Probing the Scalar Hair of Rotating Horndeski Black Holes through Thick Disk Images

TL;DR

This work develops an analytical, horizon-scale thick-disk model around a rotating Horndeski black hole with scalar hair, to predict 230 GHz imaging signatures. By solving null geodesics and performing general-relativistic radiative transfer with a thermal synchrotron emissivity, it shows that scalar hair primarily enhances gravitational redshift, dimming the direct emission while dark-lens photons preserve the photon-ring prominence. A key result is that the maximal interferometric diameter of the first photon ring, $d_+$, is highly sensitive to the hair parameter $h$ and remarkably robust against accretion-flow details, making it a promising observable for constraining Horndeski hair with space VLBI (e.g., BHEX). The findings highlight how combining $d_+$ with flux and ring/shadow morphology could tighten constraints on deviations from Kerr, with implications for tests of gravity in the strong-field regime.

Abstract

Horizon-scale images of black holes provide a potential probe of fundamental physics, including tests of gravity and black hole hair. To assess the impact of scalar hair on accretion-flow imaging self-consistently, we construct an analytical model of a geometrically thick, magnetized disk around a rotating hairy black hole in Horndeski theory and analyze its 230 GHz image morphology. We find that scalar hair modestly alters the inflow and magnetic-field structure but strengthens gravitational redshift, markedly reducing the total flux and lensed ring brightness through relativistic transfer and spectral-shift effects. Moreover, we highlight a previously unexplored channel: the maximum interferometric diameter of the first photon ring responds strongly to the hair parameter but shows little dependence on accretion-flow details, making it a promising observable for constraining black-hole hair with future space-based interferometry.

Figures (8)

• Figure 1: Radial profiles of the particle number density $n_\text{e}$ (top panel) and electron temperature $T_\text{e}$ (bottom panel) in the equatorial plane., where $n_+ = 10^6\,\mathrm{cm}^{-3}$ and $T_+ = 10^{11}\,\mathrm{K}$ represent the corresponding values at the event horizon $r_+$. The parameters are set to $a=0.5$, $z=20$, $\sigma=0.2$ and $E=\mu$. The unit of length is the gravitational radius $GM/c^2$.
• Figure 2: This figure shows the magnetic field structure $B^\mu$ in an free-falling flow with $E=\mu$, in the BL coordinate for a spin parameter of $a=0.9$. In this diagram, the black area represents the event horizon, the gray region denotes the ergosphere, and the blue dashed line indicates the critical radius where magnetic field lines transition from retrograde to prograde rotation.
• Figure 3: Imaging results of the thick accretion disk with a free-falling flow ($E=\mu$) around a rotating Horndeski black hole viewed at an inclination angle of $\theta_o=17^\circ$. The solid blue curve marks the critical curve, traced by photons orbiting the black hole infinitely many times. The dashed white curve denotes the inner shadow boundary, corresponding to the direct image of the event horizon Chael:2021rjo. The disk parameters are $z = 20, \sigma=0.2$ and $\Omega_F=0.2\,\Omega_H$. The bottom two rows display the intensity profiles along the $x$-axis and $y$-axis for different black hole parameters.
• Figure 4: Variation of typical observables in black hole image with the hair parameter $h$. The top row of panels depicts the variation of the normalized total flux $F_\text{tot}/F_0$ with parameter $h$ for different parameters at an observational inclination angle of $\theta_o=17^\circ$, where $F_0$ is the total flux with $a=0.001$, $h=0$, $\sigma=0.5$, $z=5$ and $\Omega_F=0$. The bottom row of panels characterizes the variation of the rescaled average radius $\bar{\rho}_i/\bar{\rho}_{i,\text{Kerr}}$ with $h$, where the dashed and solid lines represent the results for the photon ring (critical curve) and the inner shadow boundary curve, respectively, and the grey dashed line indicates the result for a Kerr black hole.
• Figure 5: Schematic diagram of black hole image and typical observables. (Top Left): Illustration of imaging features in an intensity cut. (Top Right): Maximum number of equatorial crossings. $N_{\text{max}} = 0, 1, 2, \ldots$ correspond to the projected horizon (inner shadow), direct emission, lensing band, etc Gralla:2019xty. (Bottom Left): Schematic of the FPR and the diameter $d_\phi$. (Bottom Right): Typical behavior of $d_{\phi}$ and $b_{\phi}$ as functions of the polar angle. In all plots, the black hole parameters are $a = 0.5$, $h=-0.5$; the thick disk's parameters are $E = \mu$, $z =20$, $\sigma=0.2$ and $\Omega_F=0.2\,\Omega_H$; the inclination angle is $\theta_o = 17^{\circ}$.