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The optically-selected 1.4-GHz quasar luminosity function below 1 mJy

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Bayesian stacking of FIRST pixels at SDSS quasar positions measures the quasar radio luminosity function to about 100 times fainter than the survey limit, exposing a flattening then steepening where radio-quiet quasars emerge.

desk verdict Competent application of bayestack to a new sample, but the sub-mJy RLF shape is conditional on optical selection effects the authors themselves flag. read the letter →

arxiv 1908.05316 v4 pith:3JU3Y73X submitted 2019-08-14 astro-ph.GA

Eliab Malefahlo , Mario G. Santos , Matt J. Jarvis , Sarah V. White , Jonathan T. L. Zwart This is my paper · ORCID
classification astro-ph.GA
keywords radioluminosityfunctionquasarsBayesianstackingFIRSTsurveySDSSradio-quietsub-mJysources1.4GHzcontinuum
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

This paper claims that the 1.4-GHz radio luminosity function (RLF) of optically selected quasars can be measured down to about two orders of magnitude below the 1 mJy detection limit of the FIRST survey by fitting parametric models to the flux densities extracted from FIRST maps at SDSS quasar positions using a Bayesian stacking approach. Reconstructed over seven redshift bins out to $z = 2.15$, the RLF of radio-loud quasars flattens below $\log_{10}[L_{1.4}/\mathrm{W\,Hz}^{-1}] \approx 25.5$ and steepens again below $\approx 24.8$, where radio-quiet quasars emerge. The steepening luminosity coincides with the luminosity where star-forming galaxies are expected to begin dominating radio source counts, suggesting that host-galaxy star formation may contribute substantially to faint quasar radio emission. The authors explicitly note that at least part of the flattening could be an artifact of the SDSS optical magnitude limit rather than a physical break, and this is the key uncertainty in the interpretation.

What carries the argument

The central mechanism is a forward-modeled Poisson likelihood for binned FIRST pixel flux densities. A parametric radio luminosity function is converted to a source-count model, the FIRST clean and snapshot biases are applied, and the predicted counts are convolved with Gaussian noise before being compared to the histogram of extracted pixel values at SDSS quasar positions. This lets sources buried in the 150-$\mu$Jy noise constrain the luminosity function rather than being averaged into a single stacked flux. The faint-end models tested are a power law, a double power law, and a log-normal power law, with the double power law winning by Bayesian evidence in all redshift bins.

What would settle it

Run the same Bayesian stacking fit on a quasar sample with a fainter optical magnitude limit or with relaxed absolute-magnitude cuts over the same redshift bins: if the flattening near $\log_{10}[L_{1.4}/\mathrm{W\,Hz}^{-1}] \approx 25.5$ and the steepening near $\approx 24.8$ stay fixed in luminosity, the shape is physical, whereas if they shift or disappear the optical selection created them.

Watch

Extended reading notes

Core claim

The central claim is that a full Bayesian stacking analysis of FIRST survey pixels can recover the 1.4-GHz radio luminosity function of optically selected quasars two orders of magnitude below the 1 mJy detection threshold, and that the recovered RLF has a distinctive faint-end shape. In every redshift bin from $0.2 < z < 2.15$, the data prefer a model with a double power-law for both the luminous and the faint populations. The bright-end function rises steeply toward lower luminosity, flattens near $\log_{10}[L_{1.4}/\mathrm{W\,Hz}^{-1}] \approx 25.5$, and then steepens again below $\approx 24.8$, the regime where the authors associate the population with radio-quiet quasars. The agreement of the low-redshift reconstruction with deep JVLA observations of the same quasars, together with the coincidence between the steepening luminosity and the expected crossover to star-forming galaxy dominance, is presented as evidence that host-galaxy star formation may contribute to the radio emission of quasars; the paper explicitly notes that at least part of the flattening could instead be imposed by the SDSS optical magnitude limit.

Load-bearing premise

The load-bearing premise is that the SDSS optical magnitude limit, after the per-bin absolute magnitude cut, does not itself imprint the observed flattening and steepening on the radio luminosity function; the paper concedes that at least some of the flattening could be due to optical incompleteness.

Editorial extensions

If this is right

  • A wide, shallow survey like FIRST is sufficient to constrain the faint end of the optically selected quasar RLF down to radio luminosities around $10^{22}\,\mathrm{W\,Hz}^{-1}$, roughly two orders of magnitude below the nominal detection threshold.
  • The double power-law description of the faint population is preferred over a power law or a log-normal in all seven redshift bins, so the faint-end shape is not confined to a single redshift range.
  • The flattening near $\log_{10}[L_{1.4}/\mathrm{W\,Hz}^{-1}] \approx 25.5$ and the steepening near $\approx 24.8$ recur in every bin, indicating that the same population change is present at all redshifts studied.
  • The steepening luminosity coincides with where star-forming galaxies are expected to dominate radio source counts, so host-galaxy star formation may contribute significantly to radio-quiet quasar emission.
  • In the lowest redshift bin the sub-mJy RLF matches deep JVLA observations of individual quasars, validating the stacking reconstruction against direct detections.

Reading between the lines

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

  • Applying the same forward-model stacking to a parent quasar sample with a fainter optical magnitude limit would directly test whether the flattening luminosity tracks the optical limit or stays fixed, distinguishing a selection artifact from a true physical break.
  • A bivariate optical-radio luminosity function, a route the authors mention, would remove the need for hard absolute-magnitude cuts and could separate accretion-driven from star-formation-driven radio emission more cleanly than the RLF alone.
  • High-resolution imaging that resolves the host-galaxy scale would allow the extended, star-formation-related radio component to be separated from the AGN core, giving a direct test of the star-formation interpretation.
  • The same noise-dominated pixel fitting could measure luminosity functions of other faint populations in current and future wide surveys, since it extracts shape information from sources buried below the detection threshold without needing deep images.
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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 applies a Bayesian stacking technique (bayestack) to 1.4-GHz FIRST flux densities extracted at the positions of SDSS DR7 quasars, fitting parametric models for the radio luminosity function (RLF) in seven redshift bins up to z=2.15. The authors claim to reconstruct the optically selected quasar RLF down to roughly two orders of magnitude below the FIRST 1 mJy detection threshold, and report that the bright-end RLF flattens below log10[L1.4/W Hz^-1] ~ 25.5 and steepens below ~24.8, where radio-quiet quasars and a possible star-formation contribution emerge. The method is tested on SKADS simulations and the lowest-redshift RLF is compared with deeper VLA data from Kellermann et al. (2016) and Condon et al. (2013).

Significance. If the reconstruction is reliable, this is a valuable technique for measuring the faint radio emission of optically selected quasars using wide, shallow surveys, and the deep low-z agreement with Kellermann et al. (2016) is an encouraging validation. The authors are careful to test the pipeline on SKADS simulations with realistic FIRST noise and to compare with independent data at low redshift; these are genuine strengths. However, the central physical claims about the RLF shape below the detection threshold rest on parametric model assumptions and on a subtle treatment of optical selection, and the paper itself acknowledges that at least part of the flattening may be caused by the optical magnitude limit. Because the claimed star-formation coincidence depends on the shape of the reconstructed faint end, the significance of that interpretation is not yet established.

major comments (4)
  1. [§5, Table 4, Eqs. 12–14] The claimed flattening at log10[L1.4/W Hz^-1] ~ 25.5 and the steepening below ~24.8 are not independent measurements but are properties of the winning double power-law model (Model B, Eq. 13): the turnover locations are essentially the break luminosities L*1 and L*2 of the fitted functions. The model-selection evidence in Table 4 only ranks Models A, B, and C relative to each other; it does not demonstrate that a turnover is required by the data, for example by testing against a model without a low-luminosity break or by computing an absolute goodness of fit. As a result, the abstract's statement of these transition luminosities, without associated uncertainties and without an explicit statement that they are posterior model parameters, is stronger than the analysis supports.
  2. [§2.1 (Eq. 1) and §6.1] The optical selection can imprint the very features the paper interprets physically. The sample is limited by i<19.1 and each redshift bin is truncated at a maximum absolute magnitude (Eq. 1); if optical and radio luminosities are correlated (as in White et al. 2017, with roughly an order of magnitude scatter), this cut differentially removes low-optical-luminosity quasars whose radio luminosities populate the regime where the flattening and steepening appear. The authors themselves state in Sec. 6.1 that 'at least some of the flattening is due to incompleteness introduced by the optical magnitude limit of the parent sample' and that they 'cannot rule out' an optical-selection origin for the bright-end flattening. Their robustness test, raising the optical limit and finding that the turnover becomes more prominent, is fully consistent with selection imprinting the shape rather than with a physical break. The SKADS validation does not address this issue because the simulated samples are cut in radio luminosity, not by an optical flux limit correlated with radio luminosity. To support the physical interpretation, the paper needs a forward-model test that injects an optical-radio correlation and an optical flux limit into the simulations and quantifies how much flattening/steepening is produced.
  3. [§3.2, Eq. 2 and §4] The White et al. (2007) bias correction (Eq. 2) was derived for sources above or near the detection threshold where noise can be neglected, and the authors correctly note this in the text. Applying this correction per source inside the likelihood, via SF = max{S/1.4, S - 0.25 mJy}, to noise-dominated sub-threshold flux densities is an extrapolation that is not validated by the SKADS simulations, which set S = SF and therefore do not include the clean or snapshot biases. Because the faint end of the RLF is precisely where this correction is applied, a mis-modeled bias could artificially produce or modify the apparent steepening below log10[L1.4/W Hz^-1] ~ 24.8. The paper should either justify the inverse correction for sub-threshold sources with simulations that include these biases or demonstrate that the results are robust to the assumed bias model.
  4. [§5.1, Table 5, and Appendix Fig. A0] The parameters that most directly control the low-luminosity shape are poorly constrained. Table 5 shows that the boundary parameters Lmin1, Lmax1, Lmin2, and Lmax2 are largely unconstrained, with many posterior intervals spanning several decades, and the appendix states that the faint-end slope beta2 is not well constrained. The authors argue that these 'have very little impact on the actual observed numbers,' but the claimed values of the flattening and steepening luminosities, and the integrated radio-quiet fractions in Table 1, are derived from these posterior components. The paper should quantify how the uncertainty in these boundary and slope parameters propagates into the claimed turnover locations and into the conclusion that the RLF 'peaks' and then 'drops rather abruptly,' rather than only presenting the MAP reconstruction.
minor comments (4)
  1. [§3.1] The description of the Bayes factor contains a typo: 'ln[ZB−ZA]' should be 'ln(ZB/ZA)'.
  2. [References] Several references are incomplete in the bibliography: Chen et al. (2017) is listed only as an arXiv preprint, Gürkan et al. (2018) has no journal or arXiv identifier, and Richards (2006) is listed as 'ArXiv Astrophysics e-prints.'
  3. [§5.1] The phrase 'the sources are volume-limited in the optical (i.e no brightness cutoff)' is misleading because the sample does have an absolute-magnitude cut (Eq. 1); the intended meaning is that there is no radio brightness cutoff, but the wording should be clarified.
  4. [Fig. 8 caption] The blue dotted and red dashed lines in Fig. 8 are described as 'an estimate of the radio-luminosity limit that corresponds to the optical limit' based on White et al. (2017); the caption should explicitly state that these are not measured limits but model-dependent extrapolations of the optical-radio correlation, given the scatter in that relation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fitted RLF features are model-based inferences, not definitional outputs, and the paper's self-citations are not load-bearing.

full rationale

The paper's derivation chain is data -> likelihood (Eqs. 6-9) -> parametric RLF model (Eqs. 12-14) -> fitted parameters and reconstructed RLF. The claimed flattening near log10[L1.4/W Hz^-1] ~ 25.5 and steepening near ~ 24.8 are properties of the best-fit double power-law, Model B, but that is ordinary statistical inference: the stacked FIRST flux-density distributions including the noise-dominated tail do constrain the parameters, and the method is validated against external SKADS simulations and against the independent low-redshift Kellermann et al. (2016) data. No equation defines one target quantity in terms of another target quantity by construction, and no fitted parameter is relabeled as a prediction. The self-citations are not load-bearing: bayestack (Zwart et al. 2015b) is a tool re-tested here, and the White et al. (2017) optical-radio correlation is used only to quantify an acknowledged selection caveat. The paper explicitly states in Sec. 6.1 that 'at least some of the flattening is due to incompleteness introduced by the optical magnitude limit of the parent sample,' which is a limitation on the physical interpretation, not a circularity in the derivation. The central measurement of the sub-mJy RLF remains an independent, externally benchmarked inference, so the circularity score is 0.

Assumptions & free parameters 11 free parameters · 9 assumptions · 0 invented entities

No new particles, forces, or entities are postulated. The central claim instead rests on 11 fitted or chosen parameters, a specific parametric model family, several domain assumptions about the FIRST noise and bias correction, and the completeness of the optical quasar selection. The most fragile assumption is that the optical magnitude limit does not imprint the faint-end shape; the authors themselves flag this as unresolved.

free parameters (11)
  • Bright-end RLF normalization, log10 Phi*1 = -8.01 to -9.02 across redshift bins (Table 5)
    Normalization of the radio-loud double power law; fitted to the FIRST stacked flux distribution in each redshift bin.
  • Bright-end break luminosity, log10 L*1 = 25.04 to 27.74 W/Hz across bins (Table 5)
    Location of the bright-end turn-over; determines the claimed flattening around log L ~ 25.5.
  • Bright-end bright slope, alpha1 = 0.27 to 2.46 across bins (Table 5)
    Fitted high-luminosity slope of the radio-loud component.
  • Bright-end faint slope, beta1 = -3.44 to 0.24 across bins (Table 5)
    Fitted low-luminosity slope of the radio-loud component; partly unconstrained.
  • Faint-end normalization, log10 Phi*2 = -6.53 to -6.97 across bins (Table 5)
    Normalization of the radio-quiet/faint component.
  • Faint-end break luminosity, log10 L*2 = 22.55 to 24.00 across bins (Table 5)
    Peak/drop luminosity of the faint component; sets where the steep low-luminosity rise occurs.
  • Faint-end bright slope, alpha2 = 0.68 to 1.67 across bins (Table 5)
    Fitted slope on the luminous side of the faint component.
  • Faint-end faint slope, beta2 = -4.21 to -1.48 across bins (Table 5)
    Fitted slope on the low-luminosity side; poorly constrained.
  • Low-luminosity boundary, log10 Lmin1/Lmin2 = 18.6 to 23.4 across bins (Table 5)
    Fitted lower bounds of the two components; largely unconstrained and prior-limited.
  • High-luminosity boundary, log10 Lmax1/Lmax2 = 24.2 to 29.9 across bins (Table 5)
    Fitted upper bounds; often unconstrained or truncated.
  • Spectral index alpha = 0.7 (assumed, not fitted)
    Adopted for K-correction and luminosity conversion (Section 1); changing it shifts derived luminosities.
assumptions (9)
  • domain assumption FIRST map noise is Gaussian with sigma_n = 150 microJy per beam at quasar positions.
    Used for the likelihood in Eq. 7; justified by random-sky stamps in Fig. 4, but not verified for every position.
  • domain assumption The bias correction S = min(1.40 S_F, S_F + 0.25 mJy), derived by White et al. (2007), holds for sub-threshold sources when included in the forward model.
    Eq. 2 in Sec. 2.3 is inverted in Sec. 3.2. This relation was originally validated for detected sources or stacked medians; its extrapolation to individual undetected sources is unverified.
  • domain assumption The optically selected SDSS DR7 sample, after the per-bin absolute magnitude cut (Eq. 1), is complete enough that the optical limit does not imprint the observed radio RLF flattening.
    Central to the physical interpretation in Sec. 6.1. The authors themselves state that at least some flattening could arise from the optical magnitude limit.
  • ad hoc to paper The radio luminosity function can be represented by the parametric family in Eqs. 12-14 (single/double power laws and log-normal power law).
    Model selection chooses Model B, but the break and slopes of the inferred RLF are contingent on this family.
  • domain assumption A single spectral index alpha = 0.7 applies to all sources for flux-to-luminosity conversion and K-correction.
    Used in Eq. 10 and in converting Kellermann et al. 6 GHz luminosities; a spread in alpha would broaden the RLF.
  • standard math Poisson likelihood and independence of flux bins (Eqs. 6-9).
    Standard statistical model for binned counts; ignores spatial clustering and confusion beyond Poisson sample variance.
  • standard math Bayes theorem and nested sampling provide unbiased posterior and evidence.
    Foundation of the bayestack framework (Sec. 3.1); relies on proper priors and sufficient sampling.
  • domain assumption SKADS simulations represent the true radio sky well enough to validate the method.
    Used in Sec. 4 to claim recovery of the RLF three orders below threshold; SKADS is semi-empirical, not real data.
  • domain assumption Lambda-CDM cosmology with H0 = 70 km/s/Mpc, Omega_m = 0.3, Omega_Lambda = 0.7.
    Assumed for luminosity distance and volume; small changes shift normalization rather than shape.

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Cite this review

Pith. "Pith review of The optically-selected 1.4-GHz quasar luminosity function below 1 mJy." pith.science (2026). https://pith.science/paper/3JU3Y73X

@misc{pith2026190805316,
  author       = {Pith},
  title        = {Pith review of: The optically-selected 1.4-GHz quasar luminosity function below 1 mJy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3JU3Y73X}},
  note         = {Machine review of arXiv:1908.05316}
}
abstract

We present the radio luminosity function (RLF) of optically-selected quasars below 1~mJy, constructed by applying a Bayesian-fitting stacking technique to objects well below the nominal radio flux-density limit. We test the technique using simulated data, confirming that we can reconstruct the RLF over three orders of magnitude below the typical $5\sigma$ detection threshold. We apply our method to 1.4-GHz flux-densities from the Faint Images of the Radio Sky at Twenty-cm survey (FIRST), extracted at the positions of optical quasars from the Sloan Digital Sky Survey (SDSS) over seven redshift bins up to $z=2.15$ {and measure the RLF down to two orders of magnitude below the FIRST detection threshold}. In the lowest redshift bin ($0.2<z<0.45$), we find that our measured RLF agrees well with deeper data from the literature. The RLF for the radio-loud quasars flattens below $\log_{10}[L_{1.4}/{\rm W\,Hz}^{-1}] \approx 25.5$ and becomes steeper again below $\log_{10}[L_{1.4}/{\rm W\,Hz}^{-1}] \approx 24.8$, where radio-quiet quasars start to emerge. The radio luminosity where radio-quiet quasars emerge coincides with the luminosity where star-forming galaxies are expected to start to dominate the radio source counts. This implies that there could be a significant contribution from star formation in the host galaxies, but additional data is required to investigate this further. The higher-redshift bins show a similar behaviour as for the lowest-$z$ bin, implying that the same physical process may be responsible.

Figures

Figures reproduced from arXiv: 1908.05316 by the authors.

Figure 2
Figure 2. The distribution of the separation between detected FIRST and SDSS quasar positions with a bin size of 0.07 arcsec. The vertical dashed line at 1.8 arcsec is the cut-off separation between FIRST and SDSS detected sources that we used in this work. given by Mi = mi − 5 log10[dL(zup)/10] − K(z), (1) where mi = 18.7 is just above the magnitude completeness limit (mi = 19.1) for DR7, dL(zup) is the luminosity distance (… view at source ↗
Figure 3
Figure 3. Comparison between the FIRST-catalogue peak flux￾densities and map-extracted flux-densities, represented by the blue points. The green crosses denote the extracted flux-densities after correction for the biases described by Eq 2. The solid black line represents the case where the extracted flux-densities would be equal to the catalogue flux-densities, and the dashed red lines indicate the 5σn threshold. the FIRST ca… view at source ↗
Figure 4
Figure 4. Histograms of the raw FIRST flux-densities (Sm) extracted from cut-outs centered at the SDSS quasar positions, with 30µJy bins. The quasars are divided into 7 redshift bins from Legacy (Shen et al. 2011). The two blue lines in each bin represent the FIRST rms σn = 150µJy and 5σn = 750µJy. The red dashed curve is a fixed Gaussian distribution, of mean flux density of zero and σn = 150µJy, which represents the expecte… view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: The posterior distributions for the full SKADS sample (20.5 < log10[L/(WHz−1 )] < 24.5) with the noise levels of 150 µJy and 15 µJy using Models A0 , B0 and C0 . The parameters Lmin2 , Lmax2 , Φ∗ 2 and L∗ 2 are in logarithmic space (log10). The dark blue and the light …
Figure 6
Figure 6. Figure 6: The SKADS radio luminosity function and the reconstruction of the RLF using bayestack for the 21.5 < log10[L/(WHz−1 )] < 24.5 (two top panels) and 20.5 < log10[L/(WHz−1 )] < 24.5 (bottom two panels) samples. The top panels and third panels from the top are the reconstr…
Figure 7
Figure 7. Figure 7: The posterior distributions for the winning model – model B, the double power-law – in the lowest redshift bin (0.2 < z < 0.45). The parameters Lmin1,2 , Lmax1,2 , L1,2∗ and Φ∗ 1,2 are presented in logarithmic space (log10). The dark blue and the light blue regions are…
Figure 8
Figure 8. Figure 8: The optically-selected quasar radio luminosity function and its evolution with redshift. The black dots are the 1/Vmax RLFs from sources above 5σn. The blue unfilled stars, red unfilled circles and the green unfilled squares respectively represent the RLFs from Kimball…

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    " write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.stat...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.