Can we have inflation with Omega > 1?

TL;DR

This paper investigates whether inflation can produce a universe with $ ext{Ω} eq 1$, focusing on closed or compact geometries. By analyzing both a toy closed quasi-de Sitter model with a step-like potential and a more realistic chaotic inflation scenario with a boundary at $ ext{φ}_0$, the authors show that achieving a modest deviation from flatness requires fine-tuning of initial conditions or parameters, such as the initial energy density near a critical value and the location of $ ext{φ}_0$, to yield around 60 $e$-folds. They find that inflation tends to be an attractor, but the probability of creation of an inflationary universe and the resulting $ ext{Ω}$ depend sensitively on $eta_0$ and $V_0$, with many histories funneling toward either rapid collapse or excessive inflation. The discussion concludes that while a short inflationary stage can suppress horizon-scale perturbations, constructing realistic closed (or compact) models remains highly fine-tuned compared to the standard flat, horizon-and-perturbation-robust predictions of inflation, though no fundamental obstacle prohibits such constructions in principle.

Abstract

It is very difficult to obtain a realistic model of a closed inflationary universe. Even if one fine-tunes the total number of e-folds to be sufficiently small, the resulting universe typically has large density perturbations on the scale of the horizon. We describe a class of models where this problem can be resolved. The models are unattractive and fine-tuned, so the flatness of the universe remains a generic prediction of inflationary cosmology. Nevertheless one should keep in mind that with the fine-tuning at the level of about one percent one can obtain a semi-realistic model of a closed inflationary universe. The spectrum of density perturbations in this model may have a cut-off on the scale of the horizon. Similar approach may be valid in application to a compact inflationary universe with a nontrivial topology.

Figures (4)

• Figure 1: Effective potential in our toy model; $V^* = {3\over 2} V$.
• Figure 2: Behavior of the scalar field in our toy model. If the field starts its motion with sufficiently small velocity, inflation begins immediately. If it starts with large initial velocity $\dot\phi_0$, due to falling from the large 'height' $V(\phi) = V^* -\Delta V$, the universe never inflates. Inflation begins at $\phi \approx \phi_0 +0.1 +0.15 \, \ln {\Delta V\over V}$.
• Figure 3: Effective potential in the model with the potential that is equal to $m^2\phi^2/2$ at $\phi < \phi_0$ and blows up at $\phi > \phi_0$.
• Figure 4: Behavior of the scalar field in the theory $m^2\phi^2/2$. If the field starts its motion with sufficiently small velocity, inflation begins immediately. If it starts with large initial velocity $\dot\phi_0$, due to falling from the large value of $V(\phi)$, the universe never inflates. Inflation begins at $\phi \approx \phi_0 - 0.05 +0.15 \, \ln {\Delta V\over V}$.