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Inflow-driven galaxy evolution - I. Revealing the physics of the fundamental metallicity relation

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper argues that the fundamental metallicity relation is not a fundamental law but a projection of a simpler ratio, and it derives the closed-form gas-flow solution that shows why.

desk verdict A clean analytic core, an honest paper, and a load-bearing assumption about eta(z) that should decide how much of the FMR interpretation you buy. read the letter →

arxiv 2608.04784 v1 pith:SF7IC5T5 submitted 2026-08-05 astro-ph.GA astro-ph.CO

Kai Wang , Carlton Baugh , N. F. Boardman , Sownak Bose , Zheng Cai , Shaun Cole , Carlos S. Frenk , Cedric Lacey
show 6 more authors
Peder Norberg Yingjie Peng Isabel Santos-Santos Amélie Saintonge Tom Theuns Enci Wang
This is my paper
classification astro-ph.GAastro-ph.CO
keywords fundamentalmetallicityrelationgas-phaseinflow-drivenregimebaryoncyclemass-loadingfactorstarformationefficiencygaseousFMRcosmologicalgasflowmodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the fundamental metallicity relation (FMR) — the tight, nearly redshift-invariant surface linking stellar mass, star formation rate, and gas metallicity — is not a deep symmetry of galaxy formation but a contingent product of how two galaxy parameters happen to depend on stellar mass and redshift. Working from two mass-continuity equations, the authors show that in the inflow-driven regime, where freshly accreted gas dominates over depletion, the gas metallicity collapses to a simple ratio, $Z_{\rm g} = (y/(1-R))\,M_\star/M_{\rm g}$, independent of how efficiently a galaxy turns gas into stars. The same physics makes the standard FMR a projection of this more basic gaseous relation, and the paper's closed-form solution (the $K_1$–$K_2$ system) lets any one of metallicity, gas fraction, or outflow strength be inferred from the other two. If the argument holds, metallicity scaling relations become precision probes of the baryon cycle — inflows, star formation, and outflows — across cosmic time.

What carries the argument

The machinery is a pair of mass-continuity equations, (2.1) and (2.2), governing the galaxy's gas reservoir and its metal content under inflow $\Phi$, star formation at efficiency $\epsilon \equiv {\rm SFR}/M_{\rm g}$, and outflows at mass-loading factor $\eta$. The central object is the equilibrium timescale $\tau_{\rm eq} = 1/[(1-R+\eta)\epsilon]$, the characteristic time for star formation plus outflows to balance inflow, whose ratio to the evolution time $t$ places every galaxy on a continuum between the inflow-driven regime ($t \ll \tau_{\rm eq}$), where $Z_{\rm g} = (y/(1-R))\,M_\star/M_{\rm g}$ exactly, and equilibrium ($t \gg \tau_{\rm eq}$), where $Z_{\rm g} \to y/(1-R+\eta)$. The closed-form system in equations (4.1)–(4.2) expresses that transition through two functions of $t/\tau_{\rm eq}$, $K_1$ and $K_2$, which tie $Z_{\rm g}$, $M_{\rm g}/M_\star$, and $\eta$ together so that any two determine the third; the calibrated cosmological model then shows that the observed FMR surface is the projection of this universal curve once $\epsilon$ and $\eta$ are given their realistic power-law dependences.

What would settle it

Measure the gas fraction $M_{\rm g}/M_\star$ and the star formation rate for a large sample of $z \approx 2$–$3$ star-forming galaxies: the inflow-driven claim predicts $M_{\rm g}/M_\star \gg (1-R+\eta)/(1-R)$ for the bulk of the population, with metallicity tracking $M_\star/M_{\rm g}$. If high-redshift galaxies instead show gas fractions close to the equilibrium expectation, or if direct wind observations at $z > 1.5$ show $\eta$ rising with redshift, the claim that the star-forming population is predominantly inflow-driven — and with it the attribution of metal-enrichment evolution to $\epsilon$ — collapses.

Watch

Extended reading notes

Core claim

The paper's central claim is that the FMR emerges from the transition between two regimes of gas-flow physics, and that its observed parameterisation encodes only the mass and redshift dependence of the star formation efficiency, $\epsilon$, and the mass-loading factor, $\eta$. In the calibrated cosmological gas flow model, galaxies collapse onto one universal sequence of $Z_{\rm g}$ against $M_\star$/SFR when $\epsilon$ and $\eta$ are constants; reintroducing their realistic power-law dependences generates exactly the offsets among mass bins and epochs that the FMR parameter $\alpha$ is then used to absorb. The gaseous FMR, $Z_{\rm g} = (y/(1-R))\,M_\star/M_{\rm g}$, is claimed to be the more fundamental relation because it holds independent of $\epsilon$ in the inflow-driven limit, with the approach to equilibrium governed by gas fraction and $\eta$ alone. The analytic solution of the ideal constant-inflow model (equations 4.1–4.2) provides closed-form functions $K_1(t/\tau_{\rm eq})$ and $K_2(t/\tau_{\rm eq})$ that relate $Z_{\rm g}$, $M_{\rm g}/M_\star$, and $\eta$, and it reproduces the full cosmological calculation, so measured metallicities and gas fractions can be turned into inferences about outflows and star formation efficiency.

Load-bearing premise

The load-bearing premise, invoked in Section 5.7 to break the degeneracy between star formation efficiency and outflow strength, is that the mass-loading factor $\eta$ does not evolve with redshift; if outflows at fixed stellar mass were substantially stronger at high redshift, an equilibrium model with $\eta(z)$ could reproduce the same redshift evolution of the mass–metallicity relation without galaxies being inflow-driven at all.

Editorial extensions

If this is right

  • The gFMR ($Z_{\rm g}$ versus $M_\star/M_{\rm g}$) is the primary relation; the standard FMR is its projection into SFR-space, so gas mass measurements should replace or augment SFR in future metallicity scaling studies.
  • The FMR parameter $\alpha$ is not universal: it is bracketed between the single-epoch collapse value $\alpha_m \approx 0.75$ and the redshift-invariance value $\alpha_z \approx 0.45$ for the fiducial model, with the observed $\alpha \approx 0.55$ lying between and encoding the mass and redshift slopes of $\epsilon$ and $\eta$.
  • Semi-analytic models with high mass-loading factors push galaxies into equilibrium and predict a nearly non-evolving mass–metallicity relation; lowering $\eta$ and $\epsilon$ lengthens $\tau_{\rm eq}$, keeps galaxies inflow-driven, and produces both the observed MZR evolution and a redshift-invariant FMR from one mechanism.
  • The gas-to-stellar metallicity difference should decline monotonically from about 0.18 dex in low-mass, high-sSFR galaxies to zero in massive systems, a trend the paper matches to integral-field survey measurements.
  • The model's fiducial calibration predicts $\epsilon \propto (1+z)^{1.1}$, a direct prediction for gas and SFR surveys at $z \approx 1$–$3$ that would confirm or rule out the inflow-driven attribution of MZR evolution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The closed-form system also inverts into a redshift test the authors do not run: with gas masses measured at $z \approx 1$–$3$, $K_1$–$K_2$ would yield the mass-loading factor directly at each epoch, providing a stacking-based check of the constant-$\eta$ premise from existing CO and dust-continuum surveys.
  • Because the inflow rate cancels out of every relation among $M_\star$, SFR, $M_{\rm g}$, and $Z_{\rm g}$ in the inflow-driven limit, the model implies that metallicity encodes the timing and duration of enrichment, not the rate of gas accretion; a testable corollary is that metallicity scatter should track variations in star-formation histories rather than scatter in halo accretion rates.
  • The bracketing of $\alpha$ between $\alpha_m \approx 0.75$ and $\alpha_z \approx 0.45$ suggests a consistency check the paper leaves implicit: measuring $\alpha$ locally together with the mass slope of $\epsilon$ would predict the redshift slope of $\epsilon$, independently verifiable from gas-depletion timescales.
  • If the gFMR is the primary relation, its near-epoch-invariance should persist even where the standard FMR breaks down at $z \gtrsim 4$; that gives JWST abundance programs a sharper target — test whether $Z_{\rm g}$–$M_\star/M_{\rm g}$ stays universal when $Z_{\rm g}$–$M_\star$/SFR does not.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a gas-flow model based on the baryon mass-continuity equations to explain the fundamental metallicity relation (FMR). It argues that the FMR is not a fundamental symmetry but arises from the transition between an inflow-driven regime and an equilibrium regime, and that the gaseous FMR (gFMR), relating Z_g to M_star/M_g, is more fundamental than the standard FMR. The authors construct a cosmological gas flow model calibrated to the mass–metallicity relation (MZR), the star-forming main sequence (SFMS), and the stellar–halo mass relation, and claim the FMR and gFMR emerge as predictions. They also derive closed-form analytic expressions (Eqs. 4.1–4.3) relating Z_g, M_g/M_star, and the mass-loading factor η, and discuss implications for previous models, degeneracies, and high-redshift extrapolations.

Significance. The analytic K1–K2 framework in Sections 4 and Appendix F is a clear and potentially useful contribution: it provides closed-form relations that interpolate between the inflow-driven and equilibrium regimes and offers a route to infer η from gas fraction and metallicity. The generalization to differential mass and metal loading (Section 5.4) is also valuable. The paper makes explicit, falsifiable predictions, including the maximum gas-to-stellar metallicity difference of about 0.18 dex (Section 5.6) and the sensitivity of high-redshift FMR offsets to the local calibration slope (Section 5.5). These strengths are substantial. However, the central empirical claim that the evolving MZR requires the inflow-driven regime is contingent on the assumption that η does not evolve with redshift, and the claim that the FMR is a non-trivial prediction is weakened by the fact that the model is calibrated to the MZR and SFMS, which jointly define the median FMR locus. The manuscript is worth publication after these load-bearing points are addressed.

major comments (4)
  1. [§3.1, Fig. 4] The FMR is presented as a non-trivial prediction because it was not a calibration target, but the model is calibrated to the median MZR and median SFMS at each redshift (§2.2.5). Together these two relations already specify, at each z, the median SFR and Z_g for every M_star, i.e. a one-dimensional locus in (M_star, SFR, Z_g) space. The median FMR surface is therefore largely inherited from the calibration data. What is genuinely predictive is the residual-level anti-correlation (e.g., galaxies with higher SFR at fixed M_star being more metal-poor) and whether a single α=0.55 aligns the redshift-dependent loci. Please state this distinction explicitly and show the residual-level prediction (for example, the model's FMR scatter after fitting only the median MZR and SFMS), rather than presenting the median surface itself as an emergent, non-trivial prediction.
  2. [§5.7, Eq. (5.15)] The degeneracy between the star formation efficiency ε(t) and the mass-loading factor η(t) is acknowledged, and the paper breaks it by assuming η has no redshift dependence. This assumption is load-bearing for the central conclusion that the evolving MZR requires the inflow-driven regime. Equation (5.15) shows that any assumed η(t) yields a unique ε(t) that reproduces the same SFR(t) and Z_g(t); an equilibrium model with η(M_star, z) can therefore reproduce the observed MZR evolution without requiring galaxies to be predominantly inflow-driven. The cited empirical support (to z~1.5) and the unpublished Wang et al. (2026, in prep.) do not establish η(z)=const at z~2–3, and the z=0 gas-mass comparison in Fig. 16 does not constrain η(z) at high redshift. The argument in §5.9 that equilibrium plus stochastic accretion cannot produce a tight, redshift-invariant FMR is an argument by elimination. Please either (i) provide direct or indirect constraints on η(z) at z~2–3, (ii) demonstrate that the central conclusions are robust to plausible η(z) variations within the z≲1.5 constraints, or (iii) explicitly qualify the inflow-driven interpretation as conditional on non-evolving η.
  3. [§3.2, Fig. 5] The gFMR is defined and displayed using α_g=0.85, yet the claimed fundamental inflow-driven relation is Z_g = y/(1−R) × M_star/M_g, which corresponds to α_g=1. The paper never derives α_g=0.85 from the K1–K2 system or from an independent fit, and the reader is left to wonder whether α_g=1 would actually collapse the model points. If α_g=1 does not produce a tight collapse across mass and redshift, then the gFMR, as operationally defined, is itself an empirical projection whose 'more fundamental' status needs qualification. Please show the α_g=1 projection, or derive the empirical α_g from the equilibrium contribution of η(M_star) in Eqs. (4.1)–(4.2), and discuss why the gFMR parameter is not unity.
  4. [§2.2.5, Table 1] The calibration is performed by manual tuning with no reported uncertainties or residuals, yet several quantitative conclusions are drawn from the fiducial parameter values: α_m≈0.75, α_z≈0.45, the bracketing in Eq. (4.7), and the inference of ε_z=1.1. Because the five parameters that 'directly affect the main results' (ε_0, ε_m, ε_z, η_0, η_m) are not accompanied by error bars, the reader cannot assess whether the agreement with the observed α≈0.55 is meaningful or a product of tuning. Please add a sensitivity analysis for these parameters, at minimum varying ε_m, ε_z, and η_m within observationally motivated ranges, and show how the predicted FMR/gFMR properties and the inferred ε_z depend on those choices.
minor comments (4)
  1. [Fig. 4] The label 'FMRfitting' in the upper panel should read 'FMR fitting'.
  2. [References] Jain et al. (2025) is listed as an arXiv paper only; since it is the primary calibration dataset for the MZR and SFMS, please provide the publication status or DOI when available.
  3. [§2.2.5] Please define more precisely what 'fitting' means in the manual calibration: Fig. 3 shows model lines and data, but no residuals, scatter statistics, or goodness-of-fit are reported, making it hard to gauge the quality of the simultaneous fit.
  4. [Appendix D] The Eddington-bias demonstration would be more convincing if the adopted Gaussian scatters (σ_M=0.2 dex, σ_SFR=0.4 dex) were justified with explicit observational references, since the resulting slope depends directly on these choices.

Circularity Check

1 steps flagged · score 4.0 of 10

Core K1–K2 derivation is self-contained, but the central inflow-driven interpretation leans on an eta(z)=const assumption supported beyond z~1.5 only by an in-prep self-citation, and the FMR 'prediction' is partly a postdiction of the calibrated MZR and SFMS.

  1. self citation load bearing [Section 5.7, 'Degeneracy between star formation efficiency and mass-loading factor'; reference list: 'Wang K., et al., 2026, in prep.']
    "We break the degeneracy by assuming that the mass-loading factor does not evolve with redshift, following the observational evidence from z∼0 to z∼1.5 (Heckman et al. 2015; Chisholm et al. 2017; Schroetter et al. 2019, 2024). This assumption is supported empirically by Wang et al. (2026), who use gas mass and metallicity observations at z∼0–1 to constrain the mass-loading factor directly and find that it evolves little over the past ∼8 Gyr."

    Equation (5.15) in the same section shows that for any assumed η(t), a corresponding ε(t) reproduces the same SFR(t) and Zg(t); the inference that galaxies are inflow-driven and that the FMR shape encodes ε(M*, z) rather than η(z) is obtained only after imposing η(z)=const. The external wind data cited reach z∼1.5, while the model is applied to z∼3; the additional support offered for the constancy is an unpublished (in prep.) paper by the present first author. The premise is therefore load-bearing and its extension beyond the verified window rests on a self-citation rather than on independent evidence. This is a self-referential support chain for the central interpretation, not an equation-level circularity.

full rationale

The analytic backbone of the paper (equations 2.1–2.19 and the K1–K2 system of equations 4.1–4.3) is derived from the mass-continuity equations and is self-contained: the inflow-driven scalings and the closed-form expressions do not assume the FMR or gFMR they are used to explain. The cosmological model is calibrated to the MZR, SFMS, and SHMR, and the FMR is then exhibited as an emergent relation; because the model also determines off-median correlations, this is not a pure fit, although it is a postdiction rather than an independent prediction. The paper is transparent about the ε–η degeneracy and states the η(z)=const assumption explicitly. The main circularity-adjacent issue is that the load-bearing assumption used to break that degeneracy is supported beyond z~1.5 only by a self-cited in-prep paper (Wang et al. 2026), and the conclusion that the FMR is inflow-driven would lose its empirical footing if η evolved with redshift. This warrants a moderate score of 4 rather than a higher one, because no central equation reduces to its inputs by construction.

Assumptions & free parameters 12 free parameters · 7 assumptions · 0 invented entities

The central claims rest on 10 calibrated model parameters (Table 1), plus the adopted concentration c and the hand-chosen projection exponent alpha_g for the gFMR. The key hidden assumptions are smooth accretion, instantaneous mixing, no recycling, power-law epsilon and eta, and a redshift-independent eta. No new physical entities are introduced.

free parameters (12)
  • lambda0 = 0.35
    Cooling efficiency normalization in Eq (2.22); calibrated to the stellar-to-halo mass relation.
  • lambda_m1 = -0.60
    Low-mass suppression slope in Eq (2.22); calibrated to the stellar-to-halo mass relation.
  • lambda_m2 = 0.20
    High-mass suppression slope in Eq (2.22); calibrated to the stellar-to-halo mass relation.
  • lambda_z = 0.20
    Redshift exponent for cooling efficiency in Eq (2.22); calibrated to the stellar-to-halo mass relation.
  • epsilon0 = 0.38 Gyr^-1
    Star formation efficiency normalization in Eq (2.23); calibrated to the SFMS and MZR.
  • epsilon_m = 0.33
    Stellar mass slope in Eq (2.23); calibrated to the SFMS and MZR.
  • epsilon_z = 1.1
    Redshift exponent in Eq (2.23); calibrated to the MZR and SFMS.
  • eta0 = 0.22
    Mass-loading normalization in Eq (2.26); calibrated to the MZR.
  • eta_m = -0.30
    Mass-loading stellar mass slope in Eq (2.26); calibrated to the MZR.
  • y = 0.012
    Metal yield, treated as free to absorb metallicity calibration systematics; calibrated to the MZR normalization.
  • c = 12
    Concentration parameter in the Wechsler et al. (2002) halo accretion history (Eq 2.21); adopted to match Fakhouri et al. (2010) means, with 0.15 dex scatter.
  • alpha_g = 0.85
    Exponent used to project the gFMR in Fig 5; chosen by hand to collapse the relation. The analytic gFMR in Eq (2.19) has slope unity, so 0.85 is an empirical compromise for the full model.
assumptions (7)
  • domain assumption Instantaneous and uniform mixing of newly produced metals in the ISM
    Section 2 states 'This description assumes that newly produced metals are instantaneously and uniformly mixed within the ISM.' This is load-bearing for the closed-form solutions, but real galaxies have imperfect mixing.
  • domain assumption Gas accretion is smooth and deterministic, with no stochasticity or mergers
    Section 2.2.1 models accretion as analytic halo growth; Section 5.10 lists stochastic accretion and mergers as omitted. The FMR scatter in real data is not addressed.
  • domain assumption Gas ejected from the galaxy is permanently removed; no recycling from the CGM
    Section 2 states: 'Higher-order processes such as gas recycling from the CGM are neglected.' Recycling is known to be important in semi-analytic models, and its omission affects the effective yield and inflow metallicity.
  • domain assumption The mass-loading factor eta does not evolve with redshift
    Section 5.7 states: 'We break the degeneracy by assuming that the mass-loading factor does not evolve with redshift...' This is load-bearing for attributing MZR evolution to the inflow-driven regime.
  • domain assumption Star formation efficiency and mass-loading factor follow power laws in stellar mass and redshift (Eqs 2.23 and 2.26)
    Section 3.3.3 states the FMR parameterization works because these dependencies are approximately power laws. The paper's analytic predictions for alpha (Eqs 4.5 and 4.6) depend on these forms.
  • domain assumption Halo mass growth follows the Wechsler et al. (2002) formula with concentration c=12
    Section 2.2.1, Eq (2.21). The inflow rate is derived from this; halo-to-halo scatter is only partially captured by a 0.15 dex log-normal spread.
  • domain assumption The one-zone mass-continuity equations (2.1) and (2.2) fully describe the baryon cycle
    The model reduces a galaxy to a single well-mixed reservoir. This is the foundation of the entire derivation; it neglects spatial structure, metallicity gradients, and multiphase gas.

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Pith. "Pith review of Inflow-driven galaxy evolution - I. Revealing the physics of the fundamental metallicity relation." pith.science (2026). https://pith.science/paper/SF7IC5T5

@misc{pith2026260804784,
  author       = {Pith},
  title        = {Pith review of: Inflow-driven galaxy evolution - I. Revealing the physics of the fundamental metallicity relation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SF7IC5T5}},
  note         = {Machine review of arXiv:2608.04784}
}
abstract

We present a unified physical framework for the fundamental metallicity relation (FMR), based on the mass-continuity equations. The FMR is not merely the anti-correlation between star formation rate (SFR) and gas metallicity ($Z_{\rm g}$) at fixed stellar mass ($M_\star$); it is a redshift-invariant surface in the $(M_\star,{\rm SFR},Z_{\rm g})$ space. We construct a minimal cosmological gas flow model, calibrated to reproduce the mass-metallicity relation, star-forming main sequence, and stellar-to-halo mass relation at $z=0-3$, and show that the FMR emerges as a prediction of the calibrated physics. Through controlled experiments that progressively simplify the model, we reveal that in a universe where both the star formation efficiency ($\epsilon$) and mass-loading factor ($\eta$) are constants, the FMR reduces to a universal scaling between $Z_{\rm g}$ and $M_\star/$SFR, whose shape traces the transition from inflow-driven regime to equilibrium. The specific parameterisation of the observed FMR is not a fundamental symmetry but a contingent consequence of how $\epsilon$ and $\eta$ depend on stellar mass and redshift. We show that the gaseous FMR (gFMR), defined in the $(M_\star,M_{\rm g},Z_{\rm g})$ space, is more fundamental than the standard FMR: in the inflow-driven limit, $Z_{\rm g}$ is proportional to $M_\star/M_{\rm g}$, and the approach to equilibrium is governed by $M_\star/M_{\rm g}$ and $\eta$ alone. We derive an analytic solution for an idealised version of the model that provides closed-form expressions relating $Z_{\rm g}$, $M_{\rm g}/M_\star$, and $\eta$, and show this framework accurately reproduces the cosmological gas flow model. By establishing the physical origin of the FMR and its connection to the more fundamental gFMR, we provide the theoretical foundation to turn metallicity scaling relations into precision probes of the baryon cycle over cosmic history.

Figures

Figures reproduced from arXiv: 2608.04784 by the authors.

Figure 1
Figure 1. Schematic illustration of the gas flow model adopted in this work. Gas is accreted from the intergalactic medium onto the dark matter halo, where it is shock-heated and added to the hot halo atmosphere. A fraction of this gas cools and settles onto the central galaxy, providing the fuel for star formation. Massive stars return a fraction 𝑅 of the newly formed stellar mass to the ISM along with freshly synthesised me… view at source ↗
Figure 2
Figure 2. Time evolution of the gas mass 𝑀g (𝑡), gas phase metallicity 𝑍g (𝑡), stellar mass 𝑀★(𝑡), and stellar metallicity 𝑍★(𝑡) for the idealised system with constant inflow rate Φ, obtained from the analytic solution in equations. (2.5)–(2.8). Each quantity is normalised properly (Φ𝜏eq for 𝑀g, 𝑦/(1 − 𝑅 + 𝜂) for 𝑍g and 𝑍★, and Φ𝜏eq (1 − 𝑅)/(1 − 𝑅 + 𝜂) for 𝑀★) and shown as a function of 𝑡/𝜏eq on logarithmic axes. The system t… view at source ↗
Figure 3
Figure 3. Calibration of the cosmological gas flow model against three observational scaling relations, each shown at six redshifts from 𝑧 ≈ 0.08 to 𝑧 = 3.30. Left panel: the gas-phase mass–metallicity relation from Jain et al. (2025). Middle panel: the star-forming main sequence from Jain et al. (2025). Right panel: the stellar mass–halo mass relation from the UniverseMachine empirical model (Behroozi et al. 2019). Solid lin… view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Gas metallicity as a function of the FMR parameter, log10 (𝑀★/M⊙ ) − 𝛼 log10 (SFR/M⊙ yr−1 ), with 𝛼 = 0.55, predicted by the cosmological gas flow model. Upper panel: results at 𝑧 = 0, colour-coded by SFR. Different SFR bins collapse onto a single sequence, consistent …
Figure 5
Figure 5. Figure 5: Gas metallicity as a function of the gFMR parameter log10 (𝑀★/M⊙ ) − 𝛼g log10 (𝑀g/M⊙ ), predicted by the cosmological gas flow model. Marker shapes distinguish four stellar mass bins from 𝑀★ ∼ 108 to 1011 M⊙, and different shading denotes redshifts 𝑧 = 0–3. All stellar…
Figure 6
Figure 6. Figure 6: Gas metallicity, 12 + log10 (O/H), versus the inverse of sSFR, log10 𝑀★/SFR, predicted by the cosmological gas flow model under five assumptions about the star formation efficiency 𝜖 ≡ SFR/𝑀g and the mass-loading factor 𝜂. Marker shapes denote stellar mass bins (𝑀★ ∼ 1…
Figure 7
Figure 7. Figure 7: Same as [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: The three functions K1, K2, and K3 that govern the analytic solution of the ideal gas flow model (equations 4.1–4.3), plotted as functions of the evolutionary stage 𝑥 ≡ 𝑡/𝜏eq. Dashed lines show the asymptotic limits. In the inflow-driven regime (𝑥 ≪ 1): K1 → 1, K2 → 𝑥/…
Figure 9
Figure 9. Figure 9: Comparison between the cosmological gas flow model (symbols) and the analytic ideal model (solid lines). Left panel: the FMR, showing gas metallicity as a function of the FMR parameter log10 (𝑀★/M⊙ ) − 𝛼 log10 (SFR/M⊙ yr−1 ) with 𝛼 = 0.55. Right panel: the gFMR, showin…
Figure 10
Figure 10. Figure 10: Gas metallicity as a function of the stellar-to-gas mass ratio 𝑀★/𝑀g, predicted by the ideal gas flow model with 𝜂 = 0. The metallicity is normalised by the maximum (outflow-free) yield 𝑦/(1 − 𝑅), and the horizontal axis is equivalently log10 𝑀★ − log10 SFR/𝜖 , showin…
Figure 11
Figure 11. Figure 11: Diagnostic of the equilibrium approximation in the gas flow model. Both panels show the ratio ( 𝜖 −1 d𝑍g/d𝑡)/𝑦, which measures the rel￾ative importance of the time-derivative term in equation (5.1) to the yield. The equilibrium approximation requires this ratio to be …
Figure 12
Figure 12. Figure 12: Geometric illustration of the distinction between total (La￾grangian) and partial (Eulerian) derivatives of metallicity with respect to stellar mass. The dashed curves show the mass–metallicity relation at two epochs separated by Δ𝑧, with the lower-redshift MZR (grey)…
Figure 13
Figure 13. Figure 13: Ratio of the sum of partial (Eulerian) derivatives to the sum of total (Lagrangian) derivatives entering the 𝛼 parameter (equation 5.3), evaluated from the cosmological gas flow model at 𝑧 = 0 , 1 , 2 , and 3 . A ratio of unity indicates that the observed slopes of th…
Figure 14
Figure 14. Figure 14: Comparison between the gas metallicity predicted by the K1–K2 framework derived in this work (solid lines) and the universal metallicity relation of Zahid et al. (2014) (dashed lines; equation H7), plotted as a function of the gas fraction 𝑀g/𝑀★ for three values of th…
Figure 16
Figure 16. Figure 16: Total cold gas mass, 𝑀g, as a function of stellar mass, 𝑀★, compared to the analytic model developed in this work. Hexagons show 𝑀g ≡ 1.36 × 𝑀HI + 𝑀H2 , the atomic plus molecular gas mass with a helium correction applied to 𝑀HI, compiled from the xGASS and xCOLD GASS …
Figure 17
Figure 17. Figure 17: Effect of varying the gas inflow rate on galaxy scaling relations. Solid lines show the fiducial model; dashed and dotted lines show models in which the cooling efficiency is reduced by a factor of 5 and increased by a factor of 5, respectively, at all halo masses and…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.