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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

Faculty of Physics, University of Tabriz, Tabriz, Iran

Received: 9 January 2017 / Accepted: 7 July 2017 / Published online: 1 August 2017

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

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) =



8π G

− ξϕ



,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

123

510 Page 2 of 9 Eur. Phys. J. C (2017) 77 :510

Table 1 Special values of the dimensionless coupling constant, (ξ), in the nonminimal coupling and their names

Amounts of ξξ= 1/6 ξ = 0 ξ1 (General case) ξ = 0

Named: coupling Conformal Minimal Strong (Standard) nonminimal

the classical action [21,22]. A nonminimal coupling term

is expected at high curvatures [23], and it has been argued

that classicalization of the universe in quantum cosmology

indeed requires ξ = 0. Moreover, the nonminimal coupling

can solve potential problems of primordial nucleosynthesis

[24] and the absence of pathologies in the propagation of ϕ-

waves seems to require conformal coupling for all nongrav-

itational scalar ﬁelds [25]. Any attempt to formulate quan-

tum ﬁeld theory on a curved spacetime necessarily leads to

modifying the Hilbert–Einstein action. This means adding

terms containing nonlinear invariants of the curvature tensor

or nonminimal couplings between matter and the curvature

originating in the perturbative expansion [26,27].

Now, let us take the incomplete action

S =



√

−g

[

F(ϕ)R +···

]

into account. Eliminating the accelerating term under integra-

tion by parts, the corresponding point-like Lagrangian reads

L = 6a ˙a

F − 6KaF + 6a

˙aF



˙ϕ +···, (1)

where K = 0, ±1. Here we assume the signs (+ - - -) for the

FRW metric components. On the other hand, in f (T ) gravity,

pursuant to the torsion form, we have no accelerating term,

so in this case, we do not have the last term in (1) in which

the derivative of the scalar ﬁeld couples with the scale factor

and its derivative. Maybe it is worth to note what happens

when we insert a term as U



ϕ, ϕ

,μ



g(T ) in the actions

of f (T ) gravity. As mentioned in the ﬁrst paragraph of the

introduction, the “Teleparallel” case is equivalent to General

Relativity (TEGR). Altogether, in many cases, the authors

construct the actions of f (T ) gravity by replacing the torsion

instead of curvature (for example, see [41,59]). However,

when the nonminimal coupling is switched on, the resulting

theory exhibits different behavior. Hence, the last term in (1)

is the inspiration for adding such a term. The main purpose

of the present work is to answer the aforementioned question

by having recourse to the Noether symmetry approach.

Symmetries play a substantial role in theoretical physics.

It can safely be said that Noether symmetries are a power-

ful implement both to select models at a fundamental level

and to ﬁnd exact solutions for speciﬁc Lagrangians. In the

literature, applications of the Noether symmetry in general-

ized theories of gravity have been abundantly studied (for

example see [28–50]). Beside this useful approach, another

lucrative approach, the Beyond Noether Symmetry approach

(B.N.S. approach), has recently been presented as an inno-

vation [51]. The B.N.S. approach may carry more conserved

currents than the Noether symmetry approach. Furthermore,

sometimes the Noether symmetry approach fails to achieve

the purpose. In such cases, utilizing the B.N.S. approach is the

ﬁrst option. Also, with this new procedure, solving an ordi-

nary differential equation system, comprising ﬁeld equations

and conserved currents, is a paved road.

The Noether theorem states that, for a given Lagrangian

L, deﬁned on the tangent space of conﬁgurations, TQ ≡

, ˙q

}, if the Lie derivative of the Lagrangian L, dragged

along a vector ﬁeld X,

X = α

(q)

∂

∂q

+˙α

(q)

∂

∂ ˙q

, (2)

where a dot means a derivative with respect to t, vanishes

[52],

L = X

= α

(q)

∂ L

∂q

+˙α

(q)

∂ L

∂ ˙q

= 0, (3)

then X is a symmetry for the dynamics and it generates the

following conserved quantity (constant of motion):



= α

∂ L

∂ ˙q

. (4)

Alternatively, utilizing the Cartan one-form

≡

∂ L

∂ ˙q

(5)

and deﬁning the inner derivative

=θ

, X (6)

we get

= 

, (7)

provided that (3) holds. Equation (7) is coordinate inde-

pendent. Using a point transformation, the vector ﬁeld X is

rewritten as

X =





∂

∂ Q









∂

. (8)

If X is a symmetry, so is

X (i.e.

XL = 0), and a point trans-

formation is chosen such that

= 1, i

= 0 (i = 1). (9)

123

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