Leading slow roll corrections to the volume of the universe and the
  entropy bound
  We make an extension to recent calculations of the probability density
\rho(V) for the volume of the universe after inflation. Previous results have
been accurate to leading order in the slow roll parameters \epsilon=\dot{H}/H^2
and \eta=\ddot{\phi}/(\dot{\phi} H), and 1/N_c, where H is the Hubble parameter
and N_c is the classical number of e-foldings. Here, we present a modification
which captures effects of order \epsilon N_c, which amounts to letting the
parameters of inflation H and \dot{\phi} depend on the value of the inflaton
\phi. The phase of slow roll eternal inflation can be defined as when the
probability to have an infinite volume is greater than zero. Using this
definition, we study the Laplace transform of \rho(V) numerically to determine
the condition that triggers the transition to eternal inflation. We also study
the average volume <V> analytically and show that it satisfies the universal
volume bound. This bound states that, in any realization of inflation which
ends with a finite volume, an initial volume must grow by less than a factor of
exp(S_{dS}/2), where S_{dS} is the de Sitter (dS) entropy.
