Continuum of discrete trajectories in eternal inflation

Vitaly Vanchurin
Phys. Rev. D 91, 023511 – Published 15 January 2015

Abstract

We discuss eternal inflation in the context of classical probability spaces defined by a triplet: sample space, σ-algebra, and probability measure. We show that the measure problem is caused by the countable additivity axiom applied to the maximal σ-algebra of countably infinite sample spaces. This is a serious problem if the bulk space-time is treated as a sample space which is thought to be effectively countably infinite due to local quantum uncertainties. However, in semiclassical description of eternal inflation the physical space expands exponentially which makes the sample space of infinite trajectories uncountable and the (future) boundary space effectively continuous. Then the measure problem can be solved by defining a probability measure on the continuum of trajectories or holographically on the future boundary. We argue that the probability measure which is invariant under the symmetries of the tree-like structure of eternal inflation can be generated from the Lebesgue measure on unit interval. According to Vitali theorem the Lebesgue measure leaves some sets without a measure which means that there are certain probabilistic questions in eternal inflation that cannot be answered.

  • Figure
  • Received 31 August 2014

DOI:https://doi.org/10.1103/PhysRevD.91.023511

© 2015 American Physical Society

Authors & Affiliations

Vitaly Vanchurin*

  • Department of Physics, University of Minnesota, Duluth, Minnesota 55812, USA

  • *vvanchur@umn.edu

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Vol. 91, Iss. 2 — 15 January 2015

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