Eur. Phys. J. C (2019) 79:530 https://doi.org/10.1140/epjc/s10052-019-7038-3 Regular Article - Theoretical Physics Cosmology in symmetric teleparallel gravity and its dynamical system Jianbo Lua, Xin Zhao, Guoying Cheeb Department of Physics, Liaoning Normal University, Dalian 116029, People’s Republic of China Received: 28 February 2019 / Accepted: 10 June 2019 / Published online: 20 June 2019 © The Author(s) 2019 Abstract We explore an extension of the symmetric telepa- symmetric teleparallel connection that is not metric compati- rallel gravity denoted the f (Q) theory, by considering a ble. This classification highlights that curvature is a property function of the nonmetricity invariant Q as the gravitational of the connection and not of the metric tensor or the mani- Lagrangian. Some interesting properties could be found in fold. It becomes a property of the metric only through the use the f (Q) theory by comparing with the f (R) and f (T ) the- of the Levi–Civita connection. GR can be equivalently for- ories. The field equations are derived in the f (Q) theory. mulated in terms of either of these connections. All of them The cosmological application is investigated. In this theory can be used to define Lagrangians of which Euler-Lagrange the accelerating expansion is an intrinsic property of the uni- equations coincide with the Einstein equations for a particu- verse geometry without need of either exotic dark energy or lar choice of contributing terms. extra fields. And the state equation of the geometrical dark In recent years teleparallel theories have gained more energy can cross over the phantom divide line in the f (Q) attention as alternative theories of gravity. While one mostly theory. In addition, the dynamical system method are inves- works in the torsion-based setting, there has been interest in tigated. It is shown that there are five critical points in the the direction of symmetric teleparallelism, where instead of STG model for taking f (Q) = Q +αQ2. The critical points curvature or torsion gravity is effectively described by non- P4 and P5 are stable. P4 corresponds to the geometrical dark metricity. It should be mentioned that the mechanism medi- wef f =− energy dominated de Sitter universe ( tot 1), while P5 ating gravity is the affine connection rather than the physi- wef f = corresponds to the matter dominated universe ( tot 0). cal manifold. This is reflected in the fact that in GR curva- Given that P4 represents an attractor, the cosmological con- ture is a property of the connection and not of the manifold stant problems, such as the fine tuning problem, could be itself, and thus can be equally described by other connection solved in the STG model. property such as nonmetricity. Symmetric teleparallel grav- ity (STG) offers an interesting geometric interpretation of gravitation besides its formulation in terms of a spacetime 1 Introduction metric and Levi–Civita connection or its teleparallel formu- lation. It describes gravity through a connection which is not The discovery of the cosmic accelerated expansion has moti- metric compatible, however is curvature-free and torsion- vated a vast number of researches on modifications of general free. This connection can be simplified to a partial derivative relativity (GR) (for recent reviews, see [1–3]) to explain this through the so-called coincident gauge [4,5]. This geometri- acceleration. A plethora of theories have been proposed in cally implies that vectors do remain parallel at long distances the literature, essentially based on specific approaches. From on a manifold [6]. By demanding that the curvature vanishes the view point of the connection, theories of gravity can be and that the connection is torsionless, the remaining gravita- classified into three broad classes. The first class uses the tional information is encoded in nonmetricity contributions Levi–Civita connection of the metric and its curvature. The [4,6–9]. second class uses the tetrads of a metric and their curvature In STG, the metricity condition of GR is relaxed and then ¨ free, metric-compatible, Weitzenbock connection with tor- produces teleparallel equivalent of general relativity [10]. sion. And the third class uses a curvature-free and torsion-free One of important properties of STG is their ability to separate gravitational and inertial effects [11] which is not possible in a e-mail: [email protected] GR. It has produced many strains on the theory such as the b e-mail: [email protected] 123 530 Page 2 of 8 Eur. Phys. J. C (2019) 79 :530 issue of defining a gravitational energy-momentum tensor sic property of the universe geometry without need of either [12]. exotic dark energy or extra fields. This theory is identified as More recently, STG was deeply analyzed in [4]. An excep- a metrical formulation of f (T ) gravity. In contrast with f (T ) tional class with a vanishing affine connection was discov- gravity, this new theory respects local Lorentz symmetry and ered. Based on this property, a simpler geometrical formula- harbour no extra degrees of freedom, since the dynamic vari- tion of GR was proposed. It leads to a purely inertial connec- able is the metric instead of the tetrad. At the same time it tion that can be completely removed by a coordinate gauge has the advantage over f (R) gravity that its field equations choice [13]. This formulation fundamentally deprives gravity are second-order instead of fourth-order and then is free of from any inertial character. The resulting theory is described pathologies. It may provide an satisfactory alternative to con- by the Hilbert action purged from the boundary term and is ventional dark energy in general relativistic cosmology and a more robustly underpinned by the spin-2 field theory. This new explanation of the acceleration of the cosmic expansion. construction also provides a novel starting point for modified The cosmological constant problem, the fine tuning problem gravity theories, and presents new and simple generaliza- and the problem of phantom divide line crossing are solved tions where analytical self-accelerating cosmological solu- or disappear. tions arise naturally in the early and late time universe. Even though the physical aspects of the dynamical non- metricity are not easy to interpret, a gauge interpretation 2 Field equations of STG was proposed from a physical point of view [11]. The equivalence principle which allows eliminating locally The symmetric teleparallel connection relies only on non- gravitation makes it to be an integrable gauge theory. In this metricity and does not possess neither curvature nor torsion approach, the metric represents the gravitational field and the which yields some interesting results. One can transform to connection corresponds to the gauge potential [14]. a zero connection gauge and thereby covariantize the partial STG was presented originally in a paper by Nester and Yo derivatives as well as split the Einstein–Hilbert action into [6], where the authors emphasize that the formulation brings the Einstein Lagrangian density and a boundary term [4,6]. a new perspective to bear on GR. However, the formulation In differential geometry, the general affine connection can is geometric and covariant. More recently, some new results always be decomposed into three independent components and important developments have been obtained. An excep- [17], namely, tional class was discovered which is consistent with a vanish- λ λ λ λ μν = μν + K μν + L μν, (1) ing affine connection [7]. Based on this remarkable property, a simpler geometrical formulation of GR was proposed. It where the first term is the Levi–Civita connection of the met- provides a novel starting point for modified gravity theories, ric gμν, given by the standard definition and presents new and simple generalizations where acceler- λ 1 λβ ating cosmological solutions arise naturally in the early and μ ν ≡ g ∂μgβν + ∂ν gβμ − ∂β gμν . (2) 2 late-time universe. λ In a generalization of STG [15], a nonminimal cou- The second term K μν is the contortion: pling of a scalar field to the nonmetricity invariant was λ 1 λ λ K μν ≡ T μν + T(μ ν), (3) introduced. The similarities and differences with analogous 2 scalar-curvature and scalar-torsion theories were discussed. λ λ with the torsion tensor defined as T μν ≡ 2 μν . The third The class of scalar-nonmetricity theories was extended by [ ] term is the disformation considering a five-parameter quadratic nonmetricity scalar λ 1 λρ and including a boundary term [16]. The equivalents for L μν ≡ g −Qμρν − Qνρμ + Qρμν , (4) GR and ordinary (curvature based) scalar-tensor theories 2 were obtained as particular cases. In a recent paper [5], which is defined in terms of the nonmetricity tensor: Qρμν ≡ STG is extended by considering a new class of theories ∇ρ gμν. We define two traces of the nonmetricity tensor: where the nonmetricity is coupled nonminimally to the mat- μ μ Qρ = Qρ μ, Qρ = Q ρμ, (5) ter Lagrangian. The theoretical consistency and motivations on this extension are established. Its cosmological applica- and introduce the superpotential tion provides a gravitational alternative to dark energy. α α α α α α 4P μν=−Q μν+2Q(μ ν) − Q gμν−Q gμν − δ(μ Qν). However, comparing to these complicated general appro- aches mentioned above, a more special and simple model is (6) more suitable to the cosmological application. In this paper, We have therefore starting from a rather simple Lagrangian a good toy model λ 1 λρ is obtained, where the accelerating expansion is an intrin- L μν ≡ g −∇μgρν −∇ν gμρ +∇ρ gμν , (7) 2 123 Eur. Phys. J. C (2019) 79 :530 Page 3 of 8 530 and can construct an invariant, the quadratic nonmetricity at first.