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A Bayesian mixture model measures the SFR–radio relation without cutting out AGN, finding clear redshift evolution but only weak stellar-mass dependence.

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2026-07-12 01:43 UTC pith:KGJ2DZFM

load-bearing objection Cleaner, less-circular SFR–radio calibration with a hierarchical mixture; redshift term is real, mass term is weak, and the spectral-index assumption is already stress-tested. the 2 major comments →

arxiv 2607.03543 v1 pith:KGJ2DZFM submitted 2026-07-03 astro-ph.GA

A MIGHTEE robust measurement of the star formation rate-radio correlation

Charlotte L. Jackson , Matt J. Jarvis , Imogen H. Whittam , James H. Matthews , Johannes Buchner , Catherine L. Hale , Joel Hamlett , S. Lyla Jung
show 4 more authors
Mara Salvato Daniel J. B. Smith Natalia Stylianou Rohan G. Varadaraj
This is my paper
classification astro-ph.GA
keywords star formation rateradio continuumSFR–radio correlationactive galactic nucleiBayesian mixture modelspectral index evolutiongalaxy evolutionMIGHTEE
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Radio luminosity is a dust-free tracer of star formation, but earlier calibrations of the SFR–radio correlation had to throw away sources that looked like AGN using hard cuts. Those cuts can bias the remaining sample. This paper replaces the cuts with a hierarchical Bayesian mixture model that treats every radio-detected galaxy as a probabilistic mixture of a star-formation-dominated sequence and a non-SF background. The model is fit to the deep MIGHTEE COSMOS catalogue while folding in the full posterior SFR distributions from multi-wavelength SED fitting. The result is a mean relation whose slope is sub-linear, whose intercept rises gently with redshift, and whose stellar-mass term is only ~0.13 dex across four decades in mass—far weaker than previous claims. The authors show that a mild steepening of the radio spectral index with redshift can fully account for the observed redshift term, and that the same relation holds inside volume-limited subsets free of Malmquist bias.

Core claim

The mean star-formation-dominated SFR–radio correlation is log10(SFR/M⊙ yr−1) = 0.790 imes (log10(L1.4/W Hz−1) − 23) + 1.244 imes (1 + z)^0.122 − 0.033 imes (log10(M*/M⊙) − 10), with an intrinsic scatter of 0.178 dex. Redshift evolution is required at high significance; stellar-mass dependence is statistically present but practically negligible.

What carries the argument

A two-component hierarchical Bayesian mixture model whose SF mean is linear in log radio luminosity, additive in a power-law redshift term and a linear mass term, and whose SF fraction is a logistic function of luminosity and stellar mass; the model is sampled with the full per-source SFR posterior.

Load-bearing premise

The conversion from observed radio flux to rest-frame 1.4 GHz luminosity assumes a spectral index that is either fixed at −0.7 or allowed only a simple linear change with redshift; any real curvature or population dependence not captured by that form is absorbed into the reported redshift evolution.

What would settle it

Matched-resolution multi-frequency radio imaging of the same MIGHTEE fields that measures the true spectral-index distribution of SF-dominated galaxies as a function of redshift; if the measured indices show no systematic steepening of order Δα ≈ −0.07 per unit redshift, the physical explanation offered for the redshift term fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Future radio surveys can convert luminosity to SFR with a relation whose mass term can be dropped at little cost and whose redshift term is now quantified.
  • Binary AGN/SFG cuts are no longer required for calibrating the relation; probabilistic membership is sufficient and less biased.
  • Claims of strong stellar-mass dependence in the IRRC or SFR–radio plane should be re-examined under mixture modelling that keeps AGN in the sample.
  • A mild spectral-index evolution of order α(z) = −0.7 − 0.068z is already enough to erase the need for an explicit redshift term in the SFR–radio plane.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the same mixture approach is applied to other SFR tracers (Hα, UV, IR), residual mass or redshift trends currently attributed to dust or IMF variations may likewise shrink once AGN contamination is treated probabilistically.
  • The logistic SF-fraction surface recovered here is itself a selection-function map; it can be used as a prior when stacking or luminosity-function modelling of the same field.
  • Once S-band data arrive, the spectral-index explanation can be turned into a joint fit of α(z) and the SFR–radio plane rather than a post-hoc test.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper derives a mean SFR–1.4 GHz radio luminosity relation for star-formation-dominated galaxies using a Bayesian hierarchical two-component mixture model applied to MIGHTEE COSMOS DR1 radio sources, with SFRs taken from full GRAHSP SED posteriors. Binary SFG/AGN cuts are avoided; instead an evolving logistic mixture fraction (in L1.4 and M*) jointly models the SF sequence and a radio-excess background. The preferred relation is log10(SFR) = 0.790 L' + 1.244 (1+z)^0.122 − 0.033 M' (L' = log10 L1.4 − 23, M' = log10 M* − 10) with intrinsic scatter 0.178 dex. Redshift evolution is required at high significance; stellar-mass dependence is weak. The redshift term can be re-absorbed into a mild spectral-index evolution α(z) = −0.7 − 0.068 z. Volume-limited bootstrap KS tests support that the relation is not driven by Malmquist bias.

Significance. If the result holds, the work supplies a practically usable, dust-agnostic SFR–radio calibration for deep radio surveys (MeerKAT, LOFAR, SKA pathfinders) that does not rely on the circular radio-excess cuts common in IRRC-based studies. The hierarchical mixture approach, full SFR-posterior marginalisation, and explicit model comparison via WAIC/PSIS-LOO-CV are methodological strengths that the community can reuse. The finding that mass dependence is much weaker than in recent IRRC work, while redshift evolution remains, is a concrete, falsifiable claim that can be tested with forthcoming multi-band MIGHTEE data. The paper is therefore of clear interest for galaxy evolution and radio continuum cosmology.

major comments (2)
  1. §2.1 and §3: photometric redshifts are adopted for a substantial fraction of the sample but their posteriors are not sampled (only SFR posteriors are). Because both L1.4 (Eq. 1) and the (1+z) term in μ_SF depend on z, residual photo-z scatter or catastrophic outliers can couple into the reported redshift evolution. A quantitative test—e.g. restricting to the spectroscopic/PAU subset, or Monte-Carlo sampling the photo-z PDFs for a representative subsample—should be shown so that the significance of γ_z = 0.122 ± 0.019 can be assessed under realistic redshift uncertainty.
  2. §4.4 / Eq. (1): the conversion from observed flux to rest-frame L1.4 assumes either fixed α = −0.7 or a simple linear α(z) = −0.7 − 0.068 z. The paper already demonstrates that the redshift term can be re-absorbed into α(z) and that the slope remains consistent, which is good. However, any spectral curvature or population-dependent spectral shape not captured by this one-parameter form will still be absorbed into the reported (1+z) term. Given that multi-frequency MIGHTEE S-band data are imminent, the manuscript should state more explicitly that the present redshift coefficient is degenerate with spectral-index assumptions and should be re-calibrated once measured α are available.
minor comments (5)
  1. Table numbering is inconsistent: the main posterior table is labelled Table 3 in the text but appears as Table 5 in the caption; the second/third-best models are Table 4. Please renumber for consistency.
  2. Fig. 2 right panel: the quantity log10(ΔSFR/SFR) is defined in the caption but the axis label is abbreviated; a short explicit definition in the figure itself would help.
  3. §3.1: the cut log10(ΔSFR/SFR) > 1 removes only 1.8 % of sources; a one-sentence statement that results are insensitive to the precise threshold would strengthen the claim of robustness.
  4. Eq. (4) and surrounding text: the decision to drop the constant intercept ζ when both mass and redshift terms are present is sensible, but a brief note that the intercept is absorbed into the normalisation of the (1+z) term would clarify the parameterisation for readers who wish to implement the relation.
  5. References: a few in-prep works (Hale et al. submitted; Stylianou et al.; Jackson et al.) are cited for catalogue construction; ensure that the public data products needed to reproduce the sample selection are clearly identified or will be released with the paper.

Circularity Check

0 steps flagged

No significant circularity: the hierarchical mixture model jointly fits free parameters for the SF mean relation and evolving background fraction; the reported coefficients are posterior results, not identities or pre-assumed cuts.

full rationale

The paper's central result is an empirical fit of a two-component hierarchical Bayesian mixture (SF-dominated Gaussian around a parametric mean µ_SF plus a background Gaussian) to the joint distribution of GRAHSP SFR posteriors and MIGHTEE L_1.4 values. All coefficients (η_L, η_z, γ_z, η_m, φ, ψ_L, ψ_m, σ_int, …) are free parameters sampled by NUTS after AIC/WAIC/LOO model selection among an exhaustive suite of functional forms (Table 1–2). The final relation (Eq. 14) is therefore the MAP/posterior mean of those free parameters, not a quantity forced by construction from an input definition or from a prior cut. The authors explicitly contrast this with the classic circular procedure of first excising radio-excess sources relative to an assumed IRRC and then re-fitting the same relation. Mild self-citations (MIGHTEE DR1 catalogues, GRAHSP SED code) supply data products and software; they do not supply a uniqueness theorem or ansatz that forces the slope, redshift exponent or mass coefficient. Spectral-index assumptions (fixed α = −0.7 or linear α(z)) are modelling choices that are tested and re-absorbed, not circular reductions. Volume-limited KS tests further confirm the relation is recovered inside complete sub-samples. Hence the derivation chain is self-contained and data-driven; circularity score is near zero.

Axiom & Free-Parameter Ledger

7 free parameters · 5 axioms · 0 invented entities

The central numerical relation is obtained by fitting a hierarchical mixture whose free coefficients are optimised to the MIGHTEE+GRAHSP data under standard cosmological and SED-modelling assumptions. No new physical entities are postulated; the free parameters are the usual slope, intercept, evolution indices and mixture weights.

free parameters (7)
  • η_L (slope of log SFR vs log L1.4) = 0.790 ± 0.010
    Fitted coefficient of the primary SF-dominated mean relation; median 0.790.
  • η_z and γ_z (redshift evolution amplitude and exponent) = η_z=1.244±0.016, γ_z=0.122±0.019
    Control the (1+z)^γ_z term; both free and jointly constrained.
  • η_m (stellar-mass coefficient) = −0.033 ± 0.009
    Additive mass term; found to be small but non-zero.
  • σ_int (intrinsic scatter of SF component) = 0.178 ± 0.004 dex
    Free Gaussian width of the SF-dominated relation.
  • ϕ, ψ_L, ψ_m (logistic mixture fraction parameters) = ϕ≈0, ψ_L≈−1.97, ψ_m≈1.29
    Control the evolving SF-dominated fraction f(L,M).
  • μ_bkg, σ_bkg (background Gaussian) = μ_bkg≈0.77, σ_bkg≈0.76
    Mean and width of the non-SF component in log SFR.
  • η_α (optional spectral-index evolution) = −0.068 ± 0.025
    Tested alternative that can absorb the redshift term; free when activated.
axioms (5)
  • domain assumption Planck 2020 flat ΛCDM cosmology (Ωm=0.31, H0=67.66)
    Used for luminosity distances throughout; standard but not re-derived.
  • domain assumption Radio spectrum Sν∝ν^α with fiducial α=−0.7 (or linear evolution)
    Converts observed flux to rest-frame L1.4 (Eq. 1); tested but still an assumption.
  • domain assumption GRAHSP SED models (delayed-τ SFH, Chabrier IMF, Bruzual-Charlot SSPs, flexible AGN torus) correctly recover SFR posteriors
    All SFR values and uncertainties come from this code; full validation deferred.
  • ad hoc to paper Two-component mixture (1-D Gaussian SF relation + 1-D Gaussian background) is an adequate description of the population
    Model choice; skewed-Gaussian background was tested and did not improve fit, but the form remains an assumption.
  • domain assumption Photometric redshifts can be treated as delta functions (posteriors not sampled)
    Stated in §2.1; introduces unquantified extra uncertainty at high z.

pith-pipeline@v1.1.0-grok45 · 28395 in / 3167 out tokens · 26057 ms · 2026-07-12T01:43:38.442598+00:00 · methodology

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read the original abstract

Determining the relationship between star-formation rate (SFR) and the radio luminosity ($L_{1.4}$) is critical if we are to trace the star-formation history of the Universe dust-agnostically using current and future radio facilities. However, until now, such work has relied on potentially biased binary classifications of sources to remove contaminating active galactic nuclei (AGN). We present a new, statistically-driven methodology for deriving the SFR -- $L_{1.4}$ relation, removing the need for problematic cuts. We use a Bayesian hierarchical mixture model fit to the radio-detected sources in the deep MIGHTEE COSMOS DR1 catalogue, incorporating the full SFR posterior probability distributions generated by state-of-the-art spectral energy distribution fitting code \texttt{GRAHSP}. This allows us to probabilistically determine a mean SFR -- $L_{1.4}$ relation for the SF dominated galaxies, whilst accounting for changing fractions of SF dominated sources across redshift, radio luminosity and stellar mass ranges. We find that the SFR -- radio luminosity correlation exhibits a significant dependence on redshift, but a stellar mass dependence that is weaker than previous studies. Our resultant SFR-radio correlation is $\log_{10}(\text{SFR}/M_{\odot}\,\text{yr}^{-1}) = 0.790\times(\log _{10}(L_{1.4}/\text{W\,Hz}^{-1})-23) + 1.244 \times(1+z)^{0.122} -0.033 \times (\log_{10}(M_*/M_{\odot})-10)$, with an intrinsic scatter of 0.178 dex. We show that this redshift evolution could be explained by a moderate evolution in the radio spectral index of SF galaxies. We attribute the lack of observed strong dependence on stellar mass, compared to recent studies, to the novel statistical approach that does not rely on cuts to remove AGN.

Figures

Figures reproduced from arXiv: 2607.03543 by Catherine L. Hale, Charlotte L. Jackson, Daniel J. B. Smith, Imogen H. Whittam, James H. Matthews, Joel Hamlett, Johannes Buchner, Mara Salvato, Matt J. Jarvis, Natalia Stylianou, Rohan G. Varadaraj, S. Lyla Jung.

Figure 1
Figure 1. Figure 1: 1.4 GHz radio luminosity as a function of redshift for the MIGHTEE DR1 (COSMOS field) sample used in this work, coloured by stellar mass. The solid and dashed lines indicate the regions used in the volume-limited test performed in Section 4.5. GRAHSP employs a Bayesian approach to constrain model parame￾ters using a nested sampling inference procedure. For this study, we adopt a delayed-𝜏 star formation hi… view at source ↗
Figure 2
Figure 2. Figure 2: (Left panel) Distribution of intrinsic 1.4 GHz radio luminosities from MIGHTEE DR1 and SFR from GRAHSP SED fitting for all sources meeting basic quality cuts described in Section 3.1, coloured by redshift. (Right panel) The SFR shown as a function of the log of the size of the SFR posterior, as defined by the difference between the 84th and 16th percentile values, scaled by the median. Most sources are wel… view at source ↗
Figure 3
Figure 3. Figure 3: The distribution of sources in log10 (𝐿1.4 )-log10 (SFR) space, as determined by the overall best performing model. Points are coloured by 𝑝(SF), which is the probability of a source belonging to the SF-dominated component, as given by equation (11). The various lines correspond to the best-fitting 𝜇SF, 𝜇SF = log10 (SFR) = 0.790𝐿 ′ + 1.244(1 + 𝑧) 0.122 − 0.033𝑀′ , with differing redshift and stellar mass v… view at source ↗
Figure 4
Figure 4. Figure 4: The posterior probability distributions of the free parameters in the optimal SFR-radio model, 𝜇SF[𝑀′+𝑧]; 𝑓 [𝐿 ′+𝑀′ ], as described by equations (14) and (15), ascertained by NUTS sampling. The median, 16 per cent and 84 per cent confidence intervals are stated [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The distribution of sources in log10 (𝐿1.4 )-log10 (SFR) space, with an evolving 𝛼 factor included in the calculation of log10 (𝐿1.4 ), of the form 𝛼 = −0.7 − 0.068𝑧. Points are coloured by 𝑝(SF), which is the probability of a source belonging to the SF-dominated component, as given by equation (11). The lines correspond to the SFR-radio relation, log10 (SFR) = 0.830𝐿 ′ [𝛼(𝑧) ] − 0.037𝑀′ + 1.329, evaluated… view at source ↗
Figure 6
Figure 6. Figure 6: The corner plot describing the best fitting combination of parameters for the evolving spectral index model, as ascertained by the NUTS sampling. The median, 16 per cent and 84 per cent confidence intervals are stated. 𝜇bkg and 𝜎bkg (values given in [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: In SFR-𝐿1.4 space, a realisation of distribution of observed sources (blue)– where their SFR value has been randomly selected from their posterior– and simulated sources (orange), calculated from observed 𝐿1.4, 𝑧 and 𝑀∗ values combined with the posterior probability distributions of the fitted parameters in our derived SFR-radio correlation. The line on each plot shows the median best fitting values of the… view at source ↗

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