[Paper Review] The distance scale and Eddington efficiency of luminous quasars

P. Teerikorpi|ArXiv.org|Oct 13, 2005

Cosmology and Gravitation Theories3 citations

TL;DR

This paper proposes a novel method to constrain the Hubble constant $H_0$ using luminous quasars radiating near the Eddington limit, leveraging the fact that Eddington ratio $L/L_{\text{Edd}}$ depends on $H_0$ while black hole mass $M_{\text{BH}}$ does not, due to distance-independent reverberation mapping calibration. The key result is that assuming $L_{\text{bol}} \leq L_{\text{Edd}}$ for high-redshift quasars implies $H_0 > 45\,\text{km/s/Mpc}$, offering a cosmology-independent alternative to the local distance ladder.

ABSTRACT

The relation between the central mass and quasar luminosity (M_BH \propto L^αFHWM^2) links a given Eddington ratio with a value of H_0, within a cosmology with fixed (Ω_m,Ω_Λ). We point out that because the relation is calibrated at low z using distance independent reverberation mapping to get the BLR size, the derived M_BH interestingly does not depend on H_0, while L/L_Edd is sensitive to H_0, but rather robust to changes of Ω_Λ in the standard flat model. This means, e.g., that enough of extragalactic objects radiating at the Eddington limit could be used to study the global Hubble constant in a new way, bypassing the local distance ladder. The method could become practical when systematic errors in derived M_BH are understood and objects with L /leq L_Edd can be independently identified. As an illustration, if we take a sample of tranquil very luminous quasars in the redshift range 0.5 < z < 1.6, and assume that they are radiating with L_bol \leq L_Edd, then the usual numeric factors used for calculating M_BH and L_bol would lead to the result that the Hubble constant must be larger than 45 km/s/Mpc.

Motivation & Objective

• To explore the connection between quasar Eddington efficiency and the global distance scale.
• To assess whether luminous quasars radiating near the Eddington limit can serve as independent distance indicators.
• To evaluate the robustness of this method against cosmological parameter variations and systematic errors.
• To derive a lower limit for $H_0$ under the assumption that high-redshift quasars do not exceed $L_{\text{Edd}}$.

Proposed method

• Uses the $R_{\text{BLR}}$–$L$ relation calibrated via distance-independent reverberation mapping to derive $M_{\text{BH}}$ without $H_0$ dependence.
• Applies the standard $M_{\text{BH}} \propto L^{\alpha} \times \text{FWHM}^2$ relation with $\alpha \approx 0.5$ to $0.7$ to estimate black hole masses.
• Calculates Eddington ratio $L/L_{\text{Edd}} \propto h^{-2}$, where $h = H_0 / 100$, showing sensitivity to $H_0$.
• Applies cosmological corrections for comoving distance and luminosity distance in a flat $\Lambda$CDM model with $\Omega_m = 0.3$, $\Omega_\Lambda = 0.7$.
• Uses observed quasar samples with $0.5 < z < 1.6$, $M_{\text{min}} < -25.5$, and $L_{\text{bol}} \leq L_{\text{Edd}}$ to infer $H_0$ bounds.
• Performs sensitivity tests on $\Omega_\Lambda$ and $\alpha$, showing minimal impact on $L/L_{\text{Edd}}$.

Experimental results

Research questions

• RQ1 Can quasars radiating near the Eddington limit be used to constrain $H_0$ without relying on the local distance ladder?
• RQ2 Why is the Eddington ratio $L/L_{\text{Edd}}$ sensitive to $H_0$ while $M_{\text{BH}}$ is not, given the calibration method?
• RQ3 How robust is the $H_0$ constraint to variations in $\Omega_\Lambda$ and the $R_{\text{BLR}}$–$L$ relation exponent $\alpha$?
• RQ4 What lower bound on $H_0$ emerges if we assume $L_{\text{bol}} \leq L_{\text{Edd}}$ for high-redshift quasars?
• RQ5 Can systematic errors in $M_{\text{BH}}$ and $L_{\text{bol}}$ estimation invalidate this method?

Key findings

• The Eddington ratio $L/L_{\text{Edd}}$ scales as $h^{-2}$, making it sensitive to $H_0$, while $M_{\text{BH}}$ is independent of $H_0$ due to distance-independent reverberation mapping.
• For a sample of luminous quasars with $0.5 < z < 1.6$ and $L_{\text{bol}} \leq L_{\text{Edd}}$, the standard mass and luminosity calibrations imply $H_0 > 45\,\text{km/s/Mpc}$.
• The method is robust to changes in $\Omega_\Lambda$ within $\pm 0.15$, with only a small shift in $L/L_{\text{Edd}}$ due to the exponent $\alpha \approx 0.5$ to $0.7$.
• Systematic errors in $f$-factor, $\alpha$, and $L_{\text{bol}}$ estimation remain a major obstacle to practical application.
• The method avoids Malmquist bias because the $R_{\text{BLR}}$–$L$ relation is calibrated against luminosity, not distance.
• The method is independent of the local distance ladder and probes the global Hubble flow, offering a complementary approach to $H_0$ determination.