The present work is devoted to investigation of non-static plane symmetric universe filled with matter and anisotropic modified holographic Ricci dark energy components within the framework of a scalar-tensor theory formulated by Saez and Ballester (Phys. Lett. A 113: 467, 1986). To get the determinate solution of the model we have used scalar expansion proportional to the shear scalar, linearly varying deceleration parameter and energy density of modified holographic Ricci dark energy. We study the equation of state parameter, deceleration parameter, squared speed of sound and and planes for our dark energy model. The equation of state parameter provides a quintom-like behavior of the universe. The plane corresponds to limit, Chaplygin and phantom regions, whereas trajectories correspond to both thawing and freezing regions.
One of the most interesting phenomenon in recent times is accelerated expansion of the universe. Initially this phenomenon has been pointed out by supernova observations (Riess et al. 1998; Garnavich et al. 1998; Perlmutter et al. 1999; Riess 2004). Later, the large scale structure (Tegmark et al. 2004) and cosmic microwave background (Spergel et al. 2003) observations also provide support this accelerated expansion phenomenon. It is believed that some unknown form of energy works behind this phenomenon which is dubbed as dark energy. This dark energy possesses strong negative pressure, but its exact nature is still unknown. The cosmological constant is one of the candidates of dark energy but it has two severe problems (Peebles 2003). To avoid these problems, two alternative ways have been adopted.
One approach is dynamical dark energy models. Among different dynamical dark energy models, holographic dark energy models (HDE), are widely used to interpret the dark energy scenario. This HDE idea comes from holographic principle (Susskind 1995), which stated that the vacuum energy density can be bounded as , where is the vacuum energy density and is the reduced plank mass. The HDE models with Hubble or particle horizons as the IR cut-off, cannot follow the current accelerated expansion (Hsu 2004) of the universe. When the event horizon is taken as the length scale, the model suffers from some disadvantage. Future event horizon is a global concept of space-time but on the other hand, density of dark energy is a local quantity. So the relation between them will enact challenges to the causality concept. These lead to the introduction of a new HDE, where the length scale is given by the average radius of the Ricci scalar curvature, . The holographic Ricci dark energy model introduced by Granda and Oliveros (2009) based on the space-time scalar curvature, is fairly good in fitting with the observational data. In this model the fine tuning and causality problems can be avoided. The coincidence problem can also be solved effectively in this model. Recently, a modified form of Ricci dark energy was studied (Chen and Jing 2009) by assuming the density of dark energy contains the Hubble parameter , the first order and the second order derivatives. The expression of the energy density of this modified holographic Ricci dark energy (MHRDE) is given by
here is the Hubble parameter, is the reduced Planck mass, , and are constants.
The another approach for explaining the accelerated expansion is called “ modified gravity approach" and in this approach the cosmic acceleration would arise not from dark energy as a substance but rather from the dynamics of modified gravity. Einstein's general theory of gravitation may not describe gravity at very high energy. The simplest alternatives to general relativity are scalar-tensor theories, in particular, Brans-Dicke (1961) and Saez-Ballester (1986) theories of gravitation. Now we pay our attention on Saez-Ballester theory in which metric is coupled to a scalar field. Here the strength of the coupling between gravity and the field was governed by a parameter . With this modification, they were able to solve a ‘ missing-mass problem'.
On this basis several authors have investigated the MHRDE models of dark energy in various theories of gravitation. Das and Sultana (2015) have investigated Bianchi type MHRDE model with hybrid expansion law in general relativity. Santhi et al. (2016a, 2016b) have discussed Bianchi type- MHRDE models in Brans-Dicke and Saez-Ballester scalar-tensor theories of gravitation. Santhi et al. (2017) have explored Bianchi type- magnetized MHRDE models in general relativity. Reddy (2017) has studied Bianchi type- MHRDE models in Saez-Ballester theory of gravitation.
We have extended this MHRDE work to Bianchi type- and MHRDE models in Saez-Ballester theory (Rao and Prasanthi 2017a) and Bianchi type- MHRDE model in self-creation theory (Rao and Prasanthi 2017b) with varying deceleration parameters. In these works, we have found that the equation of state (EoS) parameter varies in both quintessence and phantom regions. Here we investigate non-static plane symmetric MHRDE model in Saez-Ballester scalar-tensor theory of gravitation using linearly varying deceleration parameter. We also discuss the EoS parameter, squared speed of sound and and planes. So far, this problem has not been considered in the literature. The paper is arranged as follows. Section-2 presents metric and Saez-Ballester field equations for ordinary matter and anisotropic dark energy. In section-3, we present the analysis of EoS, deceleration and squared speed of sound parameters and and planes. Finally, we summarize the results in section-4.
Figure 1 explains the behavior of scalar field versus cosmic time . It can be observed that decreases as time increases and vanishes for large values of time. Figures 2 and 3 depict the energy density for matter and dark energy versus cosmic time respectively. We observed that both and are positive decreasing functions of time and converges to zero for sufficiently large time. From Fig. 4, it can be observed that skewness parameter is positive decreasing function of time and at present epoch. Therefore the anisotropy of MHRDE in our model vanishes and also becomes isotropic at present epoch.
From equations (25)-(28), we observe that volume ( ) is zero at initial epoch i.e., while the expansion scalar, the Hubble parameter and the shear scalar are infinite, which is big bang scenario. As time , volume becomes infinite whereas remaining parameters approach to zero. From equations (28) and (27), is a constant, which indicate that our model is anisotropic throughout the evolution. Also, from equation (29) it clear that anisotropic parameter is constant, which shows that the model is anisotropic. Cosmic microwave background radiations have pointed out that there exist certain amount of anisotropy in our universe.
Deceleration parameter
The behavior of deceleration parameter explore whether the model inflates or not. The positive sign of indicate the decelerating expansion of model whereas the negative sign correspond to accelerating model. The modern observations from various experiments like SNe a and CMBR favor accelerating models and value of deceleration parameter lies somewhere is the range . The deceleration parameter for our model is given in equation (15).
The plot of deceleration parameter is shown in Fig. 5 against cosmic time , which shows early deceleration ( ) and the present acceleration ( ) phase of the universe. We can observe from the figure that the model starts accelerating at Gyr, i.e., Gyr ago from today and the present value of deceleration parameter . Therefore these values are consistent with the current understanding of our universe.
We plot the EoS parameter versus cosmic time for three different values of as shown in Fig. 6. This parameter represents initially the deceleration phase and then crosses the phantom divide line (PDL) ( ) (for ) indicating the accelerating phase of the universe. It can be observed from the figure that the EoS parameter starts from matter dominated region and then finally goes towards quintessence region. In particular, the trajectories for and enters phantom region ( ) by crossing the PDL and finally attains a constant value in quintessence region ( ).
The stability of dark energy models can be check by the squared speed of sound parameter. It is defined by and helps in describing the stability of models with the help of its signature. The positive value leads to stable behavior while negative behavior gives instability. The behavior of against cosmic time is shown in Fig. 7. The trajectories express negative behavior initially and then becomes positive giving stability of the model.
Caldwell and Linder (2005) have firstly proposed plane, which is useful tool for distinguishing different dark energy models through trajectories on its plane. This approach has been applied on quintessence model which leads to two types of its plane, that is the area occupied by the region ( ) corresponds to thawing region while area under the region ( ) implies the freezing region. Later, many authors have applied this plane analysis to other well-known dynamical dark energy models such as phantom (Chiba 2006), quintom (Guo et al. 2006), polytropic dark energy (Malekjani and Mohammadi 2012) and pilgrim dark energy (Sharif and Jawad 2014) models. Here we use this analysis to discuss these regions. By taking the derivative of equation (??) with respect to , we obtain the expression for as
We can obtain the plane by plotting versus as shown in Fig. 8. It can be observed that for three values of parameter the curves correspond to both freezing and thawing regions. However, for larger values of parameter the trajectories mostly vary in freezing region (i.e., ). It is suggested from observations that the expansion of the universe is comparatively more accelerating in freezing region.
Statefinder parameters
Many dark energy models have been proposed in order to explain the accelerating expansion of the universe. However a sensitive test is required, which can differentiate between these models. For this purpose Sahni et al. (2003) have introduced two new parameters called statefinders. For , show the CDM and CDM limits, while the regions ( and ) correspond to the phantom and quintessence dark energy. The statefinders of our model are obtained as
The statefinders plane can be obtained by plotting versus as shown in Fig. 9. It can be observed from the plane that our model corresponds to model. The plane corresponding to our model also coincide with the behavior of phantom dark energy ( ) and Chaplygin gas ( ) models. We observe that the statefinders plane of our MHRDE model is the combination of all existing well-known dark energy regions, which is an interesting quality.
In this work, we have studied the non-static plane symmetric MHRDE model in Saez-Ballester scalar-tensor theory of gravitation. We have assumed dark energy candidate as modified version of holographic Ricci dark energy and constructed the model using the linearly varying deceleration parameter. Here we have evaluated cosmological parameters (deceleration, EoS, squared sound speed) as well as and planes. We have explored the behavior of these parameters through cosmic time.
It is observed from the Figs. 1-3 that , and are positive decreasing functions of time and vanishes for large values of time. It is found that the skewness parameter is positive decreasing function of time and at present epoch. Therefore the anisotropy of dark energy in our MHRDE model vanishes at present epoch (Fig. 4). It is clear from the plot of deceleration parameter (Fig. 5) against that the model exhibits a smooth transition from early deceleration to the present acceleration phase. We have also observed that the transition occurs at Gyr and the present value of deceleration parameter , which are consistent with the modern cosmological observations. It has been observed from Fig. 6 that EoS parameter starts from dust-like matter, passes the quintessence and phantom regions by crossing PDL and finally attain a constant value in quintessence region. Hence the EoS parameter crosses PDL and transits from quintessence to phantom i.e., behaves like quintom. The trajectories of squared speed of sound exhibit negative behavior initially and then become positive giving stability of the model in near future (Fig. 7).
We also develop the plane for our MHRDE model as shown in Fig. 8. From these planes, we can say that the curves correspond to both freezing and thawing regions. However, the trajectories maximum vary in freezing region as suggested by observational data (the expansion of the universe is comparatively more accelerating in freezing region). It is also observed from the statefinders plane (Fig. 9) that the trajectories correspond to model and also they coincide with the behavior of phantom dark energy and Chaplygin gas models. This combination of all existing well-known dark energy regions is an interesting quality of our MHRDE model.