Modified Gravity and Cosmology

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

Over recent decades, the role of torsion in gravity has been extensively investigated along the main direction of bringing gravity closer to its gauge formulation and incorporating spin in a geometric description. Here we review various torsional constructions, from teleparallel, to Einstein-Cartan, and metric-affine gauge theories, resulting in extending torsional gravity in the paradigm of f (T) gravity, where f (T) is an arbitrary function of the torsion scalar. Based on this theory, we further review the corresponding cosmological and astrophysical applications. In particular, we study cosmological solutions arising from f (T) gravity, both at the background and perturbation levels, in different eras along the cosmic expansion. The f (T) gravity construction can provide a theoretical interpretation of the late-time universe acceleration, alternative to a cosmological constant, and it can easily accommodate with the regular thermal expanding history including the radiation and cold dark matter dominated phases. Furthermore, if one traces back to very early times, for a certain class of f (T) models, a sufficiently long period of inflation can be achieved and hence can be investigated by cosmic microwave background observations-or, alternatively, the Big Bang singularity can be avoided at even earlier moments due to the appearance of non-singular bounces. Various observational constraints, especially the bounds coming from the large-scale structure data in the case of f (T) cosmology, as well as the behavior of gravitational waves, are described in detail. Moreover, the spherically symmetric and black hole solutions of the theory are reviewed. Additionally, we discuss various extensions of the f (T) paradigm. Finally, we consider the relation with other modified gravitational theories, such as those based on curvature, like f (R) gravity, trying to illuminate the subject of which formulation, or combination of formulations, might be more suitable for quantization ventures and cosmological applications.

f(T) teleparallel gravity and cosmology

Kaluza-Klein gravity

Spontaneous compactification with quadratic and cubic curvature terms

The authors study consistent deformations of the classical differential calculus on algebras of functions (and, more generally, commutative algebras) such that differentials and functions satisfy nontrivial commutation relations. For a class of such calculi it is shown that the deformation parameters correspond to the spacings of a lattice. These differential calculi generate a lattice on a space continuum. The whole setting of a lattice theory can then be deduced from the continuum theory via deformation of the standard differential calculus. In this framework one just has to express the Lagrangian for the continuum theory in terms of differential forms. This expression then also makes sense for the deformed differential calculus. There is a natural integral associated with the latter. Integration of the Lagrangian over a space continuum then produces the correct lattice action for a large class of theories. This is explicitly shown for the scalar field action and the action for SU(m) gauge theory.

https://iopscience.iop.org/article/10.1088/0305-4470/26/8/019/pdf

Noncommutative differential calculus and lattice gauge theory

The most general gravity Lagrangian in more than four dimensions is considered which leads to field equations with at most second derivatives of the metric. It consists of a series of dimensionally continued Euler forms and allows spontaneous compactification. The field equations are elaborated for the usual Kaluza-Klein cosmology ansatz and solved in the special case where the extra dimensions form a sphere with constant radius. The dimensional reduction of the theory to four dimensions is discussed as well.

https://iopscience.iop.org/article/10.1088/0264-9381/3/4/020/pdf

Dimensionally Continued Euler Forms, {Kaluza-Klein} Cosmology and Dimensional Reduction

Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a ‘‘reduction’’ of the ‘‘universal differential algebra’’ and this allows a systematic exploration of differential algebras on a given set. Associated with a differential algebra is a (di)graph where two vertices are connected by at most two (antiparallel) arrows. The interpretation of such a graph as a ‘‘Hasse diagram’’ determining a (locally finite) topology then establishes contact with recent work by other authors in which discretizations of topological spaces and corresponding field theories were considered which retain their global topological structure. It is shown that field theories, and in particular gauge theories, can be formulated on a discrete set in close analogy with the continuum case. The framework presented generalizes ordinary lattice theory which is recovered from an oriented (hypercubic) lattice graph. It also includes, e.g., the two‐point space used by Connes and Lott (and others) in models of elementary particle physics. The formalism suggests that the latter be regarded as an approximation of a manifold and thus opens a way to relate models with an ‘‘internal’’ discrete space (a la Connes et al.) to models of dimensionally reduced gauge fields. Furthermore, a ‘‘symmetric lattice’’ is also studied which (in a certain continuum limit) turns out to be related to a ‘‘noncommutative differential calculus’’ on manifolds.

Discrete differential calculus graphs, topologies and gauge theory

This is a comprehensive study of spatially homogeneous and SO(3)-isotropic exact solutions of the 10-parameter Lagrangian of the 'Poincare gauge theory'. Some sets of new exact solutions are presented. In particular, all solutions following from the so-called modified double quality ansatz are obtained, up to integration of some familiar ordinary differential equations. For certain classes of solutions, the occurrence of torsion singularities is discussed in detail. Furthermore, the authors investigate whether some solutions without metric singularity can provide reasonable cosmological models.

Folkert Müller-Hoissen

Papers

Spontaneous compactification with quadratic and cubic curvature terms

Noncommutative differential calculus and lattice gauge theory

Dimensionally Continued Euler Forms, {Kaluza-Klein} Cosmology and Dimensional Reduction

Discrete differential calculus graphs, topologies and gauge theory

Spatially homogeneous and isotropic spaces in theories of gravitation with torsion