PILGRIM DARK ENERGY IN f(T) GRAVITY
M. Sharif* S. Rani**
Department of Mathematics, University of the Punjab Lahore-54590, Pakistan
Received January 31, 2014
We discuss the interacting f(T) gravity with pressureless matter in an FRW spacetime. We construct an f(T) model by following the correspondence scheme incorporating a recently developed pilgrim dark energy model and taking the Hubble horizon as the IR cutoff. We use constructed model to discuss the evolution trajectories of the equation-of-state parameter, the wr — phase plane, and state-finder parameters in the evolving universe. It is found that the equation-of-state parameter gives a phantom era of the accelerated universe for some particular range of the pilgrim parameter. The ut —ui'T plane represents freezing regions only for an interacting framework, while the ACDM limit is attained in the state-finder plane. We also investigate the first and second laws of thermodynamics assuming equal temperatures at and inside the horizon in this scenario. Due to the violation of the first law of thermodynamics in f(T) gravity, we explore the behavior of the entropy production term. The validity of a generalized second law of thermodynamics depends on the present-day value of the Hubble parameter.
DOI: 10.7868/S0044451014070104
1. INTRODUCTION
There is increasing evidence of dark energy (DE) over the last few years, which is assumed to be responsible for the accelerated expansion of the universe. This has been confirmed by a variety of observational constraints in the framework of different observational schemes [1]. The standard cosmology has been remarkably successful, but there remain some serious unresolved issues including the search for the best DE candidate. The origin and nature of DE is still unknown except in some particular ranges of the equation-of-state (EoS) parameter ui. In the absence of any solid argument in favor of a DE candidate, various approaches have been adopted such as dynamical DE models, and modified and higher-dimensional gravities.
The f(T) theory of gravity [2] (the generalized teleparallel gravity, with T being the torsion scalar) attracted many people to explore it in different cosmo-logical scenarios. This theory deals with torsion via the Weitzenbock connection (having zero curvature) instead of the Levi-Civita connection in general relativity, which is responsible for curvature. The f(T)
E-mail: msharif.math'fflpu.edu.pk
** E-mail: shamailatoor.math'fflyahoo.com
gravity has been studied extensively in application to many phenomena, e.g., the accelerated expansion of the universe [3], the correspondence (via quintessence, t achy on, /\-essence. and dilaton scalar fields) carried out to discuss the dynamics of scalar fields as well as scalar potentials [4, 5] and to distinguish the f(T) model from the ACDM model, state-finder diagnostics in a specific f(T) model [6], validity/'violation of the first and second laws of thermodynamics using the Wald entropy, corrected-entropy versions and magnetic field scenarios [7 9], and many more.
The search for a viable DE model is the basic keyleading to the reconstruction phenomenon, particularly in modified theories of gravity. The corresponding energy densities are compared to construct the modified function in the underlying gravity. In this manner, the family of holographic reconstruction of DE models attains a significant place in discussing the accelerated expansion of the universe. Different f(T) models were reconstructed via holographic DE (HDE) and new age-graphic DE (original and entropy corrected) models in [10]. The authors concluded that the corresponding EoS parameter gives consistent results in entropy-corrected models. In [11], an f(T) model corresponding to the HDE model was obtained in a slightly different way. The authors found that the reconstructed model gives the phantom behavior as well as a unification of
DE and dark matter. In [12], the reconstruction scheme was extended to a general (m,n)-type HDE in f(T) as well as f(R) gravity. The viability and cosmography of the obtained models were also discussed there.
Holographic DE has been attributed to the formation of black holes. Recent observations regarding the accelerating expansion of the universe are in favor of a phantom-dominated universe with no expectation of black holes. The idea of pilgrim DE (PDE) having the key point of a phantom-like universe to prevent the black hole formation was proposed in [13]. Recently, the behavior of interacting PDE models corresponding to the Hubble, event, and conformal age of the universe via different cosmological parameters such as EoS, u) — ui' and state-finders was analyzed in [14]. The authors found consistent results for positive and negative values of the PDE parameter for these parameters.
In this paper, we construct the pilgrim f(T) model via the reconstruction scheme and explore the EoS parameter, the lot —oj't phase plane, and state-finder parameters. We also investigate thermodynamic laws for this model in f(T) gravity for same temperature of the universe. This paper is arranged as follows. In Sec. 2, we briefly describe f(T) gravity and its field equations, and construct a pilgrim f(T) model. Section 3 is devoted to examining the evolution trajectories of some cosmological parameters. The validity of first and second laws of thermodynamics is investigated in this scenario in Sec. 4. In the last section, we summarize the results.
2. /(T) GRAVITY AND PILGRIM DE MODEL
In this section, we first briefly discuss f(T) gravity and its field equations, and then construct the pilgrim f(T) model via the correspondence scheme.
2.1. The field equations
The action for f(T) gravity [2] is defined as 2
777 1
J d4xh(f(T) + Cm), (1)
where rn2 = (8ttL/P1 is the reduced Planck IIlclSS with Q being the gravitational constant,
h= v^=det(/i°),
where g is the determinant of metric coefficients, is the tetrad field, and £m is the Lagrangian density of matter in the universe. The tetrad field is related to the metric tensor as gtll, = r^h
where ;/Qf, = diag(l, — 1, — 1, — 1) is the Minkowski space metrix, the indices (u.b) represent tangent space coordinates, (fi.v) are the coordinate indices on the manifold, and all these indices range 0,1,2,3. The variation of action (1) with respect to the tetrad yields the field equations
[}r1dll(hSa'1")+hXaTpllxSp"'1] fT +
+ Sa>"'dp(T)fTT + \kj = y2KV'. (2)
where fr = df/dT, far = d2f/dT2, and Tp is the energy momentum tensor of perfect fluid. The antisymmetric torsion and superpotential tensors are
rv = K(d»K ~ W-
S,ll'p = i ^T'il'p+Tl''ip+T'il'p+26^T(,l'„ ^2 ,
which are used to define the torsion SCctlctr as T =
_ rpp q ¡IV
— ■Lfw'Jp
For a spatially flat Fricdmann Robertson Walker (FRW) universe, a straightforward choice of the tetrad
hp = diag(l, u(t), u(t), u(t)),
where u(t) is a scale factor. This leads to the expression for the torsion scalar T = —6H2, where H = u/u is the Hubble parameter and a dot represents the time derivative. The corresponding modified field equations are
12 H2fT + f = 2 m;2 (k (3) 4SH2HfTT - (12H2 + I//)/, - / = 2rn~2p. (4)
Here, p and p denote the total energy density and pressure of the universe, satisfying the energy conservation equation
f) + 3H(p + p) = 0. (5)
Equations (3) and (4) can be rewritten in terms of the usual Fricdmann equations as
H'2 = ¿2 (Pm+pr), H = (Pm +PT+PT), (6)
where pm is the matter contribution of energy density with pressureless matter (pm =0), and torsion contributions pr and pt take the form
2
777
pT=-^(2TfT-f-T), (7)
UK
Pr=-^(-SHTfTT +
+ (2T-4H)fT-f-T + 4H). (8)
In terms of fractional energy densities, the first equation in Eq. (C) can be expressed as
1 = n.
n7
0 m —
Pn
3rn2H2 '
nT =
Pt
:',mpl'2
■ 0)
The nature and properties of DE and dark matter constitute one of the central problems in modern astrophysics. Dark energy as the most dominant component in the energy budget of the universe, having the possibility of nongravitational coupling to other components of the universe, in particular, to dark matter. This coupling results in modifying the background evolution of the dark sector, permitting any type of interaction to be constrained. There is no serious evidence presented up to now against this coupling. Here, we assume that pressureless matter (cold dark matter) interacts with the torsion component [15], and the corresponding non-conservation equations are given by
Pm + 3 Hpm = Q, Pt + 3HpT(l + iot) = -Q,
(10) (ID
where lot = Pt/pt is the EoS parameter for the interacting f(T) gravity and Q represents the interaction term that exchanges the energy between the torsion component and pressureless matter. In general, Q can be an arbitrary function of the energy densities of DE and pressureless matter as well as the Hubble parameter. Commonly, it induces simple choices of interaction [15, 16] such as
Q = ZdHpm, Q = 'MHpd(:, Q = MH(pm+pd(:),
where d is the coupling constant (interaction parameter). Some of these interactions are used for mathematical simplicity, while others have been proposed within some plicnomcnological approaches. The case d = 0 represents the nonintcracting scenario. The sign of d is important in the sense that it reveals an exchange of energy: d > 0 implies that DE decays in dark matter, while d < 0 means that dark matter decomposes into DE. The positive coupling constant is favorable for the validity of thermodynamic laws. However, it was observed in [17] that Q must change its sign during the evolution of the universe from the deceleration to acceleration phase. Unfortunately, these choices for Q do not change their signs during the evolution, and this requires new interacting terms.
A new form of Q was introduced in [18] as
Q = 3dH(pdc - pm)-
(12)
As the universe evolves from the decelerated to accelerated regime, this interaction term changes its sign from negative to positive. Also, this form remain
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