Dynamical system analysis of Bianchi III Universe with $\boldsymbol{f(R,T)}$ gravity theory

Homogeneity and isotropy are the two basic cosmological principles that form the basis of standard cosmology, often known as the $\Lambda$CDM cosmology. This cosmological model provides answers to many questions about our understanding of the world through the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, together with the energy-momentum tensor in the conventional perfect fluid form [1]. Researchers are encouraged to expand this traditional formalism to identify alternative theories or modifications to general relativity (GR) for a number of reasons, including the Universe’s current rapid expansion [2, 3, 4], non-observational evidence about dark matter (DM) [5], and dark energy (DE) [6]. Therefore, a good number of modified formalisms of GR have been developed to refute the idea of the Universe’s exotic matter and energy contents used to describe respectively its missing mass and current rapid expansion, usually referred to as the modified theories of gravity (MTGs).

The most basic MTG is the $f(R)$ theory of gravity [7, 8, 9], which substitutes a function of the Ricci scalar $f(R)$ for the Ricci scalar $R$ in the Einstein-Hilbert action of GR. This theory shows its versatility in cosmology, black hole physics [10, 11, 12, 13, 14], cosmic ray physics [15, 16], and other theoretical research fields [17, 19, 18]. The Refs. [20, 21, 22, 23, 24] contain a comprehensive collection of cosmological studies for both isotropic and anisotropic Universes using the $f(R)$ theory of gravity. The $f(R,T)$ gravity [9, 25], in which the gravitational Lagrangian density is the arbitrary function of both the Ricci scalar and the trace $T$ of the energy-momentum tensor $T_{\mu\nu}$, is another well-liked MTG in cosmology as well as other areas of gravitational physics and related domains. The source term that represents the change in the energy-momentum tensor with respect to the metric is the foundation of the $f(R,T)$ theory. The matter Lagrangian density $\mathcal{L_{M}}$ serves as the basis for the general formulation of this source term. The field equations generated by each $\mathcal{L_{M}}$ option are distinct [26, 27, 28]. Numerous studies have been conducted using the $f(R,T)$ theory of gravity, some may be found in Refs. [25, 29, 30, 31, 32, 33]. In addition to these MTGs, some other theories, such as teleparallel gravity theories, like $f(T)$ gravity theory [34, 35, 36, 37, 38], $f(Q)$ gravity theory [39, 40, 42, 41, 43, 44, 45], and some standard model extension (SME) like bumblebee gravity theory [46, 47, 48], which are usually termed together as the alternative theories of gravity (ATGs), have gained popularity among cosmological scholars in recent years. Comparison of these MTGs and ATGs to data has yielded encouraging results [26, 33, 38, 48, 45].

There are several shortcomings when taking into account common assumptions like isotropy and homogeneity in cosmological research, in addition to the absence of direct empirical proof of the existence of exotic matter and energy contents as mentioned. Despite these presumptions, researchers have been able to explain the majority of cosmological aspects using the standard $\Lambda$CDM model, which includes the Hubble tension [49, 50], the $\sigma_{8}$ tension [49], the coincidence problem [51], and others. Nonetheless, many reliable observational data sources, such as WMAP [52, 53, 54], SDSS (BAO) [55, 56, 57], and Planck [58, 49], have revealed some departures from the conventional cosmological principles and raise the possibility that the Universe contains some anisotropies. Additionally, studies indicate that the Universe’s shape is large-scale planar symmetric. Without changing the higher-order multipole of temperature anisotropy in the CMB angular power spectrum, the eccentricity of order $10^{-2}$ can match the quadrupole amplitude with observational evidence [59]. The existence of asymmetry axes in the Universe is confirmed by polarization studies of electromagnetic radiation traveling great distances [60]. Thus, not all cosmic features can be adequately explained by the principles of homogeneity and isotropy alone.

A metric with a homogeneous background and an anisotropic feature is required to describe the anisotropic nature of the Universe. In order to extract anisotropy information for cosmological investigations, researchers often choose Bianchi types I, III, V and IX, which are among the eleven types of anisotropic metrics that Luigi Bianchi suggested [61]. The majority of the cosmological studies in these metrics are mostly confined to type I, and relatively little research has been done in this area. Some works on anisotropic cosmological investigations in Bianchi type I metric can be found in Refs. [62, 63, 64, 65, 66, 67, 68, 69, 70]. However, as mentioned already the study in Bianchi type III (BIII) metric is very much limited. A few works in the BIII metric are mentioned here as follows. A. Moussiaux et al. in 1981 had found an exact particular solution to the Einstein field equations for the vacuum which includes a cosmological constant in the BIII metric [71]. D. Lorenz put up a model that incorporates dust and a cosmological constant in the BIII metric in 1982 [72]. Ref. [73] has made another study that suggested a viscous cosmological model with the variable $\Lambda$ and gravitational constant $G$. J. P. Singh et al. had examined a work with the variable $G$ and $\Lambda$ in the presence of a perfect fluid, assuming a conservation rule for the energy-momentum tensor in the BIII metric in 2007 [74]. In Ref. [75], a BIII model with a perfect fluid, the time-dependent $\Lambda$, and constant deceleration parameter were discovered. P. S. Letelier in 1980 had studied certain two-fluid cosmological models that have symmetries similar to those of the BIII models. In these models, an axially symmetric anisotropic pressure is produced by the independent four-velocity vectors of the two non-interacting perfect fluids [76].

In our work, we try to study the BIII Universe as a dynamical system to understand its evolution and the possibility of various phases of it. Dynamical system analysis is a powerful and elegant way to study the dynamics of the Universe through recasting the cosmological equations as a dynamical system [77]. The dynamical system technique has been used to investigate the stability of systems employing a variety of cosmological models, including the canonical [78, 79] and non-canonical scalar-field models [80, 81, 82], the scalar-tensor theories [83, 84], the $f(R)$ gravity theory [85, 86, 87], and others [88, 89, 90, 91]. Ref. [92] provides a contemporary overview of the dynamical system method. A study on dynamical system analysis of isotropic Universe in $f(R,T)$ theory has been found in Ref. [93]. The cosmic dynamical system in $f(Q)$ gravity theory has also been the subject of a few studies [94, 95]. However, there aren’t many comparable studies using the Bianchi metric in the $f(R,T)$ theory. Thus we intend to investigate it using the BIII metric to learn more about the anisotropic Universe in $f(R,T)$ gravity theory. Here we are considering the BIII metric in $f(R,T)$ gravity theory to understand the Universe as a dynamical system. In this work, we are using constrained values of various cosmological parameters from Ref. [96] which are constrained through various observational data like Hubble data, Pantheon plus data, BAO data, etc. through powerful the Bayesian inference technique, and plot the phase portraits of dynamical variables to understand the evolution of the Universe for different $f(R,T)$ gravity models.

The current article is organized as follows. Starting from the introduction part in Section I to explain the importance of the BIII Universe and MTGs, specifically the $f(R,T)$ gravity theory from various litterateurs, we discuss the general form of field equations in $f(R,T)$ gravity theory in Section II. In Section III, we develop the required field equations and continuity equation for the BIII metric for two different $f(R,T)$ models. In Section IV, the dynamical system analyses for two different $f(R,T)$ models have been performed. In Section V we study a special $f(R,T)$ model and test its physical viability. Finally, the article has been summarized with conclusions in Section VI.

For the $f(R,T)$ theory of gravity, the modified Einstein-Hilbert action may be expressed as [25]

where $\kappa=8\pi G$. The field equations for the theory can be obtained by varying the action (1) with respect to the metric tensor $g_{\mu\nu}$ and can be written as

where $f_{R}(R,T)$ and $f_{T}(R,T)$ are the derivatives of $f(R,T)$ with respect to Ricci scalar $R$ and trace $T$ of the energy-momentum tensor respectively, and the energy-momentum tensor $T_{\mu\nu}$ can be written as

The tensor term $\Theta_{\mu\nu}$ can be expressed as [25]

For the conventional diagonal energy-momentum tensor, the matter Lagrangian can be considered as $\mathcal{L_{M}}=-\,p$ [25]. Thus, the $\Theta_{\mu\nu}$ reduces to

With the help of the general form of field equations (2), we derive field equations for the BIII metric for the conventional perfect fluid in the next sections for some specific $f(R,T)$ gravity models.

The BIII metric in our study has the form:

Here, $a_{1}$, $a_{2}$ and $a_{3}$ are the directional scale factors along $x$, $y$ and $z$ directions respectively and $m$ is a constant. Thus, we have three directional Hubble parameters $H_{1}=\dot{a_{1}}/a_{1}$, $H_{2}=\dot{a_{2}}/a_{2}$ and $H_{3}=\dot{a_{3}}/a_{3}$ along $x$, $y$ and $z$ directions respectively. The average scale factor for the metric is $a=(a_{1}a_{2}a_{3})^{\frac{1}{3}}$ and the average Hubble parameter can be written as

In our work, we consider the conventional perfect fluid model of the Universe and for this model the energy-momentum tensor $T_{\mu}^{\nu}=diag(-\,\rho,p,p,p)$, and hence the components of $\Theta_{\mu\nu}$ from equation (5) can be obtained as $\Theta_{00}=2\rho+p$, $\Theta_{11}=a_{1}^{2}\,p$, $\Theta_{22}=a_{2}^{2}\,e^{-2mx}p$, and $\Theta_{33}=a_{3}^{2}\,p$. We are now in a position to derive field equations for the $f(R,T)$ gravity models under consideration in the BIII metric. We consider the following two forms of $f(R,T)$ models in our study as suggested in Ref. [25]:

With these two forms of models, we proceed as follows.

In this section, we analyse the BIII Universe as a dynamical system by using the above mentioned $f(R,T)$ gravity models. Here we try to assess the system stability by identifying the critical points and their characteristics.

In this section, we are interested in studying a special model $f(R,T)=f_{1}(R)+f_{2}(R)f_{3}(T)=(\zeta+\eta\,\tau\,T)R$ which shows inconsistency with the standard cosmological evolution as predicted by $\Lambda$CDM model [96]. The field equations for the considered model can be written as [96]

Here $\zeta,\eta,\tau$ are model parameters, and $\xi$ is the directional anisotropy parameter ($\gamma$ in previous models). The Friedmann-like equation can be obtained from equation (42) and it takes the form:

Again, from equation (43) we can obtain

With the help of equation (44) and (45) we construct the dynamical system equations as

Both equations (46) and (47) are not only dependent on dynamical variables $X$ and $Y$ like previously mentioned models but also depend on other variables like Ricci scalar $R$ and the trace of the energy-momentum tensor $T$ which themselves are the functions of $X,Y,Z$ and $S$. Thus, the situations are more complicated in this model. However, from Ref. [96] we found that the constrained values of $\eta=0.000013$ and $\tau=0.00000033$ are very small and hence we can write $(1-3\,\zeta\tau R)\sim 1$, $(1-4\,\zeta\tau R)\sim 1$ and $(\zeta+\eta\,\tau\,T)\sim\zeta$. With these approximations, we can rewrite the equation (46) and (47) as

We have compiled the fixed-point solutions for the considered model in Table 3 and drawn the phase portrait for $X$ versus $Y$ for the constrained set of model parameters from Ref. [96] in Fig. 6. $\omega_{eff}$ and $q$ in this table are obtained by using equations (29) and (30) respectively with replacing $\gamma$ by $\xi$ as directional anisotropic parameter.