Eur. Phys. J. C (2017) 77:510
DOI 10.1140/epjc/s10052-017-5050-z
Regular Article - Theoretical Physics
Studying the intervention of an unusual term in f (T) gravity via
the Noether symmetry approach
On a new term for f (T) gravity actions
Behzad Tajahmad
a
Faculty of Physics, University of Tabriz, Tabriz, Iran
Received: 9 January 2017 / Accepted: 7 July 2017 / Published online: 1 August 2017
© The Author(s) 2017. This article is an open access publication
Abstract As has been done before, we study an unknown
coupling function, i.e. F(ϕ), together with a function of
torsion and also curvature, i.e. f (T ) and f (R), generally
depending upon a scalar field. In the f (R) case, it comes
from quantum correlations and other sources. Now, what if
beside this term in f (T ) gravity context, we enhance the
action through another term which depends upon both scalar
field and its derivatives? In this paper, we have added such an
unprecedented term in the generic common action of f (T )
gravity such that in this new term, an unknown function of
torsion has coupled with an unknown function of both scalar
field and its derivatives. We explain in detail why we can
append such a term. By the Noether symmetry approach, we
consider its behavior and effect. We show that it does not
produce an anomaly, but rather it works successfully, and
numerical analysis of the exact solutions of field equations
coincides with all most important observational data, par-
ticularly late-time-accelerated expansion. So, this new term
may be added to the gravitational actions of f (T ) gravity.
1 Introduction
Various astronomical and cosmological observations of the
last decade, including CMB studies [1], supernovae [2,3] and
large scale structure [4], have provided a picture of the uni-
verse with accelerating expansion. This profound mystery
leads us to the prospect that either about 70% of the uni-
verse is made up of a substance known as dark energy[5],
about which we have almost no knowledge at all, or that
General Relativity (GR) is modified at cosmological scales
[6–8]. A simple candidate for the dark energy is the cos-
mological constant with the equation of state (EoS) param-
eter ω =−1. However, the cosmological constant model is
a
e-mail: behzadtajahmad@yahoo.com
subject to the fine-tuning and coincidence problems [9]. In
order to solve these problems, various dynamical dark energy
models have been proposed: quintessence [10,11], phan-
tom [12,13] and quintom [14–16]. Because the quintessence
type of matter could not give the possibility that ω<−1,
extended paradigms (i.e. phantom and quintom) were pro-
posed [17]. Beside this unknown-nature dark energy, a sec-
ond way, concerning various gravitational modification the-
ories like f (R), f (T ) and scalar-tensor theories, has been
addressed. One of the modifications of the matter part of
the Einstein–Hilbert action is f (T ) gravity as an extension
of teleparallel gravity. Teleparallel Gravity (TG), demonstra-
bly equivalent to general relativity, was initially introduced
by Einstein for the sake of unifying gravity and electromag-
netism. In TG we use the Weitzenböck connection instead
of the Levi-Civita connection, so we have torsion in lieu of
curvature only. The field equations in this theory are second-
order differential equations, while for the generalized f (R)
theory they are of fourth order, thus it is simpler to analyze
and elaborate the cosmic evolution [18].
The actions of this context are likely to contain several
scalar fields, but it is normally assumed that only one of
these fields remains dynamical for a long period. We always
see the coupling function of f (R) and also f (T ) in the form
of a function; that is, F(ϕ) f (R) or F(ϕ) f (T ), depending
upon the scalar field only. The motivation for the nonmini-
mal coupling, F(ϕ)R in which F(ϕ) =
1
2
1
8π G
− ξϕ
2
,in
the gravitational Lagrangian comes from many directions.
However, this explicit nonminimal coupling was originally
introduced in the context of classical radiation problems [19],
and also it is required by renormalizability in curved space-
time [20]. For different values of ξ , we have the following
table (Table 1).
However, the values of ξ in renormalizable theories
depend upon the class of theory [56,57]. A nonzero ξ is
generated by first loop corrections even if it is absent in
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