The conventional viewpoint regarding the future evolution of a matter-dominated universe can be summarised by the following pair of statements:
[A] If the universe is spatially open or flat then it will expand for ever.
[B] Alternatively, if the universe is spatially closed then its expansion will be followed by recollapse.
In a -dominated universe [A] and [B] are replaced by
[A'] A spatially open or flat universe ( = 0, -1) will recollapse if the -term is constant and negative.
[B'] A closed universe ( = +1) with a constant positive -term can, under certain circumstances, expand forever.
Recent CMB observations indicate that our universe is close to being spatially flat, therefore, if we wait long enough we will find that the expansion of our universe rapidly approaches the de Sitter value H = H = [/3]1/2 = H0[1 - m]1/2, while the density of matter asymptotically declines to zero m a-3 0.
Density perturbations in such a universe will freeze to a constant value if they are still in the linear regime, but the acceleration of the universe will not affect gravitationally bound systems on present scales of R < 10h-1 Mpc (which includes our own galaxy as well as galaxy clusters). The universe at late times will therefore consist of islands of matter immersed in an accelerating sea of dark energy: `'.
In such a universe the local neighborhood of an observer from which he/she is able to receive signals will eventually contract and shrink so that even those regions of the universe which are currently observable to us will eventually be hidden from view. As an illustration consider an event at (r1, t1) which we wish to observe at our location at r = 0, then
An observer at r = 0 will be able to receive signals from any event (after a suitably long wait) provided the integral in the RHS of (19) diverges (as t ). For a t p, this implies p < 0, or a decelerating universe. In an accelerating universe the integral converges, signaling the presence of an event horizon. As a result one can only receive signals from those events which satisfy
For de Sitter-like expansion a = a1 expH(t-t1), H = [/3]1/2, we get r1 rH and R = a1rH = H-1 where R is the proper distance to the event horizon. In such a universe light emitted by distant objects gets redshifted and declines in intensity (an analogous situation occurs for an object falling through the horizon of a black hole.) As a result comoving observers once visible, gradually disappear from view as the universe accelerates under the influence of . One consequence of this interesting phenomenon is that at any given instant of time, t0, one can determine a redshift zH, which will define for us the `sphere of influence' of our civilization. Celestial objects with z > zH will always remain inaccessible to signals emitted by our civilization at t t0. For a -dominated universe with m = 0.3, = 0.7 one finds zH = 1.80 . More generally, horizon's exist in a universe which begins to perpetually accelerate after a given point of time [54, 55]. (To this general category belong models of dark energy with equation of state -1 < w < - 1/3, as well as `runaway scalar fields'  which satisfy V, V', V" 0 and V'/V, V"/V 0 as .) Since the conventional S-matrix approach may not work in a universe with an event horizon, such a cosmological model may pose a serious challenge to a fundamental theory of interactions such as string theory. Possible ways of cutting short `eternal acceleration' (thereby avoiding horizons) involve scalar fields with non-monotonic potentials. For instance a flat potential with a local minimum will have a negative equation of state during slow roll, which will increase to non-negative values after the cessation of slow roll and the commencement of oscillations (provided the potential is sufficiently steep in the neighborhood of the minimum). An example is provided by a massive scalar field V() = m2 2/2 for which the epoch of accelerated expansion is a transient. Other possibilities are discussed in [55, 57, 43, 44, 58, 59].