Axial and polar gravitational wave equations in a de Sitter expanding universe by Laplace transform

In this paper we study the propagation in a de Sitter universe of gravitational waves generated by perturbating some unspecified spherical astrophysical object in the frequencies domain. We obtain the axial and polar perturbation equations in a cosmological de Sitter universe in the usual comoving coordinates, the coordinates we occupy in our galaxy. We write down the relevant equations in terms of Laplace transform with respect to the comoving time $t$ instead of the usual Fourier one that is no longer available in a cosmological context. Both axial and polar perturbation equations are expressed in terms of a non trivial mixture of retarded-advanced metric coefficients with respect to the Laplace parameter $s$ (complex translation). The axial case is studied in more detail. In particular, the axial perturbations can be reduced to a master linear second-order differential equation in terms of the Regge-Wheeler function $Z$ where a coupling with a retarded $Z$ with respect to the cosmological time $t$ is present. It is shown that a de Sitter expanding universe can change the frequency $\omega$ of a gravitational wave as perceived by a comoving observer. The polar equations are much more involved. Nevertheless, we show that also the polar perturbations can be expressed in terms of four independent integrable differential equations.