Noncommutative Cosmology

The following sections are included:

• Preface
• Acknowledgments
• Contents

The main idea behind noncommutative geometry (see [Connes (1994)] for an extensive treatment of the subject) lies in the correspondence between spaces and algebras of functions. When one thinks of spaces as topological spaces, with some reasonable assumption about the topology (locally compact and Hausdorff), and of functions as continuous functions, the equivalence is expressed in terms of the Gelfand–Naimark correspondence. Assigning a locally compact Hausdorff topological space X is the same thing as assigning its C*-algebra of continuous functions C₀(X), in the sense that the space can be reconstructed from the algebra (with points corresponding to characters, that is, C*-algebra homomorphisms to 𝕔) and isomorphic algebras correspond to homeomorphic spaces. All C*-algebras of the form C₀(X), for some locally compact Hausdorff topological space X, are in particular commutative algebras, since the product is the ordinary pointwise product of functions, which is clearly commutative. However, the world of C*-algebras is rich with many interesting noncommutative examples. These cannot be directly interpreted as algebras of functions on an ordinary space. However, geometric properties of a space that can be formulated purely in terms of its algebra of functions may continue to make sense when one drops the commutativity assumption. One can then think of a noncommutative C*-algebra as the algebra of continuous functions on a noncommutative space. Unlike ordinary spaces, which can be defined in two a priori independent ways, as spaces or as algebras of functions, and then shown to be equivalent by the Gelfand–Naimark correspondence, noncommutative spaces can only be defined through their (noncommutative) algebra of functions. For simplicity, when not stated otherwise, we will assume our topological spaces X are compact and Hausdorff, with algebra of functions C(X). For a general introduction to C*-algebras, we refer the reader to [Davidson (1996)]…

In this chapter we look more closely at the dependence on the energy scale Λ of the modified gravity model derived from the asymptotic expansion of the spectral action, and how different possible assumptions about this dependence can lead to different scenarios for Early Universe models in cosmology. We begin by recalling some properties of the renormalization group flow in different particle physics models…

The results in this chapter are based on [Marcolli, Pierpaoli, Teh (2011)], [Marcolli, Pierpaoli, Teh (2012)], and on [Ćaćić, Marcolli, Teh (2012)]. The main problem we consider is whether, and to what extent, the spectral action functional can distinguish between different cosmic topologies. The main idea behind this approach is the fact that, while the asymptotic expansion of the spectral action recovers the usual kind of local terms like the Einstein–Hilbert action for gravity and its higher derivative generalization (conformal, Gauss–Bonnet, etc.), the spectral action in its non-perturbative form also contains global information, coming from the fact that the Dirac operator on a manifold carries global information about the manifold. It is clear that the usual Einstein equations for gravity can only detect geometry but not distinguish between different topologies with the same local geometry (a 3-sphere from a spherical space form; a 3-torus from an infinite at Euclidean 3-space, or from a Bieberbach manifold). However, a model of gravity derived from the spectral action may be able to provide information, which is in principle testable in an observational setting, which can distinguish the possible cosmic topologies…

This chapter is based on the results of [Manin, Marcolli (2014)].

The content of this chapter is based on [Manin, Marcolli (2015)] and on the earlier work [Manin, Marcolli (2002)] and [Marcolli (2007)]. Section 5.3 is based on [Estrada, Marcolli (2013)].

This chapter is based on the results of [Fan, Fathizadeh, Marcolli (2015)] and [Fan, Fathizadeh, Marcolli (2015b)]…

This chapter is based on the results of [Fathizadeh, Marcolli (2016)]. We show that the coefficients a_2n of the spectral action expansion of the Robertson–Walker spacetimes can be expressed in terms of periods of motives of algebraic varieties…

In this chapter we analyze cosmological models exhibiting fractal and multifractal structures and we approach their mathematical modeling using Noncommutative Geometry methods. We first consider the fractal Packed Swiss Cheese Cosmology model, and we study it from the point of view of the spectral action as an action functional for gravity. The results for this part of the chapter are based on [Ball, Marcolli (2016)]. We then consider multifractal models of eternal ination based on the “Eternal Symmetree” of [Harlow, Shenker, Stanford, Susskind (2012)], [Susskind (2012)], which we again revisit from the point of view of operator algebras and Noncommutative Geometry. This second part of the chapter is based on [Marcolli, Tedeschi (2014)].

In this chapter we discuss various aspects of the relation between the spectral action model of gravity and the Hartle–Hawking approach to quantum cosmology. We first review some basic ideas of the Hartle–Hawking formulation of Euclidean quantum gravity and quantum cosmology, and how these ideas can be transposed in a model of gravity based on the spectral action functional. We then describe a construction of categories and algebras of geometries with dynamical evolutions based on gravity action functionals and other physical and topological invariants, based on [Marcolli, Zainy al- Yasry (2008)]. We then use a discretization of this construction to enrich the formalism of spin networks and spin foams in quantum gravity with topological data that render the background topology dynamical, through monodromy data, in addition to the holonomy data describing the metric structure. We again consider categories and algebras of these topological spin networks and foams and their dynamical evolution, based on [Denicola, Marcolli, Zainy al-Yasry (2010)]. We present also a discretization of almost commutative geometries, via gauge networks, consisting of systems of finite spectral triples on a graph, and we show that they recover the usual spectral geometry and Dirac operator in a suitable continuum limit, based on [Marcolli, van Suijlekom (2014)]. Finally we present an approach to path integrals of noncommutative geometries, based on finite spectral triples, developed in [Barrett (2015)], [Barrett, Glaser (2015)].

The following sections are included:

• Bibliography
• Index