We give a rigorous and mathematically clear presentation of the Covariant and Gauge Invariant theory of gravitational waves in a perturbed Friedmann-Lemaitre-Robertson-Walker universe for Fourth Order Gravity, where the matter is described by a perfect fluid with a barotropic equation of state. As an example of a consistent analysis of tensor perturbations in Fourth Order Gravity, we apply the formalism to a simple background solution of R^n gravity. We obtain the exact solutions of the perturbation equations for scales much bigger than and smaller than the Hubble radius. It is shown that the evolution of tensor modes is highly sensitive to the choice of n and an interesting new feature arises. During the radiation dominated era, their exist a growing tensor perturbation for nearly all choices of n. This occurs even when the background model is undergoing accelerated expansion as opposed to the case of General Relativity. Consequently, cosmological gravitational wave modes can in principle provide a strong constraint on the theory of gravity independent of other cosmological data sets.
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