Abstract. 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 slowroll 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 Poincaré 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 Poincaré 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.

We show that, in a model of modified gravity based on the spectral action functional, there is a nontrivial coupling between cosmic topology and inflation, in the sense that the shape of the possible slowroll inflation potentials obtained in the model from the nonperturbative form of the spectral action are sensitive not only to the geometry (flat or positively curved) of the universe, but also to the different possible non-simply connected topologies. We show this by explicitly computing the nonperturbative spectral action for some candidate flat cosmic topologies given by Bieberbach manifolds and showing that the resulting inflation potential differs from that of the flat torus by a multiplicative factor, similarly to what happens in the case of the spectral action of the spherical forms in relation to the case of the 3-sphere. We then show that, while the slow-roll parameters differ between the spherical and flat manifolds but do not distinguish different topologies within each class, the power spectra detect the different scalings of the slow-roll potential and therefore distinguish between the various topologies, both in the spherical and in the flat case.

We compute the spectral action of SU (2)/Γ with the trivial spin structure and the round metric and find it in each case to be equal to 1. We do this by explicitly computing the spectrum of the Dirac operator for SU (2)/Γ equipped with the trivial spin structure and a selection of metrics. Here Γ is a finite subgroup of SU (2). In the case where Γ is cyclic, or dicyclic, we consider the one-parameter family of Berger metrics, which includes the round metric, and when Γ is the binary tetrahedral, binary octahedral or binary icosahedral group, we only consider the case of the round metric.Abstract. We compute the spectral action of SU (2)/Γ with the trivial spin structure and the round metric and find it in each case to be equal to 1. We do this by explicitly computing the spectrum of the Dirac operator for SU (2)/Γ equipped with the trivial spin structure and a selection of metrics. Here Γ is a finite subgroup of SU (2). In the case where Γ is cyclic, or dicyclic, we consider the one-parameter family of Berger metrics, which includes the round metric, and when Γ is the binary tetrahedral, binary octahedral or binary icosahedral group, we only consider the case of the round metric.

Abstract. We consider a model of modified gravity based on the spectral action functional, for a cosmic topology given by a spherical space form, and the associated slow-roll inflation scenario. We consider then the coupling of gravity to matter determined by an almost-commutative geometry over the spherical space form. We show that this produces a multiplicative shift of the amplitude of the power spectra for the density fluctuations and the gravitational waves, by a multiplicative factor equal to the total number of fermions in the matter sector of the model. We obtain the result by an explicit nonperturbative computation, based on the Poisson summation formula and the spectra of twisted Dirac operators on spherical space forms, as well as, for more general spacetime manifolds, using a heat-kernel computation. Contents

Spectral triples and quantum statistical mechanical systems are two important constructions in noncommutative geometry. In particular, both lead to interesting reconstruction theorems for a broad range of geometric objects, including number fields, spin manifolds, graphs. There are similarities between the two structures, and we show that the notion of type III σspectral triple, introduced recently by Connes and Moscovici, provides a natural bridge between them. We investigate explicit examples, related to the Bost-Connes quantum statistical mechanical system and to Riemann surfaces and graphs.