The Big Rip
Note: This almost certainly will not happen, but is too much fun not to mention....
The previous discussion all supposes a standard cosmological constant.
Remember how we describe the equation of state that connects pressure
(P) and energy density (rho):
P = w rho c2.
The equation of state parameter determines how the energy density of dark energy evolves with time:
rho = R-3(1+w)
So if w=-1, which current data suggests is true, the rho=constant, so
the energy density of dark matter is constant. Since the volume of
space is increasing, we get more DE density as time goes by, which
leads to the exponential behavior of R(t).
But if w<-1, then rho increases with R, leading to super-exponential
behavior of R(t). If this is true even objects which are currently
gravitationally bound will not survive!
The Big Rip timeline for w=-1.5 (from Caldwell, Kamionkowski, and Weinberg 2003):