We develop a theoretical frame for the study of classical and quantum gravitational waves based on the properties of a nonlinear ordinary differential equation for a function $sigma(eta)$ of the conformal time $eta$, called the auxiliary field equation. At the classical level, $sigma(eta)$ can be expressed by means of two independent solutions of the master equation to which the perturbed Einstein equations for the gravitational waves can be reduced. At the quantum level, all the significant physical quantities can be formulated using Bogolubov transformations and the operator quadratic Hamiltonian corresponding to the classical version of a damped parametrically excited oscillator where the varying mass is replaced by the square cosmological scale factor $a^{2}(eta)$. A quantum approach to the generation of gravitational waves is proposed on the grounds of the previous $eta-$dependent Hamiltonian. An estimate in terms of $sigma(eta)$ and $a(eta)$ of the destruction of quantum coherence due to the gravitational evolution and an exact expression for the phase of a gravitational wave corresponding to any value of $eta$ are also obtained. We conclude by discussing a few applications to quasi-de Sitter and standard de Sitter scenarios.
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