Assumptions of Randomness in Cosmology Models

TL;DR

In infinite cosmology models with non-compact symmetries, randomness cannot fully break symmetries because non-compact groups have no invariant probability distributions, complicating Copernican random placement of an observer. A remedy proposed here is to fix a designated observance point and build an infinite universe around it using the topology of pointed spaces, thereby avoiding the need for a uniform distribution on infinite domains. The discussion treats the observer and external random events as integral model ingredients that determine the observed data, which may themselves be nonobservably random depending on the observer's placement. It also engages with algorithmic randomness results, noting that the Gacs_Kucera theorem implies every sequence is computable from a random one, while Vyugin randomness results show randomized algorithms can generate positive-probability uncomputable sequences not equivalent to any random sequence. Overall, the pointed-space approach aims to render probabilities and observables well-defined in cosmology, potentially clarifying predictions and addressing paradoxes arising from partial observability and approximate physical laws.

Abstract

Non-compact symmetries cannot be fully broken by randomness since non-compact groups have no invariant probability distributions. In particular, this makes trickier the "Copernican" random choice of the place of the observer in infinite cosmology models. This problem may be circumvented with what topologists call Pointed Spaces. Then randomness will be used only in building (infinite) models around the pre-designated "observance point", that thus would not need to be randomly chosen. Additional complications come from the original randomness possibly being hidden. P. Gacs and A. Kucera proved that every sequence can be algorithmically generated from a random one. But Vladimir V'yugin discovered that randomized algorithms can with positive probability generate uncomputable sequences not algorithmically equivalent to any random ones.