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Noether Gauge Symmetry of Modified Teleparallel Gravity Minimally Coupled
with a Canonical Scalar Field
Adnan Aslam,1, ∗ Mubasher Jamil,1, 2, † Davood Momeni,2, ‡ and Ratbay Myrzakulov2, §
1Center for Advanced Mathematics and Physics (CAMP),
National University of Sciences and Technology (NUST), H-12, Islamabad, Pakistan
2Eurasian International Center for Theoretical Physics,
Eurasian National University, Astana 010008, Kazakhstan
Abstract: This paper is devoted to the study of Noether gauge symmetries of f(T ) grav-
ity minimally coupled with a canonical scalar field. We explicitly determine the unknown
functions of the theory f(T ), V (φ),W (φ). We have shown that there are two invariants for
this model, one of which defines the Hamiltonian H under time invariance (energy conser-
vation) and the other is related to scaling invariance. We show that the equation of state
parameter in the present model can cross the cosmological constant boundary. The behavior
of Hubble parameter in our model closely matches to that of ΛCDM model, thus our model
is an alternative to the later.
PACS numbers: 04.20.Fy; 04.50.+h; 98.80.-k
∗Electronic address: adnangrewal@yahoo.com
†Electronic address: mjamil@camp.nust.edu.pk
‡Electronic address: d.momeni@yahoo.com
§Electronic address: rmyrzakulov@csufresno.edu
2
I. INTRODUCTION
Observational data from Ia supernovae (SNIa) show that currently the observable Universe is
in accelerated expansion phase [1]. This cosmic acceleration has also been confirmed by different
observations of large scale structure (LSS) [2] and measurements of the cosmic microwave back-
ground (CMB) anisotropy [3]. The essence of this cosmic acceleration backs to “dark energy”,
an exotic energy which generates a large negative pressure, whose energy density dominates the
Universe (for a review see e.g. [4]). The astrophysical nature of dark energy confirms that it is
not composed of baryonic matter. Now from cosmological observations we know that the Universe
is spatially flat and consists of about 70% dark energy, 30% dust matter (cold dark matter plus
baryons) and negligible radiation. But the nature of dark energy as well as its cosmological origin
remains mysterious at present.
One of the methods for constructing a dark energy model is to modify the geometrical part of the
Einstein equations. The general paradigm consists in adding into the effective action, physically
motivated higher-order curvature invariants and non-minimally coupled scalar fields. But if we
relax the Riemannian manifold, we can construct the models based on the torsion T instead of
the curvature. The representative models based on this strategy are termed ‘modified gravity’
and include f(R) gravity [5], Horava-Lifshitz gravity [6–8], scalar-tensor gravity [9, 10] and the
braneworld model [11, 12] and the newly f(T ) gravity [13].
The f(T ) theory of gravity is a meticulous class of modified theories of gravity. This theory
can be obtained by replacing the torsion scalar T in teleparallel gravity [14] with an arbitrary
function f(T ). The dynamical equations of motion can be obtained by varying the Lagrangian
with respect to the vierbein (tetrad) basis. Wu et al have proposed viable forms of f(T ) that can
satisfy both cosmological and local gravity constraints [15]. Further Capozziello et al discussed
the cosmographical method for reconstruction of f(T ) models [16]. But the f(T ) model has some
theoretical problems. For example it’s not locally Lorentz invariance, possesses extra degrees of
freedom, violate the first law of thermodynamics and inconsistent Hamiltonian formalism [17].
In the past, the use of scalar fields in certain physical theories, especially particle physics, has
been explored. This led to study the role of scalar fields in cosmology as well. Recently some of
the present authors investigated the behavior of scalar fields in f(T ) cosmology. In [18], we intro-
duced a non-minimally conformally coupled scalar field and dark matter in f(T ) cosmology. We
investigated the stability and phase space behavior of the parameters of the scalar field by choosing
an exponential potential and cosmologically viable form of f(T ). We found that the dynamical
3
system of equations admit two unstable critical points; thus no attractor solution existed in that
model. In another investigation [19], we studied the Noether symmetries (which are symmetries
of the Lagrangian) of f(T ) involving matter and dark energy. In that model, the dark energy was
considered as a canonical scalar field with a potential. The analysis showed that f(T ) ∼ T 3/4
and V (φ) ∼ φ2. Therefore it becomes meaningful to reconstruct a scalar potential V (φ) in the
framework of f(T ) gravity. It was demonstrated that dark energy driven by scalar field, decays
to cold dark matter in the late accelerated Universe and this phenomenon yields a solution to the
cosmic coincidence problem [20]. In this paper, we study the Noether gauge symmetries (NGS) of
the model, which provide a more general notion of the Noether symmetry. This approach is useful
in obtaining physically viable choices of f(T ), and has been previously used for the f(R) gravity
and generalized Saez-Ballester scalar field model as well [21].
The plan of this paper is as follows: In Section II, we present the formal framework of the f(T )
action minimally coupled with a scalar field. In section III, we construct the governing differential
equations from the Noether condition and solve them in an accompanying subsection. In section
IV, we study the dynamics of the present model. Finally we conclude this work. In all later
sections, we choose units c = 16piG = 1.
II. f(T ) GRAVITY
If we limit ourselves to the validity equivalence principle, we must work with a gauge theory
for gravity and such a gauge theory is possible only on curved manifold. Construction of a gauge
theory on Riemannian manifolds is only one option and may be the simplest one. But it’s possible
to write a gauge theory for gravity, with metric, non-metricity and torsion can be constructed easily
[22]. Such theories are defined on a Weitzenbo¨ck spacetime, with globally non zero torsion but with
vanishing local Riemannian tensor. In this theory, which is called teleparallel gravity, people are
working on a non-Riemannian spacetime manifold. The dynamics of the metric is defined uniquely
by the torsion T . The basic quantities in teleparallel or the natural extension of it, namely f(T )
gravity, is the vierbein (tetrad) basis eiµ [23]. This basis of vectors is unique and orthonormal and
is defined by the following equation
gµν = e
a
µe
b
νηab, a, b = 0, 1, 2, 3
4
This tetrade basis must be orthonormal and ηab is the flat Minkowski metric, e
a
µe
µ
b = δ
a
b . One
suitable form of the action for f(T ) gravity in Weitzenbo¨ck spacetime is [24]
S =
∫
d4xe
(
(T + f(T )) + Lm
)
, (1)
where f(T ) is an arbitrary function of torsion T and e = det(eiµ). The dynamical quantity of the
model is the scalar torsion T and the matter Lagrangian Lm. The equation of motion derived from
the action, by varying with respect to the eiµ, is given by
e−1∂µ(eS
µν
i )(1 + fT )− e
λ
i T
ρ
µλS
νµ
ρ fT + S
µν
i ∂µ(T )fTT −
1
4
eνi(1 + f(T )) = 4pie
ρ
i T
ν
ρ .
As usual Tµν is the energy-momentum tensor for matter sector of the Lagrangian Lm. It is a
straightforward calculation to show that this equation of motion is reduced to Einstein gravity
when f(T ) = 0. Indeed, this is the equivalence between the teleparallel theory and the Einstein
gravity.
III. OUR MODEL
We take a spatially flat homogeneous and isotropic Friedmann-Lemaˆıtre-Robertson-Walker
(FLRW) spacetime
ds2 = −dt2 + a2(t)[dr2 + r2(dθ2 + sin2 θdϕ2)] . (2)
We add in the action (1) a scalar field with an unknown potential function V = V (φ) (sometimes
also known as Saez-Ballester model [25]). However a slight redefinition of (Φ(φ) =
φ∫
dφ
√
±V (φ),
where +/− correspond to non-phantom/phantom phase, respectively [26]). The total action reads:
S =
∫
d4x e
(
T + f(T ) + λ(T + 6H2) + V (φ)φ;µφ
;µ −W (φ)
)
. (3)
Here trace of the torsion tensor is T = −6H2, e = det(eµi ), λ is the Lagrange multiplier and H =
a˙
a
is the Hubble parameter. For our convenience, we keep the original Saez-Ballester scalar field in
our effective action. Varying (3) with respect to T , we obtain
λ = −(1 + f ′(T )). (4)
Here prime denotes the derivative with respect T . By substituting (4) in (3), and integrating over
the spatial volume we get the following reduced Lagrangian:
L(a, φ, T, a˙, φ˙) = a3
[
T + f(T )− [1 + f ′(T )]
(
T + 6(
a˙
a
)2
)
+ V (φ)φ˙2 −W (φ)
]
. (5)
5
For the Lagrangian (5), the equations of motion read as follows
fTT (T + 6H
2) = 0, (6)
a¨
a
= −
1
4(1 + fT )
[
f − TfT −
T
3
(1 + fT ) + V (φ)φ˙
2 −W (φ) + 4HT˙fTT
]
, (7)
φ¨+ 3Hφ˙+
1
2V
(V ′φ˙2 +W ′) = 0. (8)
Equation (6) indicates two possibilities: (1) fTT = 0, which gives the teleparallel gravity and we
are not interested in this case. (2) Another possibility is T = −6H2 which is the standard definition
of the torsion scalar in f(T ) gravity. In the next section we will investigate the Noether gauge
symmetries of the newly proposed model in (5).
IV. NOETHER GAUGE SYMMETRY OF THE MODEL
To calculate the Noether symmetries, we define it first. A vector field [21]
X = T (t, a, T, φ)
∂
∂t
+ α(t, a, T, φ)
∂
∂a
+ β(t, a, T, φ)
∂
∂T
+ γ(t, a, T, φ)
∂
∂φ
, (9)
is a Noether gauge symmetry corresponding to a Lagrangian L(t, a, T, φ, a˙, T˙ , φ˙) if
X[1]L+ LDt(T ) = DtB, (10)
holds, where X[1] is the first prolongation of the generator X, B(t, a, T, φ) is a gauge function and
Dt is the total derivative operator
Dt =
∂
∂t
+ a˙
∂
∂a
+ T˙
∂
∂T
+ φ˙
∂
∂φ
. (11)
The prolonged vector field is given by
X[1] = X+ αt
∂
∂t˙
+ βt
∂
∂T˙
+ γt
∂
∂φ˙
, (12)
where
αt = Dtα− a˙DtT , βt = Dtβ − T˙DtT , γt = Dtγ − φ˙DtT . (13)
If X is the Noether symmetry corresponding to the Lagrangian L(t, a, T, φ, a˙, T˙ , φ˙), then
I = T L+ (α− T a˙)
∂L
∂a˙
+ (β − T T˙ )
∂L
∂T˙
+ (γ − T φ˙)
∂L
∂φ˙
−B, (14)
6
is a first integral or an invariant or a conserved quantity associated with X. The Noether condition
(10) results in the over-determined system of equations
Ta = 0, Tφ = 0, TT = 0, αT = 0, (15)
γT = 0, BT = 0, 2a
3V γt = Bφ, (16)
6(1 + f ′)αφ − a
2V γa = 0, (17)
12a(1 + f ′)αt +Ba = 0, (18)
3V α+ aV ′γ + 2aV γφ − aV Tt = 0, (19)
(1 + f ′)(α + 2aαa − aTt) + af
′′β = 0, (20)
3a2(f − Tf ′ −W )α− a3Tf ′′β − a3W ′γ + a3(f − Tf ′ −W )Tt = Bt. (21)
We obtain the solution of the above system of linear partial differential equations for f(T ),
V (φ), W (φ), T , α, β and γ. We have:
f(T ) =
1
2
t0T
2 − T + c2, (22)
V (φ) = V0φ
−4, (23)
W (φ) =W0φ
−4 + c2, (24)
T = t+ c1, (25)
α = a, (26)
β = −2T, (27)
γ = φ, (28)
where t0, V0,W0, c2 and c1 are constants. It is interesting to note that quadratic f(T ) = T
2 has been
used to model static wormholes in f(T ) gravity [27]. Also the scalar potential is proportional to
φ−4 which has been previously reported in [21] for f(R)-tachyon model. Recently Iorio & Saridakis
[28] studied solar system constraints on the model (22) and found the bound |t0| ≤ 1.8 × 10
4m2.
Further a T 2 term can cure all the four types of the finite-time future singularities in f(T ) gravity,
similar to that in F (R) gravity [29]. The quadratic correction to teleparallel model is quite vital
as a next approximation in astrophysical context. Hence the Noether gauge symmetry approach
generates a cosmologically viable model of f(T ) gravity.
It is clear from (22)-(28) that the Lagrangian (5) admits two Noether symmetry generators
X1 =
∂
∂t
, (29)
X2 = t
∂
∂t
+ a
∂
∂a
− 2T
∂
∂T
+ φ
∂
∂φ
. (30)
7
The first symmetry X1 (invariance under time translation) gives the energy conservation of the
dynamical system in the form of (31) below, while the second symmetry X2 (scaling symmetry)
and a corresponding conserved quantity of the form (32) below. The two first integrals (conserved
quantities) which are
I1 = −
1
2
t0T
2a3 + 6t0Taa˙
2 − V0a
3φ−4φ˙2 −W0a
3φ−4, (31)
I2 = −
1
2
t0tT
2a3 + 6t0tTaa˙
2 − V0ta
3φ−4φ˙2 −W0ta
3φ−4
− 12t0Ta
2a˙+ 2V0a
3φ−3φ˙. (32)
Also the commutator of generators satisfies [X1,X2] = X1 which shows that the algebra of gener-
ators is closed.
V. COSMOLOGICAL IMPLICATIONS
We rewrite Eq. (5) using the solutions for f(T ), V (φ), W (φ) as obtained in the previous section
in the following form
L =
a˙4
a
+
a3(V0φ˙
2 −W0 − c2φ
4)
φ4
, (33)
where t0 is an arbitrary constant. We choose t0 =
1
18 for simplification and further we take c2 6= 0.
The Euler-Lagrange equations read
a¨
a
=
H2
4
+
1
4aφ4H2
(
V0φ˙
2 −W0 − c2φ
4
)
, (34)
φ¨+ 3Hφ˙−
2φ˙2
φ
=
2W0
V0φ
. (35)
Their evolutionary behavior is obtained by numerically solving the Euler-Lagrange equations (34)-
(35) for an appropriate set of the parameters and the initial conditions. To obtain the equa-
tion of state parameter numerically, first we note that for f(T )-Saez-Ballester theory, the energy-
momentum tensor reads
Tµν = V (φ)
[
gµνφ;αφ
;α − 2φ;µφ;ν
]
− gµνW (φ). (36)
It is easy to calculate the total energy density and averaged pressure
ρ = V (φ)φ˙2 −W (φ), (37)
p = −V (φ)φ˙2 −W (φ). (38)
8
FIG. 1: (Left) Cosmological evolution of H(t), φ(t) vs time t. The model parameters chosen as V0 = 1,
W0 = 2V0. Various curves correspond to (solid, H(t)), (dots, φ(t)). (Right) Variation of w vs time t. The
model parameters are chosen as α0 = 2, β =
ρm0
V0
, ω = 1. In both figures, we chose the initial conditions
a˙(0) = H0, φ(0) = 0.1, a(0) = 1, φ˙(0) = 1.
The EoS parameter is constructed via
w ≡
p
ρ
=
V (φ)φ˙2 +W (φ)
−V (φ)φ˙2 +W (φ)
. (39)
Putting the potential functions (23) and (24) in (39), we get
w =
φ˙2 + βφ4 + α0
−φ˙2 + βφ4 + α0
, β =
c2
V0
, α0 =
W0
V0
. (40)
The numerical simulation of w, w˙ is drawn in the figure which shows that w behaves like the
phantom energy for a brief period of time. This conclusion is exciting since there exists convincing
astrophysical evidence that the observable Universe is currently in the phantom phase [30].
VI. CONCLUSION
In this paper, motivated by some earlier works on Noether symmetry in f(T ) gravity, we
introduced a new model containing a canonical scalar field with a potential. Firstly we showed
that this model obeys a quadratic term of torsion, with potential proportional to φ−4 which also
appears for tachyonic field in f(R) model. It is also interesting to note that quadratic f(T ) = T 2
has been used to model static wormholes in f(T ) gravity in literature.
We mention our key results and comments below:
9
• Our numerical simulations show that there happens a phantom crossing scenario for a brief
period in this toy model, after which the state parameter evolves to cosmological constant
asymptotically.
• The behavior of Hubble parameter in our model closely mimics to that of ΛCDM model,
thus our model is an alternative to the later.
• Since at the same time ΛCDM model cannot explain the phantom crossing as is observed
from the empirical astrophysical results, one should prefer alternatives to ΛCDM model
such as the present one. It is curious to note that such a result is obtained from a Lorentz
invariance violating f(T ) theory, however, the same theory is consistent with solar system
tests, contains attractor solutions, and is free of massive gravitons.
• The results reported here are significantly different from [18, 19] since there we calculated
the ‘Noether symmetries’ while here only ‘Noether gauge symmetries’ are obtained. As one
can see, the results obtained here are different from the ones previously obtained in the
literature.
• The most important feature of f(T ) gravity which differs it from any other “curvature in-
variant” model like f(R) theory, is its irreducibility to a scalar model in the Jordan frame
unlike f(R). As we know, f(R) gravity can be reduced to a scalar field by a simple iden-
tification between the scalar field and the gravity sector of the action in the Jordan frame.
But here, since the Lorentz symmetry is broken locally, and also for leakage of finding such
formal transformation between the scalar field and the Torsion action, these two models are
not equivalent. So unless the f(R), here proposition of the scalar field is not artificial and
do not add any additional degree of freedom to the model. From another point of the view,
the f(T ) gravity is not conformal invariant, so reduction of the gravity sector to the scalar
matter is not possible. So indeed by introducing the scalar field in the action, we avoided
from a similar formal extensions like two scalar components models, like Quintessence. Note
that proposition of the scalar field to f(T ) is completely new irreducible action but to f(R)
is just the quintom model or in it’s extreme form reduces to the multi-scalar models with
less symmetry than the original action.
10
Acknowledgment
We would gratefully thank the anonymous referee for useful criticism on our paper.
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