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

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.

💡 Research Summary

The paper presents a fully relativistic treatment of the redshift and blueshift of photons emitted by massive particles on circular orbits around a Schwarzschild‑de Sitter (SdS) black hole, and observed by a distant, receding detector. Starting from the SdS line element, the authors derive the four‑velocity of a test particle at radius rₑ and the four‑momentum of a photon traveling along a null geodesic to a detector at radius r_d. Conserved quantities associated with the timelike and axial Killing vectors give the particle’s energy E and angular momentum L_φ, while the photon’s energy E_γ and impact parameter b=L_γ/E_γ are introduced. The redshift is defined as 1+Z = (k·U)_e / (k·U)_d, leading to a compact expression (Eq. 2.11) that contains three contributions: a purely gravitational term (mass of the black hole), a kinematic term (azimuthal motion of the emitter), and a term proportional to the cosmological constant Λ. By expanding for M/r_d → 0 and Λ r_d² → 0, the authors show that the Λ‑dependent part reduces to the familiar Hubble law Z_Λ = H₀ r_d, with H₀ = √(Λ/3). Thus, the SdS metric naturally embeds the cosmological redshift without the need to add a separate recession velocity.

The theoretical framework is then applied to five water‑megamaser galaxies (UGC 3789, NGC 5765b, NGC 6264, NGC 6323, CGCG 074‑064) observed by the Megamaser Cosmology Project. These systems provide precise measurements of the maser spots’ line‑of‑sight velocities (red‑ and blueshifted components) and their angular positions Θ on the sky. Using a Bayesian model, the authors infer three parameters for each galaxy: the mass‑to‑distance ratio (M/D), the product H₀ · D, and the angular position Θ. Markov Chain Monte Carlo (MCMC) sampling yields posterior distributions for these quantities. An additional analysis imposes a Gaussian prior on H₀ (centered at 73 km s⁻¹ Mpc⁻¹) to simultaneously estimate the black‑hole mass M, the distance D, and H₀ itself.

Results show that the inferred M/D values agree with previous Keplerian fits, while the H₀ · D product is tightly constrained, allowing an independent determination of the Hubble constant at low redshift. When the prior on H₀ is removed, the data alone still favor H₀ in the range 70–75 km s⁻¹ Mpc⁻¹, demonstrating that the relativistic redshift model can extract cosmological information without recourse to traditional distance ladders. The angular positions Θ recovered from the fits match the VLBI imaging, confirming the internal consistency of the model.

The authors emphasize that their approach differs from standard methods because the cosmological expansion is built directly into the spacetime geometry, eliminating the need for an ad‑hoc recession term. The SdS metric couples the black‑hole gravity and the de Sitter expansion, so both the gravitational redshift and the kinematic Doppler shift carry an imprint of Λ. This provides a “general‑relativistic Hubble law” that can be tested with high‑precision maser observations.

Limitations are acknowledged: the analysis assumes perfectly circular, edge‑on, thin disks; neglects the gravitational influence of surrounding matter; and treats Λ as a constant. Future work could extend the model to Kerr‑de Sitter (including spin), incorporate realistic disk thickness, and explore non‑circular orbits. Applying the method to a larger sample of megamaser galaxies could refine the H₀ measurement and contribute to resolving the current Hubble tension between local and early‑universe determinations.