Dynamical Black Hole in the accelerating Universe approaching the future singularity -- Possible origin of (super-)massive black holes

We construct and investigate the dynamical black hole spacetime embedded in the expanding universe filled with cosmic fluid, such as dark energy. When the equation of state (EoS) parameter of the fluid is a constant, we find exact solutions of the Einstein equation where the Schwarzschild black hole is embedded in the expanding universe. This solution differs from the well-known McVittie metric, where the EoS parameter is not a constant but rather depends on the radial coordinate. It is shown that a dynamical black hole grows with the expansion of the universe. If primordial black holes are created before or during inflation, above dynamical black holes might be the origin of the supermassive black holes at the centre of galaxies, massive black holes suggested by the GW231123 event, and also the dark matter. The case where the cosmic fluid EoS is more general is also considered so that the universe enters the epoch of finite-time future singularity. Thermodynamics and the behaviour of black holes around different future singularities are carefully investigated. It is then demonstrated that the black hole horizon enhances the tidal force, but near the horizon, the tidal force works to press the extended object, which is in contrast with a massive body near to future singularity. We also propose a new type of future singularity where the singularity inside the black hole is a sphere with a finite radius. When the radius of the spherical singularity becomes larger than the radius of the black hole horizon, it becomes naked. The universe may end up with a cosmic doomsday when the radius of the singularity becomes infinite.