Noether symmetries in Gauss–Bonnet-teleparallel cosmology

The European Physical Journal C, Nov 2016

A generalized teleparallel cosmological model, \(f(T_\mathcal {G},T)\), containing the torsion scalar T and the teleparallel counterpart of the Gauss–Bonnet topological invariant \(T_{\mathcal {G}}\), is studied in the framework of the Noether symmetry approach. As \(f(\mathcal {G}, R)\) gravity, where \(\mathcal {G}\) is the Gauss–Bonnet topological invariant and R is the Ricci curvature scalar, exhausts all the curvature information that one can construct from the Riemann tensor, in the same way, \(f(T_\mathcal {G},T)\) contains all the possible information directly related to the torsion tensor. In this paper, we discuss how the Noether symmetry approach allows one to fix the form of the function \(f(T_\mathcal {G},T)\) and to derive exact cosmological solutions.

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Noether symmetries in Gauss–Bonnet-teleparallel cosmology

Eur. Phys. J. C Noether symmetries in Gauss-Bonnet-teleparallel cosmology Salvatore Capozziello 0 1 2 5 Mariafelicia De Laurentis 1 3 4 5 Konstantinos F. Dialektopoulos 1 2 0 Gran Sasso Science Institute (INFN) , Via F. Crispi 7, 67100 L' Aquila , Italy 1 INFN Sezione di Napoli, Complesso Universitario di Monte S. Angelo , Edificio G, Via Cinthia, 80126 Napoli , Italy 2 Dipartimento di Fisica “E. Pancini”, Universita' di Napoli“Federico II”, Complesso Universitario di Monte S. Angelo , Edificio G, Via Cinthia, 80126 Napoli , Italy 3 Laboratory of Theoretical Cosmology, Tomsk State University of Control Systems and Radioelectronics (TUSUR) , 634050 Tomsk , Russia 4 Institute for Theoretical Physics, Goethe University , Max-von-Laue-Str. 1, 60438 Frankfurt , Germany 5 Tomsk State Pedagogical University , 634061 Tomsk , Russia A generalized teleparallel cosmological model, f (TG , T ), containing the torsion scalar T and the teleparallel counterpart of the Gauss-Bonnet topological invariant TG , is studied in the framework of the Noether symmetry approach. As f (G, R) gravity, where G is the Gauss-Bonnet topological invariant and R is the Ricci curvature scalar, exhausts all the curvature information that one can construct from the Riemann tensor, in the same way, f (TG , T ) contains all the possible information directly related to the torsion tensor. In this paper, we discuss how the Noether symmetry approach allows one to fix the form of the function f (TG , T ) and to derive exact cosmological solutions. 1 Introduction Extended theories of gravity are semi-classical approaches where the effective gravitational Lagrangian is modified, with respect to the Hilbert–Einstein one, by considering higher-order terms of curvature invariants, torsion tensor, derivatives of curvature invariants and scalar fields (see for example [1–4]). In particular, taking into account the Ricci, Riemann, and Weyl invariants, one can construct terms like R2, Rμν Rμν , Rμνδσ Rμνδσ , W μνδσ Wμνδσ , that give rise to fourth-order theories in the metric formalism [5, 6]. Considering minimally or nonminimally coupled scalar fields to the geometry, we deal with scalar–tensor theories of gravity [7, 8]. Considering terms like R R, R k R, we are dealing with higher-than fourth-order theories [9, 10]. f ( R) gravity is the simplest class of these models where a generic funca e-mail: b e-mail: c e-mail: tion of the Ricci scalar R is considered. The interest for these extended models is related both to the problem of quantum gravity [2] and to the possibility to explain the accelerated expansion of the universe, as well as the structure formation, without invoking new particles in the matter/energy content of the universe [4–15]. In other words, the attempt is to address the dark side of the universe by changing the geometric sector and remaining unaltered the matter sources with respect to the Standard Model of particles. However, in the framework of this “geometric picture”, the debate is very broad involving the fundamental structures of gravitational interaction. Just to summarize some points, gravity could be described only by metric (in this case we deal with a metric approach), or by metric and connections (in this case, we are considering a metric-affine approach [16]), or by a purely affine approach [17]. Furthermore, dynamics could be related to curvature tensor, as in the original Einstein theory, to both curvature and torsion [18], or to torsion only, as in the so-called teleparallel gravity [19]. Starting from these original theories and motivations, one can build more complex Lagrangians, by using different combinations of curvature scalars and their derivatives, or topological invariants, such us the Gauss–Bonnet term, G, as well as the torsion scalar T . Many theories have been proposed considering generic functions of such terms, like f (G), f (T ), f ( R, G), and f ( R, T ) [20–44]. However, the problem is how many and what kind of geometric invariants can be used, and, furthermore, what kind of physical information one can derive from them. For example, it is well known that f ( R) gravity is the straightforward extension of the Hilbert– Einstein case which is f ( R) = R, and f (T ) is the extension of teleparallel gravity which is f (T ) = T . However, if one wants to consider the whole information contained in the curvature invariants, one has to take into account also combinations of Riemann, Ricci, and Weyl tensors.1 As discussed in [26], assuming a f ( R, G theory means to consider the whole curvature budget and then all the degrees of freedom related to curvature. Assuming the teleparallel formalism, a f (TG , T ) theory, where TG is the torsional counterpart of the Gauss–Bonnet topological invariant, means to exhaust all the degrees of freedom related to torsion and then completely extend f (T ) gravity. It is important to stress that the Gauss–Bonnet invariant derived from curvature differs from the same to (...truncated)


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Salvatore Capozziello, Mariafelicia De Laurentis. Noether symmetries in Gauss–Bonnet-teleparallel cosmology, The European Physical Journal C, 2016, pp. 629, Volume 76, Issue 11, DOI: 10.1140/epjc/s10052-016-4491-0