Abstract
In this chapter, we have studied a spatially homogeneous and anisotropic Kantowski-Sachs universe in the presence of Barrow Holographic Dark Energy in the background of Saez-Ballester scalar-tensor theory of gravitation. To find the exact solution of the SB field equations, we have assumed that the shear scalar is directly proportional to the expansion scalar. This assumption leads to relation between metric potentials of the models. We have discussed non-interacting and interacting cosmological models. Moreover, we have discussed several cosmological parameters such as energy densities of DM and DE (
Keywords
- Kantowski-Sachs
- Barrow holographic
- scalar-tensor theory
- dark energy model
- theory of gravitation
1. Introduction
The modern cosmological evidence [1, 2, 3, 4] indicated that there is an accelerated expansion. The responsible cause behind this accelerated expansion is a miscellaneous element having exotic negative pressure termed as Dark Energy (DE). The nature and the cosmological origin of DE are still enigmatic. To describe the phenomenon of DE, several models have been presented. According to several findings, DE should behave like a fluid with ‘negative pressure, counterbalancing the action of gravity, and speeding up the universe’ [5, 6]. The general methodology is to define the dynamics of the universe by assuming the source of DE is represented as a non-zero “cosmological constant
Hooft [16] has proposed a new dark energy model, known as the Holographic Dark Energy (HDE) model, which was based on the Holographic Principle (HP) and some features of quantum gravity theory. The holographic principle states that the number of degrees of freedom of a gravity-dominated system must vary along with the area of the surface bounding the system [17, 18]. For a system with size
2. Body of the manuscript
Barrow [29, 30] has recently found the possibility that the surface of a black hole could have a complex structure down to arbitrarily tiny due to quantum-gravitational effects. The above potential impacts of the quantum-gravitational space-time form on the horizon region would therefore prompt another black hole entropy relation, the basic concept of black hole thermodynamics. In particular
Here
where
Nandhida and Mathew [39] have considered the Barrow Holographic Dark Energy as a dynamical vacuum, with Granda-Oliveros (GO) length as IR cut-off and studied the evolution of cosmological parameters with the best-estimated model parameters extracted using the combined data-set of supernovae type Ia pantheon (SN-Ia) and observational Hubble’s data. Bhardwaj et al. [40] have studied statefinder hierarchy model for the BHDE. Adhikary et al. [41] have constructed a BHDE in the case of non-at universe in particular, considering closed and open spatial geometry and observed that the scenario can describe the thermal history of the universe, with the sequence of matter and DE epochs. Considering BHDE Sarkar and Chattopadhyay [42] reconstruct modified gravity as the form of background evolution and point out that the equation of state can have a transition from quintessence to phantom with the possibility of Little Rip singularity. Saridakis [43] has studied modified cosmology through spacetime thermodynamics and Barrow horizon entropy. Koussour et al. [44, 45] have investigated Bianchi type
Shamir and Bhatti [46] have analyzed anisotropic DE Bianchi type
3. Metric and SB field equations
We consider a homogeneous and anisotropic KK Universe described by the line-element
where
We assume that the Universe is filled by a DM without pressure with energy density
where
where
and energy conservation equations are
where
The SB field Eq. (5), for KK line-element Eq.(3) with the help of Eq.(4), can be written as
We can write the conservation Eq.(7) of the DM and BHDE as
where overhead dot (.) denotes ordinary differentiation with respect to cosmic time
The SB field eqs. (8)–(11) form a system of four (4) non-linear equations with seven (7) unknowns;
The average scale parameter of the KK Universe is given by
The spatial volume of the universe
The average Hubble parameter
The Deceleration Parameter (DP)
4. Solution of the field equations and cosmological models
Hence to find the exact solution of the field equations, we have to use some physically viable conditions; The shear scalar (
where
where
Now using eqs. (16), and (18), we get the exact solution
where
From eq. (2), the energy density of BHDE is
Thus, the metric corresponding to the metric potentials (20) can be written as
From eqs. (10), (11), (20), and (21), we found the skewness parameter (
The scalar field
where
The plot of DP (
4.1 Non-interacting BHDE in the SB cosmology
First, we consider that two fluids (DM and BHDE) do not interact with each other. Hence the conversation eq. (14), of the fluids may be conserved separately. The conservation eq. (14) of barotropic fluid leads to
whereas the conservation eq. (14) BHDE leads to
From eq. (21) by using eqs. (20), (21), and (23), we get the EoS parameter
where
The evolution of DM and BHDE densities with redshift (
In Figure 4, we have plotted the behavior of skewness parameter (
In Figure 5, we observed the dynamics of the EoS parameter (
4.2 ω b − ω b ' plane
Caldwell and Linder [56] have pointed out that the quintessence phase of DE can be separated into two distinct regions, that is, thawing (
where
Figure 6 shows the
4.3 Stability analysis
We analyze now the stability of the obtained BHDE (non-interacting and interacting) models.
For our non-interacting BHDE model, squared speed sound
where
For the non-interacting model, Figure 7 shows the evolution of the SSS in terms of redshift (
4.4 Interacting BHDE in the SB cosmology
In this case, we focus on the interaction between two dark fluids. Since the nature of both BHDE and DM is still unknown, there is no physical argument to exclude the possible interaction between them. Recently, some observational data shows that there is an interaction between dark sectors [57, 58]. Several authors [59, 60, 61] have investigated the signature of interaction between DE and DM by using optical,
whereas the conservation eq. (14) BHDE leads to
where
Now, from eqs. (20), (21), (23), (24), and (32), we found that the EoS parameter is
where
For interacting BHDE model, the EoS parameter (
Figures 11–13 show the
For our interacting BHDE model, Figure 14 shows the evolution of the SSS in terms of redshift (
4.5 Statefinder diagnostics
In this section, we focus on the diagnosis of the statefinder. The Hubble parameter
Figure 15 shows the evolutionary trajectories in
4.6 Om-diagnostic
As a complementary to the statefinder parameters
where
The trajectory of Om diagnostics versus redshift (z) is shown in Figure 17. The trajectory reveals that the BHDE model shows initially a positive slope of the trajectory indicating that our model has phantom behavior and the negative slope of the trajectory indicates that our model behavior is quintessence in late time.
5. Conclusions
In this chapter, we have investigated the accelerated expansion by assuming the BHDE Universe within the framework of SB scalar-tensor theory of gravity. We have investigated various cosmological parameters to analyze the viability of the models and our conclusions are the following:
The deceleration parameter (
For our non-interacting model, the energy densities of DM and BHDE are positive and increasing function of redshift
For interacting BHDE model, the EoS parameter starts from the matter dominated era, then it moves to the quintessence region (
It can be observed from Figures 5, 8–10 that the EoS parameter of our models in both non-interacting and interacting cases lie within the above observational limits which shows the consistency of our results with the above cosmological data. We have observed that our interacting BHDE model lies in the freezing region (
The behavior of
Finally, we can state that some of the preceding conclusions in KK BHDE model are good agreement with recent observations.
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this chapter.
Data availability statement
This chapter has no associated data or the data will not be deposited.
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