Cosmological redshift of a Schwarzschild-de Sitter black hole: Towards estimating the Hubble constant

TL;DR

This work develops a general relativistic framework using the Schwarzschild-de Sitter (Kottler) spacetime to connect black hole parameters, cosmic expansion, and observables in megamaser disks. It derives a full redshift expression that includes gravitational, kinematic, and cosmological contributions, with a measurable imprint of the Hubble constant through $Z_\Lambda = H_0 r_d$. The authors apply Bayesian MCMC to five megamaser galaxies to estimate the mass-to-distance ratio $M/r_d$ and the product $H_0 r_d$, achieving high precision and revealing correlations between parameters; a joint analysis with a Gaussian prior on $H_0$ further constrains the system. The results illustrate that cosmic expansion is embedded in the gravitational redshift in SdS spacetime, offering a GR-based alternative to the standard Hubble law and suggesting avenues for refining cosmological inferences with megamaser data.

Abstract

In this work we estimate the parameters of several astrophysical black holes hosted at the core of active galactic nuclei by studying the kinematics of test objects in their accretion disk. First, we derive expressions for the redshift and blueshift of photons emitted by a massive particle circularly orbiting a Schwarzschild-de Sitter black hole, and detected by a distant receding observer. The frequency-shift depends on the mass and distance of the black hole, the orbital radius of the photon source, as well as the Hubble constant, directly relating these quantities to astrophysical observables, namely, the redshift and the angular position of the emitting particle on the sky. We apply for the first time this theoretical model, which accounts for the universe expansion through the Schwarzschild-de Sitter metric, to real astrophysical systems using megamaser galaxies within the Hubble flow, namely UGC 3789, NGC 5765b, NGC 6264, NGC 6323, and CGCG 074-064. Bayesian inference based on Markov Chain Monte Carlo methods is employed to estimate the mass-to-distance ratio, the product of the Hubble constant with the distance, and the black hole angular position. Additionally, by assuming a Gaussian prior on the Hubble constant, the mass, distance, and the Hubble constant are also estimated. Furthermore, we find that cosmic expansion is embedded in the gravitational contribution of the frequency-shift within this spacetime metric. Therefore, our results introduce a general relativistic framework that accounts for cosmic expansion and differs from the standard empirical Hubble law.

Figures (5)

• Figure 1: A photon source at $r_e$ circularly orbiting the SdS black hole and an observer under the action of the cosmological constant $\Lambda$ radially receding away from the black hole. This observer is located along the LOS at a distance $r_d$, where it measures the frequency shift ($Z_{obs}$) and the position on the sky ($\Theta$) of the photon source. Megamaser observations show three groups of maser features on the sky: those about the midline, and the ones along the LOS.
• Figure 2: Plots of the redshift (a) and blueshift (b) for SdS (red), for Schwarzschild composed with the cosmological redshift (green), and for the Hubble law (dotted line).
• Figure 3: PDFs and confidence regions for the individual fits. The estimated parameters are the mass-to-distance ratio, the black hole angular position, and the product of the Hubble constant with the distance. The contour levels of the confidence regions correspond to 1$\sigma$ and 2$\sigma$.
• Figure 4: PDFs and confidence regions for the joint fit, where 20 parameters are estimated, corresponding the the mass, distance and position of each of the five AGN black holes, as well as the Hubble constant.
• Figure 5: Plot of the redshift for a static particle in the SdS metric (purple), in the Schwarzschild metric with the Hubble law (blue), and the Hubble law (dotted line).