Continuum of discrete trajectories in eternal inflation

Vitaly Vanchurin

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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.

Original languageEnglish (US)
Article number023511
JournalPhysical Review D - Particles, Fields, Gravitation and Cosmology
Issue number2
StatePublished - Jan 15 2015

Bibliographical note

Publisher Copyright:
© 2015 American Physical Society.


Dive into the research topics of 'Continuum of discrete trajectories in eternal inflation'. Together they form a unique fingerprint.

Cite this