The spectral action functional, considered as a model of gravity coupled to
matter, provides, in its non-perturbative form, a slow-roll potential for
inflation, whose form and corresponding slow-roll parameters can be sensitive
to the underlying cosmic topology. We explicitly compute the non-perturbative
spectral action for some of the main candidates for cosmic topologies, namely
the quaternionic space, the Poincare' dodecahedral space, and the flat tori. We
compute the corresponding slow-roll parameters and see we check that the
resulting inflation model behaves in the same way as for a simply-connected
spherical topology in the case of the quaternionic space and the Poincare'
homology sphere, while it behaves differently in the case of the flat tori. We
add an appendix with a discussion of the case of lens spaces.Comment: 55 pages, LaTe
