Higher dimensional Bianchi type-III string cosmological models with dark energy in Brans-Dicke scalar-tensor theory of gravitation

Highlights

• We have investigated five dimensional Bianchi type-III string cosmological models with dark energy within the framework of Brans-Dicke scalar-tensor theory of gravitation.

• The solution of field equations gives two models of the universe i.e. power law model and exponential model.

• We have obtained the physical parameters for both the models.

• Dark energy dominates the string density.

• The value of EoS parameter shows compatibility with the current observational data.

Abstract

The Five dimensional anisotropic Bianchi type-III cosmological model has been investigated within the framework of Brans-Dicke (Phys. Rev. 124:925, 1961) scalar-tensor theory of gravitation. Dark Energy fluid and cosmic strings are considered as a source of energy-matter field and minimal interacting. To determine the solutions of field equations we have used Berman’s special law of variation, power law relation between scalar field $\varphi$ and average scale factor $a$ and proportionality between scalar expansion and shear scalar. It gives two model of the universe i.e. power law model and exponential model. Physicals and kinematical parameters of the model are also discussed for both the scenarios.

Introduction

We are in the era of accelerating expansion of the universe driven by an exotic negative pressure called dark energy. It has been detected by many cosmological tests though not yet convincing. Dark energy dominates the universe today but it is insignificant at the high redshift. Dark Energy models with higher dimensions have generated a lot of research interest to understand the current and future aspects of the universe. Dark energy has become one of the central troubles in cosmology. The nature and formation of dark energy are still mysterious. The accelerated expansion of the universe is being caused by the dark energy having negative pressure that has confirmed by Type 1a SupernovaRiess et al. (1998); Perlmutter et al. (1999) and Cosmic Microwave Background Radiation (CMBR)Spergel et al. (2003); Komatsu et al. (2009). Peebles and RatraPeebles and Ratra (2003) have given a brief review on the cosmological constant and dark energy. Further, WMAP predicts that approximately $73\%$, $23\%$, and $4\%$ of our universe is occupied with dark energy, dark matter and normal matter respectively. Cosmological constant $\Lambda$ and quintessence are two conservative possibilities for dark energy. $\Lambda$-Cold-dark-matter ($\Lambda$CDM) model is able to give a perfect description of the expanding universe, but it has some issues given by cosmological constant fine- tuning problem and cosmic coincident problem Copeland et al. (2006). Scalar field models namely, quintessenceCaldwell et al. (1998); Liddle and Scherrer (1998); Steinhardt et al. (1999), phantomCaldwell (2002), k-essenceArmendariz-Picon, Mukhanov, Steinhardt, 2000, Armendariz-Picon, Mukhanov, Steinhardt, 2001, TachyonSen (2002), QuintomFeng et al. (2005) have arisen a lot of attention due to such issues in $\Lambda$CDM model. The Dark Energy can be represented by the equation of state (EoS) $\omega =\frac{p}{\rho }$ where $p$ is the pressure and $\rho$ is energy density. It has become mathematically equivalent to the cosmological constant $\Lambda$ in the case of $(\omega =-1)$.

Brans and Dicke theory (1961) referred to as the scalar-tensor theory of gravitation, is a modified version of Einsteins general theory of relativity found on Mach’s principle by introducing a scalar field $\varphi$ coupled to the mass density of the universe Brans and Dicke (1961). The theory states that scalar field is reciprocal of time varying gravitational constant G, but this theory does not permit the scalar field to cooperate with elementary particles and photons.

BD field equations for the combined scalar and tensor fields are given by

$$\begin{array}{c}{G}_{ij}=-8\pi {\varphi }^{-1}-\omega {\varphi }^{-2}({\varphi }_{,i}{\varphi }_{,j}-\frac{1}{2}{g}_{ij}{\varphi }_{,k}{\varphi }^{,k})\hfill \\ -{\varphi }^{-1}\left({\varphi }_{i,j}{g}_{ij}{\varphi }_{;k}^{,k}\right)\hfill \end{array}$$

and

$$\varphi ={\varphi }_{;k}^{,k}=8\pi {(3+2\omega )}^{-1}T$$

where $G_{ij}=R_{ij}-\frac{1}{2}{g}_{ij}R$ is an Einstein tensor, R is the Ricci tensor, $\varphi$ is the Brans-Dicke scalar field, $\omega$ is the dimensionless constant, and $T_{ij}$ is the energy momentum tensor.

The equation of motion is given by

$$T_{,j}^{ij}=o$$

It is a consequence of the field Eqs. (1) and (2). NariaiNariai (1972), Belinskii and KhalatnikovBelinskii et al. (1973), Reddy and RaoReddy and Rao (1981), Banerjee and SantosBanerjee and Santos (1982), Singh and Rai Singh and Rai (1983), Singh et al.Singh et al. (1983), Shri ramRam (1983), Berman et al.Berman et al. (1989), ReddyReddy (2003) and Adhav et al.Adhav et al. (2007) are some of the authors who worked on the several aspects of this theory. Bianchi type-III cosmological models in the presence of dark energy have been studied by several authors. Yadav and Yadav Yadav and Yadav (2011) studied Bianchi type-III model anisotropic dark energy models with constant deceleration parameter. Pradhan and Amirhaschi Pradhan and Amirhashchi (2011) have investigated anisotropic Bianchi type-III DE model with variable EoS parameter. Also, Naidu et. el.Naidu et al. (2012), Amirhaschi et. al.Amirhashchi et al. (2013) and Samatha Samanta (2013) have constructed Bianchi type-III models with dark energy in alternative theories of gravitation. Shamir and BhatiShamir and Bhatti (2012), Neelima and RaoNeelima and Rao (2019) have investigated Bianchi type-III models with dark energy within the framework of Brans-Dicke theory of gravitation. Srivastva et. al.Srivastava et al. (2019) have investigated Bianchi-III new holographic dark energy model with k-essence. Recentely, Raju et. al.Raju et al. (2020) have studied Bianchi type-III dark energy model with massive scalar meson field in general relativity.

Furthermore, Dark energy models are valuable to get a better understanding of why our universe is expanding. Higher dimensions play a significant role in the development of string theories. Unification of all fundamental forces is possible in the presence of higher dimensions. This fact has attracted many researchers in order to investigate the cosmological models with higher dimensionsWitten (1984); Appelquist et al. (1987); Chodos and Detweiler (1980). So, it would be an interesting theoretical approach to add extra dimensions in field equations and see how they affect our universe by analyzing various cosmological parameters. Lorentz-PetzoldLorenz-Petzold (1985), Ibanez and VerdaguerIbáez and Verdaguer (1986), Gleiser and DiazGleiser and Diaz (1988), Khadekar and GaikwadKhadekar and Gaikwad (2001) have formulated multi-dimensional cosmological models in general relativity and in other alternative theories of gravitation. Adhav et. al.Adhav et al. (2010), Biswal et. al.Biswal et al. (2012), Mete et. al.Mete et al. (2013) and Mishra and BiswalMishra and Biswal (2014) are some authors who have investigated various Bianchi models with binary mixture of perfect fluid and dark energy in higher dimensions. Reddy et. al.Reddy et al. (2012) have investigated higher dimensional kaluza-klien dark energy model in Saez-Ballester theory of gravitation. Reddy et. al.Reddy et al. (2016), Aditya and ReddyAditya and Reddy (2018), Singh and SinghSingh and Singh, 2019 have studied higher dimensional dark energy models in Brans-Dicke scalar tensor theory of gravitation. It has been found that our universe had gone through a number of phase transitions at the very initial stage of the evolution Kibble (1976). After the big bang explosion, our universe had cooled down during these phase transitions and it can be explained using unified theories Zel’dovich et al. (1974); Kibble (1980); Everett (1981); Vilenkin (1981). Moreover, these phase transitions may have become the reason for the formation of topological defects. These defects are in the form of domain walls, strings or vortices. String theory is able to give valid explanation of the unication of all fundamental forces and can act as a gravitational lens. Also, string theory gives a perfect explanation regarding structure formation. A higher dimensional universe is needed to apply string theory. Mishra et al. Mishra et al. (2017) have investigated accelerating dark energy models in two non interacting fluids within the framework of General Relativity with hybrid scale factor. Mishra et al.Mishra et al. (2018) studied the Bianchi-V string cosmological model with dark energy anisotropy in order to describe the role of anisotropic components on the dark energy and dynamics of the accelerating universe. Vinutha et. al.Vinutha et al. (2018) have constructed Bianchi type-I and Bianchi type-III dark energy models containing one dimensional cosmic strings in Saez-Ballester theory of gravitation. Further, Vinutha et. al.Vinutha, Shekar, Satyanarayana, 2019, Vinutha, Rao, Mengesha, 2021 studied various dark energy cosmological models with strings in different theories of gravitation. The consequences of these models are consistent with the present observational data.

Motivated by the above investigations, we have constructed a five dimensional Bianchi-III model with two fluid scenarios in Brans-Dicke theory. This paper is organized as follows. Section 2 represents field equations of the Brans-Dicke theory of gravitation containing dark energy and cosmic string. Section 3 contains solution of field equation obtained in section 2, using some conditions. Physical parameters of the power-law and exponential models are given in subsections 3.1 and 3.2. In section 4 discussions and conclusions are given.