The premature recollapse problem in closed inflationary universes

We discuss whether closed universes can avoid recollapsing before inflation ensures. We show that in general closed universes are not equivalent to recollapsing universes or positive curvature universes. Closed universes will not in general recollapse if the matter content violates the strong energy condition. This violation is also a necessary condition for inflation to occur. When the strong energy condition holds closed universes can only recollapse if they possess S3 or S2xS1 spatial topology. Even when the topology is S3 and the strong energy condition holds it is not known whether anisotropic closed universes do all recollapse. We give examples to show that closed universes which begin in an extremely anisotropic state cannot recollapse until they are close to isotropy. This suggests that if the initial conditions prior to inflation are sufficiently anisotropic then the universe cannot recollapse until it has been isotropized by inflation. We also discuss the existence of inflation in isotropic cosmological models in R+R2 lagrangian theories of gravity and extend a result of Whitt to show that such theories are conformally equivalent to general relativity plus a scalar field with an asymmetric potential. (orig.)