Shadow of the Scalar Hairy Black Hole with Inverted Higgs Potential

Abstract

We study the imaging of a hairy black hole (HBH) in the Einstein-Klein-Gordon theory, where Einstein gravity is minimally coupled to a scalar potential $V(\phi)=-\Lambda \phi^4 + \mu \phi^2$ with $\Lambda$ and $\mu$ are constants. As a consequence, a nontrivial scalar field at the event horizon $\phi_H$ allows the HBH to evade the no-hair theorem, bifurcate from the Schwarzschild black hole by acquiring some new properties, which can affect the shadow of the HBH received by a distant observer. The framework of ray-tracing is adopted to investigate the optical appearance of the HBH, thus the trajectories of light rays around the HBH can be classified into three emissions: direct, lensed and photon ring. Employing three models of optically and geometrically thin accretion disk, we compare the differences between the Schwarzschild black hole and HBH with same horizon radius in a specific model, and find that the size of the shadow and accretion disk increases as $\phi_H$ increases, but the brightness of the rings remain nearly unaffected, this implies our HBH can potentially mimic the Schwarzschild black hole if we vary the horizon radius of the HBH. Finally, we also constraint the parameter $\Lambda$ from the observations of supermassive black holes in the galactic center of M87 and Sgr A$^{*}$, which could offer new insights for imaging of black holes and astrophysical observations.