TY - JOUR

T1 - Continuum of discrete trajectories in eternal inflation

AU - Vanchurin, Vitaly

N1 - Publisher Copyright:
© 2015 American Physical Society.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015/1/15

Y1 - 2015/1/15

N2 - 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.

AB - 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.

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U2 - 10.1103/PhysRevD.91.023511

DO - 10.1103/PhysRevD.91.023511

M3 - Article

AN - SCOPUS:84921445747

VL - 91

JO - Physical Review D - Particles, Fields, Gravitation and Cosmology

JF - Physical Review D - Particles, Fields, Gravitation and Cosmology

SN - 1550-7998

IS - 2

M1 - 023511

ER -