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 ρ(V ) for the volume of the universe after inflation. Previous results have been accurate to leading order in the slow roll parameters $$ \in \equiv \overset{\cdot }{H}/{H}^2 $$ and $$ \eta \equiv \overset{\cdot \cdot }{\phi }/\left(\overset{\cdot }{\phi }H\right) $$ , 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 ϵN c , which amounts to letting the parameters of inflation H and $$ \overset{\cdot }{\phi } $$ depend on the value of the inflaton ϕ. 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 ρ(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 $$ {e}^{S_{\mathrm{dS}}/2} $$ , where S dS is the de Sitter (dS) entropy.