Bubble Nucleation Rate In Fourth-Order Gravity

We calculate the bubble nucleation rate in a fourth-order gravity theory whose action contains an $R^2$ term, under the thin-wall approximation, by two different methods. First the bounce solution is found for the Euclidean version of the original action. Next, we use a conformal transformation to transform the theory into general relativity with an additional scalar field and then proceed to find the bounce solution. The same results are obtained from both calculations.