Scalar fields on SL(2,R) and H^2 x R geometric spacetimes and linear perturbations

Using appropriate harmonics, we study the future asymptotic behavior of massless scalar fields on a class of cosmological vacuum spacetimes. The spatial manifold is assumed to be a circle bundle over a higher genus surface with a locally homogeneous metric. Such a manifold corresponds to the SL(2,R)-geometry (Bianchi VIII type) or the H^2 x R-geometry (Bianchi III type). After a technical preparation including an introduction of suitable harmonics for the circle-fibered Bianchi VIII to separate variables, we derive systems of ordinary differential equations for the scalar field. We present future asymptotic solutions for these equations in a special case, and find that there is a close similarity with those on the circle-fibered Bianchi III spacetime. We discuss implications of this similarity, especially to (gravitational) linear perturbations. We also point out that this similarity can be explained by the "fiber term dominated behavior" of the two models.