We study cosmologies in modified theories of gravity considering Lagrangian density $f(R)$ which is a polynomial function of scalar curvature ($R$) in the Einstein-Hilbert action in vacuum. The field equation obtained from the modified action corresponding to a Robertson-Walker metric is highly non-linear and not simple enough to obtain analytic solution. Consequently we adopt a numerical technique to study the evolution of the FRW universe. A number of evolutionary phases of the universe including the present accelerating phase are found to exist in the higher derivative theories of gravity. The cosmological solutions obtained here are new and interesting. We study modified theory of gravity as a toy model to explore the past, the present and predict the future evolution. It is found that all the models analyzed here can reproduce the current accelerating phase of expansion of the universe. The duration of the present accelerating phase is found to depend on the coupling constants of the gravitational action. The physical importance of the coupling parameters those considered in the action are also discussed.

23 pages, 11 figs. accepted in Physical Review D