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:
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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)