Riemann tensor and Gauss–Bonnet density in metric-affine cosmology

Abstract

We analytically derive the covariant form of the Riemann (curvature) tensor for homogeneous metric-affine cosmologies. That is, we present, in a cosmological setting, the most general covariant form of the full Riemann tensor including also its non-Riemannian pieces which are associated to spacetime torsion and non-metricity. Having done so we also compute a list of the curvature tensor by-products such as Ricci tensor, homothetic curvature, Ricci scalar, Einstein tensor etc. Finally we derive the generalized metric-affine version of the usual Gauss–Bonnet density in this background and demonstrate how under certain circumstances the latter represents a total derivative term.