dependence of the saturation metallicity on SFR, a possible rise of the low-mass slope $\gamma$ with SFR that was insecure because of covariance between $\gamma$ and the knee-width parameter, poor sampling of the low-mass high-SFR regime, and sensitivity to the chosen metallicity diagnostics; FMR prediction uncertainties grow to about 0.3 dex in the low-mass, high-SFR region not sampled by local galaxies; and aperture corrections, metallicity gradients, and selection effects may shape the inferred relations.

Problem Statement

Determine the physical origin of the secondary dependence of gas-phase metallicity on star formation rate at fixed stellar mass — the dependence that gives the local $M_\star$–SFR–metallicity relation (the Fundamental Metallicity Relation) its reduced scatter — together with its regime dependence, on an abundance scale tied to the direct Te method. Concretely: (1) establish whether the metallicity–SFR anticorrelation at fixed $M_\star$ is driven by dilution from inflowing pristine gas, by metal-enriched galactic outflows, by variations in gas fraction or non-equilibrium evolutionary state, or by a combination, and identify which mechanism dominates in each stellar-mass regime — from the dwarf regime ($\log(M_\star/M_\odot)\lesssim9$, where the canonical FMR has now been reported absent) through the intermediate-mass regime (where the SFR dependence is strongest) to the massive regime ($M_\star\gtrsim10^{10.5}M_\odot$, where the anticorrelation has been reported to invert); (2) establish the actual structure of the Te-based relation in the previously unsampled low-mass, high-SFR region, and the robustness of the reported variation of the low-mass slope $\gamma$ with SFR once parameter covariance, choice of metallicity diagnostics, aperture corrections, metallicity gradients, and selection effects are controlled. These components jointly serve one objective: a physically explained, observationally adjudicated characterization of the local $M_\star$–SFR–metallicity relation on the Te abundance scale, in which the mechanism attribution (1) explains the empirically secured relation structure (2) rather than merely fitting it. Identifying a more fundamental underlying variable (gas fraction, size, gravitational potential, stellar metallicity) counts as an answer only if it quantitatively accounts for the observed SFR signal and its regime dependence, not merely replaces it.

The problem comprises both components: the mechanism attribution (1) and the secured relation structure (2).

References

1. Curti, M., Mannucci, F., Cresci, G., Maiolino, R. (2020). The mass–metallicity and the fundamental metallicity relation revisited on a fully Te-based abundance scale for galaxies. Monthly Notices of the Royal Astronomical Society 491, 944. DOI: 10.1093/mnras/stz2910. https://doi.org/10.1093/mnras/stz2910 (arXiv:1910.00597, https://arxiv.org/abs/1910.00597)
2. Mannucci, F., Cresci, G., Maiolino, R., Marconi, A., Gnerucci, A. (2010). A fundamental relation between mass, star formation rate and metallicity in local and high-redshift galaxies. Monthly Notices of the Royal Astronomical Society 408, 2115. DOI: 10.1111/j.1365-2966.2010.17291.x. https://doi.org/10.1111/j.1365-2966.2010.17291.x
3. Kumari, N., Maiolino, R., Trussler, J., Mannucci, F., Cresci, G., Curti, M., Marconi, A., Belfiore, F. (2021). The extension of the fundamental metallicity relation beyond the BPT star-forming sequence: Evidence for both gas accretion and starvation. Astronomy & Astrophysics 656, A140. DOI: 10.1051/0004-6361/202140757. https://doi.org/10.1051/0004-6361/202140757 (arXiv:2108.12437, https://arxiv.org/abs/2108.12437)
4. Baker, W. M., Maiolino, R., Belfiore, F., Curti, M., Bluck, A. F. L., Lin, L., Ellison, S. L., Thorp, M., Pan, H.-A. (2022). The metallicity's fundamental dependence on both local and global galactic quantities. Monthly Notices of the Royal Astronomical Society 519, 1149. DOI: 10.1093/mnras/stac3594. https://doi.org/10.1093/mnras/stac3594 (arXiv:2210.03755, https://arxiv.org/abs/2210.03755)
5. Lin, Y., Zu, Y. (2023). Constraints on galactic outflows from the metallicity–stellar mass–SFR relation of EAGLE simulation and SDSS galaxies. Monthly Notices of the Royal Astronomical Society 521, 411. DOI: 10.1093/mnras/stad502. https://doi.org/10.1093/mnras/stad502 (arXiv:2212.01402, https://arxiv.org/abs/2212.01402)
6. Langan, I., Zabl, J., Bouché, N. F., Ginolfi, M., Popping, G., Schroetter, I., Wendt, M., Schaye, J., Boogaard, L., Freundlich, J., Richard, J., Matthee, J., Mercier, W., Contini, T., Guo, Y., Cherrey, M. (2023). MusE GAs FLOw and Wind (MEGAFLOW) IX. The impact of gas flows on the relations between the mass, star formation rate, and metallicity of galaxies. Monthly Notices of the Royal Astronomical Society 521, 546. DOI: 10.1093/mnras/stad357. https://doi.org/10.1093/mnras/stad357 (arXiv:2301.10779, https://arxiv.org/abs/2301.10779)
7. Ma, C., Wang, K., Wang, E., Peng, Y., Jiang, H., Yu, H., Jia, C., Chen, Z., Li, H., Kong, X. (2024). Revisiting the Fundamental Metallicity Relation with Observation and Simulation. The Astrophysical Journal Letters 971, L14. DOI: 10.3847/2041-8213/ad675f. https://doi.org/10.3847/2041-8213/ad675f (arXiv:2407.21716, https://arxiv.org/abs/2407.21716)
8. Sánchez-Menguiano, L., Sánchez, S. F., Sánchez Almeida, J., Muñoz-Tuñón, C. (2024). More fundamental than the fundamental metallicity relation. Astronomy & Astrophysics 682, L11. DOI: 10.1051/0004-6361/202348423. https://doi.org/10.1051/0004-6361/202348423
9. McClymont, W., Tacchella, S., Smith, A., Kannan, R., Garaldi, E., Puchwein, E., Isobe, Y., Ji, X., Shen, X., Wang, Z., Belokurov, V., Borrow, J., D'Eugenio, F., Keating, L., Maiolino, R., Monty, S., Vogelsberger, M., Zier, O. (2026). The THESAN-ZOOM project: mystery N/O more – uncovering the origin of peculiar chemical abundances and a not-so-fundamental metallicity relation at $3<z<12$. Monthly Notices of the Royal Astronomical Society 548, stag016. DOI: 10.1093/mnras/stag016. https://doi.org/10.1093/mnras/stag016 (arXiv:2507.08787, https://arxiv.org/abs/2507.08787)
10. Laseter, I. H., Maseda, M. V., Bunker, A. J., Cameron, A. J., Curti, M., Simmonds, C. (2026). An Investigation into the Low-mass Fundamental Metallicity Relation in the Local and High-z Universe. The Astrophysical Journal 1007, 26. DOI: 10.3847/1538-4357/ae7d25. https://doi.org/10.3847/1538-4357/ae7d25 (arXiv:2510.15024, https://arxiv.org/abs/2510.15024)
11. Carnevale, L., Garcia, A. M., Torrey, P., Ellison, S. L., Jain, S., Zhu, P., Grasha, K., Sanders, R. L., Hernquist, L., Kewley, L. J., Scudder, J. M. (2026). The Mass Dependence of the Fundamental Metallicity Relation in Observations and Simulations. arXiv:2608.24826. https://arxiv.org/abs/2608.24826
12. Wang, K., Baugh, C. M., Boardman, N. F., Bose, S., Cai, Z., Cole, S., Frenk, C. S., Lacey, C., Norberg, P., Peng, Y., Santos-Santos, I., Saintonge, A., Theuns, T., Wang, E. (2026). Inflow-driven galaxy evolution – I. Revealing the physics of the fundamental metallicity relation. arXiv:2608.04784. https://arxiv.org/abs/2608.04784
13. Khostovan, A. A., Sanders, R. L., Shapley, A. E., Topping, M. W., Reddy, N. A., Garcia, A. M., Berg, D. A., Clarke, L., Cullen, F., Ellis, R. S., Förster Schreiber, N. M., Glazebrook, K., Jones, T., McLeod, D. J., Pahl, A., Pettini, M., Torrey, P. (2026). The AURORA Survey: The Mass–Metallicity and Fundamental Metallicity Relations at $z\sim2.3$ Based Purely on Direct Te Metallicities. The Astrophysical Journal Letters 1003, L41. DOI: 10.3847/2041-8213/ae63d2. https://doi.org/10.3847/2041-8213/ae63d2 (arXiv:2512.16989, https://arxiv.org/abs/2512.16989)
14. Lam, N., Clarke, L., Shapley, A. E., Sanders, R. L., Topping, M. W., Brammer, G. B., Reddy, N. A., Karthikeyan, S. (2026). The JADES Mass-Metallicity and Fundamental Metallicity Relations at $z\gtrsim2$ Using New High-Redshift Metallicity Calibrations. arXiv:2605.30513. https://arxiv.org/abs/2605.30513
15. He, X., Li, Z., Wang, X., Cai, Z., Jones, T., Treu, T., Vulcani, B., Malkan, M. A., Glazebrook, K. (2026). Revisiting the mass–metallicity relation and the fundamental metallicity relation of dwarf galaxies at cosmic noon with NIRISS. Monthly Notices of the Royal Astronomical Society 550, stag1235. DOI: 10.1093/mnras/stag1235. https://doi.org/10.1093/mnras/stag1235 (arXiv:2605.20810, https://arxiv.org/abs/2605.20810)