LRS Bianchi I model with strange quark matter and Λ(t) in $f(R,T)$ gravity

Highlights

• LRS Bianchi-I model is studied with strange quark matter and $\Lambda \left(t\right)$ by assuming a special law of Hubble parameter in $f(R,T)=R+2\lambda T$ gravity.

• The model presents a transition from early deceleration to late time acceleration.

• In the absence of $f(R,T)$ gravity and SQM, the variable cosmological term accelerates the universe.

• In the absence of $\Lambda \left(t\right)$, the SQM accelerates the universe, even in the absence of $f(R,T)$ gravity. Hence, the SQM could be an alternative to dark energy.

• The model evolves isotropically and becomes anisotropic in the course of evolution but finally approaches isotropy again at late times.

Abstract

A spatially homogeneous and anisotropic locally-rotationally-symmetric Bianchi type-I spacetime model with strange quark matter (SQM) and a variable cosmological term $\Lambda \left(t\right)$ is studied in $f(R,T)$ theory. The exact solutions are obtained for a particular form of $f(R,T)=R+2\lambda T$ under the law of variable deceleration parameter proposed by Banerjee et al (2005). The model presents a cosmological scenario of transition from early deceleration to late time acceleration. In the absence of $f(R,T)$ gravity and SQM, $\Lambda \left(t\right)$ accelerates the universe. In a specific case when $\Lambda \left(t\right)$ vanishes at late times, the SQM accelerates the universe, even in the absence of $f(R,T)$ gravity. The model shows the possibility that SQM could be an alternative to dark energy. The physical viability of the model costs the observational inconsistency.

Introduction

Modern cosmology rests ultimately on observational data. These observations indicate that about two thirds of the critical energy density in the universe seems to be stored in a form of an unknown component called ‘dark energy’ (DE), which causes the accelerated expansion of the universe due to the influence of its gravitationally repulsive nature (for detail see the reviews Copeland, Sami, Tsujikawa, 2006, Frieman, Turner, Huterer, 2008). In the absence of better information about DE, a cosmological constant $\Lambda$ is supposed the simplest candidate for DE. The Big-Bang model based on Einstein’s general relativity (GR) with $\Lambda$ is known as the $\Lambda$ CDM (cold dark matter) model (Sahni and Starobinsky, 2000). This is the most widely accepted model to justify the observed phenomena of the universe (Komatsu, et al., 2009, Ade, et al., 2016), except the smallness of the observed cosmological constant, which is known as the fine-tuning problem. If the cosmological constant is not a real constant but a function of time, the fine-tuning problem can be resolved. A dynamically decaying $\Lambda$ could be large in the early universe and decays in the course of the expansion of the universe to its present small value by creating massive or massless particles. Several authors have advocated a variable $\Lambda$ to account for this fact (see Abdussattar, 1999 and references therein).

Another view to adjudicate the problem of DE is the modification of Einstein’s GR. There exist numerous proposals which are the modifications in some way or the other of the Einstein–Hilbert (EH) gravitational action (De Felice, Tsujikawa, 2010, Sotiriou, Faraoni, 2010). At the beginning, the interest in modified theories was focused on the modification of the geometric part of the EH action. In 2011, Harko et al. (2011) introduced $f(R,T)$ gravitational theory by proposing a general non-minimal coupling between matter and geometry. The effective gravitational Lagrangian consists of an arbitrary function of the Ricci scalar $R$ and the trace $T$ of the energy-momentum tensor. An unusual feature of $f(R,T)$ gravity is the non-vanishing covariant derivative of the stress-energy tensor. As a consequence, the equations of motion show the presence of an extra-force acting on a test particle. Consequently, the motion is non-geodesic, and an extra acceleration is always present in $f(R,T)$ gravity not only from a geometrical contribution, but also from the matter content. This extraordinary behavior of $f(R,T)$ gravity has attracted many researchers to explore this theory in different contexts of cosmology and astrophysics (see Singh, Beesham, 2018, Singh, Beesham, 2020 for extensive list of references). The early considerations are mainly investigated in a spatially flat homogeneous and isotropic universe described by the Friedmann–Lemaitre–Robertson–Walker (FLRW) metric.

Although the universe at present is approximated as homogeneous and isotropic at sufficiently large scales, there are theoretical arguments with the support of observational evidences for the existence of an anisotropic phase at early stages of evolution. The observational data of the Cosmic Microwave Background (CMB) (Netterfield et al., 2002) and the Wilkinson Microwave Anisotropy Probe (WMAP) Bennett et al. (2013); Hinshaw et al. (2013) reveal that the universe on small scales is somewhat inhomogeneous and anisotropic. The recent outcomes of the Planck Collaboration (2020) also evidence tiny anisotropies in temperature.

The small-scale anisotropies in the CMB radiation are dominated by the acoustic oscillations created during recombination. Therefore, it is theorised that the initially small-amplitude CMB anisotropies at recombination seeded the large-scale discrete structures, e.g., galaxy clusters, filaments, and voids that we see today. Consequently, the anisotropic models play a substantial role to describe the early state of our present universe. Therefore, the study of anisotropic models gains a lot of attention. The $f(R,T)$ theory of gravity is no exception, several studies have been carried out in anisotropic spacetimes (see Singh and Beesham, 2019 and references therein). Amongst the various families of homogeneous and anisotropic cosmological models, the study of the possible effects of anisotropy at early times makes the Bianchi type-I (BI) model as the prime alternative as it is the simplest. Particularly, the Locally-Rotaionally-Symmetric (LRS) BI models represent the simplest generalisation of the flat FLRW models.

On the other hand, most of the investigations on present accelerating expansion are considered within the homogeneous and isotropic background. The existence of the analytical solutions of the field equations in most of the cases justifies this over-simplification of the geometry of space-time. In flat FLRW cosmology, the violation of strong energy condition is what uniquely characterizes accelerated expansion. With this requirement many solutions of the Einstein equations, different from the isotropic one, may describe an accelerating universe. This holds true in particular for spatially homogeneous and non-isotropic models as in most of the cases anisotropic models also tend to isotropic one at late times. Consequently, non FLRW models have been studied by several researchers as cited in the following discussion to determine whether these models can describe accelerating universe at late times.

As far as the matter content of the universe is concerned, quarks are the most fundamental particles known as the building blocks of the baryonic matter in the standard model of particle physics. Quark matter (QM) is composed of up and down quarks whereas strange quark matter (SQM) is composed of up, down and strange quarks. SQM could be the true state of hadronic matter (Itoh, 1970, Bodmer, 1971, Farhi, Jaffe, 1984, Witten, 1984) (for a detailed review of SQM and its properties, see Ref. Cheng et al., 1998). QM is thought to have originated when the universe underwent a quark gluon phase transition for a few microseconds just after the big-bang. After the quark gluon phase transition, another transition called quark-hadron phase, occurred in which the Quark Gluon Plasma (QGP) got transformed into hadron gas at the temperature $T=200\text{MeV}$. There are two approaches to SQM creation: one is the aforesaid phase transition and the other is strange matter made from neutron stars at ultra-high density. SQM is believed to exist at the center of neutron stars (see Ref. Aktas and Yilmaz, 2011 for a rich list of references), in strange stars (Drake et al., 2002), or even as small pieces of strange matter (Weber, 2005). If the SQM hypothesis is correct, then all observed pulsars may actually be strange stars but not neutron stars, due to the contamination process by strange nuggets in the universe (see Refs. Aktas, Yilmaz, 2011, Geng, Huang, Lu, 2015 and references therein).

Many authors have studied isotropic and anisotropic cosmological models with SQM in the context of general relativity as well as various modified theories of gravitation. Mak and Harko (2004) investigated charged SQM in the presence of conformal motion in isotropic homogeneous spacetime. Aktas and Yilmaz (2011) showed that the structure of the first few seconds of the early universe was homogeneous and isotropic when QM existed. Yilmaz et al. (2012) have discussed FLRW and Bianchi type I and V models filled with quark and SQM in $f\left(R\right)$ gravity. Khadekar and Shelote (2012) have obtained higher dimensional cosmological models in the presence of quark and SQM. Boeckel and Bielich (2012) have studied little inflation in the presence of quark-gluon matter. Holdon (2011) has explored DE in respect of confinement. Rahaman et al. (2012) have shown the possibility that, at the galactic level, QM behaves like dark matter and, at the global level, like DE. The dynamics of a gravitating gluon condensate in an FLRW universe have been studied in the context of acceleration and modified gravity by Klinkhamer (2010). Thomas and Zhitnitsky (2012) have examined Casimir type effects as a source of DE. Adhav et al. (2015) have considered Kantowski–Sachs cosmological models in $f\left(R\right)$ gravity.

Many researchers have employed anisotropic models for the unified description of past deceleration and present acceleration of the universe. In the present study, our purpose is to investigate whether these models are physically viable for describing a unified cosmic evolution. Also, if these models are physically viable then whether they are observationally consistent too. For this purpose, we choose an LRS BI model containing SQM in $f(R,T)$ gravity with a time dependent $\Lambda$ term. Exact solutions of the modified field equations have been generated explicitly for the specific choice $f(R,T)=R+2f\left(T\right)$, where $f\left(T\right)=\lambda T$, $\lambda$ an arbitrary constant. The work is organised as follows. The basic formalism of $f(R,T)$ gravity is presented in Section 2. The field equations of the LRS BI spacetime filled with SQM with a variable cosmological parameter are modelled in Section 3. The significance of $f(R,T)$ gravity and SQM is discussed in Section 4. The sum up of the findings are accumulated in the concluding Section 5.