Higher powers in gravitation

We consider the Friedmann-Robertson-Walker cosmologies of theories of gravity that generalize the Einstein-Hilbert action by replacing the Ricci scalar R with some function f(R). The general asymptotic behavior of these cosmologies is found, at both early and late times, and the effects of adding higher and lower powers of R to the Einstein-Hilbert action is investigated. The assumption that the highest powers of R should dominate the Universe's early history, and that the lowest powers should dominate its future is found to be inaccurate. The behavior of the general solution is complicated, and while it can be the case that single powers of R dominate the dynamics at late times, it can be either the higher or lower powers that do so. It is also shown that it is often the lowest powers of R that dominate at early times, when approach to a bounce or a Tolman solution are generic possibilities. Various examples are considered, and both vacuum and perfect fluid solutions are investigated.