An empirical model of the extragalactic radio background

Fangyou Gao; Tao Wang; Yijun Wang

REVIEW 3 major objections 8 minor 76 references

An empirical model of the extragalactic radio background

T0 review · 3 major / 8 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Building the extragalactic radio background from a mock galaxy catalog, this paper predicts that roughly half of the radio sources SKA will detect at z ~ 4–6 will be too faint for Rubin's deepest optical survey.

desk verdict Useful new mock catalog with convincing z<4 validation; the z>4 SKA/Rubin prediction is an explicit extrapolation that needs a sensitivity test. read the letter →

arxiv 2412.08995 v1 pith:WNUP4BXI submitted 2024-12-12 astro-ph.GA

Fangyou Gao , Tao Wang , Yijun Wang This is my paper · ORCID

classification astro-ph.GA

keywords extragalacticradiobackgroundluminosityfunctionmockgalaxycatalogstar-forminggalaxiesAGNSKALSST/Rubinsourcecounts

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

Recent deep radio surveys across 0

What carries the argument

The carrying object is the EGG mock galaxy catalog (Schreiber et al. 2017), a near-infrared-selected simulated sky containing active and passive galaxies with stellar masses, SFRs, and SEDs down to z~10. Onto it the paper layers two radio channels: star-forming galaxies receive 1.4 GHz luminosities by converting their IR luminosities with the mass- and redshift-dependent qIR calibration of Delvecchio et al. (2021), while radio AGNs are assigned with probability functions p(L1.4GHz|M*, z) from Wang et al. (2024), restricted to hosts above $10^{10}$ M_sun, with luminosities drawn from a pure density evolution luminosity function. A Soneira–Peebles hierarchical clustering algorithm redistributes the massive host galaxies so that the mock radio AGNs reproduce the observed angular two-point correlation function. These components convert ordinary galaxy properties into a realistic radio sky whose multi-wavelength connections can be queried directly.

What would settle it

Measure the 1.4 GHz radio luminosity function of AGNs at 4<z<6 with a deep survey, or measure the fraction of SKA-detected z~4–6 sources with r>27.5 in early SKA/Rubin overlap; a deviation from the extrapolated pure-density-evolution model, or a fraction far from ~50%, would refute the central prediction.

Watch

Extended reading notes

Core claim

The paper's central claim is that the extragalactic radio background can be built from the bottom up: instead of drawing sources from measured radio luminosity functions, it assigns radio emission to every galaxy in the EGG mock catalog based on that galaxy's star formation rate and on a probability, calibrated at z<4, that it hosts a radio AGN of a given 1.4 GHz luminosity. The synthetic catalog then reproduces the observed 1.4 GHz radio luminosity functions of both AGNs and star-forming galaxies, the differential number counts at 150 MHz, 1.4 GHz, and 3 GHz, and the angular clustering of radio AGNs. Since the radio flux is attached to physical galaxies, the catalog directly links radio detection to host properties such as stellar mass, redshift, and photometric magnitudes. Its headline prediction is that 45–56% of the radio sources SKA will detect at z>4 will be too faint at r-band (r~27.5) to be seen by Rubin's LSST survey, making deep radio observations essential for finding dust-obscured early-universe galaxies.

Load-bearing premise

The high-redshift predictions assume that the radio-AGN probability functions and the luminosity-function shape measured at z<4 continue to hold at 4<z<6 without new data; if those functions change, the 'half invisible' result changes.

Editorial extensions

If this is right

The released 4 deg^2 mock catalog lets survey planners compute, for any radio detection, the expected optical/NIR magnitude and stellar mass of its host.
SKA's 5-hour pointings should detect star-forming galaxies down to ~10^10 M_sun at z~6, extending galaxy formation studies to the early universe.
At z>4, 45–56% of SKA-detected radio sources will lack an LSST r-band (or z-band) counterpart, so SKA will be the primary window onto dust-obscured early galaxies.
Radio AGN and SFG contributions to the extragalactic radio background separate cleanly in the model, but the AGN contribution is sensitive to a single bright source per square degree at 150 MHz.

Reading between the lines

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

The same catalog-based technique could be used to predict X-ray or sub-mm emission from the same mock galaxies, yielding a fully multi-wavelength mock sky for survey planning.
The 'half invisible' prediction would shift if high-redshift radio AGNs are hosted by lower-mass galaxies than assumed; SKA pathfinder data at 4<z<6 could test the extrapolated AGN probabilities within a few years.
If the extrapolation is wrong, the model would still reproduce z<4 observables, meaning the high-z prediction should be read as the main falsifiable output rather than the calibrations.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Worth a look if you build mocks or plan radio-optical survey synergies. The core idea is genuinely new relative to S3 and TRECS: instead of drawing radio sources from RLFs, the authors start from the EGG galaxy catalog and assign radio emission via SFR, qIR, and a radio-AGN probability. Every mock radio source therefore comes with host stellar mass, SFR, and multi-band magnitudes—exactly what you need for survey design and for splitting the ERB by galaxy type. The z<4 validation is convincing: source counts at 150 MHz, 1.4, 3, 10, and 15 GHz match independent data, and the SFG RLF is derived from SFR-M* plus qIR rather than fitted to the RLF.

The soft spots are real but mostly where you'd expect. The AGN RLF validation is partly circular: mock AGNs are generated using the Wang et al. (2024) probability and RLF shapes (Eqs. 2 and 5), then compared to Wang et al. (2024) data, with two co-authors in common. That doesn't make it worthless—the source counts are independent checks of normalization and spectral handling—but the AGN RLF figure is a consistency check more than a validation. The clustering treatment also includes tuned fractions (15% active, 25% passive in the Soneira-Peebles redistribution) fitted to reproduce the target TPCF; fine as a modeling choice, but not a prediction.

The bigger concern, and the stress-test note has it right, is the z>4 result. Section 5.1(4) explicitly says Eqs. 2 and 5 are extrapolated to 44 invisible to Rubin's r~27.5—depends directly on the high-z AGN abundance, which is unmeasured. There is no sensitivity test, e.g., freezing evolution at z=4, so we don't know if the fraction is 40% or 70%. This doesn't break the method; it means that specific number is a motivated extrapolation with unquantified uncertainty, not yet a planning-grade constraint.

The paper is honest about its limitations (extended radio sources, low-mass AGN host threshold, passive galaxy fractions at z>4), the citations look appropriate, and the mock catalog is promised public. Take the z<4 catalog seriously; treat the z>4 fraction with caution; ask for a sensitivity test in revision. I'd send it to a referee—the method is a step forward and the issues are fixable.

Referee Report

3 major / 8 minor

Summary. This paper constructs an empirical model of the extragalactic radio sky by starting from an EGG-generated NIR-selected mock galaxy catalog and assigning radio continuum emission to every galaxy: star-forming galaxies receive 1.4 GHz luminosities from their SFRs through qIR, while radio AGNs are drawn from the Wang et al. (2024) probability functions and assigned luminosities following a pure density evolution RLF. The model also adds Soneira-Peebles clustering for massive galaxies and empirically calibrated sizes for SFGs. The authors validate the resulting AGN and SFG 1.4 GHz RLFs, differential source counts at 150 MHz, 1.4 GHz, and 3 GHz, and the angular two-point correlation function of radio AGNs. They then use the catalog to predict SKA source statistics and LSST cross-matching, finding that roughly half of SKA sources at z>4 will be fainter than r=27.5, and they describe the columns of the released 4 deg^2 mock catalog.

Significance. If the model is correct, it provides a useful physical link between radio flux densities and galaxy host properties, and the released mock catalog can support survey design and multiwavelength follow-up planning. The SFG RLF and the source-count checks use independent data, and the paper is commendably explicit about its limitations, including the extrapolation to z>4, the missing extended radio sources, and the ad hoc 10^10 M_sun host-mass threshold. The main weakness is that the AGN population is generated from the same Wang et al. (2024) functions that are used as the primary AGN validation, so the independent evidence for the high-redshift AGN behavior is sparse; consequently, the headline SKA/LSST prediction requires a sensitivity analysis before it can be regarded as robust.

major comments (3)

[§2.2, Eqs. (2)–(3)] The calculation of fAGN is not dimensionally consistent as written. In Eq. (2), p(L|M,z) is written as a dimensionless power law, but Eq. (3) integrates it over dL_radio. Since p is proportional to L^-0.77, the integral from L_limit to infinity diverges at the upper end, and even with a finite upper cutoff fAGN would have units of luminosity rather than being a fraction. If p is instead a probability per logarithmic luminosity, Eq. (3) must integrate over dlog L and the normalization in Eq. (2) must be interpreted accordingly. This step controls the total number of radio AGNs in every mass and redshift bin, so it should be fixed and stated explicitly.
[§3.1, Fig. 1 vs. §2.2] The agreement between the mock AGN RLF and the Wang et al. (2024) data points in Fig. 1 is partly circular: the mock AGNs are generated from the same best-fit functions in Eqs. (2) and (5), so Fig. 1 largely demonstrates that the sampling code recovers its input. The genuinely informative tests are Figs. 2 and 3, which use independent compilations. I request a quantitative comparison in the overlapping luminosity and redshift bins, for example residuals or a chi-square statistic, and a clear separation in the text between internal consistency checks and external validation when summarizing the claim that the AGN RLF is successfully recovered.
[§4.1 and §5.1(4)] The headline prediction that 56% of SKA-detected sources at z>4 are invisible in the LSST r~27.5 survey is driven by radio AGNs generated with Eqs. (2) and (5) extrapolated from z<4, as the text acknowledges, and the high-redshift validation data are sparse. No sensitivity test is provided. I request a robustness check that, at minimum, freezes the AGN probability and RLF evolution at z=4, and ideally also varies the 10^10 M_sun host-mass threshold of §2.2 and the qIR extrapolation for SFGs beyond z~4.5, and reports the resulting range for the claimed 45–56% optically invisible fraction. Without such a test, the main SKA/LSST prediction is not yet demonstrated.

minor comments (8)

[§2.2, Eq. (2)] The pivot values M0, L0, and z0 in Eq. (2) are not defined; please state them explicitly.
[§3.1, Figs. 2–3] The flux limits, selection criteria, and redshift binning of the literature data used in Figs. 2 and 3 are described in the text but would be easier to assess if summarized in the captions or a table.
[§5.1(4)] The statement that the extrapolations 'succeeded in recovering RLFs of both radio AGNs and SFGs at z>4' is stronger than the sparse z>4 data warrant; please soften it or add quantitative uncertainties.
[§4.1] The LSST r band is approximated with the SDSS r band; the potential filter and k-correction differences should be stated or estimated.
[Table 3] The mock catalog is described as publicly released, but no URL or DOI is given; please add access details.
[§2.5, Table 1] The z>2 effective-radius fit uses only 19 SFGs and the resulting A and alpha values have large uncertainties; this should be noted wherever the size model is applied.
[§5.2] The heading 'Coution' is a typo for 'Caution'; other typos include 'wavelenghths' in the introduction and 'simuations' in §4.2.
[§4.3, Eq. (8)] The constant C in Eq. (8) is not defined or used; please define it or remove it.

Circularity Check

1 steps flagged · score 4.0 of 10

The AGN RLF 'recovery' is a self-consistency check on the authors' own Wang et al. (2024) input functions, while the SKA/Rubin z>4 prediction is an admitted extrapolation rather than a circular construction.

fitted input called prediction [Section 2.2 (Eqs. 2 and 5) vs. Section 3.1 (Fig. 1)]
"We take advantage of recent work by Wang et al. (2024), which gathered deep VLA 3 GHz observations ... They carefully investigated AGN probability as a function of redshift, stellar mass and radio luminosity using maximum likelihood fitting and derived functional forms for radio AGNs in active and passive host galaxies respectively. ... Fig. 1 compares the RLF of our mock AGNs (red dots) and SFGs (blue dots) to the data points (magenta and cyan stars respectively) in Wang et al. (2024). ..."

The mock AGN population is generated from the same fitted functions used in the comparison: Eq. (2) gives p(L_1.4GHz|M*,z) from Wang et al. (2024) and Eq. (5) gives the AGN RLF shape from Wang et al. (2024). Sections 2.2 and 3.1 therefore describe a forward model whose AGN component is re-sampled from the very RLF/data it is then said to recover. The agreement in Fig. 1 is a numerical self-consistency test of the EGG host catalog and the integration in Eq. (3), not an independent observational validation of the AGN population. This is the classic fitted-input-called-prediction pattern: applying a fitted probability/RLF to a mock catalog and then 'recovering' the same RLF is forced by construction for the AGN component.

full rationale

The paper is largely a forward-model construction, not a derivation from first principles, and most of its output is genuinely independent of the input fits. The EGG parent catalog is built from empirical stellar mass functions and SED libraries; SFG radio luminosities are obtained from SFR-MS and qIR calibrations; and the source counts at 150 MHz-15 GHz are checked against many external surveys (Franzen, Mandal, Bondi, Condon, Smolčić, Matthews, Vernstrom, van der Vlugt, Jiménez-Andrade), with good agreement. These tests are not circular: the radio fluxes are not fitted to those counts. The one clear circular step is the AGN RLF 'recovery' in Section 3.1. The mock radio AGNs are assigned luminosities using Equations (2) and (5), both taken directly from Wang et al. (2024), and the validation in Fig. 1 compares the resulting RLF to the same Wang et al. (2024) data and best-fit models. The agreement is therefore a self-consistency check: the AGN component is re-sampled from the same fitted probability density and RLF shape it is later said to recover. The overlap of two co-authors (Tao Wang, Yijun Wang) strengthens the concern that this is self-citation load-bearing, although the cited work itself is an external empirical measurement and not a mere assertion. The high-redshift SKA/Rubin claim (56% optically invisible at z>4) is not circular in the strict sense. Section 5.1(4) explicitly states that Equations (2) and (5) are extrapolated to 4<z<6 because Wang et al. only constrained them at z<4. The prediction is a genuine extrapolation with independent content; the problem is robustness, not circularity. No sensitivity test (e.g., freezing the AGN evolution at z=4) is presented, so the headline fraction should be treated as model-dependent. The clustering section also tunes 15%/25% active/passive fractions to a clustering target before 'reproducing' radio-AGN clustering, but this is peripheral and explicitly fails at large scales. Overall, the central derivation is not equivalent to its inputs; one validation reduces to a re-sampling of the authors' own fitted functions, so a moderate score of 4 is appropriate.

Assumptions & free parameters 10 free parameters · 9 assumptions · 0 invented entities

The model rests on a chain of empirical relations from prior work: EGG mass functions and SEDs, Schreiber et al. SFR-MS, Bell 2003 SFR-radio, Delvecchio et al. qIR, and the same group's Wang et al. 2024 AGN probability/RLF. Several of these are extrapolated beyond their calibration range, and the clustering fractions are tuned to the validation target. No new physical entities are introduced; the main uncertainties are the extrapolations and the partial circularity of the AGN validation.

free parameters (10)

Radio AGN probability parameters, active hosts (Eq. 2a) = normalization 10^-0.79, M* slope 1.06, L slope -0.77, redshift slope 3.08
Fitted by Wang et al. (2024) to VLA 3 GHz AGN samples; used here to compute f_AGN and assign AGNs.
Radio AGN probability parameters, passive hosts (Eq. 2b) = normalization 10^-0.70, M* slope 1.41, L slope -0.60, redshift slope 2.47
Fitted by Wang et al. (2024) to passive-host AGNs; used in the same way as the active-host parameters.
Radio AGN RLF shape parameters (Eq. 5) = Phi*_0=10^-5.5, L*_0=10^24.59, alpha=1.27, beta=-0.49, alpha_D=-0.77 z + 2.69
Used to assign 1.4 GHz luminosities to radio AGNs; the shape comes from Wang et al. (2024), so the mock AGN LF is not an independent prediction.
Stellar mass threshold for radio AGN hosts = 10^10 Msun
Imposed in Section 2.2 to avoid overproducing low-mass AGNs; the paper notes this is an assumption and retains it at all redshifts.
Peak qIR for passive galaxies = 2.3
Measured from 890 COSMOS passive galaxies in Section 2.3 with no redshift binning; used as the Gaussian center for passive galaxy radio luminosities.
qIR Gaussian scatter = 0.26 dex
Adopted from Yun et al. (2001); applied to both active and passive galaxies when drawing qIR values.
qIR low-side cutoff = 0.43 dex below peak
From Delvecchio et al. (2021); galaxies with qIR below this cutoff are treated as AGN-contaminated and discarded.
Clustering redistribution fractions = 15% active, 25% passive
Chosen after multiple trials in Section 2.4 to reproduce the angular TPCF of active and passive galaxies; this is fitting to the validation target.
Effective radius parameters A and alpha (Eq. 6, Table 1) = A: 2.093/2.415/1.737, alpha: 0.092/0.029/0.211 in three z bins
MCMC fits to GOODS-S 3 GHz SFG sizes; used only for simulated images and not for the ERB or RLF results.
Median spectral slopes and scatter = SFG: -0.8 at 1.4/3 GHz, -0.5 at 150 MHz; AGN: -0.5; sigma=0.1
Assumed from literature and used to extrapolate 1.4 GHz luminosities to 150 MHz, 350 MHz, 10 GHz, and 15 GHz.

assumptions (9)

domain assumption The EGG mock catalog (Schreiber et al. 2017), including stellar mass functions, UVJ-based SEDs, SFR-M* main sequence, and passive residual SFRs, accurately represents the galaxy population, including extrapolation to z~10.
Used throughout Section 2.1 as the base catalog; EGG itself is a prior empirical model and is not re-verified here.
domain assumption A flat LCDM cosmology with Omega_M=0.286, H0=69.3 km/s/Mpc, and a Salpeter IMF are adopted.
Standard assumptions stated at the end of Section 1; they set volumes, luminosities, and magnitudes.
domain assumption The Bell (2003) relation converts SFR_MS to a 1.4 GHz radio luminosity lower limit for the AGN probability integral.
Invoked in Section 2.2 near Eq. 4 to set the lower integration limit; a biased conversion would shift f_AGN and the AGN population.
domain assumption The Delvecchio et al. (2021) qIR(M*,z) relation and its scatter describe SFG radio emission at all masses and redshifts up to z>4.5.
Used in Section 2.3 to convert IR luminosity to 1.4 GHz luminosity for active galaxies; the original calibration is z<4.5.
domain assumption The Wang et al. (2024) radio AGN probability functions and RLF shape remain valid when extrapolated to 4<z<6.
Explicitly extrapolated in Section 5.1(4); this underpins the z>4 SKA/Rubin prediction.
ad hoc to paper Radio AGNs only reside in galaxies with stellar mass above 10^10 Msun.
Assumed in Section 2.2 and retained at all redshifts to avoid overproducing low-mass AGNs; no independent evidence is given for the threshold at high z.
domain assumption Passive galaxies have a redshift-independent qIR distribution peaked at 2.3.
Measured from a small sample without redshift binning in Section 2.3; assumed to hold at all redshifts.
domain assumption The Soneira-Peebles algorithm with R=3', eta=5, lambda=6, and assigned fractions 15%/25%, reproduces the clustering of massive galaxies and radio AGNs.
Adopted in Section 2.4; the fractions are tuned to match the TPCF, so the clustering validation is not fully independent.
domain assumption A single median spectral slope applies per population across luminosity and redshift, with 0.1 dex Gaussian scatter.
Used in Sections 3 and 4.2 to extrapolate 1.4 GHz luminosities to 150 MHz-15 GHz; real spectra vary with luminosity and redshift.

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Cite this review

Pith. "Pith review of An empirical model of the extragalactic radio background." pith.science (2026). https://pith.science/paper/WNUP4BXI

@misc{pith2026241208995,
  author       = {Pith},
  title        = {Pith review of: An empirical model of the extragalactic radio background},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WNUP4BXI}},
  note         = {Machine review of arXiv:2412.08995}
}

abstract

Radio observations provide a powerful tool to constrain the assembly of galaxies over cosmic time. Recent deep and wide radio continuum surveys have improved significantly our understanding on radio emission properties of AGNs and SFGs across $0 < z < 4$. This allows us to derive an empirical model of the radio continuum emission of galaxies based on their SFR and the probability of hosting an radio AGN. We make use of the Empirical Galaxy Generator (EGG) to generate a near-infrared-selected, flux-limited multi-wavelength catalog to mimic real observations. Then we assign radio continuum flux densities to galaxies based on their SFRs and the probability of hosting a radio-AGN of specific 1.4 GHz luminosity. We also apply special treatments to reproduce the clustering signal of radio AGNs.Our empirical model successfully recovers the observed 1.4 GHz radio luminosity functions (RLFs) of both AGN and SFG populations, as well as the differential number counts at various radio bands. The uniqueness of this approach also allows us to directly link radio flux densities of galaxies to other properties, including redshifts, stellar masses, and magnitudes at various photometric bands. We find that roughly half of the radio continuum sources to be detected by SKA at $z \sim 4-6$ will be too faint to be detected in the optical survey ($r \sim 27.5$) carried out by Rubin observatory. Unlike previous studies which utilized RLFs to reproduce ERB, our work starts from a simulated galaxy catalog with realistic physical properties. It has the potential to simultaneously, and self-consistently reproduce physical properties of galaxies across a wide range of wavelengths, from optical, NIR, FIR to radio wavelengths. Our empirical model can shed light on the contribution of different galaxies to the extragalactic background light, and greatly facilitates designing future multiwavelength galaxy surveys.

Figures

Figures reproduced from arXiv: 2412.08995 by the authors.

**Figure 2.** Comparisons of RLFs of our mock AGNs (red) to literature data (McAlpine et al. 2013; Padovani et al. 2015; Smol [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Comparisons of RLFs of our mock SFGs (blue) to literature data (Padovani et al. 2011; McAlpine et al. 2013; Novak et al. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: a: Comparisons of differential source counts at 150 MHz to observations in Franzen et al. (2016); Mandal et al. (2021); Bondi et al. (2023) along with S 3 and TRECS simulations (Wilman et al. 2008; Bonaldi et al. 2019). b: Comparisons of differential source counts at 1…

**Figure 5.** Figure 5: left: Angular TPCF of mock EGG galaxies with stellar mass > 1010 M⊙ at different redshift bins. A straight line with power law slope of −1 is also plotted. middle: Angular TPCF of redistributed mock EGG galaxies with stellar mass > 1010 M⊙ at different redshift bins. A…

**Figure 6.** Figure 6: left: The 90% completeness of stellar mass of galaxies that can be observed above 5σ by SKA as a function of redshift, assuming 5 hour observation time for a single pointing. right: Red squares denote the fraction of SKA detected galaxies that are above 27.5 mag in the…

**Figure 7.** Figure 7: Top: Simulated 2′ × 2 ′ JWST-F444W images at 0.25 < z < 0.75 and 0.75 < z < 1.25. Bottom: Simulated 4′ × 4 ′ JWSTF444W images at 1.75 < z < 2.25 and 2.75 < z < 3.25. Blue circles represents sources detected under VLA observation (with rms noise level of 4.23 µJy beam−…

**Figure 8.** Figure 8: Predicted source counts at 350 MHz, 10 GHz and 15 GHz respectively. Theoretical works from Mancuso et al. (2017) and [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: Source counts from radio AGNs (solid lines) and SFGs [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 10.** Figure 10: Contribution from radio AGNs (red) and SFGs (blue) [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

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