Advances in High Energy Physics

Volume 2019, Article ID 6582470, 16 pages

https://doi.org/10.1155/2019/6582470

## Stability of the Universe Model Coupled with Phantom and Tachyon Fields

Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan

Correspondence should be addressed to M. Sharif; kp.ude.up@htam.firahsm

Received 14 May 2019; Revised 8 July 2019; Accepted 15 July 2019; Published 31 July 2019

Academic Editor: Sally Seidel

Copyright © 2019 M. Sharif and Saadia Mumtaz. 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

In this paper, we study the phase space analysis of locally rotationally symmetric Bianchi type I universe model by taking interactions between dark matter and scalar field models. We define normalized dimensionless variables to develop an autonomous system of equations. We also find the corresponding critical points in order to study the dynamics of the system. The dynamical analysis indicates that all the critical points correspond to accelerated cosmic expansion for tachyon coupled field. We observe that positive values of provide more stable future attractors as compared to its negative values. We also analyze the behavior of power-law scale factor which shows different cosmological phases. It is found that the region for decelerated expansion gets larger for the phantom coupled matter by increasing while this region decreases for tachyon coupled field.

#### 1. Introduction

Recent observations suggest that our universe is expanding at an accelerating rate [1–3]. These observational probes indicate two cosmic phases, i.e., the cosmos phase before radiation and ultimately the current cosmic era. Many substantial attempts have been taken to explore the facts behind the current cosmic acceleration. An unusual form of energy having large negative pressure was proposed by the physicists, known as dark energy (DE). This energy dominates over the matter content undergoing current cosmic expansion.

There have been a massive literature devoted to study the ambiguous nature of DE. The cosmological constant () is taken as the elementary candidate having two conventional problems, i.e., fine-tuning and cosmic coincidence. Several alternative dynamical models can be considered as a substitute of like quintessence [4, 5], phantom model [6], tachyon field [7], and k-essence [8] that also show the expanding nature of cosmos. The extension of general barotropic equation of state (EoS) to some exotic canditates like Chaplygin gas [9] and its generalization [10] also represents the DE candidates.

The astonishing idea of scalar field, with an EoS parameter other than , has been proved to be useful to interpret the universe evolution due to its applications in different cosmological problems like cosmic acceleration as well as coincidence problems. The scalar field candidates also yield the inflationary era of cosmic evolution. We can adopt scalar field models (in particular, phantom and tachyon) as an alternative of DE candidates which interact with DM to resolve the coincidence problem. Researchers have paid an extensive attention to the tachyon cosmology in which the tachyon is fundamentally attributed by string theory [11, 12]. Gibbons studied the cosmological impact of tachyon field turning down to its ground phase [13]. The tachyon matter may execute inflationary era and eventually a new form of DM at late times [14]. A phantom field was also presented as an alternative of DE that constitutes large negative pressure with EoS parameter and plays an important role for accelerated expansion of the universe [15–18]. One of the significant features of the phantom model reveals that the universe will end with a big-rip (future singularity). The interaction of DE (phantom or tachyon) and DM may alleviate the coincidence problem [19, 20].

A phase space represents various states including position as well as momentum corresponding to each point of a dynamical system. This analysis yields dynamical characterization of a cosmological model by reducing complexity of the system of equations. It has always been helpful to study different stages of evolution by converting the system of equations to an autonomous one. This investigates the influence of initial data on the stability of a system by checking whether the respective system will remain stable or not [21].

Copeland et al. [22] discussed phase space analysis of the inflationary model that was unable to solve density problem. Guo et al. [23] studied stability of FRW universe model filled with barotropic fluid and phantom scalar field. They found that phantom dominated solution is a stable late-time attractor. Guo et al. [24] used this analysis to study the dynamical behavior of cosmos by interacting phantom energy with DM. Chen et al. [25] discussed a detailed phase space analysis of various phantom cosmological models for which the DE sector interacts with the DM. Acquaviva and Beesham [26] explored this analysis for FRW model and found that the nonlinear extension of viscous cosmic model provides accelerated expansion of the universe. We studied the influence of nonlinear electrodynamics on stability of accelerated expansion in the background of FRW cosmology [27]. Shahalam et al. [28] presented phase space analysis for various combinations of the coupling between phantom and tachyon fields taking FRW universe. We also discussed phase space analysis of FRW universe model by considering a power-law model for bulk viscosity coefficient [29].

Bianchi universe models have also been discussed in literature to explore primordial anisotropy and some large angle anomalies identified by CMBR which give rise to the negligence of statistical cosmic isotropy [30]. Belinkskii and Khalatnikov [31] applied phase plane approach to Bianchi type I (BI) model and studied the effects of shear and bulk viscosity. Coley and Dunn [32] used this technique to investigate the dynamics of Bianchi type V model by taking a viscous fluid. Burd and Coley [33] discussed the influence of shear as well as bulk viscosity on stability of Bianchi universe models. Goliath and Ellis [34] studied dynamical evolution of Bianchi universe model by including the cosmological constant. Sharif and his collaborators discussed phase space analysis of Bianchi I universe in Brans-Dicke gravity with chameleon scalar field [35] as well as gravity taking nonphantom, vacuum, and phantom phases [36]. Chaubey and Raushan [37] studied phase space analysis of LRS BI model in the presence of scalar field.

This paper aims to investigate the stability of LRS BI universe model by taking linear interactions of phantom and tachyon fields coupled with DM through phase space analysis. The plan of the paper is as follows. In Section 2, we provide some basic formalism for the evolution equations. An autonomous system of equations is developed by defining normalized dimensionless variables. Section 3 deals with the dynamical analysis of the universe model by taking three different forms of interactions between phantom energy and DM. We also study the phase space analysis of tachyon field coupled with DM in Section 4. Section 5 deals with the formulation of power-law scale factor. Finally, we summarize our findings in the last section.

#### 2. General Equations

Bianchi universe models are the simplest extensions of FRW universe with the addition of anisotropic effects. The LRS BI model is described by the line element [38, 39]:where and are the cosmic scale factors. The mean Hubble parameter, in terms of the directional Hubble parameters ( and ), can be defined asWe consider a power-law relation , , where represents a constant anisotropic parameter that gives the deviation of anisotropy from the isotropic model. This leads to the following relation:This assumption can be rationalized by the velocity redshift expression for extragalactic sources through which the Hubble cosmic expansion may achieve isotropy when shear to expansion scalar ratio is constant [40].

We assume a cosmic fluid by interacting phantom field with matter such that the conservation of energy yieldswhere is the interaction term, dot corresponds to the derivative with respect to time, , , , , , and represent energy densities and pressures of the matter and phantom energy, respectively. It should be noted that the transfer of energy between two components depends on the sign of interaction term. The energy passes from phantom to matter for while yields the opposite. The conservation equations clearly show that . The constraint and Raychaudhuri equations are formulated from the field equations aswhere and . Due to many arbitrary parameters, it seems difficult to find analytical solution of the evolution equation. For this purpose, we define the following normalized dimensionless quantities:that transform the dynamical equations into an autonomous system. Differentiating and with respect to , we haveThe Raychaudhuri and conservation equations through these dimensionless parameters becomewhere is a constant. The constraint equation can be written in the formThe effective EoS for the cosmic fluid and phantom field are given by

#### 3. Dynamics of Interacting Phantom Energy

In this section, we discuss dynamics of LRS BI universe by the phase space analysis taking an interaction between phantom energy and matter. In this context, we impose the condition to solve the system of (10) and (11) and also determine the corresponding critical points . The stability of universe model depends on the behavior of critical points and their eigenvalues. We take three different types of coupling between phantom field and matter as follows.

##### 3.1. Coupling

Firstly, we consider as an interaction model for the universe where both phantom field and DM are coupled. Equation (13) for this coupling takes the formwhere . The autonomous system of equations becomesWe can find the eigenvalues by the Jacobian matrixwhere suffix defines the values at critical points . The fixed point is called a source or unstable point (resp., a sink or stable point) if both eigenvalues consist of positive (resp., negative) real parts. The real parts of the eigenvalues having opposite signs correspond to a saddle point of the system. We evaluate the following critical points in this case. For , the eigenvalues of Jacobian matrix are obtained asWe are interested to explore the influence of physical parameters ( and ) on the stability of cosmos coupled with scalar field. We plot the dynamical characteristics of critical points for in the vicinity of different values of and as shown in Figure 1. We find negative eigenvalues which show the point as a stable future attractor for in the physical phase space except for at which the system becomes undetermined. For , , this point behaves as an unstable node.