Gravitational lensing outside and inside of a marginally unstable photon sphere at a $Z_2$-symmetric wormhole throat in strong deflection limits

Abstract: The deflection angles of rays diverge logarithmically near outside and inside of a photon sphere and they could be detected by near-future space observations on black hole shadows. We show that the deflection angles of the rays near a photon sphere at the throat of an example among some commonly used types of wormholes or one of the simple wormholes, which is often studied by the Event Horizon Telescope collaborations, does diverge not logarithmically but in power. This is because it has a marginally unstable photon sphere at a $Z_2$-symmetric wormhole throat and the wormhole has not a common nor simple but a special property. We investigate numerically gravitational lensing of rays with the deflection angles diverging in power outside and inside of a marginally unstable photon sphere at $Z_2$-symmetric wormhole throat in strong deflection limits. We apply our numerical method to the commonly-used simple wormhole spacetime, a Damour-Solodukhin wormhole spacetime, and a Simpson-Visser black-bounce spacetime. Our numerical results imply universal property of the deflection angle by the marginally unstable photon sphere on $Z_2$-symmetric wormhole throat in the strong deflection limits as similar to the deflection angles of ray bent by a marginally unstable photon sphere off a throat in the strong deflection limits which have a different power. We also correct the coefficients of power-divergent terms of deflection angles of the rays bent nearly outside of the marginally unstable photon sphere at the $Z_2$-symmetric wormhole throat in a semi-analytic approach investigated by the author and others.