Under normal conditions gravity is always attractive. Under certain circumstances it can be repulsive. Alan Guth and Andrei Linde [behind the iron curtain] came up with an idea to see what could happen to a soliton, in a quantum scalar field, if the solitons local gravitational field reflected the circumstance where the pressure term dominates over all other energy components of the metric [or the SEM tensor]. From there they evaluated what components of this event could lead to an inflation signature in the CMBR. Following with theoretical predictions for the signatures. The birth of cosmology as a predictive science. Good job following the derivation. It was well done. The following is a relativistic version
v = at/[1+(at)^2]^1/2
A fairly easy to read paper on Eternal Inflation.
http://arxiv.org/abs/hep-th/0702178