Abstract
Abstract It has recently been shown that in single field slow-roll inflation the total volume cannot grow by a factor larger than $ {e^{{S_{dS}}/2}} $ without becoming infinite. The bound is saturated exactly at the phase transition to eternal inflation where the probability to produce infinite volume becomes non zero. We show that the bound holds sharply also in any space-time dimensions, when arbitrary higher-dimensional operators are included and in the multi-field inflationary case. The relation with the entropy of de Sitter and the universality of the bound strengthen the case for a deeper holographic interpretation. As a spin-off we provide the formalism to compute the probability distribution of the volume after inflation for generic multi-field models, which might help to address questions about the population of vacua of the landscape during slow-roll inflation.
Highlights
These problems are relevant in the framework of the landscape
It has recently been shown that in single field slow-roll inflation the total volume cannot grow by a factor larger than eSdS/2 without becoming infinite
As suggested by string theory, quantum gravity possesses a landscape of vacua, our Universe may be doomed to deal with eternal inflation
Summary
It has been shown in [15] that many information about the phase transition from noneternal to eternal slow-roll inflation are encoded in a rather simple formula. Where at leading order in the slow-roll approximation, f is the solution of the following differential equation (see [19]). The meaning of the differential equation is more manifest when rewritten in terms of φ as follows: This is a modified Fokker-Planck equation: the first term is the normal dispersion term due to the quantum fluctuations of the inflaton field in de Sitter space (∆φ2/∆t = H3/(4π2)); the second term is the drift induced by the tilt of the scalar potential; the last term encodes the volume growth from the de Sitter expansion. When rewritten in terms of τ , the differential equation (2.2) only depends on the single dimensionless parameter Ω — a combination of the rate of quantum fluctuations (∆φ2/∆t), the classical rolling (φ) and the Hubble expansion (3H) — which controls the different phases of slow-roll inflation.
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