Geodesics of McVittie spacetime with a phantom cosmological background

We investigate the geodesics of a Schwarzschild spacetime embedded in an isotropic expanding cosmological background (McVittie metric). We focus on bound particle geodesics in a background including matter and phantom dark energy with constant dark energy equation-of-state parameter w <-1 involving a future big rip singularity at a time t_*. Such geodesics have been previously studied in the Newtonian approximation and found to lead to dissociation of bound systems at a time t_rip<t_*, which for a fixed background w depends on a single dimensionless parameter ω_¯0 related to the angular momentum and depending on the mass and the size of the bound system. We extend this analysis to large massive bound systems where the Newtonian approximation is not appropriate and we compare the derived dissociation time with the corresponding time in the context of the Newtonian approximation. By identifying the time when the general-relativistic analog of the effective potential V_eff minimum disappears due to the repulsive force of dark energy, we find that the dissociation time of bound systems occurs earlier than the prediction of the Newtonian approximation. However, the effect is negligible for all existing cosmological bound systems and it would become important only in hypothetical bound extremely massive (10²⁰M_⊙) and large (100 Mpc) bound systems. We verify this result by explicitly solving the geodesic equations. This result is due to an interplay between the repulsive phantom dark energy effects and the existence of the well-known innermost stable orbits of Schwarzschild spacetimes.