The Continuum of Discrete Trajectories in Eternal Inflation

Abstract

We discuss eternal inflation in context of classical probability spaces defined by a triplet: sample space, $\sigma$-algebra and probability measure. We show that the measure problem is caused by the countable additivity axiom applied to the maximal $\sigma$-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.

12 pages

References in corpus (20)

• The Measure Problem in Cosmology
• Holographic probabilities in eternal inflation
• Micromanaging de Sitter holography
• A Holographic Framework for Eternal Inflation
• Predicting the cosmological constant with the scale-factor cutoff measure
• Holographic No-Boundary Measure
• Measure and Probability in Cosmology
• Eternal Symmetree
• Holographic Multiverse
• Complementarity in the Multiverse
• Geodesic measures of the landscape
• Eternal observers and bubble abundances in the landscape
• How unitary cosmology generalizes thermodynamics and solves the inflationary entropy problem
• Eternal inflation predicts that time will end
• Unitary Evolution and Cosmological Fine-Tuning
• Measures on transitions for cosmology from eternal inflation
• Is there any coherent measure for eternal inflation?
• Eternal Inflation, Global Time Cutoff Measures, and a Probability Paradox
• Dynamical systems of eternal inflation: A possible solution to the problems of entropy, measure, observables and initial conditions
• Geocentric cosmology: a new look at the measure problem

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