Testing non-circular black hole spacetime with X-ray reflection

Abstract

X-ray reflection spectroscopy is a powerful tool for testing the Kerr hypothesis and probing the strong gravity regime around accreting black holes. Most tests of General Relativity (GR) assume that the spacetime around a black hole is circular, meaning the metric possesses a specific symmetry structure common to the Kerr solution. However, deviations from circularity are predicted by various modified gravity theories and non-vacuum General Relativity solutions. In this work, we test a specific non-circular metric constructed based on a locality principle, where the deviation from the Kerr spacetime is driven by the local spacetime curvature. To accurately model the reflection spectrum in this background, we implement a relativistic ray-tracing code in horizon-penetrating (ingoing Kerr) coordinates, which are favored for their ability to avoid introducing curvature singularities at the horizon in non-circular spacetimes. We apply this model to the high-quality \textit{NuSTAR} spectrum of the Galactic black hole binary EXO 1846--031. Our spectral analysis reveals a source with a high inclination angle ($ι\approx 76^{\circ}$) and a near-extremal spin parameter ($a_* \approx 0.98$). While we identify a global minimum in the parameter space suggesting a non-zero deformation ($\ell_{\mathrm{NP}} \approx 0.12$), the 99\% confidence interval fully encompasses the Kerr limit ($\ell_{\mathrm{NP}}=0$). We conclude that the current X-ray reflection data for EXO 1846--031 are consistent with the Kerr hypothesis. This work demonstrates the feasibility of using X-ray reflection spectroscopy to constrain non-circular metrics and establishes a framework for future tests.

Figures (6)

• Figure 1: Left panel: The horizon radius $r_H(\theta)$ is plotted as a function of the polar angle $\theta$ as a solid blue line for a spin parameter $a_*=0.99$ and deformation parameter $\ell_{\textrm{NP}}=0.1137$ (the maximum $\ell_{\textrm{NP}}$), while the horizon radius for a Kerr black hole is plotted as a red dashed line for comparison. Right panel: Contour levels for the values of the ISCO radius $R_\mathrm{ISCO}$ as a function of spin parameter $a_*$ and deformation parameter $\ell_{\textrm{NP}}$. The white region is excluded from our study because the spacetime possesses a naked singularity for these parameter combinations.
• Figure 2: This diagram shows each component of the electromagnetic spectrum in the disk-corona model: thermal emission from the cold disk (red arrows), a Comptonized spectrum from the hot corona (blue arrows), and a reflection spectrum from the disk (green arrows). Figure from Ref. Bambi:2024hhi.
• Figure 3: Impact of the deformation parameter $\ell_{\mathrm{NP}}$ on the iron line shape. The spacetime is described by the metric in Eqs. \ref{['eqn:non_circular_metric']} and \ref{['eqn:non_circular_mass']} with the spin parameter $a_*=0.8$. The left panel corresponds to a system with inclination angle $\iota=30^{\circ}$, while the right panel corresponds to $\iota=80^{\circ}$. Three cases with deformation parameters $\ell_{\mathrm{NP}}=0, 0.166, 0.332$ are plotted. Note that $\ell_{\mathrm{NP}}=0.332$ is the highest allowed value of the deformation parameter for $a_*=0.8$.
• Figure 4: Same as Fig. \ref{['fig: ironlineplots']}, but for the spin parameter $a_*=0.99$. Here, $\ell_{\mathrm{NP}}=0.1137$ is the highest allowed value of the deformation parameter.
• Figure 5: The upper panel shows the total best-fit model in black with the individual components: the thermal spectrum (diskbb) is blue-dashed line, the relativistic reflection spectrum (relxillionCp_nk) is red-dashed line, and the Comptonized spectrum from the corona (nthComp) is yellow-dashed line. The lower panel shows the data-to-best-fit model ratio, where the cyan and magenta colors represent the data from the FPMA and FPMB sensors. Data is rebinned here for visual clarity.