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Black-hole mass estimation through accretion disk spectral fitting for high-redshift blazars

G. Kyriopoulos, M. Petropoulou, G. Vasilopoulos, D. Hatzidimitriou

TL;DR

This work presents a uniform, Bayesian MCMC framework to estimate black hole masses and accretion rates for 23 high-redshift blazars by fitting a Shakura–Sunyaev disk model to infrared-to-ultraviolet photometry, explicitly accounting for attenuation by intergalactic neutral hydrogen. The analysis shows that neglecting IGM attenuation biases masses high and Eddington ratios low, especially at large redshift, and reveals a significant spin-induced degeneracy that can mimic a wide range of $M_{ m BH}$ and $\dot M$ while preserving the same disk spectrum. The results span $M_{ m BH} \sim 3\times10^{8}$–$10^{10}\,M_\odot$ and $\lambda_{Edd} \sim 0.04$–1, with $\dot M$ from $2$ to $200\,M_\odot\,\text{yr}^{-1}$, and indicate accretion rates sufficient to reach high masses across redshifts without a clear mass–redshift trend. Moreover, inverting growth under Eddington-limited accretion provides broad seed-mass inferences, from very light to very massive seeds, highlighting how growth history and potential mergers shape the early SMBH population. Overall, the paper demonstrates the importance of uniform disk-based mass estimates and invites improved multiwavelength coverage and independent spin constraints to break degeneracies in SMBH growth studies.

Abstract

High-redshift ($z>2$) blazars, with relativistic jets aligned toward us, probe the most powerful end of the active galactic nuclei (AGN) population. We aim at determining the black hole masses and mass accretion rates of high-$z$ blazars in a common framework that utilizes a Markov Chain Monte Carlo (MCMC) fitting method and the Shakura-Sunayev multi-temperature accretion disk model, accounting also for attenuation due to neutral hydrogen gas in the intergalactic medium (IGM). We compiled a sample of 23 high-redshift blazars from the literature with publicly available infrared-to-ultraviolet photometric data. We performed a Bayesian fit to the spectral energy distribution (SED) of the accretion disk, accounting for upper limits, and determined the black hole masses and mass accretion rates with their uncertainties. We also examined the impact of optical-ultraviolet attenuation due to gas in the IGM. We find that neglecting IGM attenuation in SED fits leads to systematically larger black-hole mass estimates and correspondingly lower Eddington ratios, with the bias becoming more severe at higher redshift. Our MCMC fits yield median black-hole masses in the range $\sim (10^{8}-10^{10})\,M_{\odot}$ and a broad distribution of median Eddington ratios ($λ_{\rm Edd}\sim0.04$ up to $\sim1$). Comparison with previous literature shows no clear method-dependent systematic offsets, although individual mass estimates can differ by up to a factor of a few. We also demonstrate that assumptions about black-hole spin introduce a systematic degeneracy. This work is to our knowledge the first systematic study to model the accretion-disk emission of a large sample of high-$z$ blazars within a single, consistent statistical framework. Our results emphasize the importance of accounting for IGM attenuation and of using uniform fitting methods when comparing disk-based black hole estimates across samples.

Black-hole mass estimation through accretion disk spectral fitting for high-redshift blazars

TL;DR

This work presents a uniform, Bayesian MCMC framework to estimate black hole masses and accretion rates for 23 high-redshift blazars by fitting a Shakura–Sunyaev disk model to infrared-to-ultraviolet photometry, explicitly accounting for attenuation by intergalactic neutral hydrogen. The analysis shows that neglecting IGM attenuation biases masses high and Eddington ratios low, especially at large redshift, and reveals a significant spin-induced degeneracy that can mimic a wide range of and while preserving the same disk spectrum. The results span – and –1, with from to , and indicate accretion rates sufficient to reach high masses across redshifts without a clear mass–redshift trend. Moreover, inverting growth under Eddington-limited accretion provides broad seed-mass inferences, from very light to very massive seeds, highlighting how growth history and potential mergers shape the early SMBH population. Overall, the paper demonstrates the importance of uniform disk-based mass estimates and invites improved multiwavelength coverage and independent spin constraints to break degeneracies in SMBH growth studies.

Abstract

High-redshift () blazars, with relativistic jets aligned toward us, probe the most powerful end of the active galactic nuclei (AGN) population. We aim at determining the black hole masses and mass accretion rates of high- blazars in a common framework that utilizes a Markov Chain Monte Carlo (MCMC) fitting method and the Shakura-Sunayev multi-temperature accretion disk model, accounting also for attenuation due to neutral hydrogen gas in the intergalactic medium (IGM). We compiled a sample of 23 high-redshift blazars from the literature with publicly available infrared-to-ultraviolet photometric data. We performed a Bayesian fit to the spectral energy distribution (SED) of the accretion disk, accounting for upper limits, and determined the black hole masses and mass accretion rates with their uncertainties. We also examined the impact of optical-ultraviolet attenuation due to gas in the IGM. We find that neglecting IGM attenuation in SED fits leads to systematically larger black-hole mass estimates and correspondingly lower Eddington ratios, with the bias becoming more severe at higher redshift. Our MCMC fits yield median black-hole masses in the range and a broad distribution of median Eddington ratios ( up to ). Comparison with previous literature shows no clear method-dependent systematic offsets, although individual mass estimates can differ by up to a factor of a few. We also demonstrate that assumptions about black-hole spin introduce a systematic degeneracy. This work is to our knowledge the first systematic study to model the accretion-disk emission of a large sample of high- blazars within a single, consistent statistical framework. Our results emphasize the importance of accounting for IGM attenuation and of using uniform fitting methods when comparing disk-based black hole estimates across samples.

Paper Structure

This paper contains 15 sections, 19 equations, 10 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The sample
  3. Methods
  4. Previous works
  5. SED Model
  6. Fitting procedure
  7. Results
  8. Discussion
  9. Effect of black-hole spin on the thin-disk spectrum
  10. Black-hole seed masses
  11. Conclusions
  12. Representative cases of problematic photometry
  13. MCMC fitting results
  14. Impact of extinction on derived quantities
  15. Black hole mass - accretion rate: empirical modeling with spin-induced covariance

Figures (10)

  • Figure 1: Effective opacity (averaged over an ensemble of line of sights) due to neutral hydrogen gas in the intergalactic medium as a function of redshift for different wavelengths corresponding to the six Swift-UVOT filters. Results obtained from the analytical model of 2014MNRAS.442.1805I are plotted with solid lines. Results adopted from Figure 3 of 2010MNRAS.405..387G are overplotted for comparison with dashed lines.
  • Figure 2: Infrared-to-ultraviolet SEDs of three high-$z$ blazars (in the observer's frame). Green points indicate publicly available photometric observations from the ASI/SSDC SED Builder Stratta2011. Upper limits are plotted with black arrows. Model fits with and without IGM extinction are plotted with blue and orange colors, respectively. Dark shaded regions indicate the 68 per cent confidence intervals. Thin solid lines indicate the range of model expectations accounting for the intrinsic scatter described by the parameter $\ln f$. All sources, except for J064632+445116, are fitted with a two-component (jet+disk) model.
  • Figure 3: Decomposition of the total (jet+disk) model SEDs presented in Fig \ref{['fig:sed']}. For clarity, individual spectral components are plotted without IGM attenuation (power law in gray and multi-temperature blackbody in orange).
  • Figure 4: Reduced corner plots for four blazars showing the posterior distributions for the pair of fitted parameters ($M_{\rm BH}, \dot{M}$) and $\ln f$. The distribution of the Eddington ratio, which is a derived quantity, is also shown for comparison. Colors have the same meaning as in Fig. \ref{['fig:sed']}.
  • Figure 5: Comparison between our MCMC estimates (including extinction) and literature values for black hole mass $M_{\rm BH}$ (left column) and Eddington ratio $\lambda_{\rm Edd}$ (right column). Top left panel: Black hole mass estimates for each source (index on x-axis) from our MCMC (black circles) and literature (colored markers, with open symbols for spectroscopic and filled for SED-fitting estimates). Error bars show 68% uncertainties. Marker shapes indicate different literature sources. Top right panel: Eddington ratio comparison, maintaining identical visual coding to left panel. Bottom left panel: Direct comparison of black hole mass estimates between our MCMC method (x-axis) and literature values (y-axis), with points colored by source redshift $z$ and marker shapes indicating literature sources. Open symbols represent spectroscopic measurements while filled symbols denote SED-fitting estimates. MCMC uncertainties (68% confidence) are displayed. Bottom right panel: Identical in format to bottom left panel, presenting Eddington ratio comparisons with the same redshift coloring and marker coding system. In the bottom panels we overplot lines to guide the eye (see inset legend). Blazars B2 1023+25 and GB6 B1428+4217 are not shown for reasons explained in Appendix \ref{['sec:Results_Fit']}. Upward-pointing arrows indicate lower limits on the MCMC Eddington ratio estimates for the sources marked in \ref{['tab:fit_params']}.
  • ...and 5 more figures