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Dark Energy Models and Cosmic Acceleration with Anisotropic Universe in f(T) Gravity
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2014 Commun. Theor. Phys. 61 482
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Commun. Theor. Phys. 61 (2014) 482–490 Vol. 61, No. 4, April 1, 2014
Dark Energy Models and Cosmic Acceleration with Anisotropic Universe in f(T )
Gravity
M. Sharif∗ and Sehrish Azeem†
Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan
(Received May 27, 2013; revised manuscript received August 20, 2103)
Abstract This paper is devoted to studing the accelerated expansion of the universe in context of f(T ) theory
of gravity. For this purpose, we construct different f(T ) models and investigate their cosmological behavior through
equation of state parameter by using holographic, new agegraphic and their power-law entropy corrected dark energy
models. We discuss the graphical behavior of this parameter versus redshift for particular values of constant parameters
in Bianchi type I universe model. It is shown that the universe lies in different forms of dark energy, namely quintessence,
phantom, and quintom corresponding to the chosen scale factors, which depend upon the constant parameters of the
models.
PACS numbers: 04.50.Kd, 98.80.Jk, 95.36.+x
Key words: f(T ) gravity, equation of state, dark energy models
1 Introduction
The current observational data shows that the present
universe is not only expanding but also accelerating due
to unknown mysterious component called dark energy(DE).[1] There are two possible ways to explain the ac-
celerating expansion of the universe. One is to introduce
DE with large negative pressure and positive density inthe context of general relativity. Another possibility is to
explain this expansion through modified theories of grav-
ity. The simplest form of DE is the cosmological constant,
which has p = −ρ. However, it has two problems, knownas fine tuning and coincidence problems. Dark energy den-
sity can be evaluated through an equation of state (EoS)
parameter, ω = p/ρ. There are several forms of dynam-ically varying DE phases related with negative behavior
of EoS parameter, i.e., −1 < ω < −1/3 corresponds to
the quintessence phase, ω < −1 relates phantom phase(the most favorable phase of DE[2−3]). The combination
of both quintessence and phantom phases is known as
quintom model, which can cross the phantom divide lineω = −1.[4−5]
The nature of DE can also be studied by using sev-
eral models such as holographic DE (HDE),[6−7] new age-graphic DE (NADE),[8] power-law entropy corrected HDE
(PLECHDE) and power-law entropy corrected NADE
(PLECNADE) models, which are frequently used in lit-erature. The HDE is defined as
ρΛ = 3c2M2pL−2, (1)
where 3c2 is dimensional constant used for convenience, L
is the infrared cutoff and Mp denotes the reduced Planckmass, Mp = (8πG)−1/2, G is the gravitational constant.
The original ADE model can be obtained by taking theage of the universe as the infrared cutoff. However, itcannot explain the matter dominated era. Therefore, theNADE model was developed when the age of the universeis replaced by the conformal time η. This shows that DEreceives from spacetime and matter field fluctuation in theuniverse.[9] Mathematically, its energy density is given by
ρΛ = 3n2M2pη−2,
where the numerical factor 3n2 is defined to parameterizesome uncertainties.
The modified theories of gravity, i.e., f(R), f(G),f(R, T ), f(T ), scalar tensor theory and Brans-Dick the-ory have gained a lot of interest to explain the natureof DE.[10−11] The f(T ) gravity is the generalization ofteleparallel theory of gravity (TPG) in which curvature-free Weitzenbock connection is used instead of torsionlessLevi–Civita connection in general relativity.[12−14] Thistheory is obtained by replacing the torsion scalar T in theLagrangian of TPG to its arbitrary function f(T ) thatyields the field equations of second order as compared tof(R) theory with fourth order field equations.[15]
Bamba et al.[16] investigated the behavior of differentf(T ) models through EoS parameter for DE. They foundthat the crossing of phantom divide line does not crossin exponential and logarithmic models but it occurs intheir combined model. Karami and Abdolmaleki[17] dis-cussed EoS parameter for HDE, NADE and their entropycorrected models in f(T ) theory. They concluded thatphantom crossing is obtained in entropy corrected modelsexcept for the first two models. The same authors[18] alsoapplied the same procedure for PLECHDE and PLEC-NADE models and analyzed that crossing of the phantom
∗E-mail: [email protected]†E-mail: [email protected]
c© 2014 Chinese Physical Society and IOP Publishing Ltd
http://www.iop.org/EJ/journal/ctp http://ctp.itp.ac.cn
No. 4 Communications in Theoretical Physics 483
divide line is consistent with recent cosmological observa-tional data. Sharif and Azeem[19] described the graphicalrepresentation of some f(T ) models and checked their cos-mological behavior with the help of EoS parameter of DE.There are many papers available in literature where f(T )gravity has been used widely.[20−24]
In this paper, we assume locally rotationally symmet-ric (LRS) Bianchi type I (BI) universe model to explorecosmic evolution for some DE models. For this purpose,we evaluate the EoS parameter of torsion contribution.The paper is organized as follows. In Sec. 2, we give basicformalism of the generalized teleparallel theory of gravity.Section 3 is devoted to discuss about its field equations forBI universe and construct EoS parameter for HDE, NADEand their power-law entropy corrected models. The graph-ical behavior of this parameter for these models are dis-cussed in Sec. 4. We summarize and conclude our resultsin the last Sec. 5.
2 Generalized Teleparallel Theory of Gravity
In this section, we briefly describe some preliminariesof f(T ) theory of gravity. The corresponding action withmatter is of the form[17−18,20]
I =1
16πG
∫
d4xe[f(T ) + Lm] ,
where T is the torsion scalar, f(T ) is its differentiablefunction, Lm is the matter Lagrangian, and e =
√−g.The torsion scalar is defined as
T = SρµνT ρ
µν , (2)
where the torsion and the antisymmetric tensors are givenas
T ρµν = Γρ
νµ − Γρµν = hρ
i (∂µhiν − ∂νhi
µ) , (3)
Sρµν =
1
2(Kµν
ρ + δµρ T θν
θ − δνρT θµ
θ) . (4)
Here Γρνµ is the Weitzenbock connection and hi
µ(x) is atetrad field. It is orthonormal set used to determine thegeometrical structure of TPG.
The relationship between metric tensor and tetrad canbe defined as
gµν = ηijhiµhj
ν , (5)
where ηij = diag(1,−1,−1,−1) is the Minkowski space-time for the tangent space. Latin alphabets (i, j, . . . =0, 1, 2, 3) are used to denote tangent space indices andGreek alphabets (µ, ν, . . . = 0, 1, 2, 3) are for spacetimeindices. The only non-zero tetrad field hi, responsible fornon-vanishing torsion tensor, can be written as
hi = hiµ∂µ, hj = hj
ν dxν , (6)
which satisfies[25−26]
hiµhj
µ = δij , hi
µhiν = δµ
ν . (7)
The corresponding field equations are obtained byvarying the action with respect to tetrad field as[27]
4[e−1∂µ(eSiµν) − hλ
i T ρµλSρ
νµ]fT + 4Siµν∂u(T )fTT
− hνi f(T ) = −2κ2hρ
i T νρ , (8)
where Siµν = hρ
i Sρµν , κ2 = 8πG = M−2
p , fT = df/dT ,fTT = d2f/dT 2. The energy-momentum tensor Tµν forperfect fluid is
T νρ = diag(ρM ,−pM ,−pM ,−pM ) , (9)
where ρM and pM denote the energy density and pressureof matter inside the universe.
3 Bianchi I Universe and Some CosmologicalParametersSpatially homogeneous cosmological models play an
important role in attempts to understand the structureand properties of the space of all cosmological solutions.It has been pointed out that some large-angle deviationsare seen in CMB radiations, which lead to the violationof isotropy of the observable universe. Thus, the universemay have effected a slight anisotropic geometry in cos-mological models regardless of inflationary model. Fora better description of these deviations, plane symmet-ric homogeneous but anisotropic universe models play avery significant role in modern cosmology. To study thepresent day accelerated expansion of the universe underthe possible effects of anisotropy, we consider BI universemodel[29]
ds2 = dt2 − A2(t)dx2 − B2(t)(dy2 + dz2) , (10)
which has one transverse direction x and two equivalentlongitudinal directions y and z. For a given metric, thereare infinite choices of tetrad fields satisfying the properties(7). However the procedure of measuring the tetrad com-ponents for a flat universe model in Cartesian coordinatesis obvious. It is just a diagonal set of tetrad with squareroot of metric coefficients.[30] Using Eqs. (5) and (10), thetetrad components[31] are
hiµ = diag(1, A, B, B) ,
hµi = diag(1, A−1, B−1, B−1) . (11)
Using Eqs. (3) and (4) in (2), the torsion scalar takesthe following form
T = −2(
2AB
AB+
B2
B2
)
, (12)
where dot denotes derivative with respect to t. The cor-responding average scale factor R, the Hubble parameterH , and the anisotropy parameter ∆ will become
R = (AB2)1/3, H =1
3
( A
A+ 2
B
B
)
,
∆ =1
3
3∑
i=1
(Hi − H
H
)2
, (13)
where H1 = A/A, H2 = B/B = H3 are directional Hub-ble parameters along x, y and z-axes respectively. Fori = 0 = ν and i = 1 = ν in Eq. (8), the correspondingfield equations turn out to be
f(T ) + 4(
2AB
AB+
B2
B2
)
fT = 2κ2ρM , (14)
484 Communications in Theoretical Physics Vol. 61
4( AB
AB+
B2
B2+
B
B
)
fT − 16B
B
[ B
B
( A
A− A2
A2
)
+(B
B− B2
B2
)( B
B+
A
A
)]
fTT + f(T ) = −2κ2PM . (15)
The expansion and shear scalars become
Θ =A
A+ 2
B
B, σ =
1√3
( A
A− B
B
)
. (16)
Observations of velocity red shift relation for extra-galactic sources suggest that Hubble expansion of the uni-verse is isotropic within about 30% range approximatelywhile red shift studies place the limit σ/H ≤ 30. In this re-spect, Collins[32] has generally discussed the physical sig-nificance of this condition for some EoS and perfect fluid.Thus, the proportionality of expansion scalar Θ to theshear scalar σ for a spatially homogeneous metric yields
A(t) = Bm(t), m > 1 . (17)
This condition usually used to find out the exact solu-tions of cosmological models. Yadav and Saha[33] foundthat the anisotropic dominance of DE represents the ac-celerated expansion of the universe with the help of thiscondition. The anisotropy parameter of expansion takesthe form
∆ = 2(m − 1)2
(m + 2)2. (18)
Notice that the isotropic behavior of the expanding uni-verse is obtained for ∆ = 0. Using Eq. (17), the fieldequations reduce to
36(2m + 1)H2
(m + 2)2fT + f(T ) = 2κ2ρM , (19)
432(2m + 1)H2H
(m + 2)3fTT − 2(18H2 + 6H)
m + 2fT − f(T )
= 2κ2PM . (20)
These equations can be written as
9(2m + 1)
κ2(m + 2)2H2 = ρM + ρT , (21)
3
(m + 2)κ2
( 9H2
m + 2+ 2H
)
= pM + pT , (22)
where
ρT =1
2κ2(2TfT − f − T ) , (23)
pT = − 1
2κ2
(−24T H
m + 2fTT +
(2(m + 2)T
2m + 1− 12H
m + 2
)
fT −f
+12
m + 2H − 3
2m + 1T
)
. (24)
The corresponding EoS parameter turns out to be
ωT = −1
+24THm+2 fTT + 12H
m+2fT − 12Hm+2 + 2(m−1)
2m+1 T (1 + fT )
2TfT − f − T. (25)
The conservation law yields
ρM + 3H(ρM + pM ) = 0 ,
ρT + 3H(ρT + pT ) = 0 .
In the next section, we construct some f(T ) gravity
models by using two different definitions of scale factor
and then calculate the corresponding EoS parameters.
4 Construction of f(T ) Models and Their Cor-responding EoS Parameters
Since there are mainly two possible ways of workingwith cosmological equations of motion: either postulat-
ing a theory with matter content of the universe and then
solving the corresponding equation to discuss the cosmo-
logical time behavior of the model under consideration orvice versa, postulating a theory with desired time behav-
ior of the model deriving information about the matter
content. We have used the second method in this paper
by assuming two types of scale factors and construct f(T )models. Power-law scale factors give consistent results
about the accelerated expansion of the universe in which
pole-like and exact power-law types are usually used inmodified gravity theories. The pole-like type scale factor
is defined by[34−35]
B(t) = b0(ts − t)−h, t ≤ ts, h > 0 .
Here ts is the finite future singularity time. At t = ts, this
scale factor indicates the superaccelerated universe with a
big-rip singularity. The pole-like scalar factor representsconsistent results as the observations indicate the phan-
tom dominated universe. Using this scale factor, it follows
that
H(t) =(m + 2)h
3(ts − t), T =
−2(2m + 1)h2
(ts − t)2,
H =m + 2
2m + 1
(−T
6h
)
, (26)
where (ts − t) is defined in terms of redshift z as
ts − t = (1 + z)1/h.
Another class of scale factor is called exact power-law
scale factor given by[34]
B(t) = b0th, h > 0 .
For h > 0, this scale factor represents an expanding uni-
verse while contracting universe relates with h < 0. Forthis type of scale factor, there is a class of cosmological
models, which dynamically solves the cosmological con-
stant problem regardless of the matter content. Here
H(t) =(m + 2)h
3t, T =
−2(2m + 1)h2
t2,
H =m + 2
2m + 1
( T
6h
)
, (27)
where t = (1 + z)−1/h. Now we consider four different
models of DE to construct f(T ) gravity models and con-
sequently EoS parameter.
4.1 Holographic Dark Energy Model
Firstly, we assume HDE model in f(T ) gravity. The
future event horizon Rh which is equal to L for the flat
No. 4 Communications in Theoretical Physics 485
universe can be defined as[7]
Rh = B(t)
∫ ∞
t
dt
B(t). (28)
For pole-like scale factor, it yields
Rh = B(t)
∫ ts
t
dt
B(t)
=(ts − t)
h=
h
h + 1
(4m + 2
−T
)1/2
. (29)
Thus the HDE density (1) becomes
ρΛ = − 3c2
2κ2(2m + 1)
(h + 1
h
)2
T . (30)
In order to get f(T ) gravity model, we take HDE density(30) as torsion density (23), i.e., ρΛ = ρT , and obtain thefollowing HDE f(T ) model as
f(T ) = ǫ√
T − (γ − 1)T , (31)
where ǫ is an integration constant and γ is defined as
γ =3c2
2m + 1
(h + 1
h
)2
.
Here γ depends upon c, which is equal to 0.818 for theflat universe.[36] This model (31) satisfies the condition
f(T ) → 0 as T → 0 to be a realistic model.[37]
The EoS parameter is obtained by using Eq. (31) in
(25)
ωT = −1 − 2
(2m + 1)h− 2(m − 1)
(2m + 1)γ
×(
2 − γ +ǫ
2√
T
)
, h > 0 . (32)
The corresponding torsion scalar in terms of redshift canbe defined as
T = −2(2m + 1)h2
(1 + z)2/h. (33)
Using Eq. (33) in (32), we obtain EoS parameter in termsof z shown in Fig. 1. We see that ωT is always less than−1 and becomes constant showing that the universe lies
in phantom region.
Fig. 1 Plot of ωT versus z for HDE model with c =0.818, h = 0.5, m = 3, ǫ = 0 for pole-like scale factor.
For the power-law scale factor, the future event hori-zon can be obtained by using Eqs. (27) and (28)
Rh =t
h − 1=
h
h − 1
(4m + 2
−T
)1/2
, (34)
where h > 1 for finite future event horizon Rh. The cor-responding energy density is
ρΛ = − 3c2
2κ2(2m + 1)
(h − 1
h
)2
T . (35)
Equating Eqs. (23) and (35), we obtain the same f(T )gravity model given in Eq. (31) with γ
γ =3c2
2m + 1
(h − 1
h
)2
.
The corresponding EoS parameter turns out to be
ωT = −1 +2
(2m + 1)h− 2(m − 1)
(2m + 1)γ
×(
2 − γ +ǫ
2√
T
)
, h > 1 . (36)
The torsion scalar for power-law scale factor in terms of zcan be defined as
T = −2(2m + 1)h2(1 + z)2/h. (37)
Replacing this value in Eq. (36), we obtain ωT whosegraphical representation is shown in Fig. 2. We see thatωT ≪ −1 for particular values of h and m, hence thismodel for power-law scale factor does not give useful re-sults.
Fig. 2 Plot of ωT versus z for HDE model with c =0.818, h = 3, m = 3, ǫ = 0 for power-law scale factor.
4.2 New Agegraphic Dark Energy Model
The energy density of NADE is defined as
ρΛ =3n2
κ2η2, (38)
where n is a constant which has value 2.716 for flat uni-verse model[38] and η is the conformal time given by
η =
∫
dt
a=
∫
da
Ha2.
In terms of pole-like scale factor, it turns out to be
η =(ts − t)h+1
b0(h + 1)=
hh+1
b0(h + 1)
(4m + 2
−T
)(h+1)/2
. (39)
The corresponding energy density becomes
ρΛ =6n2b2
0(h + 1)2
2κ2
( −T
(4m + 2)h2
)h+1
. (40)
Consequently, we obtain new agegraphic f(T ) gravitymodel
f(T ) = ǫT 1/2 + T +γ
2h + 1T h+1, (41)
486 Communications in Theoretical Physics Vol. 61
where ǫ is an integration constant and γ takes the follow-
ing form
γ =6n2b2
0(h + 1)2
(−(4m + 2)h2)h+1.
It also satisfies the condition for realistic model similar to
the HDE f(T ) model. Inserting Eq. (41) in (25), the EoS
parameter becomes
ωT = −1 − 2(h + 1)
(2m + 1)h+
2(m − 1)
(2m + 1)γT h
×(
2 +(h + 1)
2h + 1γT h +
ǫ
2√
T
)
, h > 0 . (42)
Using Eq. (33) in the above equation, we get the EoS pa-
rameter ωT for pole-like scale factor in terms of redshift
z. Its graphical behavior is shown in Fig. 3. The EoS pa-
rameter shows that the universe lies in the region in which
the value of ωT is not more than −1, so it converges to
the phantom era.
Fig. 3 Plot of ωT versus redshift z for NADE modelwith n = 2.716, h = 0.5, m = 3, b0 = 1, ǫ = 0 forpole-like scale factor.
The conformal time for power-law scale factor can be
expressed as
η =t1−h
b0(1 − h)=
h1−h
b0(1 − h)
(4m + 2
−T
)(1−h)/2
,
0 < h < 1 , (43)
where h must be less than 1 due to finite η. The corre-
sponding energy density is
ρΛ =6n2b2
0(1 − h)2
2κ2
( −T
(4m + 2)h2
)1−h
. (44)
The resulting f(T ) gravity model is
f(T ) = ǫT 1/2 + T +γ
1 − 2hT 1−h,
γ =6n2b2
0(1 − h)2
(−(4m + 2)h2)1−h. (45)
This satisfies the realistic model condition f/T → 1 as
T → ∞ at high redshift and is responsible for the accel-
erated expansion of the universe. This is consistent with
cosmic microwave background constraints and the primor-
dial nucleosynthesis.[39]
Fig. 4 Plot of ωT versus redshift z for NADE model (forpower-law scale factor) with n = 2.716, m = 3, b0 = 1,ǫ = 0, h = 0.2 in (a) and h = 0.212 in (b).
The EoS parameter for NADE model (for power-lawscale factor) turns out to be
ωT = −1 +2(1 − h)
(2m + 1)h+
2(m − 1)T h
(2m + 1)γ
×(
2 +(h − 1)
2h − 1γT−h +
ǫ
2√
T
)
, 0 < h < 1 . (46)
The graphical behavior of ωT (z) is shown in Fig. 4. Ini-tially, the behavior of ωT is positive and stays in radiation(ωT = 1/3) and then matter (ωT = 0) dominated era forvery small region. After the short interval, it enters inquintessence era (ωT > −1) that shows the acceleratingexpansion of the universe as z increases for 0 < h ≤ 0.211.As the value of h exceeds from 0.211, ωT crosses the phan-tom divide line and enters in phantom phase as shown inFig. 4(b).
4.3 Power-Law Entropy Corrected Holographic
Dark Energy Model
The PLECHDE density is
ρΛ =3c2
κ2R2h
− β
κ2Rαh
, (47)
where α and β are constants. The HDE density is recov-ered for β = 0. Using Eq. (29) in (47), the PLECHDEdensity can be written as
ρΛ = − 3c2
2κ2
(1 + h
h
)2( T
2m + 1
)
− β
2κ2
(1 + h
h
)2( −T
2m + 1
)α/2
. (48)
No. 4 Communications in Theoretical Physics 487
The corresponding PLECH f(T ) gravity model for pole-like scale factor is
f(T ) = (1 − γ)T + ǫ√
T +δ
1 − α(−T )α/2, (49)
where
γ =3c2
2m + 1
(h + 1
h
)2
,
δ = 2β( h + 1
√
2(2m + 1)h
)α
,
and ǫ is an integration constant and satisfies the conditionto be a realistic model.[37] Replacing Eq. (49) in (25), ωT
takes the following expression
ωT = −1 − 1
(2m + 1)h
(2Tγ + (−T )α/2αδ
Tγ + (−T )α/2δ
)
− (m − 1)
2m + 1
(2(2 − γ)T + (−T )α/2αδ(1−α) + ǫT 1/2
Tγ + (−T )α/2δ
)
. (50)
This can be calculated in terms of redshift z by inserting Eq. (33) in the above equation. The evolution of ωT as afunction of z is shown in Fig. 5. The behavior of ωT depends upon the constant parameters of the model, which are tobe taken properly. As z increases, it crosses the phantom divide line from quintessence phase to phantom phase andremains in it that causes the accelerating expansion of the universe for α ≥ 3 shown in Fig. 5(a). For α ≤ 2, ωT showsthe same behavior as phantom HDE model shown in Fig. 5(b). The universe is also lying in phantom phase for smalland positive values of β with α ≤ 2.
Fig. 5 Plot of ωT versus z for PLECHDE model (pole-like scale factor) with c = 0.818, β = −0.5, h = 2, m = 3, ǫ = 0,α = 3 in left graph and α = 2 in right graph.
Fig. 6 Plot of ωT versus z for PLECHDE model (power-law scale factor) with c = 0.818, β = −0.90, h = 2, m = 3,ǫ = 0, α = −7 in (a) and α = 7 in (b).
For power-law scale factor, the energy density for PLECHDE is
ρΛ = − 3c2
2κ2
(h − 1
h
)2( T
2m + 1
)
− β
2κ2
(h − 1
h
)2( −T
2m + 1
)α/2
. (51)
Using Eqs. (24) and (51), we obtain the same f(T ) gravity model as (49) with
γ =3c2
2m + 1
(h − 1
h
)2
, δ = 2β( h − 1
√
2(2m + 1)h
)α
.
The corresponding EoS parameter is
ωT = −1 +1
(2m + 1)h
(2Tγ + (−T )α/2αδ
Tγ + (−T )α/2δ
)
− (m − 1)
2m + 1
(2(2 − γ)T + (−T )α/2αδ(1−α) + ǫT 1/2
Tγ + (−T )α/2δ
)
, (52)
488 Communications in Theoretical Physics Vol. 61
which can be expressed in terms of z by replacing Eq. (37) in (52). The cosmological behavior of this parameter for
positive and negative value of α is shown in Fig. 6. The similar behavior of ωT with negative α for the range z > 1
is obtained for PLECHDE model (in Fig. 6(a)) as for HDE (power-law model). However for z < 1, it represents
the increasing behavior and induces the value −1.5 < ωT < −1, which is the phantom era. For positive α, the EoS
parameter indicates the opposite behavior as shown in Fig. 6(b), i.e., it shows the phantom era for z > 2 whereas no
useful results are obtained for z < 2 correspond to accelerated expansion of the universe.
4.4 Power-Law Entropy Corrected New Agegraphic Dark Energy Model
For this model, the PLECNADE density is defined as
ρΛ =3n2
κ2η2− β
κ2ηα. (53)
This reduces to Eq. (38) (NADE density) for β = 0. For pole-like scale factor, we obtain PLECNADE density as
ρΛ =3n2b2
0(h + 1)2
κ2(2(2m + 1)h2)h+1(−T )h+1 − βbα
0 (h + 1)α
κ2(√
2(2m + 1)h)α(h+1)(−T )α(h+1)/2. (54)
The corresponding PLECNADE f(T ) gravity model can be obtained as follows
f(T ) = ǫ√
T + T +λ
1 + 2h(−T )h+1 +
δ
1 − α(h + 1)(−T )α(h+1)/2, (55)
where
λ =6n2b2
0(h + 1)2
(2(2m + 1)h2)h+1, δ =
2βbα0 (h + 1)α
(√
2(2m + 1)h)α(h+1).
This model represents a realistic model by satisfying the condition f(T ) → 0 as T → 0. The corresponding EoS
parameter is
ωT = −1 − 1 + h
(1 + 2m)h
(−2λ + (−T )(α/2−1)(h+1)αδ
−λ + (−T )(α/2−1)(h+1)δ
)
+m − 1
2m + 1
×(4T + ǫ
√T + 2(1+h)(−T )h+1λ
1+2h + (1+h)(−T )(h+1)α/2αδ1−(1+h)α
(−T )h+1λ − (−T )(h+1)α/2δ
)
. (56)
Figure 7 can be plotted in terms of redshift z. The behavior of ωT is the same as that of NADE model (for pole-like
scale factor) and the universe always stays in phantom phase without crossing the phantom divide line.
Fig. 7 Plot of ωT versus redshift z for PLECNADE model (for pole-like scale factor) with n = 2.716, α = −0.4,β = −1.99, h = 2, m = 3, ǫ = 0.
For power-law scale factor, the energy density for PLECNADE model is
ρΛ =3n2b2
0(1 − h)2
κ2(2(2m + 1)h2)1−h(−T )1−h − βbα
0 (1 − h)α
κ2(√
2(2m + 1)h)α(1−h)(−T )α(1−h)/2. (57)
No. 4 Communications in Theoretical Physics 489
Fig. 8 Plot of ωT versus z for PLECNADE model (for power-law scale factor) with c = 0.818, β = −0.90, h = 0.4,m = 3, ǫ = 0, α = 4 in (a) and α = −4 in (b).
The resulting f(T ) model can be expressed as
f(T ) = ǫ√
T + T +λ
1 − 2h(−T )−h+1 +
δ
1 − α(1 − h)(−T )α(1−h)/2, (58)
where
λ =6n2b2
0(1 − h)2
(2(2m + 1)h2)1−h, δ =
2βbα0 (1 − h)α
(√
2(2m + 1)h)α(1−h).
Similar to the NADE f(T ) model (power-law scale factor), the PLECNADE f(T ) model is also realistic model, whichsatisfies the corresponding condition.[39] Inserting Eq. (58) in (25), we obtain ωT
ωT = −1 − h − 1
(1 + 2m)h
(−2λ + (−T )(α/2−1)(1−h)αδ
−λ + (−T )(α/2−1)(1−h)δ
)
+m − 1
2m + 1
×(4T + ǫ
√T + 2(h−1)(−T )1−hλ
2h−1 + (1−h)(−T )(1−h)α/2αδ1−(1−h)α
(−T )1−hλ − (−T )(1−h)α/2δ
)
, (59)
where 0 < h < 1. This can also be written in terms of zby using torsion scalar (37) and is shown in Fig. 8. Forpositive increment in α, ωT always rests in quintessencephase causing the accelerating expansion of the universeas shown in Fig. 8(a). ωT evolves from phantom phase tonon-phantom phase by crossing the phantom divide lineand remains in non-phantom phase with negative valuesof α shown in Fig. 8(b).
5 Summary and ConclusionThe main purpose of this paper is to investigate the
cosmological evolution of EoS parameter ωT in the contextof f(T ) gravity for BI universe model. We have recon-structed f(T ) models with the correspondence of its en-ergy density with that of different DE models i.e., HDE,NADE, PLECHDE, and PLECNADE. All these modelssatisfy the conditions to be a realistic model. We have dis-played the EoS parameter in terms of redshift z for pole-like and power-law scale factors. To study the behaviorof ωT , the appropriate values of constants are used. Thegraphical representation of the phantom and non-phantomphases of the expanding universe is examined. The behav-ior of these models can be summarized as follows:
• The HDE f(T ) gravity model shows that the phaseof the universe depends upon the scale factors.For pole-like scale factor, the universe converges tophantom era for particular value of m and h, while
power-law scale factor does not show any useful re-sults for accelerated expansion.
• For NADE f(T ) model, ωT remains in phantomphase for pole-like scale factor, which is consistentwith recent observations. The universe behaves likequintessence DE era for power-law scale factor inthe range 0 < h < 0.212, otherwise, it indicates thecrossing of phantom divide line.
• In PLECHDE f(T ) model, ωT (z) shows that theuniverse transits from the non-phantom to phantomphase for α ≥ 3 with pole-like scale factor. The be-havior of ωT remains the same as that of HDE forpole-like scale factor with α ≤ 2, β < 0 and smallpositive β. On the other hand, for power-law scalefactor, ωT (z) shows the phantom era for z < 1 andz > 2 taking negative and positive values of α re-spectively. Beyond this range, EoS parameter doesnot give fruitful results.
• For PLECNADE f(T ) model, ωT (z) has the samebehavior as that of NADE model whereas the ωT (z)(for power-law scale factor) shows transition fromphantom to quintessence phase with α < 0 by usingappropriate values of constant parameters.
We conclude that the cosmological evolution of EoSparameter is one of the best attempts in observations like
490 Communications in Theoretical Physics Vol. 61
SNIa and WMAP data today. All the above models yield
different behavior of accelerating expansion of the universe
in DE era for arbitrary values of parameters and integra-
tion constant. It is worthwhile to mention here that our
results correspond to the FRW universe model.
Acknowledgments
We would like to thank Miss Shamaila Rani for usefuldiscussions.
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