Percolation of 'Civilisation' in a Homogeneous Isotropic Universe

TL;DR

This paper models the spread of civilization in different idealized universe types, revealing logistic growth patterns and how cosmic expansion affects colonization times and reachability of planets.

Contribution

It introduces a framework extending percolation theory to cosmological scales, analyzing colonization dynamics in static, dark energy-, and matter-dominated universes.

Findings

Logistic growth fits colonization data across universe types.

Dark energy universes exhibit divergence in colonization time due to shrinking Hubble sphere.

Matter-dominated universes allow finite colonization times with specific Hubble parameter dependencies.

Abstract

In this work, we consider the spread of a 'civilisation' in an idealised homogeneous isotropic universe where all the planets of interest are habitable. Following a framework that goes beyond the usual idea of percolation, we investigate the behaviour of the number of colonised planets with time, and the total colonisation time for three types of universes. These include static, dark energy-dominated, and matter-dominated universes. For all these types of universes, we find a remarkable fit with the Logistic Growth Function for the number of colonised planets with time. This is in spite of the fact that for the matter- and dark-energy dominated universes, the space itself is expanding. For the total colonisation time, $T$, the case for a dark energy-dominated universe is marked with divergence beyond the linear regime characterised by small values of the Hubble parameter, $H$. Not all planets in a spherical section of this universe can be 'colonised' due to the presence of a shrinking Hubble sphere. In other words, the recession speeds of other planets go beyond the speed of light making them impossible to reach. On the other hand, for a matter-dominated universe, while there is an apparent horizon, the Hubble sphere is growing instead of shrinking. This leads to a finite total colonisation time that depends on the Hubble parameter characterising the universe; in particular, we find $T\sim H$ for small $H$ and $T\sim H^2$ for large $H$.