Advances in High Energy Physics

Volume 2019, Article ID 5393491, 12 pages

https://doi.org/10.1155/2019/5393491

## Parameterizing Dark Energy Models and Study of Finite Time Future Singularities

^{1}Department of Mathematics, Adani Institute of Infrastructure Engineering, Ahmedabad-382421, Gujarat, India^{2}Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, India

Correspondence should be addressed to Tanwi Bandyopadhyay; moc.liamg@biwnat

Received 1 November 2018; Revised 1 March 2019; Accepted 21 March 2019; Published 9 April 2019

Academic Editor: Antonio J. Accioly

Copyright © 2019 Tanwi Bandyopadhyay and Ujjal Debnath. 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.

#### Abstract

A review on spatially flat D-dimensional Friedmann-Robertson-Walker (FRW) model of the universe has been performed. Some standard parameterizations of the equation of state parameter of the dark energy models are proposed and the possibilities of finite time future singularities are investigated. It is found that certain types of these singularities may appear by tuning some parameters appropriately. Moreover, for a scalar field theoretic description of the model, it was found that the model undergoes bouncing solutions in some favorable cases.

#### 1. Introduction

In order to avoid the initial singularity problem [1], a competitive alternative structure is proposed to the standard inflationary description. This is called the big bounce scenario ([2–10]). In this framework, the universe initially contracts continuously up to an initial narrow state with a nonzero minimal radius and then transforms to an expanding phase [11]. This means that the initial singularity is replaced by a bounce and consequently the cosmic equation of state (EoS) parameter crosses the phantom divide line (from to ).

On the other hand, various cosmological observations indicate the current cosmic acceleration ([12–21]). To explain this phenomenon in the homogeneous and isotropic universe, it is necessary to assume a component of matter with large negative pressure, called dark energy. Since the inception of this concept, a wide variety of phenomenological models have been proposed as the most suitable candidate of dark energy. Among them, the cosmological constant provides plausible answers but suffers from the fine-tuning problem. This issue has forced researchers to probe models having time-dependent equation of state parameter for the dark energy component. A simple classification of these models could be as follows:

Cosmological constant ()

Dark energy with constant [cosmic strings (), domain walls ()].

Dynamical dark energy ( constant) (i.e., quintessence, Chaplygin gas, k-essence, braneworld) ([22–30])

Dark energy with [scalar-tensor theory, phantom models] ([31, 32])

The reconstruction problem for dark energy is reviewed in [33] in much detail (see also the references therein). Recently, in the literature [34], the possibilities of future singularities have been studied in the framework of Modified Chaplygin Gas filled universe and conditions of bounce were investigated. In this work, we provide more general results in terms of the possibilities of finite time future singularities in the context of a D-dimensional Friedmann-Robertson-Walker universe by imposing five standard parameterizations of the dark energy equation of state parameter. Different aspects of dark energy models have been studied so far to reconcile standard cosmological scenario with observations but no work has been done to the parameterization of these models in view of investigating the finite time future singularities.

The paper is organized as follows. In Section 2, we state the governing equations of the metric in the D-dimensional universe. In Section 3, the descriptions for the major physical quantities are provided in the backdrop of five different parameterization models both analytically and graphically. In Section 4, the possibilities of future singularities are examined for these five models and the possible restrictions for certain type of singularity are shown in tabular form. In Section 5, the bouncing universe is described in the five parameterization models and, finally, we end with a short discussion in Section 6.

#### 2. D-Dimensional Friedmann-Robertson-Walker Universe

We consider the D-dimensional spatially flat Friedmann-Robertson-Walker universe, given by [34]

where is the scale factor and is the metric for the -dimensional space. The field equations together with the energy conservation equation can be obtained as (assuming )and

Here denotes the Hubble parameter. In the following section and subsequent subsections, we will examine different parameterizations of dark energy models and attain analytical as well as graphical expressions of some physical entities involved in the process.

#### 3. Dark Energy as Scalar Field for Various Parameterization Models

By considering scalar field with the potential , the effective Lagrangian [34] is given bySo the energy density and pressure take the formThe kinetic energy for the field is given by

where . One can always write , where the prime denotes differentiation with respect to redshift . With the help of (2), we have

The scalar potential associated with the field is given by

In the next subsections, we will investigate the scalar field and its potential in different well-known parameterization models. A detailed review of different dark energy models can be found in [35].

##### 3.1. Model I: Linear Parameterization

Here we assume the linear equation of state [36] . Here and are constants. Using (4), the energy density of the model then gives rise to

Here is the present value of the energy density. The pressure of the fluid then becomes

Then (9) reduces to

which on further integration gives rise to the scalar field as

where . Also (10) changes to

We draw the variations of energy density , pressure , EoS parameter , Hubble parameter , potential , and the scalar field with the variation in in Figures 1(a)–1(f), respectively. Figure 1(g) shows the variation of with . We observe that , , and decrease as decreases. and decrease from positive level to negative level as decreases. first increases up to certain value of and then decreases as decreases. Also decreases as increases.