Stability of FRW cosmology in higher order gravity

We analyze the behavior of radiation-filled, homogeneous, and isotropic cosmological solutions to a generalized higher order gravity theory which is derived from a gravitational Lagrangian that is an arbitrary function of the scalar spacetime curvature $f(R)$. We give necessary and sufficient conditions for the existence and stability of general relativistic $\ensuremath{\sigma}=\ifmmode\pm\else\textpm\fi{}1, 0$ FRW solutions within this general theory. We show that under some general conditions any homogeneous and isotropic solution of general relativity is also an exact solution of the $f(R)$ theory, and every radiation solution (not necessarily isotropic) in general relativity is an exact solution in higher order gravity provided there are no nonzero constants and the Einstein term is present in the gravitational Lagrangian of our theory. We then prove that nonflat FRW solutions of general relativity are generically unstable and so do not approach the corresponding ones in higher order gravity for large times. This may be interpreted as an indication that homogeneous and isotropic solutions of higher order gravity cannot be obtained from the corresponding nonflat FRW solutions of general relativity via perturbation theory. However, we find a stable regime for flat FRW solutions of general relativity in higher order gravity. In particular, under fairly general circumstances, flat FRW solutions of general relativity are stable against homogeneous and isotropic perturbations in higher order gravity and always approach their corresponding ones in the generalized theory at the large time limit. The requirements for stability of the flat FRW solutions in higher order gravity coincide with well-known constraints for the absence of tachyons and the existence of complex instanton solutions in the theory, and are exactly those needed to produce bouncing, regular solutions on approach to the singularity.