Abstract
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 $f(R)$ 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.
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