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 $t_{\ast}$. Such geodesics have been
previously studied in the Newtonian approximation and found to lead to
dissociation of bound systems at a time $t_{rip}<t_{\ast}$ which for fixed
background $w$, depends on a single dimensionless parameter $\bar{\omega}_0$
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 effective potential 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 ($10^{20} M_\odot$) and large ($100Mpc$) bound systems.
We verify this result by explicit solution of 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.
