Advances in High Energy Physics

Volume 2015, Article ID 292767, 10 pages

http://dx.doi.org/10.1155/2015/292767

## Quintessence and Holographic Dark Energy in Gravity

Department of Mathematics, COMSATS Institute of Information Technology, Lahore 54000, Pakistan

Received 22 September 2014; Revised 21 December 2014; Accepted 2 January 2015

Academic Editor: Filipe R. Joaquim

Copyright © 2015 M. Zubair. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We regard theory as an efficient tool to explain the current cosmic acceleration and associate its evolution with the known dark energy models. The numerical scheme is applied to reconstruct theory from dark energy model with constant equation of state parameter and holographic dark energy model. We set the model parameters and as describing the different evolution eras and show the distinctive behavior of each case realized in theory. We also present the future evolution of reconstructed and find that it is consistent with the recent observations.

#### 1. Introduction

The development of cosmology and gravitation can be seen as one of the scientific triumphs of the twentieth century. Since 1998, when observations of supernova type Ia [1, 2] pointed accelerated cosmic expansion, various observational measurements [3, 4] have affirmed such evolutionary change in the history of the universe. In spite of tremendous efforts, late cosmic acceleration is certainly a major challenge for cosmologists. In the current scenario, the unknown form of energy component, usually named as dark energy (DE), is said to be responsible for such mechanism. DE is recognized by its distinctive nature from ordinary matter sources having negative pressure which may lead to cosmic expansion counterstriking the gravitational pull.

DE has appeared as enigmatic cosmic ingredient and interpretation of its gravitational effects is a dynamic research field. There are two representative directions to address the issue of cosmic acceleration. Introducing an exotic cosmic fluid in the framework of the Einstein gravity is one direction to deal with such issue. The most likely theoretical campaigner of DE is the cosmological constant characterized by constant EoS [5]. A number of alternative models have been proposed in this perspective to explain the role of DE in the present cosmic acceleration [6–11]. The issue of cosmic acceleration can also be counted on the basis of modified theories of gravity like gravity [12], gravity ( is the torsion scalar) [13], gravity ( is the trace of energy-momentum tensor ) [14–17], gravity (where ) [18–20], Gauss-Bonnet gravity [21], and scalar-tensor theories [22].

Recently, gravity a generalization of the action of teleparallel gravity theory has gained special attention. This theory assumes Weitzenbck connection instead of the Levi-Civita connection which has no curvature but nonzero torsion. theory has gained more importance to explain the accelerated cosmic expansion without introducing DE components [23, 24]. Wu and Yu [25] discussed the dynamical behavior of theory for a concrete power law model that has a stable de Sitter phase along with an unstable radiation dominated phase and an unstable matter dominated one. In recent literature, various aspects have been discussed such as local Lorentz invariance [26], static spherically symmetric solutions [27], wormhole solutions [28], Noether symmetries [29, 30], cosmographic analysis [31], thermodynamics [32], and phase space analysis [33]. Another significant feature in modified theories of gravity is the choice of Lagrangian such as or function. Wu and Yu [34] proposed two new models in theory to explain the phantom crossing of effective equation of state parameter. They also constrain the model parameters according to recent observations.

Cosmological reconstruction has appeared as one of the promising aspects in modified theories. In this perspective, different schemes have been proposed for known cosmic evolutions to find the corresponding particular Lagrangian [35–44]. Nojiri et al. [35] developed a general reconstruction scheme for gravity and formulated different epochs in FRW cosmology including matter dominated phase, transition from deceleration to acceleration, accelerating epoch, and CDM phase. In this study, one interesting way is to consider the known cosmic evolution and use the field equations to find particular form of Lagrangian that can reproduce the given evolution background. Nojiri et al. [36] executed such reconstruction scheme in order to find some realistic models in theory which was applied in and modified GaussBonnet theories [37]. Capozziello et al. [39] proposed an efficient scheme to reconstruct gravity from the known Hubble parameter in terms of redshift. In this procedure, one needs to know the relation of for a given DE model so that Lagrangian can be reconstructed implying the same dynamics. This scheme is also applied for a class of holographic dark energy (HDE) models [40–42]. The finite-time future singularities may appear for the universe dominated by dark energy and these are classified in four types [43]. Such singularities have been discussed in and other modified theories [44, 45].

Bamba et al. [45] reconstructed power-law, exponential and logarithmic cosmological models with realizing early inflation, CDM, little rip, and pseudo rip cosmology. The issue of finite-time future singularities is also discussed in gravity and it is shown that power-law type correction terms can be useful to get rid of such singularities. Setare and Mohammadipour [46] developed dynamical system to reconstruct function for a background CDM cosmology. They investigated the stability condition with respect to the homogeneous scalar perturbations in different cosmological eras. Cosmological reconstruction corresponding to holographic DE (HDE) model has also been studied in the literature. Daouda et al. [47] developed the reconstruction of gravity according to HDE by imposing the initial condition on constant parameter. The holographic gravity model with power-law entropy correction is also studied in [48], where model parameters are fitted using the latest observational data including type Ia supernovae, baryon acoustic oscillations, cosmic microwave background, and Hubble parameter data. It is shown that in the power-law entropy-corrected holographic gravity model, the universe begins a matter dominated phase and approaches a de Sitter regime at late times. The equation of state parameter has appeared as a significant tool and various people discussed the properties of models regarding the crossing of phantom divide line. In [49], Wu and Yu proposed two models realizing the phantom crossing and discussed the observational parameters. Bamba et al. [50] presented the evolution of effective equation of state for the models involving exponential and logarithmic functions. They also discussed the model with combination of exponential and logarithmic functions and it was found that this model can cross the phantom divide line. The cosmological observations favor such feature of models.

In this paper we are interested in reconstructing some models for known cosmic evolution which has been discussed in the literature. Following [39], we apply the numerical reconstruction scheme to develop the models corresponding to quintessence and HDE models. The future evolution of function is also presented for different choices of essential parameters. The paper has the following format: in the next section, we overview the theory and present the modified field equations. We reconstruct the theory as an effective description to known DE models and present their corresponding evolution. Section 3 summarizes our findings.

#### 2. Reconstruction of Theory

The action of gravity with matter Lagrangian readswhere , , and is the torsion scalar defined by the relationwhereand is the contorsion tensorwhich equals the difference between Weitzenböck and Levi-Civita connections. The variation of the action with respect to the vierbein vector field presentswhere is the matter energy momentum tensor and the prime denotes the differentiation with respect to .

We take the homogeneous and isotropic flat FRW metric defined aswhere represents the scale factor and corresponding tetrad components are . Using relations (2)–(4), we obtain the torsion scalar , where is the Hubble parameter. We consider the matter content described by an energy momentum tensor of perfect fluid.

The substitution of and in (5) results in the following Friedmann equations:where and are the energy density and pressure of matter components. One can rewrite these equations as the usual form of effective Einstein equationswhere is the Hubble parameter andIn this discussion, we consider the pressureless matter and is determined from the conservation equation . We combine (8) and continuity equation to a single equation:Using the relation , (10) can be expressed aswhere is defined in terms of and its derivatives asOne can see that an expression of is needed to reconstruct . Consequently, the respective theories can be determined for a known predicted by a given DE model. To figure out the evolution of , we set the boundary conditions [31]In the following we consider two well-known models of DE to numerically reconstruct the corresponding function.

##### 2.1. Quintessence Model

In the first place, we consider the familiar quintessence model of DE with constant equation of state parameter and . The corresponding Hubble parameter is defined as [39]In fact it is the generalization of CDM model with EoS . One can figure out other DE models depending on such as quintessence and phantom , violating the null energy condition. has been constrained through distinct observations: the Planck and WMAP9 observations set the constraints for as [51] and [52], respectively. In this study, we set the present day values of and as and [51]. We choose three different values of as representing quintessence, CDM, and phantom regimes of the universe. Equations (11) and (14) can be solved numerically using conditions (13) as well as . The reconstructed function is presented on plane in Figure 1, where and . For the sake of comparison, we also show the evolution trajectories on plane in Figure 2. The evolution trajectories show distinct behavior for and coincide for . Figures 1 and 2 indicate the role of to determine the evolution of in remote past. To further explain the effect of , we consider the future evolution of . Figure 3 shows the future variation of . As is expected for , the DE dominates over the matter and would take infinitely large values. For , we have the cosmological constant regime which is also assured from the corresponding curve in Figure 3. For , vanishes in the future. We also plot the future evolution of reconstructed function which more effectively reflects the difference in choice of parameter . For , decreases and attains a constant value as shown in Figure 4(a). In case of , Figure 4(b) shows the linear dependence of on up to the present day value and in future evolution it approaches definite value preferring the CDM model. For the phantom evolution with , the corresponding evolution of function is shown in Figure 4(c). It can be seen that decreases initially and after the recent epoch it starts increasing leading to the corresponding function .