Cosmology within Noncommutative Spectral Geometry
  Close to the Planck energy scale, the quantum nature of space-time reveals
itself and all forces, including gravity, should be unified so that all
interactions correspond to just one underlying symmetry. In the absence of a
full quantum gravity theory, one may follow an effective approach and consider
space-time as the product of a four-dimensional continuum compact Riemanian
manifold by a tiny discrete finite noncommutative space. Since all available
data are of a spectral nature, one may argue that it is more appropriate to
apply the spectral action principle in this almost commutative space. Following
this procedure one obtains an elegant geometric explanation for the most
successful particle physics model, namely the standard model (and
supersymmetric extensions) of electroweak and strong interactions in all its
details, as determined by experimental data. Moreover, since this gravitational
theory lives by construction at very high energy scales, it offers a perfect
framework to address some of the early universe cosmological questions still
awaiting for an answer.
  After introducing some of the main mathematical elements of noncommutative
spectral geometry, I will discuss various cosmological and phenomenological
consequences of this theory, focusing in particular on constraints imposed on
the gravitational sector of the theory.
