The Limits of Bound Structures in the Accelerating Universe
  According to the latest evidence, the Universe is entering an era of exponential expansion, where gravitationally bound structures will get disconnected from each other, forming isolated `island universes'. In this scenario, we present a theoretical criterion to determine the boundaries of gravitationally bound structures and a physically motivated definition of superclusters as the largest bound structures in the Universe. We use the spherical collapse model self-consistently to obtain an analytical condition for the mean density enclosed by the last bound shell of the structure (2.36 times the critical density in the present Universe, assumed to be flat, with 30 per cent matter and 70 per cent cosmological constant, in agreement with the previous, numerical result of Chiueh and He). $N$-body simulations extended to the future show that this criterion, applied at the present cosmological epoch, defines a sphere that encloses $\approx 99.7$ per cent of the particles that will remain bound to the structure at least until the scale parameter of the Universe is 100 times its present value. On the other hand, $(28\pm 13)$ per cent of the enclosed particles are in fact not bound, so the enclosed mass overestimates the bound mass, in contrast with the previous, less rigorous criterion of, e.~g., Busha and collaborators, which gave a more precise mass estimate. We also verify that the spherical collapse model estimate for the radial infall velocity of a shell enclosing a given mean density gives an accurate prediction for the velocity profile of infalling particles, down to very near the centre of the virialized core. 
Rolando D\"unner, Pablo A. Araya, Andr\'es Meza and Andreas   Reisenegger