Table Of ContentStudying the Intervention of an Unusual Term in f(T) Gravity via Noether
Symmetry
Behzad Tajahmad∗
Faculty of Physics, University of Tabriz, Tabriz, Iran
As usual, we observe an unknown coupling function, i.e. F(ϕ), with a function of torsion and
also curvature, i.e. f(T) and f(R), generally depending on a scalar field. In f(R) case, it comes
from quantum correlations and other reasons. Now, what if beside this term in f(T)-gravity
context, we enhance the action through another term which depends on both scalar field and its
derivatives? In this paper, we have added such an unprecedented term in the generic common
7 action of f(T)-gravity such that in this new term, an unknown function of torsion has coupled
1 with an unknown function of both scalar field and its derivatives. We have explained why we can
0 append such term, in the introduction in details. By the use of Noether symmetry approach, we
2 have considered its behavior and effect. We have shown that it does not produce an anomaly, but
ratheritworkssuccessfullyandnumericalanalysisoftheexactsolutionsoffieldequationscoincide
r
a with all most important observational data particularly late-time-accelerated expansion. So, this
M new term may be added to the gravitational actions of f(T)-gravity.
7
2
I. INTRODUCTION it is simpler to analyze and elaborate the cosmic
] evolution [18].
c
q
- Various astronomical and cosmological observations The actions of this context are likely to contain sev-
gr of the last decade, including CMB studies [1], super- eral scalar fields, but it is normally assumed that only
[ novae [2, 3] and large scale structure [4], have provided oneofthesefieldsremainsdynamicalforalongperiod,
a picture of the universe with accelerating expansion. eye-catchingly. We always see the coupling function
2 This profound mystery leads us to the prospect that, of f(R) and also f(T) in the form of a function;
v
either about 70% of the universe is made up of a sub- that is, F(ϕ)f(R) or F(ϕ)f(T), depending on the
0
2 stance known as dark energy[5], about which we have scalar field only. The motivation for the non-minimal
6 almost no knowledge at all, or that General Relativity coupling, F(ϕ)R in which F(ϕ) = 1(cid:0) 1 −ξϕ2(cid:1), in
2 8πG
1 (GR)ismodifiedatcosmologicalscales[6–8]. Asimple the gravitational Lagrangian comes from many direc-
0 candidate for the dark energy is the cosmological tions. However, this explicit non-minimal coupling
1. constant with the equation of state (EoS) parameter was originally introduced in the context of classical
0 ω = −1. However, the cosmological constant model is radiation problems [19], and also it is requested
7 subjecttothefine-tuningandcoincidenceproblems[9]. by renormalizability in curved space-time [20]. For
1 In order to solve these problems, various dynamical different amounts of ξ, we have the following table.
v: dark energy models have been proposed consisting However, the values of ξ in renormalizable theories
i quintessence [10, 11], phantom [12, 13] and quintom depend on the class of theory [56, 57]. A nonzero ξ is
X
[14–16]. Since the quintessence type of matter could generated by first loop corrections even if it is absent
r not give the possibility that ω < −1, so the extended intheclassicalaction[21,22]. Anon-minimalcoupling
a
paradigms (i.e. phantom and quintom) are proposed term is expected at high curvatures [23], and it has
[17]. Besidethisunknown-naturedarkenergy,asecond been argued that classicalization of the universe in
way which is about various gravitational modification quantum cosmology indeed requires ξ (cid:54)= 0. Moreover,
theories like f(R), f(T) and scalar-tensor have been the non-minimal coupling can solve potential problems
lionized. One of the modifications of the matter part of primordial nucleosynthesis [24] and the absence of
of the Einstein-Hilbert action is f(T) gravity as an pathologies in the propagation of ϕ-waves seems to
extension of teleparallel gravity. Teleparallel Gravity require conformal coupling for all non-gravitational
(TG), demonstrably equivalent to general relativity, scalar fields [25]. Any attempt to formulate quantum
was initially introduced by Einstein for the sake of field theory on a curved space-time necessarily leads
unifying the gravity and the electromagnetism. In to modifying the Hilbert-Einstein action. This means
TG we use Weitzenbck connection instead of the adding terms containing non-linear invariants of the
Levi-Civita connection, so we have torsion in lieu of curvature tensor or non-minimal couplings between
curvature only. The field equations in this theory matter and the curvature originating in the perturba-
are second-order differential equations, while for the tive expansion [26, 27].
generalized f(R) theory they are fourth-order, thus
Now, let us take the following incomplete action
(cid:90) √
S = d4x −g[F(ϕ)R+...].
∗Electronicaddress: [email protected]
2
Table - 1
Amounts of ξ ξ=1/6 ξ=0 (cid:107)ξ(cid:107)(cid:29)1 (general case) ξ(cid:54)=0
Named as ... coupling conformal minimal strong (standard) non-minimal
intoaccount. Eliminatingtheacceleratingtermbyinte- [52]
gral by parts, the corresponding point-like Lagrangian
reads ∂L ∂L
L L=X Lµ =αi(q) +α˙i(q) =0, (3)
X µ ∂qi ∂q˙i
L=6aa˙2F −6KaF +6a2a˙F(cid:48)ϕ˙ +..., (1)
then X isasymmetryforthedynamicsanditgenerates
where K = 0,±1. Here we assume the signature the following conserved quantity (constant of motion)
(+−−−) for the FRW’s metric components. On
the other hand, in f(T)-gravity, pursuant to torsion’s ∂L
Σ =αi . (4)
form, we have no accelerating term, so in this case, 0 ∂q˙i
we have no the last term in (1) in which derivative
of the scalar field couples with scale factor and its Alternatively, utilizing the Cartan oneform
derivative. Maybe, it is worth to note what happens
whenweinsertthetermas U(ϕ,∇ ϕ∇µϕ)g(T) inthe ∂L
µ θ ≡ dqi (5)
actions of f(T)-gravity context. As mentioned in the L ∂q˙i
first paragraph of the introduction, the “Teleparallel”
equivalent of General Relativity (TEGR). Altogether, and defining the inner derivative
in many cases, the authors construct the actions
of f(T)-gravity via replacing the torsion insisted of iXθL =(cid:104)θL,X(cid:105) (6)
curvature (for example, see [41, 59]). However, when
the non-minimal coupling is switched on the resulting we get,
theory exhibits different behavior. Hence, the last
i θ =Σ , (7)
term in (1) is the inspiring for adding such term. The X L 0
main purpose of the present work is to answer the
provided that (3) holds. The Eq. (7) is coordinate
aforementioned question by having recourse to the
independent. Using a point transformation, the vector
Noether symmetry approach.
field X is rewritten as
Symmetries play a substantial role in the theoretical (cid:20) (cid:21)
physics. It can safely be said that Noether symmetries X˜ =(cid:0)i dQk(cid:1) ∂ + d (cid:0)i dQk(cid:1) ∂ . (8)
X ∂Qk dQk X ∂Q˙k
are a powerful implement both to select models at
a fundamental level and to find exact solutions for
If X isasymmetry,sois X˜ (i.e. X˜L=0),andapoint
specific Lagrangians. In the literature, applications
transformation is chosen such that
of the Noether symmetry in generalized theories of
gravity have been superabundantly studied (for exam-
i dQ1 =1, i dQi =0 (i(cid:54)=1). (9)
ple see [28–50]). Beside this useful approach, another X X
lucrative approach as B.N.S. approach has recently
It follows that
been innovated [51]. B.N.S. approach may carry more
conserved currents than Noether symmetry approach. ∂ ∂L
Furthermore, sometimes Noether symmetry approach X˜ = , =0, (10)
∂Q1 ∂Q1
lacks achieving the purpose. In such cases, utilizing
the B.N.S. approach is hobson’s choice. Also, with this therefore, Q1 is a cyclic coordinate and the dynamics
new procedure, solving process of ordinary differential
can be reduced. However, the change of coordinates is
equation’s system, comprised of field equations and
not unique and a clever choice would be advantageous
conserved currents, is a paved road.
[53].
The Noether theorem states that for a given la-
The structure of the paper is as follows. In sec-
grangian L, defined on the tangent space of configu-
tion(II)weintroducethemodelandextractthepoint-
rations TQ ≡ {q ,q˙ }, if the Lie derivative of the La-
i i like lagrangian and field equations. In section (III) we
grangian L, dragged along a vector field X,
present the Noether symmetries, invariants and exact
∂ ∂ solutions of the model. Moreover, by data analysis, we
X=αi(q) +α˙i(q) , (2) demonstrate that the observational data corroborate
∂qi ∂q˙i
our findings. In section (IV) we sum up the obtained
where dot means derivative with respect to t, vanishes graceful results.
3
II. THE MODEL in which the τ denotes a differentiation with respect
to the torsion T. So, the point-like Lagrangian corre-
Regarding the mentioned points in the second and sponding to the action (11) becomes
thirdparagraphsoftheintroduction(I),wewanttoin-
1
vestigatethefollowinggravitationalactioninextended L=fTa3−hϕ˙ga3−(f−hϕ˙gτ)[Ta3+6a˙2a]− ωϕ˙2a3+Va3.
2
gravity context
(16)
(cid:90) (cid:20) Hence, the Euler-Lagrange equations for the scale fac-
S = d4xe f(ϕ)T− U(ϕ,ϕ ϕ,µ)g(T)
,µ tor a would be
(cid:124) (cid:123)(cid:122) (cid:125)
The Unusual Term
(11) (cid:18)a¨(cid:19) (cid:18)a˙2(cid:19)2
4 (f −hϕ˙gτ)+2 (f −hϕ˙gτ)
(cid:21) a a2
ω(ϕ)
− 2 ϕ,µϕ,µ+V(ϕ) , +4(cid:18)a˙(cid:19)(cid:16)f(cid:48)ϕ˙ −hgτϕ¨−hgττϕ˙T˙ −h(cid:48)gτϕ˙2(cid:17) (17)
√ a
where e = det(ei) = −g with ei being a vierbein
ν ν 1
(tetrad) basis, f(ϕ) is the generic function describing +(Tgτ −g)hϕ˙ − ωϕ˙2+V =0,
2
the coupling between the scalar field and scalar tor-
sion T, ϕ,µ indicates the covariant derivative of ϕ, where the prime indicates a derivative with respect to
U(ϕ,ϕ,µϕ,µ) is the unknown coupling function which ϕ. Forthescalarfield, ϕ,theEuler-Lagrangeequation
we hypothesize it, in general, to depend on scalar field takes the following form
and gradients of it. This function coupled with an un-
known function of torsion g(T). Here, ω(ϕ) and V(ϕ) (cid:18)a¨(cid:19)(cid:18)a˙(cid:19) (cid:18)a˙(cid:19)3 (cid:18)a˙(cid:19)2(cid:16) (cid:17)
are the coupling function and scalar potential, respec- 12hgτ +6hgτ +6 f(cid:48)+hgττT˙
a a a a
tively. Note that the scalars here are caused by confor-
(cid:18) (cid:19)
a˙ 1
mal symmetry [58]. We presume that the geometry of −3 (hg−hgτT +ωϕ˙)+hgττTT˙ −ωϕ¨− ω(cid:48)ϕ˙2−V(cid:48) =0,
spacetimeisdescribedbytheflatFRWmetricwhichis a 2
consistent with the present cosmological observations (18)
ds2 =dt2−a2(t)(cid:2)dx2+dy2+dz2(cid:3), (12) which is the Klein-Gordon equation. The energy func-
tionwhichisthe (cid:0)0(cid:1)-Einsteinequation,associatedwith
where the scale factor a is a function of time. With 0
the point-like Lagrangian (16) is found as
this background geometry, the scalar torsion takes the
form T = −6a˙2/a2. First, for simplifying the action,
(cid:18)a˙(cid:19)2 1
we set the following form by assuming that two main (12hgτϕ˙ −6f) − ωϕ˙2−V =0. (19)
parts of U are separable a 2
U(ϕ,ϕ,µϕ,µ)=h(ϕ)Φ(ϕ˙)=h(ϕ)ϕ˙, (13) And,theEuler-Lagrangeequationforthetorsionscalar
T reads
where h(ϕ) is an unknown function of the scalar field
ϕ,andthedotrepresentsadifferentiationwithrespect (cid:18) a˙2(cid:19)
to t. Wecannotpresentanyphysicalargumentbehind a3hϕ˙gττ T + =0. (20)
a2
such choice for the unknown function Φ(ϕ˙), rather re-
lies on the fact that it works fairly and the Hessian FromEq. (20),therearethreepossibilities: (1) gττ =0
determinant turns out to be zero through this choice
implyinglinearityfor g(T) whichisnotinteresting,for
of function after finding the other unknown coupling
we have linear form of torsion in the action (11), (2)
functionsviaNoetherapproach. Moreover,speakingof
ϕ˙ = 0 which leads to the linear form for the scalar
the last term in (1) and mentioned points in the intro-
field, so it is not suitable and (3) the possibility of
duction, it is better we fit the second main part as ϕ˙
atfirst. Using(11), (13)andtheLagrange’smethodof a˙2
undetermined coefficients, the action (11) can be writ- T =−6 ,
a2
ten as
(cid:90) (cid:20) which is the definition of scalar torsion for flat FRW.
S = d4xe f(ϕ)T −h(ϕ)ϕ˙g(T)
(cid:18) a˙2(cid:19) ω(ϕ) (cid:21)
−λ T +6 − ϕ˙2+V(ϕ) , III. EXACT SOLUTIONS VIA NOETHER
a2 2 SYMMETRY APPROACH AND DATA
(14) ANALYSIS
where the Lagrange multiplier λ is derived by varying
In this section, we utilize the Noether symmetry ap-
the action (14) with respect to T
proach for solving Eqs. (17)-(20). The configuration
λ=f −hϕ˙gτ, (15) spaceofthepoint-likeLagrangian(16)is Q={a,ϕ,T}
4
whosetangentspaceis TQ={a,a˙,ϕ,ϕ˙,T,T˙}. Theex- 1+ξϕ2 +ζϕ4, but on the other hand ω must be di-
istence of the Noether symmetry implies the existence mensionless, so n should be equal to 2. According to
of a vector field X, (26), symmetry generator turns out to be
∂ ∂ ∂ ∂ ∂ ∂ ∂ −3 ∂ ∂ −3 ∂
X=α∂a+β∂ϕ+γ∂T +α,t∂a˙ +β,t∂ϕ˙+γ,t∂T˙, (21) X=a∂a + 2 ϕ∂ϕ +a˙∂a˙ + 2 ϕ˙∂ϕ˙. (27)
where Hence corresponding conserved current is found as
y =y(a,ϕ,T) (cid:16) ω (cid:17)
I=3α a2ϕ 2f ϕa˙ −4f aϕ˙ + 0aϕ˙ . (28)
∂y ∂y ∂y 0 0 0 2
−→ y =a˙ +ϕ˙ +T˙ ; y ∈{α,β,γ},
,t ∂a ∂ϕ ∂T
Since the form of g(T) has been specified (26) and
such that γ = 0, and on the other hand, regarding the third
option in Eq. (20), it is an ineffective shot to tow T in
LXL=0 Q. Therefore, the configuration space reduces to two
∂L ∂L ∂L ∂L ∂L ∂L Q = {a,ϕ}. This means we can rewrite the point-like
−→α +β +γ +α +β +γ =0.
∂a ∂ϕ ∂T ,t∂a˙ ,t∂ϕ˙ ,t∂T˙ Lagrangian (16) free of T by substituting the form of
torsion T. Now, by assuming that X is a symmetry,
This condition yields the following system of linear we seek a point transformations on the vector field X
partial differential equations such that
∂α ∂α ∂β ∂β
i dz =1, i dp=0, (29)
=0, =0, =0, =0, (22) X X
∂b ∂T ∂a ∂T
whereas i : (a,ϕ) → (z,p) in which z = z(a,ϕ) and
(cid:18) (cid:19) p = p(a,ϕ). Here z is a cyclic variable. If we were
∂α
3αV +βaV(cid:48) =0, 2af +αf +aβf(cid:48) =0, to keep T, then it had to be mapped to itself, i.e. i :
∂a
(a,ϕ,T)→(z,p,T). Solving the Eq. (29) leads to
(cid:18) (cid:19)
∂β
2aω +3αω+aβω(cid:48) =0,
∂ϕ p=a3/2ϕ, z =a3/2ϕ+ln(a). (30)
(23)
So, the corresponding inverse transformation is
(cid:18)∂α(cid:19) (cid:18)3 (cid:19)
6αhgτ +6aβh(cid:48)gτ+12ahgτ a(p,z)=exp(z−p), ϕ(p,z)=pexp (p−z) .
∂a 2
(cid:18) (cid:19) (31)
∂β
+6hagτ +6ahγgττ =0, It is clear that (30) is an arbitrary choice since more
∂ϕ
general conditions are possible. The point-like La-
(24)
grangian (16) can be rewritten in terms of cyclic vari-
ables by (31) as
(cid:18) (cid:19)
∂β
(aThgτ −ahg) +aThγgττ +3αThgτ p˙2 (cid:18)3 (cid:19) (cid:18)3 (cid:19)
∂ϕ L= p+1 −p p˙z˙−V p , (32)
2 4 8 0
+βaTgτh(cid:48)−3αhg−βh(cid:48)ga=0.
(25) in which we put ω =1 and f =3/32 or equivalently
0 0
h = 3/16. The Euler-Lagrange equations relevant to
0
This system of linear partial differential equations can
(32) are
be solved by using the separation of variables. Hence,
one may obtain d d
(pp˙)=0 ; (pp˙)≡I (33)
dt dt
f(ϕ)=f ϕn, h(ϕ)=h ϕn−1, V(ϕ)=V ϕn,
0 0 0
g(T)=(−6T)1/n, ω(ϕ)=ω0ϕn−2, α=α0a, (26) 3 (cid:18) 3 (cid:19) 3
β =β0ϕ, γ =0, 8pz¨−p¨ 1+ 4p − 8p˙2−2V0p=0. (34)
in which f , ω , V , h , α , and β are constants
0 0 0 0 0 0 Note that Eq. (33) is equivalent to I which is given by
of integration and β = −3α /n and h = nf . This
0 0 0 0 Eq. (28), so rewriting Eq. (28) in terms of cyclic vari-
solution holds for n = 2, only. Of course, n = 2 has
ables does not add a new equation. And Hamiltonian
a physical nature as well, because the common form
constraint reads
of f(ϕ) can be given by the non-minimal coupling as
f(ϕ)= M2 (cid:0)1 −ξϕ2(cid:1), however, certain Grand-Unified 1 (cid:18)3 (cid:19) 3
2 κ p˙2 p+1 − z˙p˙p+V p2 =0. (35)
theories lead to a polynomial coupling of the form 2 4 8 0
5
One can write const. in right side of Eq. (35) instead firstdeceleratinguniverse, q >0,thenacceleratinguni-
of zero, as in general, we have E = const.. Solving verse, q < 0, and the present value of it, is q = −1,
L 0
Eqs. (33)-(34) through (35) leads to as we expect. Limpidly, q = 0 renders the inflection
point (i.e. shifting from decelerated to accelerated ex-
(cid:112)
p(t)= 2(c t+c ), (36) pansion in figure 1(A)). So, at redshift z = 0.712
1 2 acce.
or equivalently t = 6.198 Gyr acceleration started
acce.
(See figure 3(B)). It coincides with the astrophysical
2 8
z(t)= ln(|c t+c |)+ V t2 data, for observational data concede that at redshift
3 1 2 3 0 z <1, or equivalently at about half the age of the
acce.
(cid:18) (cid:19)
+ c − 16V0c2 t+(cid:112)2(c t+c )+c , universeaccelerationcommenced. Itiswell-knownthat
3 3 c 1 2 4 scalar field must decrease with time. As figure 4(A)
1
(37) shows,ourscalarfieldisconsistentwiththispoint. Fig-
ure 4(B) indicates the manners of the scalar potential
where {c ; i = 1,...,4} are constants of integration. V(ϕ), coupling function U(ϕ,ϕ ϕ,µ) and h(ϕ) ver-
i ,µ
Doing inverse transformations by the use of Eq. (31) sustime. Thebehaviorofscalarpotentialisadmissible
give the solutions of Eqs. (17)-(19) and I=0 (See Eq. (i.e. detractive in face of time). With an eye to action
(28)) as (11), we found that the terms U and V have different
sign. So, by applying the minus sign of U we learn
that both V and U are positive and have subtractive
(cid:18) (cid:20) (cid:21) (cid:19)
8 16V c
a(t)=(c t+c )2/3exp V t2+ c − 0 2 t+c , behaviors versus time. A marked difference between
1 2 3 0 3 3 c1 4 those is that U falls sharply than V . It is obvious
(38) that h has a narrow band around zero, and may this
restrictsthebehaviorof U. Inaddition, allthesethree
(cid:16) (cid:104) (cid:105) (cid:17)
ϕ(t)= exp −4V0t2− 32 c3− 136Vc01c2 t− 32c4 . (39) fduiffnecrtieonncsetfurronmouthtattoobfetahlemporsetsceonntsttiamntes. wBitehfoarelittetrle-
(cid:112)
2(c t+c )
1 2 minating this section, we would like to investigate the
Om-diagnostic analysis. In this manner, we present
Therefore, these solutions carry two conserved cur-
the figure 5 which shows the Om-diagnostic parameter
rents; E (Eq. (19))and I (Eq. (28))thatcorrespond
L versusredshift. TheOm-diagnostic isanimportantge-
to two symmetry generators X = ∂/∂t and X (Eq.
1 ometrical diagnostic proposed by Sahni et al. [54], in
(27)), respectively.
ordertoclassifythedifferentdarkenergy(DE)models.
The Om is able to distinguish dynamical DE from the
For illustrating the descriptions of late-time-
cosmological constant in a robust manner both with
acceleratedexpansionfromtheperspectiveofthestud-
and without reference to the value of the matter den-
ied model, we single out the constants as below
sity. It is defined as [55]
c =1, c =0.6068, c =−0.0162,
1 2 3 (cid:104) (cid:105)2
(40) H(z) −1
c4 =−1.8340, V0 =0.0006. Om(z)≡ H0 . (41)
(1+z)3−1
We present five figures with data analysis with time
unit 1 Gyr ≡ 1. The behavior of the scale factor a Fordarkenergywithaconstantequationofstate(EoS)
versus time and redshift in figure 1 imply; first, the ω, it reads
scale factor with increasing nature expressing initially
the decelerated and then accelerated expansion of the (1+z)3(1+ω)−1
Om(z)=Ω +(1−Ω ) , (42)
universe (See figure 1(A)), second, the scale factor ver- m0 m0 (1+z)3−1
sus redshift plot 1(B), confirms that the present value
of the scale factor is exactly 1, so, from figure 1(A) we so, Om(z) = Ωm0 states the ΛCDM model, there-
learn that the age of universe is t0 = 13.816 Gyr. As fore the regions Om(z) > Ωm0 and Om(z) < Ωm0
usual,ignoringsmallvariationoftheprefactor,wecon- correspond with quintessence (ω > −1) and phantom
sider that the CMBR temperature falls as a−1, then, (ω < −1), respectively. The figure 5 indicates phase
the present value of it is, T = 2.725 K (See figure crossing from quintessence (Om(z)(cid:38)0.3) to phantom
0
2(A)).TheevolutionoftheHubbleparameter(notpre- (Om(z)(cid:46)0.3). Hence, now we are in phantom phase.
sented) gives its present value, H = 7.238 × 10−11
0
yr−1 ≡70.82 Km.s−1.Mpc−1. Itisaniceresult,since
the recent observational data tell H = 72.25±2.38
0
Km.s−1.Mpc−1 [60]. Therefore H ×t = 1, which IV. CONCLUSION
0 0
fits the observational data with high precision, as plot-
ted in figure 2(B). Figure 3(A) indicates deceleration Owing to the mentioned fact in the introduction (I),
parameter, q = −(aa¨)/a˙2, versus time plot. It shows weappliedtheterm U(ϕ,ϕ ϕ,µ)g(T) withinageneric
,µ
a pass from negative to positive values which states action in f(T)-gravity context in FRW background
6
FIG. 1: Plot (A) indicates the scale factor a(t) versus time t at the time range [0.0006,13.816] while plot (B) shows the
scale factor a(t) versus redshift z at the redshift range [0,7].
FIG. 2: Plots (A) and (B) indicate the temperature T(t) and t×H(t) versus redshift z at the redshift range [0,3.5],
respectively.
space-time. Whereas the resulting system was overde- entering accelerated expansion epoch, became a
termined,soforsimplifyingtheactionwedidasuitable victim of Friedmann-like matter dominated era
choicefortheunknownfunction U =h(ϕ)ϕ˙ byassum- for quite a long time. Thus, acceleration dawns
ingtheseparationoftwomainparts. Thenbyapplying at the redshift value z = 0.712 which is
acce.
the Noether symmetry approach and using cyclic vari- equivalent to t = 6.198 Gyr that is about
acce.
ablesmethod,wecouldreachatniceresultsasourdata half the age of the universe.
analysis showed deep compatible results with observa-
4. The present value of the Hubble parameter is
tional data.
H =7.238×10−11 yr−1 or equivalently
In a nutshell, some of our noteworthy findings were, 0
H =70.82 Km.s−1.Mpc−1.
0
1. The present value of the scale factor is a = 1
0
(i.e. (z ,a ) = (0,1)), so the present value of
0 0
temperature may be found as T0 =2.725 K. 5. H0×t0 =1.
2. Age of the universe t =13.816 Gyr. 6. The present value of the deceleration parameter
0
is q =−1.
0
3. The resulting model can give Late-time-
accelerated expansion. The universe before 7. Om-diagnostic shows phase crossing from
7
FIG.3: Plot(A)indicatesthedecelerationparameter q(t) versustime t atthetimerange [1,13.816] whileplot(B)shows
the deceleration parameter q(t) versus redshift z at the redshift range [0,3.5].
FIG.4: Plot(A)indicatesthescalarfiled ϕ(t) versustime t atthetimerange [0.0006,13.816] andplot(B)showscoupling
function U(ϕ,ϕ ϕµ), h(ϕ) and the scalar potential V(ϕ) versus time t at the same time range.
µ
quintessence to phantom phase. effect (at least in studied case) but it may give desired
results.
The added extra function, U(ϕ,ϕ ϕ,µ), affected like
,µ
the scalar potential V(ϕ) with the difference that
Acknowledgment
U decayed sharply than V . Moreover, around the
present time, their amount had negligible difference.
Considering the deep compatible results with astro-
physical and observational data, we conclude that I thank Professor Sergei D. Odintsov for useful
entering such term, U(ϕ,ϕ ϕ,µ)g(T), to the scope of remarks.
,µ
theactionsof f(T)-gravity,notonlyhasnoanomalous
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