PHYSICAL REVIEW D 100, 084007 (2019) D = 11 cosmologies with teleparallel structure † ‡ Christian G. Böhmer,1,* Franco Fiorini,2, P. A. González,3, and Yerko Vásquez 4,§ 1Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom 2Departamento de Ingeniería en Telecomunicaciones and Instituto Balseiro, Centro Atómico Bariloche (CONICET), Avenida Ezequiel Bustillo 9500, CP8400 S. C. de Bariloche, Río Negro, Argentina 3Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Avenida Ej´ercito Libertador 441, Casilla 298-V Santiago, Chile 4Departamento de Física y Astronomía, Facultad de Ciencias, Universidad de La Serena, Avenida Cisternas 1200, La Serena, Chile (Received 22 August 2019; published 4 October 2019) The presence of additional compact dimensions in cosmological models is studied in the context of modified teleparallel theories of gravity. We focus the analysis on eleven dimensional spacetimes, where the seven dimensional extra dimensions are compactified. In particular, and due to the importance that global vector fields play within the conceptual body of teleparallel modified gravity models, we consider the additional dimensions to be products of parallelizable spheres. The global vector fields characterizing the different topologies are obtained, as well as the equations of motion associated to them. Using global dynamical system techniques, we discuss some physical consequences arising because of the existence of the extra dimensions. In particular, the possibility of having an early inflationary epoch driven by the presence of extra dimensions without other matter sources is discussed. DOI: 10.1103/PhysRevD.100.084007 I. INTRODUCTION quantum theory of the gravitational field, it is well known that one of the candidates, string theory (M-theory), can be Despite being somewhat counterintuitive, the possible consistently constructed in spaces with extra six (seven) existence of extra spatial dimensions has a distinguished history in theoretical physics and can be considered a well- spatial dimensions [5,6]. established idea by now. Since their introduction in the At a purely gravitational level, the study of theories including (or formulated on) manifolds with additional early 1920s [1] as a tentative approach of unifying ’ electrodynamics and gravitation under a common geomet- spatial dimensions has been worked since Lovelock s rical context, there has been a considerable and growing expansion was discovered [7] (see also [8]). It emphasized interest in physical models involving more dimensions the fact that general relativity (GR) does not seem to be the than the three spatial dimensions and time which seem to most natural theory of gravity when the number D of govern our daily experience. These interests rapidly went spacetime dimensions is bigger than four. According to the far beyond the unifying purposes present in the original original philosophy surrounding GR, if we remain within models, for it was demonstrated that the inclusion of extra the metric description of gravity, Lovelock’s Lagrangian dimensions could solve several long standing problems is the only one assuring second order field equations in theoretical physics. For instance, it was argued that by which are automatically conserved. If D>4 the expansion extending the number of dimensions, two of the most necessarily contains higher order terms in the curvature; important hierarchy problems could find an elegant reso- for instance, in five and six spacetime dimensions, the lution; the Higgs mass hierarchy problem [2,3] and the Lagrangian density is not just the Hilbert-Einstein term, but problem of the cosmological constant [4]. Another area in it contains a specific quadratic combination of curvature which the existence of extra dimensions seems to play an terms given by the Gauss-Bonnet term. This quadratic piece important role is quantum gravity. Even though no general is “harmless” in D ¼ 4 in the sense that it is a topological consensus exists toward the formulation of a consistent invariant, the Euler density, which does not contribute to the field equations. However, when D>4 the Gauss- Bonnet term not only becomes dynamical, it also arises as *[email protected] † [email protected]. the curvature correction to the Einstein-Hilbert Lagrangian ‡ [email protected] coming from supersymmetric string theory [9]. This seems §[email protected] to indicate that the study of classical gravitation in extra 2470-0010=2019=100(8)=084007(15) 084007-1 © 2019 American Physical Society BÖHMER, FIORINI, GONZÁLEZ, and VÁSQUEZ PHYS. REV. D 100, 084007 (2019) dimensions is well motivated from a theoretical point with cases up to D ¼ 7 where the spatial extra dimensions M of view. in were products of spheres up to dimension three. Research of the previous decades has contrived a number In this paper, in turn, we extend the analysis by focusing ’ 11 M of modifications and extensions of Einstein s original on D ¼ , and considering in as a seven dimensional theory, and many are being studied extensively in a variety manifold constructed out of products of parallelizable of different contexts. Among the many theories developed, spheres. This restriction of considering only the paralleliz- the so called fðTÞ-gravity or modified teleparallel gravity able spheres simplifies the subsequent analysis as there are has attracted much attention in recent years. Originally only three such spheres S1, S3 and S7. On one hand, the [10], models of this type were proposed as a high energy number ns of different products of arbitrary spheres grows modification of GR in relation to the existence of strong rapidly as the number N of extra dimensions increases; we curvature singularities in cosmological models. It became actually have ns ¼ PðNÞ, the partition of N. In the general clear soon after that fðTÞ theories exhibit interesting case, this makes the problem hard to deal with for D ¼ 11, late time cosmological implications as well, as witnessed where one would have ns ¼ Pð7Þ¼15 distinct cases to in the study of missing matter problems, and concerning deal with. On the other hand, in view of the close the current acceleration stage of the universe without relationship between the 1-form fields EaðxÞ and the introducing any exotic matter content [11–13]. Recent parallelizations underlying a given manifold, the structure a M developments on fðTÞ-gravity concerning cosmological of E ðxÞ turns out to be easier to deal with when in is implications can be found in [14–18], for instance. itself a product of parallelizable submanifolds; the EaðxÞ M The subject of extra dimensions in fðTÞ gravity was simply inherit the product structure of in. scarcely presented in the literature [19–23], see also The paper is organized as follows: In Sec. II we present a Refs. [24,25]. The reason for this relative absence of concise account on fðTÞ gravity. Section III is devoted to the contributions in the area is clearly understood when the structure of EaðxÞ for the four different cases under con- structure of the fðTÞ field equations is considered more sideration arising by imposing the parallelizability condition M carefully. Unlike many of the modified gravity theories in on every member of the product in in. Albeit technical and vogue, fðTÞ gravity is formulated in Weitzenböck space- cumbersome, this section is crucial for finding the proper set time which is characterized by a set of 1-forms EaðxÞ¼ of field equations in every case, which can be seen as the a μ Eμdx producing torsion instead of curvature; the vielbein main contribution of this work. Some consequences of field EaðxÞ, which encodes the dynamics of the gravita- the field equations are obtained in Sec. IV, where a global tional field, determine the structure of the spacetime by dynamical system analysis is performed on one of the means of a parallelization process. This means, among cases exposed. More specifically Sec. III discusses the case M 7 other things, that the field equations are not locally Lorentz in ¼ S . Finally, we discuss our results in Sec. V.Various invariant, or at least they are not in the usual sense [26],an Appendices are required to present some of the lengthy issue that is not fully settled yet. Of course, fðTÞ modified equations and certain technical details which are not gravity includes GR as a limit when fðTÞ¼T, in which essential in the main body. Appendix B briefly discusses case one speaks of the teleparallel equivalent of GR. the remnant symmetries, which are related to the different Therefore the Lorentz covariance is fully restored in those parallelizations admitted by a given manifold. Appendix C M regimes or scales where the gravitational field is correctly contains a further example in which in is not constructed described by GR. However, near spacetime singularities, as a product of parallelizable spheres. for example, the structure of the fields EaðxÞ is fixed only Throughout the paper, we will adopt the signature a up to a certain subgroup of the Lorentz group which is ðþ; −; −; −; ÁÁÁ; −Þ, Latin indices a∶ð0Þ; ð1Þ; … in EμðxÞ characteristic of the spacetime under consideration. This refer to tangent-space objects while Greek μ∶0; 1; … μ poses an additional technical complication at the time of denote spacetime indices. Dual vector basis ea ¼ ea∂μ solving the field equations. The field equations are second a ν δν a μ δa are defined according to Eμea ¼ μ and Eμeb ¼ b. order partial differential equations which determine the full components of the vielbein EaðxÞ, and not only those a b II. BRIEF NOTES ON MODIFIED associated to the metric tensor gμν ¼ EμEν η . ab TELEPARALLEL GRAVITY In [27] we started the program of characterizing multi- dimensional cosmological models with fðTÞ structure by The extended gravitational schemes with absolute par- assuming that the compact extra dimensions consisted of allelism, often referred to as fðTÞ theories, take as a starting topological products of spheres.