Cosmological gravitational wave in a gravity with quadratic order curvature couplings

We present a set of equations describing the cosmological gravitational wave in a gravity theory with quadratic order gravitational coupling terms which naturally arise in quantum correction procedures. It is known that the gravitational wave equation in the gravity theories with a general $f\left(R\right)\mathrm{}$ term in the action leads to a second order differential equation with the only correction factor appearing in the damping term. The case for a $R^{\mathrm{ab}}{R}_{\mathrm{ab}}$ term is completely different. The gravitational wave is described by a fourth order differential equation both in time and space. However, curiously, the contributions to the background evolution are qualitatively the same for both terms.