Probability ofR2inflation

Abstract

The Gibbons-Hawking-Stewart canonical measure is applied to classical Friedmann-Robertson-Walker cosmologies with an R+\ensuremath{\epsilon}${R}^{2}$ Lagrangian. Both the inflationary solutions and the noninflationary solutions have infinite measure, and the ratio is ambiguous. All but a finite measure of the k=-1 solutions have an arbitrarily long period of near spatial flatness, but for k=+1 there is also an infinite measure for solutions which have a small maximum radius, unlike the case with Einstein gravity coupled to a massive scalar field. For k=-1 there is a finite positive measure for complete nonsingular solutions with a nonzero minimum radius; all other k=-1 solutions expand from zero to infinite radius. For k=+1 all solutions but a set of measure zero expand from zero radius and eventually recollapse to zero radius. However, there is also an apparently fractal set of discrete (zero measure) k=+1 solutions which have no singularities but rather expand and recontract perpetually, with an arbitrary number of oscillations of the scalar curvature between each successive bounce.