Cosmology within Noncommutative Spectral Geometry

Abstract

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.

Fixing extensions to general relativity in the nonlinear regime

“…Moreover, in this case there is no signature of acausality because the modes' velocity is bounded in any frequency regime. Notice also that the solutions s(k) behave well if we assume λ < 0, which is precisely the case being explored in the literature [49]. Nevertheless, the blowing-up of low frequency modes is indeed an issue per se, and thus we employ this model as a convenient example of the application of the Israel-Stewart line of thought in a gravity context.…”

“…As an example of the latter, Ref. [49] explores constraints on the gravitational sector of the theory, and derives equations of motion for the metric g ab from a spectral action. To the best of our knowledge, this theory has not been explicitly shown to not admit a well posed initial value formulation.…”

Spectral Geometry

“…66,[76][77][78]81,95 Spectral triples associated to manifolds with boundary have been considered in. 14,18,18,58,59,61 The main difficulty is precisely to put nice boundary conditions to the operator D to still get a selfadjoint operator and then, to define a compatible algebra A.…”