Liouvillian solutions of the Klein–Gordon equation on D-dimensional de Sitter spacetime

We apply the well known Kovacic algorithm to generate Liouvillian (i.e., closed-form) solutions to the ordinary differential equation that governs the radial part of the separable, coupled to scalar curvature, massive scalar wave equation on a D-dimensional de Sitter background spacetime. The radial equation is simply related to the famous hypergeometric equation, and our work is carried out in this general setting. We find several families of infinite sets of special parameter values for which Liouvillian solutions exist, and we demonstrate how to generate these solutions recursively. The solutions are related to hypergeometric polynomials and in some cases can be constructed explicitly. For some of the families, the Liouvillian solutions exactly satisfy the boundary conditions for quasinormal modes, a fundamental set of damped, outgoing wave-functions that depend only on the background spacetime and scalar field parameters. In these families we obtain agreement with published results on the scalar quasinormal modes of the pure de Sitter spacetime. For other families, we establish conjectures regarding the factorization of the parameter constraints required for existence of Liouvillian solutions. Our results are consistent with published results on the hypergeometric equation and the well known Pöschl–Teller equation. Our new contributions are to demonstrate how to generate the hypergeometric Liouvillian solutions recursively, to completely solve the quasinormal mode problem with Liouvillian solutions, and to analyze the Liouvillian parameter constraints for the hypergeometric equation, including the case of the physical de Sitter parameters. This work contributes new knowledge to the fields of special functions and general relativity.