Innermost and outermost stable circular orbits in the presence of a positive cosmological constant

Normally one thinks of the observed cosmological constant as being so small that it can be utterly neglected on typical astrophysical scales, only affecting extremely large-scale cosmology at gigaparsec scales. Indeed, in those situations where the cosmological constant only has a quantitative influence on the physics, a separation of scales argument guarantees the effect is indeed negligible. The exception to this argument arises when the presence of a cosmological constant qualitatively changes the physics. One example of this phenomenon is the existence of outermost stable circular orbits (OSCOs) in the presence of a positive cosmological constant. Remarkably the size of these OSCOs is of a magnitude to be astrophysically interesting. For instance, for galactic masses the OSCOs are of order the intergalactic spacing; for galaxy cluster masses the OSCOs are of order the size of the cluster.