On the perturbative formalism and a possible quantum discrete spectrum for the Regge-Wheeler equation of a de Sitter spacetime

In this paper we study the perturbative regime in the static patch of de Sitter metric in the Regge-Wheeler formalism. After realizing that perturbative regime in a de Sitter spacetime depicted in terms of usual spherical coordinates cannot be extended up to the cosmological horizon, we study perturbative equations, in particular the axial ones, in terms of the tortoise coordinate $r_{\ast }$. We show that perturbative regime can be extended up to the cosmological horizon, provided that suitable boundary conditions are chosen. As an application, we explore the Regge-Wheeler equation at short distances by performing a taylor expansion. In order to study some possible quantum effects at short distances, we impose to the equation so obtained the same boundary conditions suitable for a quantum 3D harmonic oscillator. As a result, a discrete spectrum can be obtained. The aforementioned spectrum is analysed and a relation with possible effects denoting quantum behavior of gravitons is suggested.