To me it seams that it is necessary that the scale factor becomes infinite in finite time, at least for a flat universe. I am not sure what happens in an universe which is not spatially flat.
It follows from this argumentation:
First, let's show that the phantom energy implies a growing Hubble parameter in a flat universe. The time derivative of the Hubble parameter is:
\dot H = \frac{d}{dt} \left( \frac{\dot a}{a} \right) = \frac{\ddot a}{a} - \left( \frac{\dot a}{a} \right)^2
Let's take the
Friedmann equations for zero spatial curvature k = 0 and insert in the relation above:
\dot H = - 4 \pi G \left( \rho + \frac{p}{c^2} \right)
If we have a phantom energy:
\frac{p}{c^2} < - \rho
then:
\dot H > 0
Second, let's go back to the first relation for \dot H. Take:
a = t^q
Inserting for a, \dot a and \ddot a and imposing the condition \dot H > 0, we get:
q (q-1) - q^2 > 0
This can only hold if q < 0, which in turn implies some asymptotic behaviour of a.
By the way, the relation:
a = t^q
with negative q does not make sense at all, because a must be zero at t = 0, but one could imagine a similar relation like e.g.
a = - a_0 + (t - t_{R})^q
which needs of a negative q to fulfil the phantom energy condition and has an asymptotic behaviour.
From such a relation you can see also that there exists another branch for the scale factor after t
R with a contraction. It is not clear, however, how to interpret this, because the universe goes through a singularity in the energy density (from the first Friedmann equation you can see how the energy density behaves).
