Causality in Inflationary Universes with Positive Spatial Curvature

TL;DR

This paper demonstrates that in positively curved inflationary universes, the horizon problem is generally not solved unless inflation begins in an almost static state with extremely high initial vacuum energy, raising implications for early universe models.

Contribution

It shows that inflation in positively curved universes does not solve the horizon problem for most initial conditions, highlighting the importance of initial state in cosmological models.

Findings

Horizon problem remains unresolved for typical initial conditions in $k=+1$ universes.

Only near-static initial states with high $\Omega_{\Lambda}$ solve the horizon problem.

Presence of event horizons and compact spaces challenges M-theory compatibility.

Abstract

We show that in the case of positively-curved Friedmann-Lema\^{\i}tre universes $(k=+1)$, an inflationary period in the early universe will for most initial conditions not solve the horizon problem, no matter how long inflation lasts. It will only do so for cases where inflation starts in an almost static state, corresponding to an extremely high value of $\Omega_{\Lambda}$, $\Omega_{\Lambda} \gg 1$, at the beginning of inflation. For smaller values, it is not possible to solve the horizon problem because the relevant integral asymptotes to a finite value (as happens also in the de Sitter universe in a $k=+1$ frame). Thus, for these cases, the causal problems associated with the near-isotropy of the Cosmic Background Radiation have to be solved already in the Planck era. Furthermore both compact space sections and event horizons will exist in these universes even if the present cosmological constant dies away in the far future, raising potential problems for M-theory as a theory of gravity.