Standardizing reverberation-mapped H$\alpha$ and H$\beta$ active galactic nuclei using radius--luminosity relations involving monochromatic and broad H$\alpha$ luminosities

Amit Kumar Mandal; Bharat Ratra; Bo\.zena Czerny; Michal Zaja\v{c}ek; Shulei Cao

REVIEW 3 major objections 4 minor 105 references

Standardizing reverberation-mapped H$\alpha$ and H$\beta$ active galactic nuclei using radius--luminosity relations involving monochromatic and broad H$\alpha$ luminosities

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A homogeneous sample of 41 low-redshift AGNs becomes standardizable through four radius–luminosity relations.

desk verdict A careful, honest extension of the RM-AGN cosmological program: the homogeneous 41-object sample is a real step, but weak cosmology and unpropagated peculiar-velocity systematics keep it from being a decisive validation. read the letter →

arxiv 2412.19665 v2 pith:5S5TEX57 submitted 2024-12-27 astro-ph.GA astro-ph.COgr-qchep-phhep-th

Shulei Cao , Amit Kumar Mandal , Michal Zajaček , Bożena Czerny , Bharat Ratra This is my paper · ORCID

classification astro-ph.GAastro-ph.COgr-qchep-phhep-th

keywords reverberationmappingactivegalacticnucleiradius-luminosityrelationbroad-lineregiontimedelaysH-alphaemissionH-betacosmologicaldistanceindicatorsAGNstandardizability

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 argues that 41 low-redshift active galactic nuclei with reverberation-mapped Hα and Hβ time delays can serve as standardizable distance indicators. The basis is four radius–luminosity relations, pairing each line delay with either the broad Hα luminosity or the monochromatic 5100 Å luminosity, whose fitted slope and intercept stay essentially unchanged across six different cosmological models. That stability is the paper's criterion for standardizability, because it shows the calibration does not secretly depend on the assumed expansion history. The cosmological constraints from these 41 sources are weak but consistent within 2σ with those from H(z)+BAO data, in contrast to an earlier, larger but less homogeneous Hβ sample that disagreed by more than 2σ. If the claim holds, consistent time-lag measurement is the key to turning reverberation-mapped AGNs into a usable cosmological probe.

What carries the argument

The load-bearing object is the radius–luminosity relation, $\log(\tau/\mathrm{day}) = \beta + \gamma \log(L/10^{44}\,\mathrm{erg\,s^{-1}})$, where $\tau$ is the measured rest-frame line delay and $L$ is the luminosity inferred from the observed flux through $L = 4\pi D_L^2 F$. Because the delay is measured and the flux is observed, a calibrated relation turns each AGN into a distance estimate whose only free ingredients are $\beta$, $\gamma$, and the assumed cosmology. The paper fits $\beta$, $\gamma$, an intrinsic scatter parameter, and cosmological parameters simultaneously in a joint likelihood; if the relation parameters are insensitive to the cosmological model, the dataset is declared standardizable. This machinery is applied to four combinations of Hα or Hβ delay with broad Hα or monochromatic 5100 Å luminosity, using measured fluxes directly rather than cosmology-dependent luminosities.

What would settle it

Recompute the four radius–luminosity fits after removing all sources with $z < 0.05$, or after replacing the adopted peculiar-velocity corrections with an independent local-flow velocity field; if the slopes or intercepts shift by more than the quoted uncertainties, or the cosmological consistency with H(z)+BAO data disappears, the standardizability claim would be falsified.

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Extended reading notes

Core claim

The authors claim that, for a sample of 41 Type 1 AGNs whose Hα and Hβ lags were all determined with the interpolated cross-correlation function, the four radius–luminosity relations of the form $\log(\tau/\mathrm{day}) = \beta + \gamma \log(L/10^{44}\,\mathrm{erg\,s^{-1}})$ have slopes $\gamma$ near 0.54–0.62 and intercepts larger for Hα than for Hβ. These parameters do not shift by more than about half a $\sigma$ when the assumed cosmology changes among flat and nonflat ΛCDM, XCDM, and ϕCDM models, which the authors take as evidence that the sources are standardizable. The measured slopes are steeper than the 0.5 predicted by a simple photoionization model and steeper than slopes from larger Hβ samples, a difference the authors attribute to the absence of high-accreting sources in their low-redshift sample. The cosmological parameters derived from these AGNs are only weakly constrained but are consistent within 2σ with those from more established probes, unlike the earlier 118-source Hβ analysis, whose constraints were in greater tension.

Load-bearing premise

The peculiar-velocity corrections applied to the low-redshift sources are assumed accurate enough that the resulting luminosity distances are unbiased, and the correction's own uncertainty is not propagated into the analysis.

Editorial extensions

If this is right

Larger homogeneous reverberation-mapping samples would yield radius–luminosity calibrations independent of the assumed cosmology, allowing low-redshift AGNs to be appended to the distance ladder.
The radius–luminosity parameters change by at most about 0.8σ when H(z)+BAO data are added, so combining these AGNs with established probes is safe and shifts cosmological constraints by only about 0.1σ.
Because Hβ lags paired with Hα luminosity show the lowest intrinsic scatter, future monitoring campaigns should prioritize homogeneous Hβ lag measurement to tighten the calibration.
The steeper-than-0.5 slopes imply that the simple photoionization scaling $R \propto L^{0.5}$ is incomplete, and samples dominated by low-Eddington-ratio sources will systematically overestimate the slope.
The earlier disagreement with standard cosmological probes is attributed to sample inhomogeneity rather than to the radius–luminosity method itself, so consistent lag determination is the enabling condition for the probe.

Reading between the lines

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

If peculiar-velocity systematics can be controlled, the calibrated radius–luminosity intercepts could provide absolute distance measurements at $z \lesssim 0.5$, offering a low-redshift anchor independent of supernovae.
Combining this low-redshift Hα/Hβ calibration with higher-redshift Mg II and C IV reverberation samples could cover a much wider redshift range with one consistent method; the paper notes the current sample is too small for such a joint fit now.
The observed negative correlation between radius–luminosity residuals and Eddington ratio suggests a testable extension: explicitly including accretion rate as a second calibration parameter might remove the steep-slope offset and identify accretion physics, not sample selection, as the driver.
The mild anti-correlation between Hα equivalent width and continuum luminosity implies that line-based and continuum-based luminosities are not perfectly interchangeable, so future calibrations may need separate slope parameters for the two luminosity definitions.

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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

Colleague, read this if you work on AGN cosmology. The useful bit is the sample construction and the negative result framing: a homogeneous 41-object Hα/Hβ reverberation sample gives R–L parameters stable across six cosmological models, and the earlier ≥2σ disagreement with H(z)+BAO from the 118-object sample goes away. That is a legitimate new application of the simultaneous R–L/cosmology fitting method, not just a rerun: they correct redshifts for peculiar velocities, propagate asymmetric lag errors, and let cosmology float instead of fixing it. The paper is honest about the weak cosmological constraints—the 2σ limits are prior-dominated—so the standardizability claim rests mainly on the model-independence of the R–L parameters, which is a real but modest test.

The soft spots are proportionate. The peculiar-velocity corrections are the weakest link: the NED calculator is applied to every object, corrected z enters D_L in Eq. (9), and no uncertainty on the correction is propagated into σ_tot. For z<0.05 objects, a 300 km/s error gives a 0.04–0.1 dex shift in log D_L, comparable to the intrinsic scatter these low-luminosity anchors help set. This does not invalidate the paper, but it needs a robustness check—rerun with a few hundred km/s peculiar-velocity dispersion added, or drop the z<0.02 objects. If the results survive, the claim is considerably stronger. Also, the text in Sec. I appears to say "we do find significant constraints against the standard model" while the abstract and results say the opposite; that sentence needs fixing.

One more caution: the steeper-slope explanation (sample lacking high-accreting sources) is plausible but not established; the Eddington-ratio correlation is weak and self-correlation-prone, as the authors themselves note. Fine as a suggestion, not a result.

Overall, the methods are clear, the likelihood and priors are specified, and the comparison with earlier work is fair. I’d cite it, and I’d send it to a referee—it deserves a serious look. The main request should be a sensitivity analysis on peculiar velocities. The paper's own flag that a larger homogeneous sample is needed is the right summary.

Referee Report

3 major / 4 minor

Summary. The paper tests whether 41 low-redshift reverberation-mapped AGNs can be used as standardizable distance indicators. It simultaneously fits the parameters of four radius-luminosity relations (Hα and Hβ time delays, combined with either broad Hα luminosity or monochromatic 5100 Å luminosity) and the parameters of six cosmological models, using a Gaussian likelihood with an intrinsic scatter term. The authors report that the R-L parameters are stable across all six models, that the slopes are steeper than the simple photoionization value of 0.5 and steeper than those of earlier larger Hβ samples, and that the cosmological constraints from the 41 AGNs are weak but consistent with H(z)+BAO data. They interpret the sample homogeneity, particularly the consistent ICCF lag determinations, as the reason the results differ from their earlier 118-object Hβ analysis.

Significance. If the central standardizability claim holds, this is a useful step toward developing low-redshift RM AGNs as a cosmological probe, complementing higher-redshift Mg II and C IV samples. The paper is methodologically transparent: the likelihood, priors, MCMC setup, and model-comparison criteria are clearly specified, and the simultaneous fitting of R-L and cosmological parameters is the appropriate route around the circularity of assuming a cosmology when computing luminosities. The comparison with the independent 118-object sample is a genuine strength. However, the evidence for standardizability is currently limited by two issues: the low-redshift peculiar-velocity corrections enter every luminosity distance without propagated uncertainty, and the data have so little cosmological constraining power that model-independence of the R-L parameters is a weak test. The physical interpretation of the steeper slopes also relies on Eddington-ratio effects that are only marginally detected in the sample itself.

major comments (3)

[Sec. III A, Table I, Eq. (12)] The NED Velocity Correction Calculator is applied to all 41 redshifts, and the corrected redshift enters the luminosity distance in Eq. (9) and hence every R-L and cosmological fit, but no uncertainty is assigned to the peculiar-velocity correction and none is propagated into sigma_tot in Eq. (12). For the eleven sources with z < 0.05 (e.g., NGC 4151, NGC 6814, Mrk 142, Arp 151, NGC 5548), a peculiar-velocity error of 300 km/s changes log D_L by roughly 0.02–0.08 dex, which is comparable to or larger than the flux errors and a substantial fraction of the fitted intrinsic scatter (sigma_int ~ 0.25 dex). These low-z, low-luminosity sources anchor the low-luminosity end of each R-L relation and therefore strongly influence the intercept and slope. A coherent error in the NED model, such as an imperfect local-flow correction, would bias beta and gamma and thus bias the distance moduli without necessarily changing the apparent model-independence of the R-L parameters. The authors should propagate a peculiar-velocity uncertainty through Eq. (12), or at minimum perform a sensitivity test in which all low-z redshifts are shifted by a plausible systematic velocity and show that the values in Tables IV and VI are stable within their quoted uncertainties.
[Sec. V, Tables IV and VII] The main evidence for standardizability is that the R-L parameters vary by at most 0.22 sigma in gamma and 0.50 sigma in beta across the six cosmological models (Table VII). This is a weak test because the 41 AGNs by themselves barely constrain cosmology: in flat LambdaCDM the H-alpha broad data give only a 1-sigma lower limit Omega_m0 > 0.424, and the 2-sigma limits are set by the priors (Table IV). When the data have essentially no cosmological constraining power, the fitted R-L parameters are expected to be nearly model-independent whether or not the sources are truly standardizable. To make the standardizability claim load-bearing, the authors should calibrate the expected spread in beta and gamma under a null hypothesis, for example by running mock catalogs of non-standardizable sources drawn from a single cosmology, or by showing that the constraints tighten in a meaningful way when H(z)+BAO data are added. Absent such a test, the statement that the sources are 'standardizable' should be softened to consistency with standardizability.
[Sec. VI A, Fig. 6] The interpretation that the steeper R-L slopes compared with the 118-object H-beta sample arise from the absence of high-accreting sources is not strongly supported by the internal diagnostic presented in Fig. 6. The anti-correlation between Delta R and Eddington ratio is weak and statistically marginal for the H-alpha relations (Spearman rho = -0.10, p = 0.55 for R_Halpha-L_Halpha; rho = -0.18, p = 0.26 for R_Halpha-L_5100), and only the H-beta relations show more suggestive correlations (p = 0.073 and p = 0.011). As the authors note, lambda_Edd scales as L/R_BLR while Delta R scales approximately as R_BLR/L^0.6, so the correlation may be partly a self-correlation. Since the slope difference versus earlier samples is one of the main physical conclusions, the manuscript should either fit an explicit Eddington-ratio term in the R-L relation to test the proposed interpretation, or clearly state that the self-correlation caveat prevents a definitive physical conclusion.

minor comments (4)

[Sec. V, Fig. 4 caption and text] The caption of Fig. 4 and the corresponding discussion refer to 'Tables IV and IV'; the second table should be Table VI.
[Sec. VI A] There is a typo in the sentence describing the comparison with Cho et al.: 'wehavecorected' should read 'we have corrected'.
[Eq. (12)] The asymmetric treatment of the time-delay uncertainties is described in prose but not in the equation; defining sigma_log_tau explicitly as a piecewise function of the residual sign would make the likelihood unambiguous.
[Sec. IV, Eqs. (13)-(14)] In the luminosity-luminosity fits, the luminosity distance is computed using posterior mean cosmological parameters, so cosmological-parameter uncertainty is not propagated in that step; this should be stated explicitly as a limitation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the R-L relations are jointly fit with cosmology and tested for model-independence, with no prediction reducing to a fitted input.

full rationale

The paper's central claim is that four RM AGN radius-luminosity relations are standardizable, meaning the fitted R-L parameters (beta, gamma, sigma_int) do not depend on the assumed cosmological model. The method (Eqs. 8-12) simultaneously fits these R-L parameters and the cosmological parameters to the same 41-object flux and time-delay data. This is a self-consistency test, not a circular reduction: changing the cosmological model changes D_L in Eq. (9), which shifts the x-coordinates of the data in a redshift-dependent way, so the R-L parameters could in principle have come out different. The paper finds differences of at most ~0.5 sigma for beta and smaller for gamma and sigma_int (Table VII), which is a genuine empirical result. No parameter is fitted to a subset and then 'predicted' for the same subset; the comparison against H(z)+BAO data is an external benchmark. The peculiar-velocity corrections from the NED calculator are applied without propagating their uncertainty, but this is a systematic-robustness limitation, not a circular step, because no equation is reduced to another by construction. Self-citations (e.g., refs. [6,7,8,19,20]) are used for methodology, prior comparison samples, and technique context, but the present R-L fits and standardizability test are computed from the present data and do not rely on the correctness of those prior results. The luminosity-luminosity LHalpha-L5100 analysis uses posterior-mean D_L values, but because both luminosities share the same D_L, the fitted slope is essentially a flux-ratio property and does not smuggle in the target result. Overall, the derivation chain is self-contained and the paper appropriately acknowledges the usual circularity issue in this kind of simultaneous fit (footnote 1) rather than importing an unsupported uniqueness claim.

Assumptions & free parameters 13 free parameters · 5 assumptions · 0 invented entities

The central claim depends on the fitted R-L intercepts, slopes, and intrinsic scatters, which are free parameters. The analysis also fits cosmological parameters in six models, but the standardizability test only requires the R-L parameters to be stable across models, so those cosmological parameters are not load-bearing for the main claim. No new particles, forces, or physical entities are introduced; the paper relies on standard AGN and cosmological assumptions.

free parameters (13)

R-L intercept β (Hα broad) = 2.336+0.096-0.086
Fitted intercept in log(τ/day) = β + γ log(L_Hα/10^44 erg/s) for Hα lags against broad Hα luminosity (flat ΛCDM, Sec. VI A).
R-L slope γ (Hα broad) = 0.571 ± 0.055
Fitted slope for Hα broad; steeper than the 0.5 photoionization prediction.
Intrinsic scatter σ_int (Hα broad) = 0.263+0.033-0.046
Fitted extra scatter in the Hα broad R-L relation.
R-L intercept β (Hα mono) = 1.601 ± 0.058
Fitted intercept for Hα lags against monochromatic L5100.
R-L slope γ (Hα mono) = 0.543 ± 0.058
Fitted slope for Hα mono.
Intrinsic scatter σ_int (Hα mono) = 0.285+0.036-0.051
Fitted extra scatter for Hα mono.
R-L intercept β (Hβ broad) = 2.184+0.095-0.087
Fitted intercept for Hβ lags against broad Hα luminosity.
R-L slope γ (Hβ broad) = 0.616 ± 0.053
Fitted slope for Hβ broad.
Intrinsic scatter σ_int (Hβ broad) = 0.248+0.032-0.046
Fitted extra scatter for Hβ broad; lowest intrinsic scatter among the four datasets.
R-L intercept β (Hβ mono) = 1.408+0.058-0.054
Fitted intercept for Hβ lags against L5100.
R-L slope γ (Hβ mono) = 0.584 ± 0.059
Fitted slope for Hβ mono.
Intrinsic scatter σ_int (Hβ mono) = 0.271+0.034-0.049
Fitted extra scatter for Hβ mono.
L_Hα-L5100 slope = 0.971 ± 0.031
Fitted slope of the luminosity-luminosity relation in flat ΛCDM (Sec. VI B), used to support the Baldwin effect interpretation.

assumptions (5)

domain assumption The R-L relation is a linear power law in log-log space, log(τ/day) = β + γ log(L/10^44), with intrinsic scatter.
Standard in RM AGN studies; invoked in Eq. (8) and throughout the analysis.
domain assumption A simple photoionization model predicts γ = 0.5.
Used as a benchmark for the fitted slopes in Sec. VI A, taken from Davidson (1972) and Karas et al. (2021).
domain assumption The six cosmological models (flat/nonflat ΛCDM, XCDM, ϕCDM) are sufficient to test standardizability.
The test of model-independence of R-L parameters uses this set (Sec. II). Other dark energy models could in principle give different results.
domain assumption Peculiar velocity corrections from the NED Velocity Correction Calculator are accurate.
Applied to all redshifts (Sec. III A); no uncertainty is propagated, which is important for low-redshift sources.
domain assumption Time delays measured with the Interpolated Cross-Correlation Function (ICCF) are unbiased and homogeneous across the sample.
Underpins the claim that homogeneity is crucial (Sec. III A). If ICCF lags are biased for some sources, the sample is not truly homogeneous.

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Cite this review

Pith. "Pith review of Standardizing reverberation-mapped H$\alpha$ and H$\beta$ active galactic nuclei using radius--luminosity relations involving monochromatic and broad H$\alpha$ luminosities." pith.science (2026). https://pith.science/paper/5S5TEX57

@misc{pith2026241219665,
  author       = {Pith},
  title        = {Pith review of: Standardizing reverberation-mapped H$\alpha$ and H$\beta$ active galactic nuclei using radius--luminosity relations involving monochromatic and broad H$\alpha$ luminosities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5S5TEX57}},
  note         = {Machine review of arXiv:2412.19665}
}

abstract

We test the standardizability of a homogeneous sample of 41 lower-redshift ($0.00415\leq z \leq 0.474$) active galactic nuclei (AGNs) reverberation-mapped (RM) using the broad H$\alpha$ and H$\beta$ emission lines. We find that these sources can be standardized using four radius$-$luminosity ($R-L$) relations incorporating H$\alpha$ and H$\beta$ time delays and monochromatic and broad H$\alpha$ luminosities. Although the $R-L$ relation parameters are well constrained and independent of the six cosmological models considered, the resulting cosmological constraints are weak. The measured $R-L$ relations exhibit slightly steeper slopes than predicted by a simple photoionization model and steeper than those from previous higher-redshift H$\beta$ analyses based on larger datasets. These differences likely reflect the absence of high-accreting sources in our smaller, lower-redshift sample, which primarily comprises lower-accreting AGNs. The inferred cosmological parameters are consistent within 2$\sigma$ (or better) with those from better-established cosmological probes. This contrasts with our earlier findings using a larger, heterogeneous sample of 118 H$\beta$ AGNs, which yielded cosmological constraints differing by $\gtrsim 2\sigma$ from better-established cosmological probes. Our analysis demonstrates that sample homogeneity$-$specifically, the use of a consistent time-lag determination method$-$is crucial for developing RM AGNs as a cosmological probe.

Figures

Figures reproduced from arXiv: 2412.19665 by the authors.

**Figure 2.** FIG. 2. One-dimensional likelihoods and 1 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (15 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: FIG. 5. Comparisons between H [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: FIG. 6. The deviations in BLR radii, based on the H [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: FIG. 7. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

**Figure 8.** Figure 8: FIG. 8. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

**Figure 9.** Figure 9: FIG. 9. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]

**Figure 10.** Figure 10: FIG. 10. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]

**Figure 11.** Figure 11: FIG. 11. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]

**Figure 12.** Figure 12: FIG. 12. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p026_12.png]

**Figure 13.** Figure 13: FIG. 13. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p027_13.png]

**Figure 14.** Figure 14: FIG. 14. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p028_14.png]

**Figure 15.** Figure 15: FIG. 15. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p029_15.png]

**Figure 16.** Figure 16: FIG. 16. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p030_16.png]

**Figure 17.** Figure 17: FIG. 17. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p031_17.png]

**Figure 18.** Figure 18: FIG. 18. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p032_18.png]

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Works this paper leans on

105 extracted references · 71 canonical work pages

[1]

P. J. E. Peebles, Astrophys. J.284, 439 (1984)

1984
[2]

Perivolaropoulos and F

L. Perivolaropoulos and F. Skara, New Astron. Rev.95, 101659 (2022)

2022
[3]

Moresco, et al., Living Rev

M. Moresco, et al., Living Rev. Relativity25, 6 (2022)

2022
[4]

Abdalla, et al., J

E. Abdalla, et al., J. High Energy Astrophys.34, 49 (2022)

2022
[5]

Hu and F.-Y

J.-P. Hu and F.-Y. Wang, Universe9, 94 (2023)

2023
[6]

Khadka, Z

N. Khadka, Z. Yu, M. Zajaček, M. L. Martinez-Aldama, B. Czerny, and B. Ratra, Mon. Not. R. Astron. Soc. 508, 4722 (2021)

2021
[7]

S. Cao, M. Zajaček, S. Panda, M. L. Martínez-Aldama, B. Czerny, and B. Ratra, Mon. Not. R. Astron. Soc. 516, 1721 (2022)

2022
[8]

Khadka, M

N. Khadka, M. L. Martínez-Aldama, M. Zajaček, B. Cz- erny, and B. Ratra, Mon. Not. R. Astron. Soc.513, 1985 (2022)

2022

Show all 105 references

[9]

Cho, et al., Astrophys

H. Cho, et al., Astrophys. J.953, 142 (2023)

2023
[10]

S. Cao, M. Zajaček, B. Czerny, S. Panda, and B. Ratra, Mon. Not. R. Astron. Soc.528, 6444 (2024)

2024
[11]

Netzer, The Physics and Evolution of Active Galac- tic Nuclei (Cambridge: Cambridge University Press) (Cambridge University Press, 2013)

H. Netzer, The Physics and Evolution of Active Galac- tic Nuclei (Cambridge: Cambridge University Press) (Cambridge University Press, 2013)

2013
[12]

Karas, J

V. Karas, J. Svoboda, and M. Zajaček, in RAGtime: Workshops on black holes and netron stars (2021) p. E1

2021
[13]

Czerny, et al., Astrophys

B. Czerny, et al., Astrophys. Space Sci.368, 8 (2023)

2023
[14]

Zajaček, et al., Space Science Reviews 220, 29 (2024)

M. Zajaček, et al., Space Science Reviews 220, 29 (2024)

2024
[15]

Du, et al., Astrophys

P. Du, et al., Astrophys. J.856, 6 (2018)

2018
[17]

Zajaček, et al., Astrophys

M. Zajaček, et al., Astrophys. J.896, 146 (2020)

2020
[18]

J.-M. Wang, J. Qiu, P. Du, and L. C. Ho, Astrophys. J. 797, 65 (2014)

2014
[19]

Khadka and B

N. Khadka and B. Ratra, Mon. Not. R. Astron. Soc. 499, 391 (2020)

2020
[20]

S. Cao, J. Ryan, N. Khadka, and B. Ratra, Mon. Not. R. Astron. Soc.501, 1520 (2021)

2021
[21]

Khadka, O

N. Khadka, O. Luongo, M. Muccino, and B. Ratra, J. Cosmol. Astropart. Phys.2021 (9), 042

2021
[22]

S. Cao, N. Khadka, and B. Ratra, Mon. Not. R. Astron. Soc. 510, 2928 (2022)

2022
[23]

S.Cao, M.Dainotti,andB.Ratra,Mon.Not.R.Astron. Soc. 512, 439 (2022)

2022
[24]

S.Cao, M.Dainotti,andB.Ratra,Mon.Not.R.Astron. Soc. 516, 1386 (2022)

2022
[25]

Cao and B

S. Cao and B. Ratra, J. Cosmol. Astropart. Phys.2024, 093 (2024)

2024
[26]

Cao and B

S. Cao and B. Ratra, arXiv e-prints , arXiv:2502.08429 (2025)

2025
[27]

Lusso, et al., Astron

E. Lusso, et al., Astron. Astrophys.642, A150 (2020)

2020
[28]

Khadka and B

N. Khadka and B. Ratra, Mon. Not. R. Astron. Soc. 502, 6140 (2021)

2021
[29]

Khadka and B

N. Khadka and B. Ratra, Mon. Not. R. Astron. Soc. 510, 2753 (2022)

2022
[30]

Petrosian, J

V. Petrosian, J. Singal, and S. Mutchnick, Astrophys. J. Lett.935, L19 (2022)

2022
[31]

Khadka, M

N. Khadka, M. Zajaček, R. Prince, S. Panda, B. Czerny, M. L. Martínez-Aldama, V. K. Jaiswal, and B. Ratra, Mon. Not. R. Astron. Soc.522, 1247 (2023)

2023
[32]

Zajaček, B

M. Zajaček, B. Czerny, N. Khadka, M. L. Martínez- Aldama, R. Prince, S. Panda, and B. Ratra, Astrophys. J. 961, 229 (2024)

2024
[33]

A. L. González-Morán, R. Chávez, E. Terlevich, R. Ter- levich, D. Fernández-Arenas, F. Bresolin, M. Plionis, J. Melnick, S. Basilakos, and E. Telles, Mon. Not. Roy. Astron. Soc.505, 1441 (2021)

2021
[34]

Cao and B

S. Cao and B. Ratra, Phys. Rev. D109, 123527 (2024)

2024
[35]

Melnick and E

J. Melnick and E. Telles, Astron. Astrophys.690, A157 (2024)

2024
[36]

J. Ooba, B. Ratra, and N. Sugiyama, Astrophys. J.869, 34 (2018)

2018
[37]

Park and B

C.-G. Park and B. Ratra, Astrophys. Space Sci.364, 82 (2019)

2019
[38]

Khadka and B

N. Khadka and B. Ratra, Mon. Not. R. Astron. Soc. 492, 4456 (2020)

2020
[39]

Di Valentino, A

E. Di Valentino, A. Melchiorri, and J. Silk, Astrophys. J. Lett.908, L9 (2021)

2021
[40]

S. Cao, J. Ryan, and B. Ratra, Mon. Not. R. Astron. Soc. 497, 3191 (2020)

2020
[41]

S. Cao, J. Ryan, and B. Ratra, Mon. Not. R. Astron. Soc. 504, 300 (2021)

2021
[42]

Arjona and S

R. Arjona and S. Nesseris, Phys. Rev. D103, 103539 (2021)

2021
[43]

Dhawan, J

S. Dhawan, J. Alsing, and S. Vagnozzi, Mon. Not. R. Astron. Soc. Lett.506, L1 (2021)

2021
[44]

Renzi, N

F. Renzi, N. B. Hogg, and W. Giarè, Mon. Not. R. As- tron. Soc.513, 4004 (2022)

2022
[45]

926, 74 (2022)

C.-Q.Geng, Y.-T.Hsu,andJ.-R.Lu,Astrophys.J. 926, 74 (2022)

2022
[46]

Mukherjee and N

P. Mukherjee and N. Banerjee, Phys. Rev. D 105, 063516 (2022)

2022
[47]

Glanville, C

A. Glanville, C. Howlett, and T. Davis, Mon. Not. R. Astron. Soc.517, 3087 (2022)

2022
[48]

Wu, J.-Z

P.-J. Wu, J.-Z. Qi, and X. Zhang, Chin. Phys. C47, 055106 (2023)

2023
[49]

de Cruz Pérez, C.-G

J. de Cruz Pérez, C.-G. Park, and B. Ratra, Phys. Rev. D 107, 063522 (2023)

2023
[50]

Dahiya and D

D. Dahiya and D. Jain, Res. Astron. Astrophys. 23, 095001 (2023)

2023
[51]

Stevens, H

J. Stevens, H. Khoraminezhad, and S. Saito, J. Cosmol. Astropart. Phys.2023, 046 (2023)

2023
[52]

Favale, A

A. Favale, A. Gómez-Valent, and M. Migliaccio, Mon. Not. R. Astron. Soc.523, 3406 (2023)

2023
[53]

J.-Z. Qi, P. Meng, J.-F. Zhang, and X. Zhang, Phys. Rev. D108, 063522 (2023)

2023
[54]

de Cruz Perez, C.-G

J. de Cruz Perez, C.-G. Park, and B. Ratra, Phys. Rev. D 110, 023506 (2024)

2024
[55]

Shimon and Y

M. Shimon and Y. Rephaeli, arXiv e-prints , arXiv:2411.00080 (2024)

2024 arXiv
[56]

Wu and X

P.-J. Wu and X. Zhang, arXiv e-prints , arXiv:2411.06356 (2024)

2024 arXiv
[57]

P. J. E. Peebles and B. Ratra, Astrophys. J. Lett.325, L17 (1988)

1988
[58]

Ratra and P

B. Ratra and P. J. E. Peebles, Phys. Rev. D37, 3406 (1988)

1988
[59]

Pavlov, S

A. Pavlov, S. Westmoreland, K. Saaidi, and B. Ratra, Phys. Rev. D88, 123513 (2013)

2013
[60]

J. Ooba, B. Ratra, and N. Sugiyama, Astrophys. J.866, 68 (2018)

2018
[61]

J. Ooba, B. Ratra, and N. Sugiyama, Astrophys. Space Sci. 364, 176 (2019). 34

2019
[62]

Park and B

C.-G. Park and B. Ratra, Astrophys. J.868, 83 (2018)

2018
[63]

Park and B

C.-G. Park and B. Ratra, Astrophys. Space Sci.364, 134 (2019)

2019
[64]

Park and B

C.-G. Park and B. Ratra, Phys. Rev. D101, 083508 (2020)

2020
[65]

Singh, A

A. Singh, A. Sangwan, and H. K. Jassal, J. Cosmol. Astropart. Phys.2019 (4), 047

2019
[66]

Khadka and B

N. Khadka and B. Ratra, Mon. Not. R. Astron. Soc. 497, 263 (2020)

2020
[67]

L. A. Ureña-López and N. Roy, Phys. Rev. D 102, 063510 (2020)

2020
[68]

Sinha and N

S. Sinha and N. Banerjee, J. Cosmol. Astropart. Phys. 2021 (4), 060

2021
[69]

S. Cao, J. Ryan, and B. Ratra, Mon. Not. R. Astron. Soc. 509, 4745 (2022)

2022
[70]

de Cruz Perez, J

J. de Cruz Perez, J. Sola Peracaula, A. Gomez- Valent, and C. Moreno-Pulido, arXiv e-prints , arXiv:2110.07569 (2021)

2021 arXiv
[71]

T. Xu, Y. Chen, L. Xu, and S. Cao, Phys. Dark Universe 36, 101023 (2022)

2022
[72]

J. F. Jesus, R. Valentim, A. A. Escobal, S. H. Pereira, and D. Benndorf, J. Cosmol. Astropart. Phys. 2022 (11), 037

2022
[73]

Cao and B

S. Cao and B. Ratra, Mon. Not. R. Astron. Soc.513, 5686 (2022)

2022
[74]

A. Adil, A. Albrecht, and L. Knox, Phys. Rev. D107, 063521 (2023)

2023
[75]

F. Dong, C. Park, S. E. Hong, J. Kim, H. S. Hwang, H. Park, and S. Appleby, Astrophys. J.953, 98 (2023)

2023
[76]

Van Raamsdonk and C

M. Van Raamsdonk and C. Waddell, J. Cosmol. As- tropart. Phys.2024, 047 (2024)

2024
[77]

D. Blas, J. Lesgourgues, and T. Tram, J. Cosmol. As- tropart. Phys.2011 (7), 034

2011
[78]

Khadka, M

N. Khadka, M. Zajaček, S. Panda, M. L. Martínez- Aldama, and B. Ratra, Mon. Not. R. Astron. Soc.515, 3729 (2022)

2022
[79]

M. L. Martínez-Aldama, B. Czerny, D. Kawka, V. Karas, S. Panda, M. Zajaček, and P. T. Życki, As- trophys. J.883, 170 (2019)

2019
[80]

Woo, et al., Astrophys

J.-H. Woo, et al., Astrophys. J.962, 67 (2024)

2024
[81]

Kaspi, P

S. Kaspi, P. S. Smith, H. Netzer, D. Maoz, B. T. Jan- nuzi, and U. Giveon, Astrophys. J.533, 631 (2000)

2000
[82]

M. C. Bentz, et al., Astrophys. J.716, 993 (2010)

2010
[83]

C. J. Grier, et al., Astrophys. J.851, 21 (2017)

2017
[84]

Li, H.-C

S.-S. Li, H.-C. Feng, H. T. Liu, J. M. Bai, R. Li, K.-X. Lu, J.-G. Wang, Y.-K. Huang, and Z.-X. Zhang, Astro- phys. J.936, 75 (2022)

2022
[85]

S. G. Sergeev, S. V. Nazarov, and G. A. Borman, Mon. Not. R. Astron. Soc.465, 1898 (2017)

2017
[86]

A. J. Barth, et al., Astrophys. J.732, 121 (2011)

2011
[87]

C. M. Gaskell and L. S. Sparke, Astrophys. J.305, 175 (1986)

1986
[88]

C. M. Gaskell and B. M. Peterson, Astrophys. J. Suppl. 65, 1 (1987)

1987
[89]

Cao and B

S. Cao and B. Ratra, Phys. Rev. D107, 103521 (2023)

2023
[90]

Note that we account for the asymmetric uncertainties in τ by separately incorporating its upper error (στ,+) and lower error (στ,−)

is ln L = − 1 2 " χ2 + NX i=1 ln 2πσ 2 tot,i # , (10) where χ2 = NX i=1 (log τobs,i − β − γ log Lbroad/mono,i)2 σ2 tot,i , (11) where τobs,i is in units of day andLbroad/mono,i is in units of 1044 erg s −1, with total uncertainty σ2 tot,i = σ2 int + σ2 log τobs,i + γ2σ2 log Fb...

2020
[91]

D’Agostini, arXiv e-prints , physics/0511182 (2005)

G. D’Agostini, arXiv e-prints , physics/0511182 (2005)

2005 arXiv
[92]

Brinckmann and J

T. Brinckmann and J. Lesgourgues, Phys. Dark Uni- verse 24, 100260 (2019)

2019
[93]

Lewis, arXiv e-prints , arXiv:1910.13970 (2019)

A. Lewis, arXiv e-prints , arXiv:1910.13970 (2019)

2019 arXiv
[94]

Davidson, Astrophys

K. Davidson, Astrophys. J.171, 213 (1972)

1972
[95]

J. A. Baldwin, Astrophys. J.214, 679 (1977)

1977
[96]

Wang and J.-H

S. Wang and J.-H. Woo, Astrophys. J. Suppl.275, 13 (2024)

2024
[97]

Du, et al., Astrophys

P. Du, et al., Astrophys. J.806, 22 (2015)

2015
[98]

Feng, et al., Astrophys

H.-C. Feng, et al., Astrophys. J.979, 131 (2025)

2025
[99]

G. T. Richards, et al., Astrophys. J. Suppl.166, 470 (2006)

2006
[100]

Fonseca Alvarez, et al., Astrophys

G. Fonseca Alvarez, et al., Astrophys. J.899, 73 (2020)

2020
[101]

B.Czerny, J.-M.Wang, P.Du, K.Hryniewicz, V.Karas, Y.-R. Li, S. Panda, M. Sniegowska, C. Wildy, and Y.-F. Yuan, Astrophys. J.870, 84 (2019)

2019
[102]

A. K. Mandal, J.-H. Woo, S. Wang, S. Rakshit, H. Cho, D. Son, and C. S. Stalin, Astrophys. J.968, 59 (2024)

2024
[103]

Chen, et al., Mon

Y.-J. Chen, et al., Mon. Not. R. Astron. Soc.522, 3439 (2023)

2023
[104]

J. A. Kollmeier, et al., arXiv e-prints , arXiv:1711.03234 (2017)

2017 arXiv
[105]

A. B. Kovacevic et al., Astrophys. J. Supp. 262, 49 (2022)

2022
[106]

Czerny, et al., Astron

B. Czerny, et al., Astron. Astrophys.675, A163 (2023)

2023

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

P. J. E. Peebles, Astrophys. J.284, 439 (1984)

1984

[2] [2]

Perivolaropoulos and F

L. Perivolaropoulos and F. Skara, New Astron. Rev.95, 101659 (2022)

2022

[3] [3]

Moresco, et al., Living Rev

M. Moresco, et al., Living Rev. Relativity25, 6 (2022)

2022

[4] [4]

Abdalla, et al., J

E. Abdalla, et al., J. High Energy Astrophys.34, 49 (2022)

2022

[5] [5]

Hu and F.-Y

J.-P. Hu and F.-Y. Wang, Universe9, 94 (2023)

2023

[6] [6]

Khadka, Z

N. Khadka, Z. Yu, M. Zajaček, M. L. Martinez-Aldama, B. Czerny, and B. Ratra, Mon. Not. R. Astron. Soc. 508, 4722 (2021)

2021

[7] [7]

S. Cao, M. Zajaček, S. Panda, M. L. Martínez-Aldama, B. Czerny, and B. Ratra, Mon. Not. R. Astron. Soc. 516, 1721 (2022)

2022

[8] [8]

Khadka, M

N. Khadka, M. L. Martínez-Aldama, M. Zajaček, B. Cz- erny, and B. Ratra, Mon. Not. R. Astron. Soc.513, 1985 (2022)

2022