### Abstract

We calculate the probability distribution for the volume of the Universe after slow-roll inflation both in the eternal and in the non-eternal regime. Far from the eternal regime the probability distribution for the number of e-foldings, defined as one third of the logarithm of the volume, is sharply peaked around the number of e-foldings of the classical inflaton trajectory. At the transition to the eternal regime this probability is still peaked (with the width of order one e-folding) around the average, which gets twice larger at the transition point. As one enters the eternal regime the probability for the volume to be finite rapidly becomes exponentially small. In addition to developing techniques to study eternal inflation, our results allow us to establish the quantum generalization of a recently proposed bound on the number of e-foldings in the non-eternal regime: the probability for slow-roll inflation to produce a finite volume larger than e ^{SdS/2}, where S _{dS} is the de Sitter entropy at the end of the inflationary stage, is smaller than the uncertainty due to non-perturbative quantum gravity effects. The existence of such a bound provides a consistency check for the idea of de Sitter complementarity.

Original language | English (US) |
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Article number | 118 |

Journal | Journal of High Energy Physics |

Volume | 2009 |

Issue number | 4 |

DOIs | |

State | Published - Jul 16 2009 |

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### Keywords

- Cosmology of theories beyond the SM
- Models of quantum gravity
- Space-time symmetries
- Spacetime singularities

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

*Journal of High Energy Physics*,

*2009*(4), [118]. https://doi.org/10.1088/1126-6708/2009/04/118