Gravitational lensing by hairy black holes in Einstein-scalar-Gauss-Bonnet theories

We investigate the weak and strong deflection gravitational lensing by the hairy black holes in the Einstein-scalar-Gauss-Bonnet (EsGB) gravity with five types of coupling functions (quadratic, cubic, quartic, inverse-polynomial, and logarithmic), which can evade the no-hair theorem. The lensing observables are found and the possibility of detecting the effects of the scalar hair is analyzed. We find that all of the shadows cast by these black holes are consistent with the measurement by the Event Horizon Telescope, while other lensing observables are beyond the present capacity. Therefore, it is currently impossible to distinguish these black holes from the Schwarzschild one and from each other merely by their lensing signatures, leaving testing the EsGB gravity by gravitational lensing until new technology is available.

Figure 1

$Q$ against $p$ (top) and $\xi$ (bottom) for $f(\varphi )=\{{\varphi }^2,{\varphi }^3,{\varphi }^4,{\varphi }^{-1},\log (\varphi )\}$.

Figure 2

By taking Sgr A* as the lens, it shows the observables of the weak deflection gravitational lensing by the hairy EsGB black holes with different coupling functions $f(\varphi )=\{{\varphi }^2,{\varphi }^3,{\varphi }^4,{\varphi }^{-1},\log (\varphi )\}$ and their deviations from those of the Schwarzschild black hole, where $\beta =0.5$. For the Schwarzschild black hole, $P_{\mathrm{tot},\mathrm{Sch}}=1.147$ mas, $\mathrm{\Delta }{P}_{\mathrm{Sch}}=0.271$ mas, $\mathrm{\Delta }{F}_{\mathrm{Sch}}/F_{\mathrm{src}}=0.994$ and $\mathrm{\Delta }{\tau }_{\mathrm{Sch}}=86.136\mathrm{s}$.

Figure 3

The relative errors of $r_{\mathrm{m}}$ and $b_{\mathrm{m}}$ between the CFA and numerical approaches.

Figure 4

The observables in the strong deflection gravitational lensing by the hairy EsGB black holes with different coupling functions $f(\varphi )=\{{\varphi }^2,{\varphi }^3,{\varphi }^4,{\varphi }^{-1},\log (\varphi )\}$ and their deviations from those of the Schwarzschild black hole are shown by taking Sgr A* as the lens. If Sgr A* is assumed to be a Schwarzschild black hole, then ${\vartheta }_{\infty ,\mathrm{Sch}}=26.4\mu \mathrm{as}$, $s_{\mathrm{Sch}}=33.0$ nas, $\mathrm{\Delta }{m}_{\mathrm{Sch}}=6.8\mathrm{mag}$, and $\mathrm{\Delta }{T}_{2,1,\mathrm{Sch}}=11.6\min$.

Figure 5

The observables in the strong deflection gravitational lensing by the hairy EsGB black holes and their deviations from those of the Schwarzschild black hole are shown by taking M87* as the lens. If M87* is assumed to be a Schwarzschild black hole, then ${\vartheta }_{\infty ,\mathrm{Sch}}=19.7\mu \mathrm{as}$, $s_{\mathrm{Sch}}=24.7$ nas, $\mathrm{\Delta }{m}_{\mathrm{Sch}}=6.8\mathrm{mag}$, and $\mathrm{\Delta }{T}_{2,1,\mathrm{Sch}}=294$ h.