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.
