What happens to spacetime as the universe exapnds depends very much on what is driving the expansion.

Under the normal assumption of a cosmological constant - one in which the energy density of the vacuum does not change and where the pressure is equal to minus the energy density - then the expanion becomes exponential with a Hubble parameter tending towards a constant value. i.e. $$ a \propto \exp[H t]$$ and the velocity of the expansion $$ \dot{a} \propto H \exp[Ht] = Ha$$

In such circumstances nothing will really happen to spacetime itself - any region of space becomes infinitely big in an infinite amount of time, but gravitationally bound structures and smaller structures held together by atomic forces for example will maintain their identities, just getting further and further apart.

However, a scneario has been motted called the big rip. This is where the pressure due to the vacuum energy is less than minus the energy density. This is known as phantom energy. Calculations using phantom energy are presented by Caldwell et al. (2003), which suggest that the Hubble parameter will become infinitely large in a finite time. This has the effect of ripping everything apart and spacetime ends in a singularity at some finite time in the future.

At present, observational data are insufficient to rule out the phantom energy scenario. But if there is to be a big rip we know at least that it must be at least many tens of billions of years into the future.

What happens to spacetime as the universe expands depends very much on what is driving the expansion.

Under the normal assumption of a cosmological constant - one in which the energy density of the vacuum does not change and where the pressure is equal to minus the energy density - then the expansion becomes exponential with a Hubble parameter tending towards a constant value. i.e. $$ a \propto \exp[H t]$$ and the velocity of the expansion $$ \dot{a} \propto H \exp[Ht] = Ha$$

In such circumstances nothing will really happen to spacetime itself - any region of space becomes infinitely big in an infinite amount of time, but gravitationally bound structures and smaller structures held together by atomic forces for example will maintain their identities, just getting further and further apart.

However, a scenario has been mooted called the big rip. This is where the pressure due to the vacuum energy is less than minus the energy density. This is known as phantom energy. Calculations using phantom energy are presented by Caldwell et al. (2003), which suggest that the Hubble parameter will become infinitely large in a finite time. This has the effect of ripping everything apart and spacetime ends in a singularity at some finite time in the future.

At present, observational data are insufficient to rule out the phantom energy scenario. But if there is to be a big rip we know at least that it must be at least many tens of billions of years into the future.