  We present some astrophysical consequences of the metric for a point mass in
an expanding universe derived in Nandra, Lasenby & Hobson, and of the
associated invariant expression for the force required to keep a test particle
at rest relative to the central mass. We focus on the effect of an expanding
universe on massive objects on the scale of galaxies and clusters. Using
Newtonian and general-relativistic approaches, we identify two important
time-dependent physical radii for such objects when the cosmological expansion
is accelerating. The first radius, $r_F$, is that at which the total radial
force on a test particle is zero, which is also the radius of the largest
possible circular orbit about the central mass $m$ and where the gas pressure
and its gradient vanish. The second radius, $r_S$, which is \approx r_F/1.6$,
is that of the largest possible stable circular orbit, which we interpret as
the theoretical maximum size for an object of mass $m$. In contrast, for a
decelerating cosmological expansion, no such finite radii exist. Assuming a
cosmological expansion consistent with a $\Lambda$CDM concordance model, at the
present epoch we find that these radii put a sensible constraint on the typical
sizes of both galaxies and clusters at low redshift. For galaxies, we also find
that these radii agree closely with zeroes in the radial velocity field in the
neighbourhood of nearby galaxies, as inferred by Peirani & Pacheco from recent
observations of stellar velocities. We then consider the future effect on
massive objects of an accelerating cosmological expansion driven by phantom
energy, for which the universe is predicted to end in a `Big Rip' at a finite
time in the future at which the scale factor becomes singular. In particular,
we present a novel calculation of the time prior to the Big Rip that an object
of a given mass and size will become gravitationally unbound.
