# Leading slow roll corrections to the volume of the universe and the entropy bound

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Journal of High Energy Physics

, 2014:60

First Online: 09 December 2014Received: 18 September 2014Accepted: 20 November 2014 Abstract

We make an extension to recent calculations of the probability density ρV 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 ight \, and 1-Nc, where H is the Hubble parameter and Nc is the classical number of e-foldings. Here, we present a modification which captures effects of order ϵNc, 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 ρV 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 SdS is the de Sitter dS entropy.

KeywordsCosmology of Theories beyond the SM Models of Quantum Gravity Stochastic Processes Space-Time Symmetries ArXiv ePrint: 1309.6705

Autor: Matthew Lewandowski - Ashley Perko