Can we use pressure analogy to understand inflation? Classically, the expansion of a gas in a container requires the gas to apply outward pressure to the walls of the container. Is it also true for the Universe during inflation? How did the inflation drive the expansion of Universe? I think this analogy is poor because it's not the expansion of a gas in a container. Nothing applies an outward pressure of the "walls" of the Universe rather it is the space that expands. 
 A: During inflation the universe was expanding de Sitter like which means it expanded exponentially. This requires repulsive gravity due to the inflaton field and implies a negative Friedmann term $\rho + 3p$ (then the the second derivative of the scale factor has a positive constant value as required). Hence the pressure $p$ must be negative because during inflation the matter density $\rho$ is zero. 
It's clear from this that we can't use the gas analogy to explain inflation.  Simply said while gas exerting positive pressure pushes negative pressure pulls. Think of a stretched rubberband. 
How do we explain negative pressure? Withdrawing a piston containing vacuum energy creates an increase of internal energy $dU = \rho_{vac} dV$. Now because of the conservation of energy this increase must according to thermodynamics equal the work done by the pressure of the vacuum $-p_{vac}dV$. So we get the equation of state $p_{vac} = -\rho_{vac}$ and as the vacuum density is positive a negative vacuum pressure. The point here is that in contrast to gas the vacuum energy density is constant while the piston is withdrawn. So the work done to inflate the universe is supplied by the vacuum energy "for free".       
