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X-ray reverberation modelling of the continuum, optical/UV time-lags in quasars

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The UV/optical time lags of 128 quasars are fully consistent with X-ray reverberation, provided the X-ray corona sits above about 40 gravitational radii.

desk verdict First real application of the K21 model to quasar continuum lags, but the 'h > 40 Rg' lower limit is not supported by the paper's own 1σ error bars. read the letter →

arxiv 2411.09681 v1 pith:H72H3HZT submitted 2024-11-14 astro-ph.GA astro-ph.HE

D. A. Langis , I. E. Papadakis , E. Kammoun , C. Panagiotou , M. Dovčiak This is my paper · ORCID
classification astro-ph.GAastro-ph.HE
keywords X-rayreverberationaccretiondiscquasartimelagscoronaheightblackholespinNovikov-ThorneUV/opticalcontinuumactivegalacticnuclei
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 tests whether the well-known wavelength-dependent delays between UV and optical variations in quasars can be produced by X-ray reverberation, the heating of the accretion disc by X-rays from a compact corona. Using archival time-lag measurements for 128 quasars from four monitoring surveys, it builds a rest-frame mean time-lag spectrum and fits it with analytic X-ray reverberation models. The data are fully consistent with the model when the corona height is larger than roughly 40 gravitational radii, for both a non-spinning black hole (best fit 42 Rg with colour correction 1.7) and a maximally spinning black hole (best fit 50 Rg with colour correction 2.4). This matters because earlier work reported quasar disc sizes that appear two to three times too large; the paper's models remove that tension without invoking enlarged or exotic discs.

What carries the argument

The load-bearing object is the K21b analytic time-lag model, built on the K21a disc response functions. These functions compute, at each disc radius, the extra time-variable UV/optical flux produced when X-rays from a point-like corona are absorbed by the disc, including relativistic light travel and bending, the disc ionisation state, and a radial-dependent ratio of external (X-ray) to internal (accretion) heating. Given a black hole mass, accretion rate, 2-10 keV luminosity, spin, colour-correction factor (the standard adjustment for electron scattering in the disc atmosphere), and corona height $h$, the model predicts the lag between two wavelengths; the authors fix all parameters from the literature and leave $h$ as the only free parameter, scanning 5 to 80 $R_{\rm g}$ (gravitational radii). A later colour-correction prescription is used to adapt the model to $f_{\rm col}$ values of 1 and 1.7 as well as the original 2.4.

What would settle it

A direct measurement of the X-ray corona height in one of these quasars — for example from an iron-K X-ray reverberation lag that locates the corona, or from microlensing size measurements — that placed the corona below about 20 gravitational radii would contradict the best-fit models, which require about 40-50 Rg. Alternatively, if future time-lag data for quasars above $10^9\,M_\odot$ (where the analytic relations are extrapolated) require heights below 40 Rg to fit, the paper's central claim would be falsified.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the continuum UV/optical time lags of luminous quasars can be explained by thermal reverberation of a standard Novikov-Thorne accretion disc, with no need for anomalously large disc radii. The mean rest-frame lag spectrum of 128 quasars is well fitted by the K21b analytic time-lag relations for both zero and maximal black hole spin, as long as the X-ray corona height is above about 40 gravitational radii. For a non-spinning black hole with colour correction 1.7, the best-fit height is 42 Rg ($\chi^2=11.4$ for 5 degrees of freedom); for a maximally spinning black hole with colour correction 2.4, it is 50 Rg ($\chi^2=11.3$). The residuals of individual quasars are consistent with a zero-mean Gaussian with the expected scatter, and the paper notes the same corona-height range independently explains the half-light radii of microlensed quasars as supporting evidence. The paper states this is the first time quasar continuum UV/optical time lags have been shown to be consistent with X-ray reverberation of standard discs.

Load-bearing premise

The load-bearing premise is that the analytic lag relations, originally calibrated for black holes at or below about $10^8\,M_\odot$ with accretion rate below 0.5, stay accurate for the quasars in this sample, which reach $10^9\,M_\odot$ and Eddington ratios up to 1; if they lose accuracy there, the inferred corona heights and the good fits would both change.

Editorial extensions

If this is right

  • The observed UV/optical quasar lags do not require disc radii larger than standard thin-disc predictions; X-ray heating of a Novikov-Thorne disc accounts for them.
  • The X-ray corona in quasars is typically located at or above about 40 gravitational radii, matching the height independently inferred from microlensed quasar half-light radii.
  • Continuum time-lag measurements can constrain corona height, and the best-fitting height is insensitive to whether the black hole is non-spinning or maximally spinning.
  • Earlier models that reported factor-of-2-3 lag discrepancies likely failed because they assumed a constant ratio of X-ray to internal heating across the disc, rather than computing it radius by radius.
  • If the flat-disc assumption is relaxed, smaller corona heights may fit equally well, so the quoted heights are upper-side estimates.

Reading between the lines

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

  • A natural extension the paper does not pursue is to split the sample by black hole mass or Eddington ratio and re-fit the corona height in each bin; a trend would reveal that height is not universal.
  • If the best-fit heights are taken at face value, the X-ray corona sits well above the disc plane in luminous quasars, which bears on models of the disc-corona interface and on jet-launching geometry; the paper does not discuss this connection.
  • Future wide-field time-domain surveys with longer, denser UV/optical light curves could apply the same stacking method in several redshift bins, and simultaneous far-UV monitoring could help break the spin-colour degeneracy the paper identifies.
  • Because the flat-disc assumption tends to push the inferred height upward, a finite disc scale height modelled with radiative transfer could bring the best-fit heights down to around 20 Rg; this is a testable extension of the same model rather than a claim of the paper.
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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 compiles archival UV/optical continuum time-lag measurements for 128 quasars, computes a rest-frame mean time-lag spectrum in six wavelength bins, and fits it with the analytical X-ray reverberation models of Kammoun et al. (2021a,b), extended to different colour-correction factors by Kammoun et al. (2023). For each of six model variants (spin a*=0 or 1 with fcol = 1, 1.7, 2.4), the only free parameter is the X-ray corona height h. The authors report that only the models with (a*=0, fcol=1.7) and (a*=1, fcol=2.4) can fit the data, with best-fit heights h=42^{+34}_{-23} R_g and h=50^{+15}_{-12} R_g respectively, both with chi^2_min/dof = 11.4/5 and p_null = 0.04. The central claim is that the observed quasar time lags are fully consistent with the X-ray reverberation hypothesis, provided the corona height is larger than about 40 R_g, and that this is consistent with microlensing disc-size constraints.

Significance. If substantiated, the result would be a notable step: it would show that standard Novikov-Thorne discs illuminated by a compact X-ray corona can reproduce the UV/optical time lags in luminous quasars, potentially resolving the apparent discrepancy between observed and predicted disc sizes without invoking non-standard disc physics. The study uses a large archival sample and a physically motivated model that includes relativistic effects and radial-dependent external heating, and it explicitly connects to independent microlensing constraints. However, the statistical support is marginal (p_null = 0.04, below the conventional 0.05 threshold), and the headline lower limit on the corona height is not supported by the confidence intervals reported in Table 1. The analysis also relies on an arbitrary error-inflation factor and on the applicability of K21b relations outside their calibrated parameter range.

major comments (4)
  1. [Abstract; Sect. 5.2; Table 1] The claim that 'the corona height is larger than ~40 Rg' is not supported by the best-fit uncertainties reported in Table 1. For model M01.7 the best fit is h=42^{+34}_{-23} Rg, so h≈19 Rg is within the 1σ lower interval; for model M12.4 h=50^{+15}_{-12} Rg, so h≈38 Rg is within 1σ. Thus the data are consistent with heights well below 40 Rg, and the statements in the abstract and in Sect. 5.2 ('the corona height should be equal or larger than 40 Rg') are overstatements. Please quote the confidence intervals and rephrase the conclusion as a best-fit range rather than a firm lower limit.
  2. [Sect. 4; Table 1] The reported p_null = 0.04 for the two acceptable models is below the conventional 0.05 threshold for rejecting the null hypothesis. The text nevertheless describes these fits as fitting 'well' and later concludes that the data are 'fully consistent' with the model. This is an internal inconsistency in the statistical interpretation. The authors should acknowledge that the fits are marginal at the 5% level, report the exact p-values, and discuss how the conclusion depends on the choice of significance threshold.
  3. [Sect. 4] The error-inflation factor of 1.1 applied to the observed mean time-lag errors is ad hoc and is not a propagation of the stated uncertainties on MBH, Lbol, and L2-10keV. Since the chi^2 values are near the rejection threshold, the choice of this factor directly affects whether a model is deemed acceptable. The authors should either propagate the input parameter uncertainties through the model to obtain model time-lag uncertainties, or at minimum perform a sensitivity analysis varying the inflation factor and reporting how chi^2_min and p_null change.
  4. [Sect. 2.2] The K21b analytic time-lag relations were developed for AGN with MBH <~ 1e8 Msun and accretion rate <~ 0.5, but the final sample extends to MBH < 1e9 Msun and lambda_Edd <= 1. The text states that the analysis was repeated on smaller subsamples with MBH < 5e8 and 2e8 Msun and that the results were 'almost identical', but no numerical results are provided. This robustness test is load-bearing for the extrapolation claim; please present the best-fit heights, chi^2 values, and p-values for these restricted samples, or justify why the K21b relations remain valid in the extrapolated regime.
minor comments (5)
  1. [Abstract] The abstract states that the model assumes 'the measured BH mass, accretion rate and X-ray luminosity', but the 2-10 keV luminosity is not measured; it is estimated from the Lusso et al. (2012) relation. Please clarify this in the text.
  2. [Section 5] The 'Conclusions' heading is followed by no text. A concise conclusions paragraph summarizing the main findings and their caveats should be added.
  3. [Table 1] For models M01 and M11 the best-fit height is listed as 80 Rg, which is the upper boundary of the searched range. Please state explicitly whether the chi^2 minimum is at the boundary and whether the fit was attempted beyond 80 Rg.
  4. [Sect. 4] The description 'we increased the error of \bar t_lag,obs by a factor of 1.1' is ambiguous: it should state whether the factor is applied to the standard error of the mean or to the individual time-lag errors before averaging.
  5. [Sect. 4; Fig. 7] The Kolmogorov–Smirnov test result is quoted as 'pnull = 0.15'. Please define this quantity explicitly and ensure the notation is consistent with the p_null used for the chi^2 fits.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the corona height is a fitted parameter and the consistency claim is a goodness-of-fit against independent archival data.

full rationale

The paper's central derivation chain is not circular. The observed mean time-lags spectrum is constructed from archival measurements (Jiang et al. 2017; Homayouni et al. 2019; Guo et al. 2022; Jha et al. 2022), which are external to the authors. The model time lags are computed from the K21a/K21b relativistic response functions and the Kammoun et al. (2023) colour-correction prescription; these are previously published physical calculations, not fits to the quasar time lags used here. The only free parameter is the X-ray corona height h, scanned over 5-80 Rg, with χ² minimized against the six binned mean lags. Thus the statement that X-ray reverberation is 'fully consistent' with the observed lags is a goodness-of-fit result with one free parameter, not an input recycled as an output. Heavy self-citation to K21a/b, Kammoun et al., Dovčiak et al., and Papadakis et al. (2022) is present, but it is not load-bearing in a circular way: the K21b equations are used as an independent model, and the Papadakis et al. microlensing comparison is a consistency check across separate observables (disc half-light radii vs. continuum time lags). No equation in the paper defines the target result in terms of itself, and no fitted parameter is renamed as a prediction. The skeptic's concern that the h > 40 Rg abstract claim is not supported by the 1σ intervals (e.g., h = 42^{+34}_{-23} Rg allowing h ≈ 19 Rg) is a statistical calibration or overstatement issue, not a circularity of the derivation chain.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on three classes of inputs: the analytic K21b model, which assumes a point-like corona, a flat Novikov-Thorne disc, and a specific treatment of relativistic effects; per-source quantities (MBH, Lbol, L2-10keV) taken from literature or empirical relations; and a single fitted corona height. No new entities are invented, but the model's validity is assumed beyond the parameter range for which it was originally developed.

free parameters (2)
  • X-ray corona height h = 42 Rg (a*=0, fcol=1.7); 50 Rg (a*=1, fcol=2.4)
    The only parameter varied in the fit; best-fit values from Table 1.
  • Error inflation factor = 1.1
    Applied to observed mean-lag errors to absorb uncertainties in MBH, lambdaEdd, and L2-10keV; chosen by hand in Sect. 4.
assumptions (5)
  • domain assumption K21b analytic time-lag relations remain valid for MBH up to 1e9 Msun and lambdaEdd up to 1, despite being developed for MBH <= 1e8 Msun and accretion rate < 0.5.
    Sect. 2.2 selects sources with MBH < 1e9 and lambdaEdd <= 1 for exactly this reason; the assumption is load-bearing for all fitted heights and p-values.
  • domain assumption X-ray corona is a point source on the BH rotation axis at height h, emitting isotropically in its rest frame.
    Section 3 describes the model basis; the disc response functions and resulting time lags depend on this geometry.
  • domain assumption The accretion disc is a standard, geometrically thin, flat Novikov-Thorne disc, with a single colour-correction factor applied to the whole disc.
    Section 3 and Sect. 5.2 caveat; the inferred corona height is explicitly stated to depend on the flat-disc assumption, and fcol = 1.7/2.4 is a global value rather than radius-dependent.
  • domain assumption 2-10 keV luminosity for every source follows the Lusso et al. (2012) relation, log(Lbol/L2-10keV) = 0.75 log(lambdaEdd) + 2.13.
    Section 4 uses this relation because no direct X-ray luminosity is available for most quasars; the model time-lags depend on L2-10keV.
  • domain assumption All quasars in the sample share the same corona height, black hole spin, and colour-correction factor.
    Section 5.2 states the analysis assumes a single height and spin; the authors note a distribution is more plausible, which would broaden the inferred parameters.

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Cite this review

Pith. "Pith review of X-ray reverberation modelling of the continuum, optical/UV time-lags in quasars." pith.science (2026). https://pith.science/paper/H72H3HZT

@misc{pith2026241109681,
  author       = {Pith},
  title        = {Pith review of: X-ray reverberation modelling of the continuum, optical/UV time-lags in quasars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H72H3HZT}},
  note         = {Machine review of arXiv:2411.09681}
}
abstract

Context: Extensive, multi-wavelength monitoring campaigns of nearby and higher redshift active galactic nuclei (AGN) have shown that the UV/optical variations are well correlated with time delays which increase with increasing wavelength. Such behaviour is expected in the context of the X-ray thermal reverberation of the accretion disc in AGN. Aims: Our main objective is to use time-lag measurements of luminous AGN and fit them with sophisticated X-ray reverberation time-lags models. In this way we can investigate whether X-ray reverberation can indeed explain the observed continuum time lags, and whether time-lag measurements can be used to measure physical parameters such as the X-ray corona height and the spin of the black hole (BH) in these systems. Methods: We use archival time-lag measurements for quasars from different surveys, and we compute their rest frame, mean time-lags spectrum. We fit the data with analytical X-ray reverberation models, using $\chi^2$ statistics, and fitting for both maximal and non spinning BHs, for various colour correction values and X-ray corona heights. Results: We found that X-ray reverberation can explain very well the observed time lags, assuming the measured BH mass, accretion rate and X-ray luminosity of the quasars in the sample. The model agrees well with the data both for non-rotating and maximally rotating BHs, as long as the corona height is larger than $\sim 40$ gravitational radii. This is in agreement with previous results which showed that X-ray reverberation can also explain the disc radius in micro-lensed quasars, for the same corona heights. The corona height we measure depends on the model assumption of a perfectly flat disc. More realistic disc models may result in lower heights for the X-ray corona.

Figures

Figures reproduced from arXiv: 2411.09681 by the authors.

Figure 1
Figure 1. Upper left panel: The rest frame wavelength of the reference band (in our work always the g−band), λre f , versus the rest frame wavelength of the secondary light curves, λS , used to measure the time lags. Upper right panel: The MBH as a function of λS for the objects in our final sample. Bottom panels: Same as the top right panel but for λEdd and L2−10keV (left and right panels, respectively). The boxes in all pan… view at source ↗
Figure 2
Figure 2. The redshift, MBH, Lbol and λEdd distributions for the sources in the final sample. 3000 4000 5000 6000 7000 S (Å) 5 0 5 10 15 20 25 Time Lag (Days) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The observed time lags as a function of λS . Red filled circles show the mean time lags (computed as explained in Sect. 2.3). tributions in other quasar samples like, for example the SDSS quasars in Stripe 82 (see e.g. Petrecca et al. 2024). We are miss￾ing quasars with BH mass larger than ∼ 109 M⊙ and Lbol> 1046 erg/s, as the optical/UV continuum time lags have not been mea￾sured yet in such objects. On the other h… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The dependence of the time lags on BH mass, accretion rate and reference wavelength (top, middle and lower panels, respectively), according to the K21a,b time-lags models (see text for details). the range of the reference wavelengths in the sample (see top left panel i…
Figure 5
Figure 5. Figure 5: The observed, mean time-lags spectrum (black circles, connected with a solid line) and the best-fit, M01.7 and M12.4 models (red and blue points, respectively, connected with dashed lines). thermore, the standard deviation of the residuals is 3.3 days. This is very sim…
Figure 6
Figure 6. Figure 6: shows that the best-fit residuals at wavelengths longer than ∼ 6200Å appear to be predominately positive, indi￾cating that the model time lags may underestimate the observa￾tions at these wavelengths. This is also shown in [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: The distribution of the best-fit residuals plotted in the lower panel of [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

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Forward citations

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