The Volume of the Universe after Inflation and de Sitter Entropy

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^(S_dS/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.

47 pages, 11 figures

References in corpus (12)

• Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology
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• Flux Compactification
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• Eternal Inflation: The Inside Story
• Lorentz violation and perpetual motion
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• Reheating-volume measure for random-walk inflation

Cited by in corpus (13)

• Micromanaging de Sitter holography
• Soft de Sitter Effective Theory
• Stochastic Inflation at NNLO
• Self-Organised Localisation
• Beyond Perturbation Theory in Inflation
• Lectures on Inflation
• Theoretical bounds on the tensor-to-scalar ratio in the cosmic microwave background
• Early-Time Measure in Eternal Inflation
• Islands and the de Sitter entropy bound
• Hubble selection of the weak scale from QCD quantum critical point
• Overlapping qubits from non-isometric maps and de Sitter tensor networks
• A Tail of Eternal Inflation
• Quantization of Yang-Mills Theory in de Sitter Spacetime