Initial conditions for R+εR2cosmology

A pure gravity cosmology based on the R+ε$R^2$ Lagrangian is known to exhibit inflation for a wide range of initial conditions. In this paper we use the wave function from quantum cosmology to describe this inflation as a chaotic inflationary phase immediately following the quantum creation of the Universe. We evaluate, compare, and discuss the distributions over initial conditions that are fixed by the two boundary-condition proposals of Hartle and Hawking (‘‘no boundary’’) and Vilenkin (‘‘tunneling from nothing’’). We find that among all classical inflationary trajectories that begin on the classical-quantum boundary, those that lead to an inflation of at least 70 e-foldings make up a fraction of ∼exp${(\mathrm{-}10}^{12}$) in the former case and ∼1-exp(-8×${10}^{10}$) in the latter. Thus, in the simplest interpretation, the observable Universe would be the outcome of a rare event for the first boundary-condition proposal and a typical event for the second.