Can we have inflation with Ω > 1?*Can we have inflation with Ω > 1?

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 δρ/ρ~ΔT/T = O(1) 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 per cent one can obtain a semi-realistic model of an inflationary universe with Ω>1. The spectrum of density perturbations in this model may have a cut-off on the scale of the horizon. A similar approach may be valid in application to a compact inflationary universe with a nontrivial topology.