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; these radii are found to be insensitive to relativistic effects. The first radius, rF, 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, rS, is that of the largest possible stable circular orbit, which we interpret as the theoretical maximum size for an object of mass m. The radius rS is typically smaller than rF by a factor of ∼1.6. In contrast, for a decelerating cosmological expansion, no such finite radii exist. Assuming a cosmological expansion consistent with a Λ cold dark matter 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 zeros in the radial velocity field in the neighbourhood of nearby galaxies, as inferred by Peirani & de Freitas 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 and the Hubble parameter become singular. In particular, we present a novel way of calculating the time prior to the Big Rip that an object of a given mass and size will become gravitationally unbound.