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Exploring Quasar Variability With ZTF at 0<z<3: A Universal Relation with Eddington Ratio

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Quasar flicker rate predicts black hole feeding pace across z=0 to z=3.

desk verdict The underlying Fvar–λEdd anti-correlation is real and the sample is solid, but Eq. 5 as a universal predictor does not survive the paper's own Table 4. read the letter →

arxiv 2505.09779 v1 pith:54UQPNA3 submitted 2025-05-14 astro-ph.GA astro-ph.HE

Hygor B. Gonçalves , Swayamtrupta Panda , Thaisa Storchi Bergmann , Edward M. Cackett , Michael Eracleous This is my paper · ORCID
classification astro-ph.GAastro-ph.HE
keywords quasarvariabilityEddingtonratiofractionalsupermassiveblackholesaccretionZwickyTransientFacilityactivegalacticnucleiphotometricmonitoring
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 uses six years of Zwicky Transient Facility g-band photometry for 915 quasars spanning redshifts 0 to 3 to ask whether the amplitude of quasar brightness variations tracks the rate at which the black hole accretes matter. It argues that the fractional variability $F_{\rm var}$ anticorrelates with Eddington ratio in every redshift bin, while correlations with luminosity and black hole mass change sign or strength with redshift. The combined sample yields a single relation, $\log \lambda_{\rm Edd} = (-0.71\pm 0.06)\log F_{\rm var} - (1.52\pm 0.06)$, which the paper presents as nearly redshift-independent. If the relation holds, photometric monitoring alone can estimate the Eddington ratio of a quasar at $0

What carries the argument

The load-bearing quantity is fractional variability $F_{\rm var}$, the excess variance of a light curve normalized by its mean flux, computed after subtracting the average measurement-error variance. It is measured from ZTF g-band light curves binned to one-day means, after removing outliers and requiring at least 100 points. Because the Eddington ratio normalizes bolometric luminosity by the Eddington luminosity, it removes most of the mass and redshift dependence carried by luminosity alone; the weighted least-squares regressions of $\log\lambda_{\rm Edd}$ against $\log F_{\rm var}$ are the machinery that produces Equation 5.

What would settle it

Extend the $z\approx2$-$3$ light curves to a rest-frame baseline comparable to low redshift, roughly 2000 days as LSST monitoring will provide, and recompute $F_{\rm var}$. If high-redshift $F_{\rm var}$ values rise substantially, the universal relation steepens or becomes redshift-dependent; if they stay put, the relation is confirmed.

Watch

Extended reading notes

Core claim

The central claim is that the Eddington ratio, not luminosity or black hole mass alone, is the physical parameter that organizes optical quasar variability across cosmic time. Over the full $0<z<3$ sample, the measured fractional variability in the g-band obeys $\log\lambda_{\rm Edd}=(-0.71\pm0.06)\log F_{\rm var}-(1.52\pm0.06)$ with $R^2=0.15$ and global correlation $r=-0.31$; the authors state that this relation is particularly useful to estimate the Eddington ratio of a general quasar at $0<z<3$ from its measured $F_{\rm var}$. They interpret the anti-correlation physically: at high accretion rates the disk radiates more efficiently and the stochastic amplitude of brightness changes is suppressed relative to the mean. The paper does not claim the relation is exact; it presents it as a robust empirical framework with redshift-dependent scatter, weakest at $2<z<3$.

Load-bearing premise

The comparison treats $F_{\rm var}$ measured at different redshifts as equivalent, even though the rest-frame baseline shrinks from about 2000 days at $z\approx0$ to about 500 days at $z\approx3$ and the g-band moves to progressively bluer rest wavelengths.

Editorial extensions

If this is right

  • Eddington ratios for quasars at $0<z<3$ can be estimated from single-band photometric variability alone, without spectra.
  • Upcoming high-cadence surveys can map accretion states over large quasar samples and cosmic time using one photometric relation.
  • The redshift dependence seen in luminosity and black-hole-mass correlations is largely a selection effect; Eddington ratio is the underlying variable.
  • The weaker high-redshift correlation points to a timescale requirement: variability metrics need rest-frame baselines long enough to capture the full fluctuation power.

Reading between the lines

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

  • If the relation survives longer baselines, variability could serve as a cheap proxy for black-hole growth rate in surveys too faint for spectroscopy, complementing X-ray and radio accretion indicators.
  • A natural test is to apply Equation 5 to r-band or i-band light curves; if the same slope holds across bands, the relation is truly wavelength-independent and not a g-band artifact.
  • One could check individual high-Eddington quasars known from single-epoch spectra: those with low $F_{\rm var}$ should systematically be the most efficiently accreting, and outliers may flag unreliable virial mass estimates.
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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 / 5 minor

Summary. The paper analyzes ZTF g-band photometric variability for 915 quasars from the AQMES-MED sample and correlates the fractional variability Fvar with bolometric luminosity, black hole mass, and Eddington ratio across five redshift bins from 0 to 3. The authors report an anti-correlation between Fvar and luminosity, a redshift-dependent Fvar-MBH relation, and a global anti-correlation between log lambda_Edd and log Fvar. This global relation is expressed as Eq. (5) and is claimed to be nearly redshift independent and useful for estimating Eddington ratios of quasars at 0<z<3.

Significance. If the claimed universal lambda_Edd-Fvar relation were established, it would provide a simple photometric estimator of Eddington ratio across a wide redshift range and a useful constraint for accretion-disk variability models. The paper contributes a carefully constructed sample, explicit checks on emission-line contamination in the photometric bands, publicly available filtering code, and a machine-readable catalog of variability and spectral properties. However, the central claim of a universal, redshift-invariant relation is not supported by the paper's own bin-dependent fits, and the global relation's predictive power is weak. The lasting value of the work is likely the descriptive correlations and the data products rather than Eq. (5) as currently stated.

major comments (4)
  1. [Section 4.4 and Table 4] The claim that the WLS fits are "all very similar" for 0<z<2 is contradicted by Table 4: the slopes are -0.99, -0.32, -0.48, -0.43, and -0.16 across the five redshift bins. The lowest-redshift slope differs from the adjacent bin by roughly a factor of three, and the 2<z<3 bin has R^2=0.01, which is effectively no within-bin correlation. Therefore Eq. (5), presented as a universal 0<z<3 relation, is not supported by the paper's own bin-resolved fits.
  2. [Section 4.4, Eq. (5)] The global regression may reflect between-bin covariance rather than an intrinsic physical relation. Higher-redshift quasars in this sample have higher lambda_Edd and lower Fvar, with the latter partly caused by the rest-frame baseline shrinking from about 2000 days at z~0 to about 500 days at z~3 (Section 4.1). The authors should report partial correlations controlling for redshift and quantify the global fit after formally excluding or down-weighting the highest-redshift bin; the statement that the correlation coefficient remained unchanged after excluding that bin is not accompanied by any quantitative result.
  3. [Section 4.4, Eq. (5)] The predictive utility claimed for Eq. (5) is not established. The relation is an in-sample fit to the same 915 objects with R^2=0.15, and no independent validation set or cross-validation is provided. With R^2=0.15, the scatter in inferred lambda_Edd is large relative to the dynamic range of the relation, so the phrase "particularly useful to estimate the Eddington ratio" needs quantitative support, such as a scatter plot of predicted versus measured lambda_Edd or a cross-validated prediction interval.
  4. [Section 4.1 and Section 4.4] The systematic redshift dependence of the rest-frame monitoring baseline and rest-frame wavelength is load-bearing for the universality claim. The paper concedes that the rest-frame interval at z>2 may be insufficient to fully capture variability. If high-z Fvar values are systematically underestimated, the observed flattening of the lambda_Edd-Fvar relation at high redshift is an expected artifact, not evidence of a universal relation. The authors should either quantify this effect by recomputing Fvar on matched rest-frame baselines or explicitly restrict the claimed generality of Eq. (5) to z<2.
minor comments (5)
  1. [Abstract] The sentence "an anti-correlation the highest redshifts" is missing the word "at" before "the highest redshifts."
  2. [References] Lu et al. (2019) and Vanden Berk et al. (2004) each appear twice in the reference list with slightly different citation details; these duplicates should be merged.
  3. [Eq. (4)] The second term in Eq. (4) uses sigma_err without a clear definition; if it denotes the standard deviation of the measurement errors, that should be stated explicitly, and the notation should be made consistent with sigma^2_err used in Eqs. (2) and (3).
  4. [Figure 11 and Section 4.4] The caption of Figure 11 describes a green shaded region as a 95% prediction interval, while the text in Section 4.4 refers to the green-shaded region as the 95% confidence interval; these statements should be aligned.
  5. [Section 2.2] The phrase "approximately 2000 MJDs in the rest frame per source" is ambiguous: it likely means roughly 2000 epochs or a baseline in MJD units, but the wording should be clarified to distinguish the number of epochs from the temporal baseline.

Circularity Check

1 steps flagged · score 6.0 of 10

Eq. (5) is the in-sample WLS regression of the same 915 quasars, so its advertised use as a predictor of Eddington ratio reduces to the fitted line; no self-citation chain or definitional circularity is present.

  1. fitted input called prediction [Section 4.4, Eq. (5), Fig. 11]
    "logλEdd = (−0.71 ± 0.06)·logFvar−(1.52 ± 0.06) (5) This relation is particularly useful to estimate the Eddington ratio of a general quasar at 0<z<3 from its measured Fvar"

    Equation (5) is not derived from an independent physical principle; it is the WLS regression of log λEdd on log Fvar for the same 915 quasars whose individual bins and overall fit are shown in Figs. 10-11 and Table 4. The suggested use, to 'estimate the Eddington ratio ... from its measured Fvar', is an evaluation of the fitted line on the training sample, so the claimed prediction is the regression equation itself. The abstract's 'or vice-versa' is likewise just the algebraic inverse of Eq. (5). No held-out sample or external benchmark is provided, so the predictive claim reduces by construction to the fitted relation.

full rationale

No load-bearing self-citations or imported uniqueness theorems appear: earlier work by the authors is cited for background and for spectral fitting tools, not to force the central relation. The λEdd-Fvar anti-correlation is an empirical fit between independently measured quantities and therefore is not self-definitional. The concrete circularity is confined to the presentation of Eq. (5): it is the in-sample regression of the 915-object sample, yet it is offered as 'particularly useful to estimate the Eddington ratio of a general quasar ... from its measured Fvar'. That is a fitted input called a prediction. Separately, and not as a circularity argument, Table 4's per-bin slopes range from -0.99 to -0.16 with R² values of 0.18, 0.03, 0.10, 0.12, and 0.01, which conflicts with the text's claim that the fits are 'all very similar' and makes the 'universal' relation statistically fragile, especially at 2<z<3. Those internal inconsistencies are correctness risks, not circularity. Because the central claim 'general equation ... enabling predictions' reduces to the fitted line, a partial-circularity score of 6 is appropriate.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central scaling Eq. 5 is an empirical fit with two free parameters, resting on the validity of virial mass estimates, the representativeness of the sample, and the comparability of Fvar across redshift. The paper has no external validation of the claimed predictive relation.

free parameters (2)
  • Slope a in Eq. 5 = -0.71 ± 0.06
    Fitted by weighted least squares to log λEdd vs log Fvar for all 915 quasars; the central claim of a universal relation depends on this number.
  • Intercept b in Eq. 5 = -1.52 ± 0.06
    Fitted intercept of the same regression; together with the slope it defines the claimed universal relation.
assumptions (4)
  • domain assumption The excess variance after subtracting measurement noise traces intrinsic quasar variability.
    Invoked in Eq. 1-3, Section 3.1. If the noise model or binning is wrong, Fvar is biased, especially for low variability sources.
  • domain assumption Single-epoch virial black hole masses and Eddington ratios from Wu & Shen (2022) are unbiased across 0<z<3.
    Section 2.1 and 4.3 note that C IV-based masses at high z are less reliable; the λEdd-Fvar relation inherits any mass bias.
  • domain assumption The AQMES sample, though not representative, is sufficient to infer a universal quasar relation.
    Section 2.1 states the sample is preferentially lower redshift and not representative, yet the paper generalizes to 'a general quasar at 0<z<3'.
  • domain assumption Fvar values measured over different rest-frame baselines and wavelength coverages are comparable.
    Section 4.1 lists the baseline shrinking with redshift; Section 4.4 concedes the high-z baseline may be insufficient to capture variability. This underpins the redshift-independent fit.

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Pith. "Pith review of Exploring Quasar Variability With ZTF at 0<z<3: A Universal Relation with Eddington Ratio." pith.science (2026). https://pith.science/paper/54UQPNA3

@misc{pith2026250509779,
  author       = {Pith},
  title        = {Pith review of: Exploring Quasar Variability With ZTF at 0<z<3: A Universal Relation with Eddington Ratio},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/54UQPNA3}},
  note         = {Machine review of arXiv:2505.09779}
}
abstract

Quasars, powered by accretion onto supermassive black holes (SMBHs), exhibit significant variability, offering insights into the physics of accretion and the properties of the central engines. In this study, we analyze photometric variability and its correlation with key quasar properties, including black hole mass ($M_{\mathrm{BH}}$) and nuclear luminosities, using 915 quasars with $0\leq z<3.0$ from the AQMES sample monitored within SDSS-V. Variability metrics were derived from approximately 6-year light curves provided by the Zwicky Transient Facility -- ZTF, while SMBH masses and luminosities were obtained from the SDSS DR16 quasar catalog of \citet{wu2022catalog}. We identify a strong anti-correlation between variability amplitude and luminosity, which strengthens with redshift, and a redshift-dependent trend for $M_{\mathrm{BH}}$: a positive correlation at low redshifts, no significant correlation at intermediate redshifts, and an anti-correlation the highest redshifts. Our main finding is a robust anti-correlation between photometric variability amplitude and Eddington ratio, consistent across redshift bins. We present a general equation encapsulating this relationship, that appears to be almost free of redshift dependence, enabling predictions of quasar variability based on accretion parameters {\bf or vice-versa}. The derived relation with the Eddington ratio provides a unified framework for interpreting variability in active galactic nuclei (AGN) and facilitates future studies of quasar variability using high-cadence surveys, such as the Vera C. Rubin Observatory's Legacy Survey of Space and Time -- LSST.

Figures

Figures reproduced from arXiv: 2505.09779 by the authors.

Figure 1
Figure 1. Properties of our filtered sample. From left to right, top row: spectroscopic redshift (z) from the DR16Q catalog; absolute magnitude in the g-band (Mg); and bolometric luminosity (log LBol, in erg s−1 ). Bottom row: black hole mass (log MBH, in M⊙); Eddington ratio (log LBol/LEdd); and apparent magnitude in the g-band (mg). The dashed red line represents the respective median values [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 2
Figure 2. Bolometric luminosity plotted against redshift (z) for the entire filtered sample. 3.2. Emission line contribution to the photometric bands To investigate the contribution of emission lines to the different filters, we selected the four quasar spectra with the best signal-to-noise ratio and the strongest emission lines from each redshift bin. One representative spec￾trum from each redshift bin is shown in [PITH_FUL… view at source ↗
Figure 3
Figure 3. Typical SDSS spectra from our sample, representative of seven redshift bins: 0–0.25, 0.25–1.0, 0.75–1.25, 1.25–1.75, 1.75–2.0, 2.0–2.50, and 2.5–3.0. The ZTF filter transmission profiles are overplotted and typical emission lines are identified. Flux densities (Fλ) are in units of 10−17 erg s−1 cm−2˚A −1 and the filters transmission fractions (T) are shown on the right y-axis [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: ZTF lightcurves in the three ZTF bands (g-, r-, and i-bands) for one of the AGNs (SDSS J225307.36+194234.6) from our filtered sample [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Flowchart illustrating the applied logical sequence for data filtering. g, r, and i filters across the redshift bins. This highlights how the filters shift with redshift and allows the verifica￾tion of possible inclusion or absence of specific emission lines listed in …
Figure 6
Figure 6. Figure 6: Relationship between bolometric luminosity (left) and black hole mass (right) with the central rest-frame wavelength at each quasar corresponding to the observed g-band central wavelength. The color gradient indicates varying redshifts, high￾lighting a decrease of both…
Figure 7
Figure 7. Figure 7: Histograms of Fvar for the sample in five redshift bins, showing both g- and r-band distributions. The total number of sources is given in the top right corner of each subplot. 4.2. Luminosity vs. Fvar [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Luminosities L1350, L3000, L5100, and bolometric luminosity versus Fvar in different redshift bins, with linear regression fits overlaid. For the highest (lowest) redshift bins, the wavelength range of the spectra did no cover L5100 (L1350). Pearson’s correlation coeff…
Figure 9
Figure 9. Figure 9: Black hole mass versus Fvar in different redshift bins, with linear regression fits overlaid. The Pearson’s cor￾relation coefficient (r), p-value (p), and the total number of points (N) are shown at the bottom [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 11
Figure 11. Figure 11: Same as [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 10
Figure 10. Figure 10: λEdd versus Fvar across different redshift bins, with linear regression fits overlaid. The Pearson’s correlation coefficient (r), p-value (p), and the total number of points (N) are displayed at the bottom. Additionally, a fit for all points, along with its correspond…

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