Reconciling inflation with openness

It is already understood that the increasing observational evidence for an open Universe can be reconciled with inflation if our horizon is contained inside one single huge bubble nucleated during the inflationary phase transition. In this frame of ideas, we show here that the probability of living in a bubble with the right $Ω_0$ (now the observations require $Ω_0 \approx .2$) can be comparable with unity, rather than infinitesimally small. For this purpose we modify both quantitatively and qualitatively an intuitive toy model based upon fourth order gravity. As this scheme can be implemented in canonical General Relativity as well (although then the inflation driving potential must be designed entirely ad hoc), inferring from the observations that $Ω_0 < 1$ not only does not conflict with the inflationary paradigm, but rather supports therein the occurrence of a primordial phase transition.

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