#### Abstract

In this article, we analyze Bianchi type–II, VIII, and IX spatially homogeneous and anisotropic space-times in the background of the Brans–Dicke theory of gravity within the framework of viscous holographic dark energy. To solve the field equations, we have used the relation between the metric potentials as and the relation between the scalar field and the scale factor as . Also, we have discussed some of the dynamical parameters of the obtained models, such as the deceleration parameter , the jerk parameter , the EoS parameter , the density parameter , Om-diagnostic, squared speed of sound , EoS plane , and statefinder plane through graphical representation, which are significant in the discussion of cosmology. Furthermore, all the models obtained and graphically presented shown an expanding and accelerating Universe, which is in better agreement with the latest experimental data. The viscous holographic dark energy models are compatible with explaining the present cosmic accelerated expansion.

#### 1. Introduction

In 1905, the theory of Special Relativity (SR) [1–3] was put forward by A. Einstein, which shown the genesis of absolute space and absolute time by surpassing the single 4D space-time, which had only an absolute meaning [4]. The perception that the gravitational field in a small neighborhood of space-time is incomprehensible from a proper acceleration in the frame of reference (principle of equivalence), has taken an upturn from Special Relativity (SR) to General Relativity (GR), where the gravitation has been adjoined to SR (holds true only in the absence of gravitation), which eventually gives a curved space-time, as the SR is generalized for the accelerating observers. As an outcome of Mach’s limitation of absolute space, as Einstein had anticipated, the idea of general covariance (the absence of an advantaged frame of reference) develops [5] and by default obeys Mach’s principle. Apparently, this was not the case, since various anti-Machian elements were discovered in GR.

Although GR is undeniably an appealing theory [6–10], it fails to offer the ultimate interpretation of gravity (a paradigm of a perfect theory), disregarding all the advantages. The theory has several conceptual issues, most of which are often overlooked, in addition to its much-discussed incompatibility with quantum mechanics. If in space, consistent with the same old epitome, where 95% of the overall constituent material continues to be missing, its miles an intimidating sign for us to doubt back to the very foundations of the theory. A significant perspective with a prominent context of alternative theories of gravitation develops from a critical study of Mach’s principle, the equivalence principle, dark energy (DE) and dark matter (DM), and so on. Over the years, alternative theories of gravity have continued to draw considerable interest, leading to the discussion of numerous theories. These theories offered the first potentially feasible alternatives to the conventional general relativistic theory of gravity as proposed by Einstein. One of them is scalar-tensor theories of gravitation, where the dynamical DE component is introduced in the right-hand side of the Einstein field equations, and the other is modified theories of gravitation, where the left-hand side of the Einstein field equations are modified. Scalar-tensor theories have emerged as some of the most well-established and well-studied alternatives to conservative gravity theories in the literature.

The Brans–Dicke theory (BDT) [11] is the most natural choice as the scalar-tensor generalization of GR, which can be considered as a pioneer in the study of scalar-tensor theories, and the inclusion of Mach’s Principle led to the advent of this theory. This can be called the first-ever theory of gravity, where the metric tensor represents the dynamics of space-time and the scalar field describes the dynamics of gravity. The BDT also gives a fair description of the early era, as well as the present phases of cosmic evolution that gives a proper explanation for the Universe’s accelerated expansion [12], as this theory justifies the experiments in the Solar System domain [13]. The gravitational constant G in this theory is to be replaced with , where purely depends on the time and is coupled to gravity with a coupling parameter . It is evident in the literature that GR can be retrieved from the BDT if is a constant and [14, 15]. As and when the coupling constant takes huge values greater than 500, GR can be deduced from BDT [16], accounting for the recent Universe’s expansion and accommodating the observational data as well [17–19]. There is a prominent position for BDT among theories of gravitation because it is capable of accounting for the properties of cosmic expansion since the early phases of inflation [20]. A generalized (or modified) version of BDT [21–23] supports the notion that the parameter depends on the scalar field . Several models, based on the BDT, have been perfectly able to explain the properties of the expanding Universe with the help of various cosmological parameters [24–27]. The models based on the generalized BDT have an extremely low value for the coupling parameter, unaccommodating the findings during the implementation of the previous versions of the theory [20, 28–30]. A recent study in the BDT [31] has been made in the FRW models with a varying -term and a dynamic deceleration parameter. Also, it has been recently justified that for huge values of , the generalization of BDT can explain the Universe’s accelerated expansion with the interaction between matter and a scalar field [32].

The anisotropy and the spatial homogeneity are the two major characterizations for the Bianchi type (BT) cosmological models. There are nine different models altogether, and these have been classified into two classes [33]. The class-A represents the BT models I, II, VII, VIII, and IX, and class-B consists of BT models III, IV, V, VI, and VII. These models are known to describe the evolution of the Universe’s early stages in the presence of various physical distributions of matter, thereby explaining the structure and space properties of all the Einstein field conditions along with cosmological arrangements, and thus, rewriting the Einstein equations in the Hamiltonian form. In this regard, Misner [34, 35] and other authors have focused much of their efforts on fabricating a fine Hamiltonian system. In spite of this, these Hamiltonian forms could not be utilized to prove the collapse speculations by Lin and Wald [36, 37]. These BT models uncover the magnitude of anisotropy in the background radiation and provide a pragmatic picture of the past eras in the history of the cosmos. Furthermore, the cosmological problem of Einstein field equations from a theoretical perspective can be addressed well through the anisotropic models, as they tend to have greater generality when compared with isotropic solutions. In particular, we are involved in studying the BT-II, VIII, and IX cosmological models in the presence of viscous holographic dark energy (VHDE) in BDT. Diverse aspects of the BT-II, VIII, and IX cosmological models have been explored by many authors [38–41].

Present-time experimental and theoretical supernovae type Ia (SNeIa) observations [42, 43], cosmic microwave background radiation (CMBR) [44, 45], and large-scale structures [45, 46] provide the most enthralling evidence for the same. Many models of the Universe have been studied assuming the existence of a mysterious component, so-called DE, which generates huge negative pressure, imparting the mechanism for the accelerated cosmic expansion. Planck’s current measurements indicate that 68.3% of the Universe’s total energy content is in the form of DE. In spite of the success of standard cosmology, it is known that there are a few unsolvable problems that include the search for accommodating DE candidates, in which researches found the cosmological constant as a primary candidate for DE that not only describes the phenomenon of DE but also the ``fine tuning” and “cosmic coincidence”. For this reason, various dynamical DE models, which include a family of scalar fields such as quintessence [47–50], phantom [51–54], quintom [55, 56], tachyon [57–59], K-essence [60], and various Chaplygin gas models like generalized Chaplygin gas, extended Chaplygin gas, and modified Chaplygin gas [61–75], have been developed.

Way back then, viscosity played an influential role in the study of cosmology, which has been extended in recent years to include the study of an accelerating Universe and has acquired an immense interest in present times for numerous reasons. The idea of perfect fluid in the study of cosmic models has shown no dissipation and helps in the study of cosmic evolution. In the existent scenario, the study of imperfect fluid models has been suggested by introducing the concept of viscosity. In particular, the bulk viscous fluids that are included in the discourse of inflation are competent for explaining late-time cosmic acceleration. The increase in the viscosity is attributed to a Universe that is expanding at a rapid rate and can be understood as an accumulation of states that are out of thermal equilibrium in a small fraction of time. For these obvious reasons, the concept of viscosity has gained popularity in the study of space.

The holographic dark energy (HDE) models have seen success in recent years, with many considering them as the appropriate candidates to explain the problems of modern cosmology. The concept of HDE was initially introduced by Li [76] in 2004 with respect to the holographic principle [77–83] to elucidate the late-time Universe’s accelerated expansion. The holographic principle, as stated by black hole thermodynamics [84, 85], says that a hologram can be completely represented as a volume of space, which agrees with a theory related to the boundary of that space [86] and the AdS/CFT (anti-de Sitter/Conformal field theory) correspondence, as it can be observed in the seminal reference [87]. In [76], a holographic principle-based cosmic acceleration model was developed for the first time. As such, the reduced Plank mass and a cosmological length scale, taken as the future event horizon of the Universe, are the two physical quantities of the boundary of the Universe on which a DE model relies on. As a consequence of an ultraviolet cutoff for a region of size *L*, where the mass of a black hole of the similar size is not exceeded by the total energy, HDE density can be stated as with being a constant, being the reduced Planck mass, and being the Newtonian gravitational constant. The holographic principle is considered as a central principle of quantum gravity because of its applications in various fields of physics, viz. cosmology [88] and nuclear physics [89] in the present era. All the generalized HDE models known as of now are the suggested ones by [90], which came only after Li. Moreover, the Nojiri–Odintsov HDE gives a detailed description of covariant theories different from Li’s HDE [91]. A more dynamical scenario for HDE in the BDT along with matter creation has been suggested instead of Einstein gravity because of the fact that a dynamical frame is necessary to accommodate the HDE density that belongs to a dynamical cosmological constant. Considering various IR cut-offs in the framework of the BDT, a number of authors [92–99] have explained the rapid expansion of the cosmos and have shown a solution to alleviate the cosmic coincidence problem. With the help of cosmic observational data, Xu et al. [100] have constructed the HDE model in BDT.

Motivated by the above discussions and investigations in the Bianchi space-times, we investigate the anisotropic BT–II, VIII, and IX space-times in the presence of VHDE. This paper is planned as, in Section 2, we explain the metric and field equations. In Section 3, we obtain the solutions of the field equations, along with some important properties of the Universe in Section 4. Lastly, the interpretations of our models are presented in the last section.

#### 2. Metric and Field Equations

The spatially homogeneous BT metrics II, VIII, and IX of the form,have been considered, where the Eulerian angles are represented as , and are a function of only. It represents the following: BT II if and ; BT VIII if and ; BT IX if and .

The action of BDT in the presence of matter with Lagrangian in the canonical form (Jordan frame) is given bywhere is the Brans–Dicke scalar field representing the inverse of Newton’s constant, which is allowed to vary with space and time, is the scalar curvature, and is the Brans–Dicke constant. Varying the action in (2) with respect to the metric tensor and the scalar field , the field equations are obtained aswhere is an Einstein tensor, is the stress-energy tensor of the matter, and is the Ricci curvature tensor.

The conservation equationis a consequence of the field equations (3) and (4).

The energy momentum tensor for the VHDE is taken as

Here, and represent matter and VHDE tensors, which are given aswhere is the energy density of the matter, and , respectively, represent the pressure and energy density of the VHDE; denotes the comoving velocity vector of the matter and VHDE, satisfying . The VHDE pressure satisfies the relation where is the bulk viscosity coefficient. Equation of state parameters for VHDE and HDE, respectively, are defined as and . The unification of viscosity and HDE is a mathematical attempt in the light of the holographic principle. Moreover, it is hypothesized that DE is pervading the whole Universe, with bulk viscosity giving the accelerated expansion of phantom types in the late time evolution of the Universe. Here, we have assumed pressure less DM and that the effective pressure of VHDE is a sum of the pressure of HDE and bulk viscosity. Eckart [101] has proposed a type of effective pressure in the context of general relativity. Recently, Singh and Srivastava [102] have considered VHDE for FRW space-time.

Now, with the help of equation (6), the field equations (3) and (4) for the metric in equation (1) can be written as and the energy conservation equation becomes

Here, the over-head “dot” denotes differentiation with respect to ‘‘. When , the field equations (8)–(11) correspond to the BT-II, VIII, and IX Universes, respectively. Now, by using the transformation , the field equations (8)–(11) can be written as

Also, the energy conservation equation leads to

The conservation equation of the matter isand for VHDE, the conservation equation is

Here, the over-head dash denotes differentiation with respect to “.”

#### 3. Solutions of the Field Equations

Now, the set of equations (13)–(16) forms a system of four independent equations with seven unknowns: , , , , , , and . Hence, to find a determinate solution to these highly nonlinear differential equations, we need at least three physically viable conditions: We assume that the relation between the metric potentials (Collins et al., [103]) as where *n* *>* 1. We consider the relation between scale factor and scalar field (Tripathy et al., [104]) as where is a positive constant. We take the bulk viscosity coefficient in the following form (Ren and Meng [105]; Meng et al., [106])where and are positive constants and is the Hubble parameter.

Now, using conditions (20) and (21) in field equations (13) and (14), we get

##### 3.1. Bianchi Type- II Cosmological Model

If , equation (23) can be written as

On solving equation (24), we getwhere, , with , whereas is an integration constant and .

The spatial volume and average scale factor are given by

From equations (27) and (21), the scalar field is given by

The pressure of the VHDE iswhere

The energy density of the matter iswhere is the constant of integration.

The energy density of the VHDE iswhere

The viscosity coefficient is given by

Now the metric (1) can be written as

##### 3.2. Bianchi Type-VIII Cosmological Model

If , equation (23) can be written as

We can solve the above equation and get the deterministic solution only for .

Thus, from the equation (36), we getwhere and with .

From the equation (37), we get

The spatial volume and an average scale factor are given by

From equations (40) and (21), the scalar field is given by

The pressure of the VHDE iswhere

The energy density of the matter has the following expression:where is an integration constant.

The energy density of the VHDE is obtained aswhere

The viscosity coefficient is given by

Hence, the metric (1) takes the following form:

##### 3.3. Bianchi Type-IX Cosmological Model

If , equation (23) can be written as

We can solve the above equation and get the deterministic solution only for .

So, from equation (49), we getwhere, and with .

From equation (49), we get

The spatial volume and average scale factor are given by

From equations (21) and (53), the scalar field is given by

The pressure of the VHDE is as follows:where

The energy density of the matter is obtained as

the energy density of the VHDE has the following expression:where

The viscosity coefficient is

Hence, the metric (1) can be written as

Thus, equations (35), (48), and (61) represent spatially homogeneous and anisotropic BT-II, VIII, and IX VHDE cosmological models, respectively, in the Brans–Dicke scalar theory of gravitation. For graphical representation we consider the following vales: , for BT-II, for BT-VIII, for BT-IX, , , , , , , , , , , , , , . The plots of VHDE pressure (*P*_{vhde}) against redshift respectively for BT-II, VIII, and IX models have been represented in Figures 1–3for different values of . Here, we observe that the behavior of for three different values of is the increasing with redshift and varies in the negative region, which indicates the cosmic expansion. Moreover, as increasing the values of , we get more acceleration of the Universe. Also, we have depicted the energy density of VHDE versus redshift in Figures 4–6 for BT–II, VIII. and IX models, respectively, and we observe that the trajectory varies in the positive region, which indicates an accelerated expansion of the cosmos. The bulk viscous coefficient has been plotted against redshift with various values of and in Figures 7–9 for BT-II, VIII, and IX models, respectively. The trajectories vary in positive regions throughout the evolution of all the three models for various values of and , which indicates an accelerated expansion of the Universe.

Some of the cosmological properties of the models are discussed as follows: The mean Hubble parameter is given by where are directional Hubble’s parameters, which express the expansion rates of the Universe in the directions of , , and , respectively. The mean Hubble’s parameter of BT-II, VIII, and IX VHDE cosmological models are, respectively, given by The anisotropic parameter of the BT-II VHDE model is given by and for the BT-VIII and IX VHDE cosmological models, we get From equations (64) and (65), we can observe that , which indicates that the BT-II, VIII, and IX models are always anisotropic throughout the evolution of the Universe with respect to VHDE. The expansion scalar () has been defined as whose expressions for the BT-II, VIII, and IX VHDE models are, respectively, given as The shear scalar () is defined by the following equation and is followed by the expressions of for BT–II, VIII, and IX VHDE models, respectively, as

#### 4. Some Other Important Properties of the Models

Now we compute the following dynamical parameters, which are significant in the physical discussion of the cosmological models presented in equations (35), (48), and (61).

##### 4.1. Deceleration Parameter ()

The parameter is defined asthat depends upon the scale factor and its derivatives. It is considered to describe the transition phase of the Universe and basically computes the expansion rate of the cosmos. Whenever the deceleration parameter shows a positive curve, it indicates the decelerated expansion of the Universe. Whereas, the negative curve implies that there is an accelerated expansion of the cosmos and at there exists the marginal inflation. The deceleration parameters of the BT-II, VIII, and IX models are, respectively, given by

The behavior of the deceleration parameter(*q*) for BT-VIII and IX models has been depicted against the redshift in Figures 10 and 11, respectively. From Figure 10, it is observed that the path of the BT-VIII curve travels from the deceleration to the acceleration phase while passing through the transition line. Whereas, from Figure 11, it is clear that the curve for BT-IX varies in an accelerated phase. However, the deceleration parameter of BT-II is independent of time. Some of the authors, namely, Berman [107], Bishi et al. [108], Santhi et al. [109], Santhi and Naidu [110], Samanta [111], Kumar and Singh [112], have attained a constant *q* in their research.

##### 4.2. Jerk Parameter (*j*)

The cosmic jerk,

can be accounted for by the transition of the Universe from the decelerating to the accelerating phase. For various models of the cosmos, there is a variation in the transition of the Universe whenever the jerk parameter lies in the positive region and the deceleration parameter lies in the negative region (Visser [113]). Rapetti et al. [114] showed that for the flat ΛCDM model, the value of jerk becomes unity.

The Jerk parameter of the BT-II, VIII, and IX models is, respectively, given by

Figures 12 and 13 represent the variations of the jerk parameter plotted against redshift (*z*) for the models BT-VIII and IX, respectively. The trajectory of the jerk parameter for the BT-VIII model in Figure 12 varies in the positive region, whereas for the BT-IX model in Figure 13, the trajectory varies in the negative region, both of which approach unity in late times. However, the jerk parameter for BT-II is independent of time. Santhi and Naidu [115], Rao et al. [116], Santhi et al. [117], Rao and Prasanthi [118], and Shaikh et al. [119] are some of the researchers who have acquired a constant jerk parameter in their work.

##### 4.3. Statefinder Pair (*r*, *s*)

As mentioned earlier, a mysterious force, the DE, may be responsible for the cosmos to undergo an accelerated expansion int the current era. But as of now, there is no adequate information about DE. Hence, it becomes necessary to identify and understand the various properties of DE and its importance in various kinds of cosmographic models. Ratra and Peebles [47], Kamenschik et al. [61], Armendariz Picon et al. [60] Dvali et al. [120] have proposed various studies to realize that different DE forms, such as quintessence, Chaplygin gas, k-essence, and brane world models, give several families of curves for scale factor *a*(*t*). As a way of categorizing the various types of DE, Sahni et al. [121] have proposed a diagnostic pair known as the “statefinder diagnostic,” defined asthat is based upon the derivatives of the scale factor *a*(*t*) and the deceleration parameter *q*. We have obtained expressions for the statefinder diagnostic pair (*r, s*) for the models BT-II, VIII, and IX, which are respectively given by

The interpretation of the statefinder pair from Figures 14 and 15 says that the (*r, s*) plane for BT–VIII and IX models starts its evolution from the quintessence and phantom regions and reaches the ΛCDM model(for *r* = 1*, s* = 0). Also, for the BT-II Universe, the statefinder plane is independent of time. Shanti et al. [122], Samanta and Mishra [123], Katore and Gore [124], and Shaikh et al. [119] are some of the authors who have obtained the statefinder parameters independent of time.

##### 4.4. EoS Parameter

To classify the phases of the inflating Universe, viz., the transition from decelerated to accelerated phases containing DE and radiation dominated eras, the EoS parameter can be broadly used, whose expression is given by .

It categorizes various epochs as follows:

Decelerated phase:(i)stiff fluid (),(ii)the radiation dominated phase () and(iii)dust fluid phase or cold dark matter ().

Accelerated phase:(i)the quintessence phase (),(ii)cosmological constant/vacuum phase ()(iii)quintom era and phantom era ()

The EoS Parameter for the BT-II VHDE cosmological model is given bywhere

The EoS parameter for the BT-VIII VHDE cosmological model is given bywhere

The EoS parameter for the BT-IX VHDE cosmological model is given bywhere

Figures 16–18 show the behavior of the EoS parameter taken against redshift () with different values of and for all the three models of BT-II, VIII, and IX, respectively. Here, we notice that the curves of for BT-II and IX models for different values of and begin from the quintessence region () and cross the nonrelativistic matter () then reaching the phantom region (), whereas for BT-VIII, in the presence of null viscosity (and ), the curve of varies in the quintessence region and as the values of and are increased, we get the quintessence to the phantom region by crossing the phantom divided line, varying more in the phantom region, which indicates accelerated expansion of the Universe. According to the obtained models, the observed EoS parameters match the Planck data [125], where the EoS parameter limits are as follows:

##### 4.5. plane

Cadwell and Linder [126] have suggested plane (where signifies differentiation w.r.t a) to interpret the accelerated expansion regions of the cosmos and to analyze the quintessence scalar field for the first time. For various values of and , the plane describes two distinct areas. The plane is described as the thawing zone for when and the freezing region for when .

For the BT-II VHDE cosmological model,where

For the BT-VIII VHDE cosmological model,where