D=11 Cosmologies with Teleparallel Structure

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; −; −; −; ; −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. We did so by considering point the teleparallel equivalent of general relativity M M cosmological manifolds of the form ¼ FLRW4 × in, (TEGR), see for instance [28,29]. We will summarize here where FLRW4 represents the four dimensional, spatially flat the basic elements needed to present the key ideas required Friedmann-Lemaître-Robertson-Walker (FLRW) spaces for the formulation of our work. For a thorough introduc- which are consistent with our present day understanding tion to fðTÞ gravity as well as its mathematical foundations of the large scale structure of the universe. In [27] we dealt the reader is referred to [30,31], for instance.

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μ The spirit underlying TEGR finds its motivation in the components eaðxÞ, and not just the subset related to the μν μ ν ηab equivalence between the Riemann and Weitzenböck for- metric tensor g ¼ eaeb . In other words, Eqs. (5) define mulations of GR, which can be summarized in the equation the spacetime structure by means of a parallelization by D μ non-null and smooth vector fields eaðxÞ. It is natural, then, −1 σν T ¼ −R þ 2e ∂νðeTσ Þ; ð1Þ that the vielbein grid so determined is sensitive to local pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi boosts and rotations Λ bðxÞ acting on it according to μ μ μ μ a e e gμν → Λ a where ¼ detð aÞ¼ detð Þ. On the left-hand side of ea eb ¼ b ea. These local Lorentz transformations Eq. (1) we have the so-called Weitzenböck invariant Λ bðxÞ have the effect of breaking the global structure a μ of eaðxÞ, turning the vielbein grid into an uncorrelated set ρ μν T ¼ S μνTρ ; ð2Þ of orthonormal bases at different points of the tangent space T ðMÞ, however leaving the metric invariant. It is impor- μν p where Tρ are the local (spacetime) components of the tant to mention that the breaking of the Lorentz covariance a a 1 a μ ν 00 torsion two-form T ¼ de ¼ 2T μνdx ∧ dx coming from is relevant at scales where f ðTÞ is considerably different λ λ a → the Weitzenböck connection Γνμ ¼ ea∂νeμ, and Sρλμ is from zero, for otherwise fðTÞ T and the full Lorentz defined according to group is restored through TEGR. Scales where jf00ðTÞj ≫ 1 are relevant concerning the manifestation of new degrees 1 1 1 ρ ρ − ρ ρ δρ σ − δρ σ of freedom [31,34]. S μν ¼ 4 ðT μν Tμν þ Tνμ Þþ2 μTσν 2 νTσμ : ð3Þ On purely mathematical terms, there exist equivalence classes of vielbein grids, because parallelizations (if they Actually, T is the result of a very specific quadratic exist) are nonunique. In the context of fðTÞ theories this combination of irreducible components of the torsion symmetry is realized by means of the remnant group of tensor under the Lorentz group SOð1; 3Þ, see [32]. Lorentz transformations of a given spacetime [35], about Equation (1) simply says that the Weitzenböck invariant which we shall briefly comment in Appendix B.Onan T differs from the scalar curvature R by a total derivative; operational level, in turn, the restricted (remnant) group therefore, both conceptual frameworks are totally equiv- usually is not enough to impose symmetries on the vielbein alent at the time of describing the dynamics of the if one only knows the symmetries of the metric tensor; this gravitational field. This also implies that T is the unique makes it complicated to anticipate the structure of the fields μ combination of quadratic torsion terms which is locally eaðxÞ by having only certain knowledge on the structure μ Lorentz invariant up to a surface term. of gμν, for infinitely many eaðxÞ’s correspond to the same μ In many ways fðTÞ gravity can be viewed as a natural metric gμν. In GR (or TEGR), the infinite set of eaðxÞ is not extension of Einstein gravity in its teleparallel form, similar a problematic issue because every pair of vielbeins is to fðRÞ gravity in the more standard metric formulation. connected by a local Lorentz transformations which are It is governed by the following action in D spacetime part of the symmetry of the theory. In stark contrast, within dimensions fðTÞ gravity, it is precisely the local orientation of the Z vielbein which becomes important. A simple discussion on 1 D the importance of parallelizability in fðTÞ gravity can be S ¼ ½fðTÞþLmattered x: ð4Þ 16πG consulted in Ref. [33]. We are now ready to construct μ suitable fields eaðxÞ for a number of different cosmological Of course GR is contained in (4) as the particular case manifolds in eleven dimensions. when fðTÞ¼T and D ¼ 4. The dynamical equations of f T ð Þ gravity theories are obtained after varying the action III. EXTRA DIMENSIONS GIVEN BY THE (4) with respect to the vielbein components. The matter TOPOLOGICAL PRODUCT OF fields couple to the metric in the usual way so that the field PARALLELIZABLE SPHERES equations become A. General considerations −1∂ μν λ ρ μν 0 μν∂ 00 ðe μðeSa ÞþeaT μλSρ Þf ðTÞþSa μðTÞf ðTÞ Let us now discuss the general structure of the vielbein 1 field when the D dimensional manifold is given by the − ν −4π λ ν 4 eafðTÞ¼ GeaTλ ; ð5Þ topological product of parallelizable submanifolds. This is a very subtle point in view of the lack of local Lorentz ν where the prime means derivative with respect to T and Tλ invariance of fðTÞ theories of gravity, a point which is is the energy-momentum tensor. particularly relevant in the strong field regime. In this As pointed out several times (see [33] for a recent section we will obtain the proper parallel one-form fields of discussion), the Eqs. (5) in the general case when the manifolds in consideration, postponing the discussion f00ðTÞ ≠ 0 are sensitive to the local orientation of the regarding their uniqueness until Appendix B. With the vielbein. This is because they determine the entire set of frame fields so obtained, we will proceed to compute the

084007-3 BÖHMER, FIORINI, GONZÁLEZ, and VÁSQUEZ PHYS. REV. D 100, 084007 (2019) field equations for the specific cases in which the seven where Tj ¼ S1 × … × S1 is the j–torus. In this way, the full extra dimensions are given by topological products of spacetime vielbein will have the structure parallelizable spheres. 0 1 In what follows, we are interested in a cosmological 00 0 B C setting where the four-dimensional space is described by B 0 00 O C B a0ðtÞ C the flat FLRW manifold with local pseudo Euclidean B C 1 2 3 B 00a0ðtÞ 0 C coordinates ðt; x; y; zÞ¼ðt; xnÞ with n ¼ ; ; . The mani- B C a B 00 0 C fold structure in D dimensions is chosen to be Eμ ¼ B a0ðtÞ C; B C B C M ¼ M × M ; ð6Þ B C D FLRW in B O M C @ Eið inÞ A M 4 4 where FLRW ¼ R with a frame EðR Þ whose components are given by ð11Þ t 4 n 4 E ðR Þ¼dt; E ðR Þ¼a0ðtÞdxn; ð7Þ M where Eið inÞ formally refers to any of the fields (10).In turn, the full space-time metric will be given by leading to the line element 2 2 − 2 2 2 2 − 2 2 2 − 2 2 2 2 ds ¼ dt a0ðtÞðdx1 þ dx2 þ dx3Þ dsin; ð12Þ dsFLRW ¼ dt a0ðtÞðdx þ dy þ dz Þ: ð8Þ 2 where dsin is the line element corresponding to the internal It is worth mentioning that the frame defined in (7) is by dimensions given by any of the four possible forms (10). no means simply a choice. Even though many other proper M In order to apply these models to a cosmological setting, tetrads exist for the description of FLRW, related to (7) we will assume a perfect fluid with energy density ρðtÞ through transformations of the remnant group, the autopar- and pressure pðtÞ as the only matter source in the field allel curves of flat Euclidean space are given by straight equations. This means we have ∂ lines which can be generated by the coordinate basis xn , μν μ ν μν whose dual cobasis is dxn. The Euclidean grid so obtained T ¼ðρ þ pÞV V þ pg ; ð13Þ is unaltered by the conformal scale factor, which only depends on time, turning (7) into the simplest and most where Vμ is the tangent vector to the congruence of curves M transparent vierbein field describing FLRW. defining the stream lines of the fluid. In the comoving Next, it will be assumed that the topological structure frame of the fluid, the energy-momentum tensor takes the M of in is simple form X μ M j1 j2 … jk − 4 in ¼ S × S × × S ; jk ¼ D ; ð9Þ Tν ¼ diagðρ; −p0; −p0; −p0; −p1; …; −pD−4Þ: ð14Þ k We proceed now to characterize the global one-form fields jk M where S is the jk-sphere. In general in is not a of any of the internal manifolds mentioned in (10), and to parallelizable manifold, but it turns out to be if at least obtain the field equations which are implied by them. one of the jk is odd [36] and, if this would be the case, it 7 was shown that explicit parallelizations can be found B. Min = T [37,38]. However, it is clear that even if (at least) one Let us begin with probably the simplest case, since T7 ¼ the jk is odd, the structure of the one-forms associated to a 1 1 M S × … × S is just the topological product of trivially parallelization of in, does not inherit the product structure of the space, because just three spheres are paralleliz- parallelizable manifolds. In the previous article [27] we 1 3 7 have analyzed the structure of the vielbein field and the able by themselves, recall that these are S ;S ;S , see [39]. relevant equations of motion for the general case given by If we fix D ¼ 11 and we work only with the three Tj, so we shall revisit the main results here and focus on T7. parallelizable spheres, we have four possible structures j M If we consider coordinates Xj on S , it becomes trivial to concerning the parallel one-form fields of in, namely parallelize the full spacetime T7 by means of the vielbein M 7 field (no summation in j) E1ð inÞ¼EðT Þ; M 4 3 j 7 E2ð inÞ¼EðT Þ × EðS Þ; E ðT Þ¼ajðtÞdXj; 1 ≤ j ≤ 7; ð15Þ 1 3 3 E3ðM Þ¼EðS Þ × EðS Þ × EðS Þ; in where a is the time dependent scale factor corresponding M 7 j E4ð inÞ¼EðS Þ; ð10Þ to each of the spheres Sj. The Weitzenböck invariant T

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4 7 3 associated to the entire manifold M11 ¼ R × T [by must now focus on the 3-sphere part of the geometry EðS Þ. means of (7) and (15)] is given by The fact that S3 has a maximum number of global, non-null vector fields, is a consequence of the fact that any three- X7 1 X7 X7 dimensional orientable manifold is parallelizable [40].An −6 2 − 2 T ¼ H0 þ H0 Hj þ 6 HiHj Hi ; explicit parallelization is, however, not that trivial to find. 1 1 1 j¼ i;j¼ i¼ The usual way is to view S3 as the unit quaternions, and ð16Þ then, to realize that it has a non-Abelian Lie group structure induced by quaternion multiplication, which in turn, _ _ 3 where we used H0 ¼ a0=a0 and Hj ¼ aj=aj for the induces a right translation on S . In this way a paralleliza- corresponding Hubble functions. The dynamics of the tion can be obtained by applying the right translation to a various scale factors is determined by the Eqs. (5) which basis of vectors at the unit element of the group. The in the present case give the Hubble constraint equation associated one-forms fields are obtained by means of the 4 standard inner product on R . If coordinates Xj 5 ≤ j ≤ 8 − 2 0 16π ρ f Tf ¼ G : ð17Þ are set up in R4, a globally defined basis on T ðS3Þ (up to a time dependent conformal factor) is Next, we have the three (identical) spatial equations coming from the standard FLRW part 5 3 E ðS Þ¼X6dX5 − X5dX6 − X8dX7 þ X7dX8; X7 2 X7 6 3 E S X8dX5 − X7dX6 X6dX7 − X5dX8; 0 ð Þ¼ þ 2f 2H0 H H0 2H0 H þ n þ þ n 7 3 n¼1 n¼1 E ðS Þ¼−X7dX5 − X8dX6 þ X5dX7 þ X6dX8: ð20Þ X7 X7 _ _ 00 _ þ 2H0 þ Hn þ 2f T 2H0 þ Hn After changing coordinates by means of (A1), we can write n¼1 n¼1 the line element of the internal dimensions as þ f ¼ −16πGp0: ð18Þ X4 2 2 2 2 Ω2 Finally there are seven additional equations which are dsin ¼ aj ðtÞdXj þ a5ðtÞd 3; ð21Þ j¼1 X7 2 X7 2 0 3 3 2 f H0 þ Hn þ Hj H0 þ Hn where dΩ3 is the line element of the three-sphere with n 1;n≠j n 1;n≠j ¼ ¼ θ1; θ2; ϕ coordinates ð Þ X7 X7 _ _ 00 _ þ 3H0 þ H þ 2f T 3H0 þ H 2 2 2 2 2 2 2 n n dΩ3 ¼ dθ1 þ sin θ1dθ2 þ sin θ1sin θ2dϕ : ð22Þ n¼1;n≠j n¼1;n≠j −16π þ f ¼ Gpj: ð19Þ Using the frame fields set up in this way, we can write the torsion scalar in the following way As in standard cosmology, Eq. (17) coming from the temporal coordinate is the constraint, which plays the role 6 X4 of a modified Friedmann equation. Note that (19) contains − 6 2 2 − 18 T ¼ 2 ðH0 þ H5Þ H5 Hi seven different equations which correspond to the seven a5 i¼0 μ different pressures pj appearing in Tν. There is no a priori X4 X4 X4 reason to assume these different pressures to be the same. þð12H5 − 4H0Þ Hi − 2 Hi Hj : This simple observation motivates the study of spaces i¼1 i¼0 j¼iþ1 made up of products of higher dimensional spheres which ð23Þ introduces few scale factors, in the case of S7 one will only introduce one additional scale factor. Note that the spacetime is characterized by six different 4 3 scale factors, one scale factor of the FLRW part, one scale C. Min = T × S factor for any of the four 1-spheres, and finally one Next we consider the case where the parallel one-forms associated with the 3-sphere. Consequently, field equations M 4 3 have the topological structure Eð inÞ¼EðT Þ × EðS Þ.If are rather involved and somewhat complicated. The final 4 the coordinates on T are Xj, we simply have, as in (15), result of those field equations begins with the Hubble j 4 j that E ðT Þ¼ajðtÞdX , with 1 ≤ j ≤ 4. This means we constraint equation

084007-5 BÖHMER, FIORINI, GONZÁLEZ, and VÁSQUEZ PHYS. REV. D 100, 084007 (2019) 6 0 2 2 f þf þ6H þ6ðH1 þH2 þH3 þH4 þ3H5ÞH0 þ2ð3H þ3ðH3 þH4ÞH5 þH3H4 a2 0 5 5

þH2ðH3 þH4 þ3H5ÞþH1ðH2 þH3 þH4 þ3H5ÞÞ−T ¼ 16πGρ: ð24Þ

Next comes the equation for the FLRW pressure term 6 0 6 2 4 3 4 _ 2 _ 2 _ 2 _ 2 _ 6 _ f þ f 2 þ H0 þ ðH1 þ H2 þ H3 þ H4 þ H5ÞH0 þ H0 þ H1 þ H2 þ H3 þ H4 þ H5 a5 2 2 2 2 2 þ 2ðH þðH2 þ H3 þ H4 þ 3H5ÞH1 þ H þ H þ H þ 6H þ H3H4 þ 3ðH3 þ H4ÞH5 1 2 3 4 5 _ 00 þ H2ðH3 þ H4 þ 3H5ÞÞ − T þ 2ð2H0 þ H1 þ H2 þ H3 þ H4 þ 3H5ÞTf ¼ −16πGp0: ð25Þ

The next four equations can be conveniently written in the form 6 X4 X4 X4 0 6 _ 2 _ 6 _ 2 6 2 3 9 2 6 2 f þ f 2 þ H0 þ Hi þ H5 þ ð H0 þ HiH0 þ H5H0 þ Hi þ H5 a5 n¼1;n≠j n¼1;n≠j n¼1;n≠j X4 X3 X4 X4 _ 00 þ 3 HiH5 þ HmHnÞ − T þ 2 3H0 þ Hi þ 3H5 Tf ¼ −16πGpj; ð26Þ n¼1;n≠j m¼1;m≠j n¼mþ1;n≠j n¼1;n≠j where j ¼ 1; 2; 3; 4. The final equation involving the pressure p5 is given by 2 0 12 2 6 2 6 _ 2 _ 2 _ 2 _ 2 _ 4 _ f þ f 2 þ H0 þ ðH1 þ H2 þ H3 þ H4 þ H5ÞH0 þ H0 þ H1 þ H2 þ H3 þ H4 þ H5 a5 2 2 2 2 2 þ 2ðH þðH2 þ H3 þ H4 þ 2H5ÞH1 þ H þ H þ H þ 3H þ H3H4 þ 2ðH3 þ H4ÞH5 1 2 3 4 5 _ 00 þ H2ðH3 þ H4 þ 2H5ÞÞ − T þ 2ð3H0 þ H1 þ H2 þ H3 þ H4 þ 2H5ÞTf ¼ −16πGp5: ð27Þ

1 3 3 D. Min = S × S × S One must take note of the common coordinate (here X5). 3 This is necessary in order to embed the six-dimensional Once a parallelization for S is obtained, as described in 7 the previous case, the characterization of S1 × S3 × S3 manifold in R . The internal metric in coordinates 1 X1; θ1; θ2; ϕ1; θ3; θ4; ϕ2 give proceeds straightforwardly. Let X be a coordinate on ð Þ T 1 the circle, it follows that a global basis for ðS Þ is given 2 2 2 2 2 2 2 1 1 1 ds ¼ a1ðtÞdX1 þ a2ðtÞdΩ þ a3ðtÞdΩ ; ð30Þ by E ðS Þ¼a1ðtÞdX . The parallelization of the remaining in 3;ð1Þ 3;ð2Þ six dimensional manifold S3 × S3 is obtained by means of two copies of the fields given in (20). Up to time dependent where we used the notation conformal factors in the corresponding 3-spheres, we have Ω2 θ2 2θ θ2 2θ 2θ ϕ2 d 3; 1 ¼ d 1 þ sin 1d 2 þ sin 1sin 2d 1; ð31Þ 2 3 ð Þ E ðS Þ¼X3dX2 − X2dX3 − X4dX3 þ X3dX4; 3 3 − − Ω2 θ2 2θ θ2 2θ 2θ ϕ2 E ðS Þ¼X5dX2 X4dX3 þ X3dX4 X2dX5; d 3;ð2Þ ¼ d 3 þ sin 3d 4 þ sin 3sin 4d 2: ð32Þ 4 3 E ðS Þ¼−X4dX2 − X5dX3 þ X2dX4 þ X3dX5; ð28Þ With the fields so obtained, we can compute the for the first 3-sphere and accordingly for the second one Weitzenböck invariant

5 3 2 2 2 −2 −2 E ðS Þ¼X6dX5 − X5dX6 − X8dX7 þ X7dX8; T ¼ −6ðH0 þ H2 þ H3 − a2 − a3 6 3 E ðS Þ¼X8dX5 − X7dX6 þ X6dX7 − X5dX8; þ H1ðH0 þ H2 þ H3Þþ3ðH0H2 þ H0H3 þ H2H3ÞÞ: 7 3 E ðS Þ¼−X7dX5 − X8dX6 þ X5dX7 þ X6dX8: ð29Þ ð33Þ

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The resulting field equations in this case begin with the Hubble constraint equation

0 −2 −2 f þ 2f ð6a2 þ 6a3 − TÞ¼16πGρ: ð34Þ

This is followed by the four evolution equations

0 _ _ _ _ f þ 2f ð2H0 þ H1 þ 3H2 þ 3H3 þð2H0 þ H1 þ 3ðH2 þ H3ÞÞð3H0 þ H1 þ 3ðH2 þ H3ÞÞÞ 00 _ þ 2f Tð2H0 þ H1 þ 3ðH2 þ H3ÞÞ ¼ −16πGp0; ð35Þ

0 _ _ _ 00 _ f þ 6f ðH0 þ H2 þ H3 þðH0 þ H2 þ H3Þð3H0 þ H1 þ 3ðH2 þ H3ÞÞÞ þ 6f TðH0 þ H2 þ H3Þ¼−16πGp1; ð36Þ

0 _ _ _ _ −2 f þ 2f ð3H0 þ H1 þ 2H2 þ 3H3 þð3H0 þ H1 þ 2H2 þ 3H3Þð3H0 þ H1 þ 3ðH2 þ H3ÞÞ − 2a2 Þ 00 _ þ 2f Tð3H0 þ H1 þ 2H2 þ 3H3Þ¼−16πGp2; ð37Þ

0 _ _ _ _ −2 f þ 2f ð3H0 þ H1 þ 3H2 þ 2H3 þð3H0 þ H1 þ 3H2 þ 2H3Þð3H0 þ H1 þ 3ðH2 þ H3ÞÞ − 2a3 Þ 00 _ þ 2f Tð3H0 þ H1 þ 3H2 þ 2H3Þ¼−16πGp3: ð38Þ

The entire set of field equations can be viewed as four dynamical equations for the four Hubble functions Hi, i ¼ 0; 1; 2; 3 subject to the constraint (34). For given fðTÞ one could attempt a dynamical systems formulation in order to understand the dynamics of such a cosmological model, this is what will be done with the final case which is studied next.

7 E. Min = S It has been known for a long time that S7 is parallelizable, it is nonetheless surprising that explicit expressions for global bases of vector fields in T ðS7Þ (or one-forms in T ðS7Þ), are rarely found in the literature (see, e.g., [41]). The procedure in order to obtain a parallelization is to view S7 as the unit octonion, and to use their multiplication rule to obtain right invariant vector fields. Due to the fact that multiplication of unit octonions is not associative, S7 is not a Lie group, however this is not 7 8 an impediment in getting a global basis of vector fields on S . Let us choose coordinates Xi, i ¼ 1; …; 8 in R , a global basis of one-forms in T ðS7Þ can be written explicitly (up to a time dependent conformal factor) as follows

1 7 E ðS Þ¼−X2dX1 þ X1dX2 − X4dX3 þ X3dX4 − X6dX5 þ X5dX6 − X8dX7 þ X7dX8; 2 7 E ðS Þ¼−X3dX1 þ X4dX2 þ X1dX3 − X2dX4 − X7dX5 þ X8dX6 þ X5dX7 − X6dX8; 3 7 E ðS Þ¼−X4dX1 − X3dX2 þ X2dX3 þ X1dX4 þ X8dX5 þ X7dX6 − X6dX7 − X5dX8; 4 7 E ðS Þ¼−X5dX1 þ X6dX2 þ X7dX3 − X8dX4 þ X1dX5 − X2dX6 − X3dX7 þ X4dX8; 5 7 E ðS Þ¼−X6dX1 − X5dX2 − X8dX3 − X7dX4 þ X2dX5 þ X1dX6 þ X4dX7 þ X3dX8; 6 7 E ðS Þ¼−X7dX1 þ X8dX2 − X5dX3 þ X6dX4 þ X3dX5 − X4dX6 þ X1dX7 − X2dX8; 7 7 E ðS Þ¼−X8dX1 − X7dX2 þ X6dX3 þ X5dX4 − X4dX5 − X3dX6 þ X2dX7 þ X1dX8: ð39Þ

7 After changing to hyperspherical coordinates ðθ1; …; θ6; ϕÞ the line element in S becomes

2 2 Ω2 dsin ¼ a1ðtÞd 7; ð40Þ where the line element of the 7-sphere is

Y6 2 2 2 2 2 2 dΩ7 ¼ dθ1 þ sin θ1dθ2 þþdϕ sin θi: ð41Þ i¼1

The torsion scalar T reads

2 2 −2 T ¼ −6ðH0 þ 7H0H1 þ 7H1 − 7a1 Þ: ð42Þ

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Finally we can state the complete set of field equations. Due 0 2 _ 00 2 _ f þ 4f ð3H þ H0Þ − 48f H H0 ¼ −16πGp0; ð48Þ to the appearance of only one additional scale factor in this 0 0 model, we expect somewhat simpler equations than in the 6 0 3 2 _ − 2 −2 − 36 00 2 _ −16π previous case. This turns out to be the case as can be seen in f þ f ð H0 þ H0 a1 Þ f H0H0 ¼ Gp1: the following. The temporal field equation simply becomes ð49Þ

0 −2 f þ 2f ð42a1 − TÞ¼16πGρ; ð43Þ Equations (47) and (48) have the same structure as the cosmological field equations of a spatially flat FLRW while the two dynamical equations are given by cosmology in four-dimensional fðTÞ gravity [11]. The only difference comes from the Weitzenbock scalar which 0 2 2 _ _ f þ 2f ð6H0 þ 35H0H1 þ 49H1 þ 2H0 þ 7H1Þ includes the constant scale factor of the extra dimensions. 00 _ Hereafter, we will refer to Eqs. (47) and (48) as the 4D þ 2f Tð2H0 þ 7H1Þ¼−16πGp0; ð44Þ reduced fðTÞ equations. 0 2 2 −2 _ _ The role of the constant a1 in the expression for T can f þ 6f ð3H þ 13H0H1 þ 14H − 2a þ H0 þ 2H1Þ 0 1 1 be easily appreciated if we consider the GR case. Taking 00 _ 0 00 þ 3f TðH0 þ 2H1Þ¼−16πGp1: ð45Þ f ¼ T;f ¼ 1 and f ¼ 0 in the system (47)–(49) we arrive at Compared with the previously discussed cases where the 8 extra dimensions contain various products of spheres, the 2 −2 H þ 7a ¼ πGρ; ð50Þ field equations for S7 are much simpler to deal. In what 0 1 3 follows we will discuss those equations in some detail and 2 8 2 _ −2 show that they contain many desirable features when H0 þ H0 þ 7a1 ¼ − πGp0; ð51Þ considering applications to realistic cosmological models. 3 3 8 2 2 _ 5 −2 − π IV. SOME CONSEQUENCES H0 þ H0 þ a1 ¼ 3 Gp1: ð52Þ OF THE FIELD EQUATIONS A. Early inflation powered by extra dimensions Despite the absence of the cosmological constant in the original action, the effect of the constant scale factor of the Let us assume for the moment that the scale factors extra dimensions is to generate a negative “effective” corresponding to the internal dimensions are constant, −2 cosmological constant given by Λ ¼ −21a1 . The sign meaning that all Hubble functions other than H0 vanish of Λ is fixed to be negative which comes from the identically. Presumably, this could represent a good 2 expression of the Weitzenbock scalar (46). There H and approximation to the final stages of the evolution where 0 a−2 enter with different signs. the additional dimensions no longer affect the universe, 1 In the general setting where the function fðTÞ is arbitrary, which is then governed solely by the scale factor a0ðtÞ of we can solve Eqs. (47) and (48) for a given matter source the four dimensional, spatially flat FLRW metric. In this case, it is not hard to see that the full system of equations (5) content, and then obtain the pressure of the internal space by decouples into two sets of equations which are very means of (49). Due to the automatic conservation of Tμν in different in structure. One of the sets correspond to the the external space, the two equations (47) and (48) are not usual fðTÞ equations associated with a four dimensional, independent. Consequently it is enough to solve (47) with −3ð1þωÞ spatially flat, FLRW cosmological model. The equations ρðtÞ given by ρðtÞ¼ρ0ða0ðtÞÞ . This factorization is within this set completely determine the scale factor a0ðtÞ possible primarily because a1 is assumed to be constant, and of the physical macroscopic dimensions. The other set of therefore it is absent in the conservation equation of the fluid. equations, on the other hand, consists of algebraic equa- In general, we have tions relating the pressures of the internal dimensions. Let XD−4 XD−4 us henceforth look into this in some detail. 7 ρ_ þ 3H0 þ Hn ρ þ 3H0p0 þ Hnpn ¼ 0; ð53Þ If we start with the S case, the imposition of constant a1 n¼1 n¼1 lead us to torsion invariant (42) which together with its time derivative read which shows, in the present situation, that the pressure p1 obtained by means of Eq. (49), has no effect whatsoever −6 2 − 7 −2 _ −12 _ T ¼ ðH0 a1 Þ; T ¼ H0H0: ð46Þ on the energy density of the 4-dimensional Universe we experience. This is so because H1 vanishes. In turn, the equations of motion (43)–(45) can be written as The above discussion extends nicely to the other topologies we considered. In the T7 case, Eqs. (17)–(19) 0 2 f þ 12f H0 ¼ 16πGρ; ð47Þ for constant scale factors ai i ¼ 1; …; 7 become the 4D

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2 reduced fðTÞ equations, together with the one additional α to the (squared) length scale a1 characterizing the size of equation the extra dimensions during the inflationary era. In other words small extra dimensions give rise to large inflation. 0 2 _ 00 2 _ f þ6f ð3H0 þH0Þ−36f H0H0 ¼ −16πGpj;j¼ 1;…;7: However, not every topology enables us to describe extra dimensions-powered inflation. Note that the 4D ð54Þ reduced fðTÞ equations (47) and (48) coalesce to the 0 2 4 3 single equation f þ 12f H0 ¼ 0, in vacuum and for con- In case of topology T × S the corresponding algebraic 7 relations are stant H0. Then, the T topology leads to the inconsistent (for a non-null H0) system 6 0 3 2 _ −72 00 2 _ −16π 1 … 4 f þ f ð H0 þH0Þ f H0H0 ¼ Gpj;j¼ ; ; : 0 2 0 2 f þ 12f H0 ¼ 0;fþ 18f H0 ¼ 0: ð61Þ ð55Þ Similarly the T4 × S3 case shows the same inconsistency by 0 2 _ −2 00 2 _ means of the equations f þ 2f ð9H0 þ 3H0 − 2a5 Þ − 72f H0H0 ¼ −16πGp5: 0 2 0 2 ð56Þ f þ 12f H0 ¼ 0;fþ 18f H0 ¼ 0; 2 −2 1 3 3 f 2f0 9H − 2a 0: The case S × S × S follows the similar lines. þ ð 0 5 Þ¼ ð62Þ Equations (34)–(38) show that, assuming the internal scale Exactly the same happens when S1 × S3 × S3 is considered. factors a1;a2 and a3 to be constants, the system leads to the The vacuum field equations turn into 4D reduced fðTÞ equations plus 12 0 2 0 18 0 2 0 0 2 _ 00 2 _ f þ f H0 ¼ ;fþ f H0 ¼ ; f þ 6f ð3H0 þ H0Þ − 72f H0H0 ¼ −16πGp1; ð57Þ 2 0 9 2 − 2 −2 0 f þ f ð H0 aj Þ¼ ; ð63Þ 2 0 9 2 3 _ − 2 −2 − 72 00 2 _ f þ f ð H0 þ H0 aj Þ f H0H0 with j ¼ 2; 3. These results seems to suggest that the −16π 2 3 ¼ Gpj;j¼ ; : ð58Þ 7-sphere S7 is clearly favored on physical grounds, at least, regarding the interpretation of the early inflationary era as It is interesting to note that, in absence of any matter, an effect produced by the presence of the extra dimensions. fðTÞ gravity is sometimes able to describe an early time de This motivates us to have a closer look at the dynamical Sitter accelerated stage for the macroscopic 4-spacetime equations of that case. which is caused by the presence of extra dimensions. As is clear from Eqs. (50)–(52), in GR this cannot be achieved. 2 −2 B. Dynamical systems analysis Equations (50) and (51) give H0 ¼ −7a1 , which is not only nonphysical, but also inconsistent with (52). Let us now have a closer look at the equations of motion 2 In vacuum, Eqs. (47)–(49) considering the S7 case with for the aforementioned model fðTÞ¼T þ αT which constant H0, reduce to the simple equations appears to have some desirable properties regarding the inflationary era. As in the above discussion, we consider 0 2 0 2 −2 the vacuum equations, this means setting ρ p f þ 12f H0 ¼ 0;fþ 6f ð3H0 − 2a1 Þ¼0: ð59Þ ¼ a ¼ pb ¼ 0. Equations (42)–(45) do contain first time deriva- −2 2 tives of the Hubble parameters H0 and H1 but do not Combining them we get a1 ¼ H0=2, which is valid for any fðTÞ model other than GR. However, the value contain second derivatives. Therefore, we can, in principle, reformulate these field equations as a first order system of of H0 (or a1) depends on the function fðTÞ. For instance, let us consider an ultraviolet deformation of the form autonomous equations of the form 2 −2 2 fðTÞ¼T þ αT . Due to the fact that a1 ¼ H0=2 we have 2 −2 dH0 dH1 that T ¼ 15H ¼ 30a [see Eq. (46)]. This means that ¼ AðH0;H1Þ; ¼ BðH0;H1Þ; ð64Þ 0 1 dt dt Eqs. (59) relate the constant α to the inflationary Hubble rate by means of where A and B are two rather complicated functions which can be stated explicitly and are given in Appendix D, see 3 2 α − −3 a1 Eq. (D1) and (D2). In the following we will set α ¼ −0.1 ¼ 2 ¼ : ð60Þ 65H0 130 since the equations are somewhat too cumbersome to deal with generically. Consequently, α should be negative and very small, by Following the standard procedure of cosmological virtue of the fact that H0 is large during inflation. This dynamical systems, see for instance [42], we begin by simple model enables to link the typical deformation scale looking for the critical points of this systems. These are the

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TABLE I. Critical points and eigenvalues of the model fðTÞ¼ T þ αT2 with α ¼ −0.1.

Point H0 value H1 value Eigenvalues λ Properties 0 719 −0 571 1 093 2 517 Aþ þ . . . . i Unstable spiral A− −0.719 þ0.571 −1.093 2.517i Stable spiral 0 679 −3 309; −1 157 Bþ þ . 0 . . Stable node B− −0.679 0 3.309; 1.157 Unstable node points where the system is in equilibrium and are defined by the vanishing of the right-hand sides of (64). The critical points can be found numerically and the four points and their properties are summarized in Table I. The equilibrium points A in Fig. 1 correspond to cosmological solutions where one of the two scale factors expands while the other one contracts. The stable spiral point A− corresponds to an ever contracting universe 7 H0 < 0 in which the extra space S expands. From a physical point of view this equilibrium point is somewhat undesirable. On the other hand, point Aþ describes an 7 expanding universe H0 > 0 (with contracting S ) which is FIG. 1. Phase portrait near the critical points. Quadratic model 2 unstable. Such a state is suitable for an early time infla- fðTÞ¼T þ αT with α ¼ −0.1. tionary model as the universe would grow rapidly but then change its behavior. Recall that unstable points can be Before proceeding we note that the system (64) is interpreted as early time attractors. invariant under the transformation t ↦ −t and H0;1 ↦ Points B are characterized by the condition that the −H0 1 which explains the observed symmetries of this 0 ; extra space becomes static H1 ¼ , we can either have an system. This will becomes particularly obvious when the expanding or a contracting universe. The value of H0 at α −0 1 global phase portrait is taken into account. these points is given in Eq. (60), with ¼ . . In order to show the global phase portrait of our Interestingly, the expanding solution which corresponds system, we follow the standard procedure, see e.g., [42], to Bþ is a stable node. The deceleration parameter for the of introducing the Poincaré sphere thereby compactifying −1 − _ 2 scale factor a0 can be expressed as q ¼ H0=H0 so for the entire phase space to the unit sphere. We introduce the _ all critical points we obtain qa ¼ −1, since H0 ¼ 0 by new variables definition. Consequently all critical points are candidates for either early time or late time accelerated expansion. H0 H1 X ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;Y¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð65Þ Going back to (60) using α −1=10 we can solve for 2 2 2 2 pffiffiffiffiffiffiffiffiffiffi ¼ 1 þ H0 þ H1 1 þ H0 þ H1 H0 and find H0 ¼ 6=13 which corresponds to the → ∞ values at the points B. Therefore, the critical point Bþ so that the region H0;H1 corresponds to the boun- corresponds to an early time inflationary state. We note dary of the unit circle. The critical points at infinity are that in this model trajectories are attracted to this state. The found by finding the roots of the function point Aþ would correspond to a dark energy dominated B − A 0 state. We also note that Aþ and B− are early time attractors Gmþ1 ¼ X mðX; YÞ Y mðX;YÞ¼ ; ð66Þ (unstable points) and it is becoming quite clear that the phase space shows an intricate structure. where Am; Bm stand for the highest order polynomial Therefore the global picture needs to be considered power in the right-hand sides of (64). However, this system before making detailed conclusions. Figure 1 suggests the is not polynomial in the variables so we expanded the existence of various critical points at infinity and it will turn system near infinity and extracted the leading order poly- out that we cannot identify trajectories in phase space nomial terms in that expansion. These are quadratic in the 2 which connect points Aþ and Bþ. variables, so m ¼ , and are given by

−15147 − 16894 sinð2θÞþ5245 sinð4θÞþ11920 cosð2θÞþ1979 cosð4θÞ A2 ¼ ; ð67Þ 8ð123 − 87 cosð2θÞþ231 sinðθÞ cosðθÞÞ

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−39345 − 43358 sinð2θÞþ14483 sinð4θÞþ32072 cosð2θÞþ4777 cosð4θÞ B2 ¼ : ð68Þ 84ð−82 − 77 sinð2θÞþ58 cosð2θÞÞ

It turns out that these terms are independent of the From a physical point of view it is important to parameter α which consequently only affects the local point out that all critical points at infinity are located critical points. Given that m ¼ 2 the function G3 of (66) in the second or fourth quadrant of the phase space. will have at most pairs of roots. Each root θnðn ¼ 1; 2; 3Þ This means all solutions approaching these points will comes with an associated root located at θn þ π.The always have one contracting and one expanding Hubble global phase portrait with critical points at infinity parameter, in the second quadrant we have H0 < 0 and including the previously discussed local critical points H1 > 0 while in the fourth quadrant we have H0 > 0 and 0 isgiveninFig.2. H1 < . Only the local critical points B are different in It appears that there are only two pairs of critical points the sense that the Hubble parameters H1 identically at infinity in Fig. 2. This has to do with the fact that two of vanishes, H1 ¼ 0. these pairs are very close to each other. The angular values This model clearly displays a very interesting dyna- θi;i¼ 1; 2; 3 which determine the locations of those mical behavior which contains various epochs where the points on the unit circle ðcosðθiÞ; sinðθiÞÞ are approxi- 4-dimensional part of the manifold expands, the right half mately θ1 ¼ 2.450; θ2 ¼ 2.962 and θ3 ¼ 2.971. As men- of the phase space shown in Fig. 2. During the expansion tioned above, these come with their associated pair located of the 4-dimensional part of the manifold, the extra at θi þ π. 7-dimensional part will eventually contract due to the This is not a numerical effect but a true feature of the location of the critical points at infinity. We can conclude dynamical system. This can be seen by showing a detailed that the extra dimensions, in general, affect the dynamics of phase portrait near these points, see Fig. 3. One sees that the 4-dimensional part of the manifold and that inflationary both critical points θ2 and θ3 are “close” to each other, epochs are naturally part of such systems. No matter however, they show distinct features. The lower point sources are required to drive the expansion (or contraction) attracts trajectories while the upper one repels them, a and consequently one could conclude that epochs of feature which is lost when the entire phase space is shown. expansion appear naturally in such models.

FIG. 3. Detailed global phase portrait near two nearby FIG. 2. Global phase portrait with critical points at infinity. critical points at infinity. Quadratic model fðTÞ¼T þ αT2 Quadratic model fðTÞ¼T þ αT2 with α ¼ −0.1. with α ¼ −0.1.

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V. CONCLUDING COMMENTS No. 11140674 (P. A. G.). Y. V. acknowledge support by the Dirección de Investigación y Desarrollo de la Modified teleparallel models of gravity were studied in Universidad de La Serena, Grant No. PR18142. C. G. B. eleven dimensions and applications to cosmology were and F. F. acknowledge support by The Royal Society, considered. The four-dimensional part of the manifold was International Exchanges 2017, Grant No. IEC/R2/ assumed to be the usual FLRW manifold with flat constant 170013. F. F. is member of Carrera del Investigador time hypersurfaces while the seven extra dimensions were 1 3 Científico (CONICET), and he is supported by assumed to be products of parallelizable spheres, that is S ;S CONICET and Instituto Balseiro. F. F. would like to thank and S7. Using these assumptions one is led to the following the hospitality at Universidad Diego Portales and at the four possible compactifications of the extra dimensions: Department of Mathematics at University College London, T7;T4 S3;S1 S3 S3 S7 × × × ,and . For each of these cases where this investigation was conceived and partially the corresponding structure of the 1-forms field was obtained. undertaken. P. A. G. acknowledge the hospitality of the These vielbeins constitute the starting point for any f T ð Þ Universidad de La Serena and Centro Atómico Bariloche cosmological model including the above mentioned com- where part of this work was done. pactifications, because they represent the basis responsible for the parallelization of the manifolds under consideration, i.e., they define the space-time structure. Endowed with this APPENDIX A: SPHERICAL COORDINATES important information, we obtained the f T cosmological ð Þ IN D DIMENSIONS equations for each of the cases in question. M In Sec. IV we analyzed the structure of the equations more In obtaining explicit parallel one-forms fields for in, closely by considering the possibility of having de Sitter-like it will be very convenient to introduce hyperspherical epochs in the four dimensional FLRW submanifold due to coordinates ðθ1;::θj−1; ϕÞ in Sj, which are related to the jþ1 the presence of the extra dimensions. In the absence of any cartesian coordinates Xj in the internal space R , accord- matter content, and fixing the Hubble factors of the extra ing to dimensions to be vanishing, we could show that not all topologies considered give rise to situations where the extra 8 > r cos θ1 if k ¼ 1 dimensions can drive a period of accelerated expansion. In > 7 <> θ Πk−1 θ 2 … − 1 fact, for the cases analyzed, S is clearly favored from a r cos k p¼1 sin p if k ¼ ; ;j theoretical point of view, in particular regarding the inter- Xk ¼ > ϕΠj−1 θ ðA1Þ > r sin p¼1 sin p if k ¼ j pretation of an early time inflationary era driven by the extra > dimensions. Due to the fact that the size of the extra : ϕΠj−1 θ 1 r cos p¼1 sin p if k ¼ j þ dimensions and the constant Hubble factor of the infla- −2 2 tionary stage satisfy a1 ¼ H0=2, it was shown that the dynamical equations favor the evolution towards the stable Now, by inverting these equations, we obtain the spherical coordinates in terms of Cartesian coordinates, i.e., node Bþ corresponding to large exponential growth given by the smallness of the 7-sphere S7. 2 2 2 It remains unclear why only the 7-sphere naturally leads r ¼ X1 þþXjþ1; to a vacuum inflationary stage, but it should be emphasized 0 1 that this property was anticipated at the end of Ref. [27].As B C θ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXk 1 … − 1 a matter of fact, something similar happens in D ¼ 7, k ¼ arccos@ P A;k¼ ; ;j ; jþ1 2 where the 3-sphere plays the role of powering vacuum i¼k Xi inflation there, see the mentioned reference. In order to 7 X prove that S is the sole topology driving inflation in ϕ ¼ arctan j ; ðA2Þ D ¼ 11, we need to consider the remaining eleven different Xjþ1 topological products of spheres possible. One of those, the case S3 × S2 × S2, is considered in the Appendix C, from In order to change coordinates, we will need the Jacobian of where it is easy to understand, again, that S7 appears to be the coordinate transformation in question. The derivatives the most natural choice. Continuing along those lines, we of the above expressions with respect to Xi are expect to develop the appropriate techniques in order to “ ” deal with the complete set of nontrivial internal space ∂r X parallelizations in the future. ¼ k ; ∂X r k 8 ACKNOWLEDGMENTS < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 PXkXl ∂θ − P δ − 1 if k≥l l jþ1 2− 2 kl jþ X2 ¼ Xi X i¼l i ðA3Þ : i¼l l This work was partially funded by Comisión Nacional ∂Xk de Ciencias y Tecnología through FONDECYT Grant 0 if k

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APPENDIX B: ON THE REMNANT GROUP OF for any of the 8–forms ϕ˜ which can be constructed by LORENTZ TRANSFORMATIONS AND THE wedge products of Ei. Solutions of the Eq. (B6) will UNIQUENESS OF THE VIELBEIN FIELD include, then, time dependent corresponding Lorentz gen- In this short section we discuss the remnant symmetries erators. For instance, if underlying fðTÞ gravity, and their impact on the set of ϕ˜ 1 ∧ … ∧ 8 parallelizations admissible for a given spacetime. Details ¼ E E ; ðB8Þ concerning the following exposition can be found in [35]. then a (time dependent) free rotation parameter σ910ðtÞ will In general, under a Lorentz transformation of the vielbein 910 0 0 σ ∝ Ea → Ea ¼ Λa Eb, the Weitzenböck invariant T in D solve Eq. (B6). This is so because d ðtÞ dt, and then b 0 ˜ 910 spacetime dimensions transform as dðE ∧ ϕÞ ∧ dσ ¼ 0. Of course, many more time dependent rotations are allowed; actually, the full time

0 −1 i1 i −2 bc a d dependent group of rotations about a certain axis is T → T ¼ T þ e dðϵ … E …E ðD Þ η Λ dΛ Þ; i1; ;iðD−2Þ;a;b d c contained in (B6). It is possible to show that certain ðB1Þ time-dependent Lorentz boost are also contained in AðEaÞ, see [35]. This infinite set of allowed 1-form fields, where the wedge product ∧ is understood. Note that T is a each of them connected by remnant symmetries, are d representative of the nonuniqueness of the parallelization scalar only under the global Lorentz group (dΛc ¼ 0). The remnant group AðEaÞ of a given spacetime process of the cosmological manifold under consideration. ðT ⋆M;EaðxÞÞ is defined as the subgroup of SOð1;D−1Þ T under which becomes a Lorentz scalar, i.e., by demanding 3 2 2 APPENDIX C: Min = S × S × S

i1 i −2 bc a d d ϵ … E …E ðD Þ η Λ dΛ 0: Here, as an example of “nontrivial” internal space ð i1; ;iðD−2Þ;a;b d cÞ¼ ðB2Þ parallelization, we proceed to show a case in which the If we consider infinitesimal Lorentz transformations product topology of the internal dimensions is not con- stituted by parallelizable spheres. As a consequence of the 2 1 nonparallelizability of S , the one-form fields have not the Λa δa σcd x M a O σ2 ; b ¼ b þ 2 ð Þð cdÞb þ ð Þ ðB3Þ block structure coming from the topological product, but instead, they will contain cross terms. Even though no 2 σcd −σdc − 1 2 global basis exist for S , it certainly exists for the three where ðxÞ¼ ðxÞ are the DðD Þ= parameters of 1 2 the transformations, and dimensional manifold S × S , which is orientable. Explicit global fields for T ðS1 × S2Þ are [43] a δaη − δaη ðMcdÞb ¼ c db d cb; ðB4Þ 1 1 2 E ðS × S Þ¼a0X3dx1 − a2ðX2dX1 − X1dX2Þ; 2 1 2 the term appearing in (B2) results E ðS × S Þ¼a0X2dx1 þ a2ðX3dX1 − X1dX3Þ; 3 1 2 − − 1 E ðS × S Þ¼a0X1dx1 a2ðX3dX2 X2dX3Þ; ðC1Þ ΛadΛd ≃− dσbdðM Þa ¼ η dσba: ðB5Þ d c 2 bd c cb 1 3 where the coordinates of S × R in which we are embed- 1 2 In this way, the condition (B2) becomes ding S × S , are ðx1;X1;X2;X3Þ. Although we have not an S1 in the internal space, the trick consists on using the

i1 i −2 ab periodic coordinate on it (here x1), to “rectify it,” and to ϵ … dðE ∧ … ∧ E ðD Þ Þ ∧ dσ ¼ 0: ðB6Þ i1; ;iðD−2Þ;a;b think about it as one of the coordinates of the external space. Clearly, this process involves the inclusion of the Recall that, in D spacetime dimensions, we have D − 1 boosts 4 1 spatial section of the FRW space, so the block structure of generators Kα ¼ M0α,and ðD − 1ÞðD − 2Þ rotations 2 the field is broken. In this way, a global basis for the entire − 1 ϵ βγ Jα ¼ 2 αβγM . eleven dimensional manifold M consists of two copies of Not much can be said about the solutions of Eq. (B6) in the sort (C1), one corresponding to S3 [see Eq. (20)], plus general. However, for the specific case under consideration, the temporal part and the remaining bulk dimension. the structure of the parallel vector fields allow us to briefly In spherical coordinates ðθ1; θ2; ϕ1; θ3; ϕ2; θ4; ϕ3Þ we explore some consequences of it. Regarding the vielbein have the internal metric components given in Eq. (11), we immediately note that, due to the fact that E0 ¼ dt,wehave 2 2 Ω2 2 Ω2 2 Ω2 dsin ¼ a1ðtÞ d 3 þ a2ðtÞd 2;ð1Þ þ a3ðtÞd 2;ð2Þ; ðC2Þ 0 ∧ ϕ˜ ∧ ϕ˜ dðE Þ¼dt d ; ðB7Þ where

084007-13 BÖHMER, FIORINI, GONZÁLEZ, and VÁSQUEZ PHYS. REV. D 100, 084007 (2019)

2 2 2 2 2 2 2 dΩ3 ¼ dθ1 þ sin θ1dθ2 þ sin θ1sin θ2dϕ1; ðC3Þ Ω2 θ2 2θ ϕ2 d 2;ð1Þ ¼ d 3 þ sin 3d 2; ðC4Þ Ω2 θ2 2θ ϕ2 d 2;ð2Þ ¼ d 4 þ sin 4d 3: ðC5Þ The invariant T is given by

2 2 2 2 −2 −2 −2 T ¼ −2ð9H1H0 þ 6H1H2 þ 6H1H3 þ 3H1 þ 6H0H2 þ 6H0H3 þ 3H0 þ 4H2H3 þ H2 þ H3 − 3a1 − a2 − a3 Þ: ðC6Þ

The fðTÞ field equations are given by:

0 −2 −2 −2 f þ 2f ð2a3 þ 2a2 þ 6a1 − TÞ¼16πGρ; ðC7Þ 3H1 3 3 3 4 00 _ 4 0 − _ 2 − f 2 þ H0 þ H2 þ H3 T þ f 2 H1H0 þ 2 H1 þ 2 H1 H0ðH2 þ H3Þ _ _ 2 _ 2 3 −2 −2 −2 − T −16π þ H0 þ H2 þ H2 þ H3 þ H3 þ a1 þ a2 þ a3 2 þ f ¼ Gp0: ðC8Þ 3H0 1 4 00 _ 4 0 − 3 2 2 _ f H1 þ 2 þ H2 þ H3 T þ f 2 H1ð H0 þ H2 þ H3ÞþH1 3 3 _ 2 _ 2 _ 2 2 −2 −2 −2 − T −16π þ 2 H0 þ 2 H0 þ H2 þ H2 þ H3 þ H3 þ a1 þ a2 þ a3 2 þ f ¼ Gp1: ðC9Þ 00 _ 0 _ 2 2f ð3H1 þ 3H0 þ H2 þ 2H3ÞT þ 2f −3H1H2 þ 3H1 þ 3H1 þ 3H0ðH0 − H2Þ 3 _ − 2 _ 2 _ 2 2 6 −2 −2 2 −2 − T −16π þ H0 H2H3 þ H2 þ H3 þ H3 þ a1 þ a2 þ a3 2 þ f ¼ Gp2: ðC10Þ 00 _ 0 _ 2 2f ð3H1 þ 3H0 þ 2H2 þ H3ÞT þ 2f −H3ð3H1 þ 3H0 þ 2H2Þþ3H1 þ 3H1 3 _ 3 2 2 _ 2 2 _ 6 −2 2 −2 −2 − T −16π þ H0 þ H0 þ H2 þ H2 þ H3 þ a1 þ a2 þ a3 2 þ f ¼ Gp3: ðC11Þ

APPENDIX D: EXPLICIT FORMS OF FUNCTIONS A AND B

_ 2 00 0 H0 ¼½b fð21ðH0 þ 2H1ÞðH0 þ 8H1Þf − f Þ 0 2 4 2 3 2 2 2 2 2 2 2 4 00 þ 6f f21ð6b H0 þ 336b H0H1 þ H0H1ð59b H0 − 46Þþ3H1ð71b H0 − 28Þ − 8H0 þ 196b H1Þf 2 0 2 0 0 2 2 00 þð14 − 3b H0ð3H0 þ 7H1ÞÞf g=½18b f ðf − 3ð21H0H1 þ 4H0 þ 14H1Þf Þ; ðD1Þ

_ 0 2 3 2 2 2 H1 ¼½6f ð−3ð1029b H0H1 þ 21H1ð27b H0 − 16Þ 2 2 2 2 2 2 4 00 2 0 þ 4H0H1ð34b H0 − 35Þþ4H0ð3b H0 − 4Þþ686b H1Þf − ð3b H1ð3H0 þ 7H1Þþ4Þf Þ 2 00 0 2 0 0 2 2 00 − b fð3ð2H0 þ 7H1ÞðH0 þ 8H1Þf ðTÞþf ðTÞÞ=½18b f ðTÞðf − 3ð21H0H1 þ 4H0 þ 14H1Þf Þ; ðD2Þ with f ¼ T þ αT2, f0 ¼ 1 þ 2αT, f00 ¼ 2α and where b is given by 84α 2 − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ 2 2 2 2 2 2 : ðD3Þ 1 þ 12αðH0 þ 7H0H1 þ 7H1Þþ 1 þ 576α ðH0 þ 7H0H1 þ 7H1Þ

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