Testing black hole space-times with the S2 star orbit: a Bayesian comparison

TL;DR

This work tests seven non-rotating black-hole space-times around Sgr A* by jointly fitting the S2 star’s astrometry, radial velocity, and GRAVITY-detected pericentre precession using a uniform MCMC framework. It advances previous studies by modeling orbital precession consistently across all spacetimes (including JNW precession and a Yukawa-like potential without forcing Ψ = Φ) and by introducing the first S2-based analysis of the Bardeen regular black hole. Bayesian evidence shows no statistically significant preference for any non-Schwarzschild spacetime given current data, though tight constraints emerge on certain parameters (e.g., the Yukawa scale λ and Horndeski q_H) and the EMd dilaton sector stays compatible with zero charge. The results underscore the power and limitations of current S2 data for testing gravity in the strong-field regime and highlight the potential gains from future high-precision astrometry and additional short-period S-stars.

Abstract

We implement a Markov Chain Monte Carlo method to obtain posterior probability distributions for the parameters of the S2 star orbit around Sagittarius A*, for seven representative non-rotating black hole space-time solutions. In particular, we consider the Schwarzschild, Reissner-Nordström, Janis-Newman-Winicour, and Bardeen black hole space-times from General Relativity, as well as a black hole solution from Einstein-Maxwell-dilaton gravity, a hairy black hole solution from Horndeski theory, and a Yukawa-like black hole from $f(\mathscr{R})$ gravity. To constrain model parameters, we use the most recent publicly available observational data of the S2 star orbit, namely astrometric measurements, spectroscopic data, and the pericentre advance measured by the GRAVITY Collaboration. We further perform a consistent Bayesian comparison of models, calculating the log-Bayes factor of each space-time with respect to the Schwarzschild solution. Our results show that the currently available data indicate no statistically significant preference among the space-times considered. The Bardeen and Yukawa-like models are indistinguishable from Schwarzschild within current uncertainties, while the Reissner-Nordström, Janis-Newman-Winicour, Horndeski and Einstein-Maxwell-dilaton geometries show at most weak and non-decisive strength of evidence under the adopted priors and likelihood choices.

Figures (10)

• Figure 1: Probability density distribution of the dilaton parameter $b$ of the EMd model. The vertical lines denote the upper limit of $b$ under 68% and 95% confidence intervals (CI), which correspond to $1.708\times 10^{6}\ \mathrm{M}_{\odot}$ and $3.555\times 10^{6} \mathrm{M}_{\odot}$, respectively.
• Figure 2: Trajectory in the sky plane and radial velocity of S2 in the different space-time models under consideration. The left part of the figure shows the best-fit trajectory and radial velocity of S2, with the corresponding observational data points, i.e., 145 astrometric positions of the S2 star from the years $\sim$ 1992 to $\sim$ 2016, measured with the SHARP and NACO instruments, and 44 radial velocities from the years $\sim$ 2000 to $\sim$ 2016, measured with the NIRC2 and SINFONI instruments. The right side of the figure shows enlarged views of the boxed regions highlighted on the left. All best-fit models predict a noticeable precession in the orbit of S2. However, no substantial differences between models are discernible.
• Figure 3: Representative example of the autocorrelation time as a function of the number of iterations, for three runs of the MCMC method under the same initial conditions and parameters, for the Schwarzschild model.
• Figure 4: Posterior distribution contours with 68% (1$\sigma$), 95% (2$\sigma$), and 99.7% (3$\sigma$) confidence levels for the parameters that model the orbit of the S2 star in Schwarzschild space-time. The gray-shaded region in the histograms corresponds to 1$\sigma$ confidence interval. The mean value and the 1$\sigma$ range of the corresponding parameter are shown above each column, and are represented in the histogram by the three vertical lines.
• Figure 5: Posterior distribution contours with 68% (1$\sigma$), 95% (2$\sigma$), and 99.7% (3$\sigma$) confidence levels for the parameters that model the orbit of the S2 star in Reissner-Nordström space-time. The gray-shaded region in the histograms corresponds to 1$\sigma$ confidence interval. The mean value and the 1$\sigma$ range of the corresponding parameter are shown above each column, and are represented in the histogram by the three vertical lines.