Some Bianchi Type Viscous Holographic Dark Energy Cosmological Models in the Brans–Dicke Theory

. 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 feld equations, we have used the relation between the metric potentials as R � S n and the relation between the scalar feld ϕ and the scale factor a as ϕ � a m . Also, we have discussed some of the dynamical parameters of the obtained models, such as the deceleration parameter ( q ) , the jerk parameter ( j ) , the EoS parameter ( ω vhde ) , the density parameter ( Ω vhde ) , Om-diagnostic, squared speed of sound ( v 2 s ) , EoS plane ( ω vhde − ω vhde ′ ) , and statefnder plane ( r − s ) through graphical representation, which are signifcant 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. Te viscous holographic dark energy models are compatible with explaining the present cosmic accelerated expansion.


Introduction
In 1905, the theory of Special Relativity (SR) [1][2][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]. Te perception that the gravitational feld 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][7][8][9][10], it fails to ofer the ultimate interpretation of gravity (a paradigm of a perfect theory), disregarding all the advantages. Te 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 signifcant 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. Tese theories offered the frst 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 feld equations, and the other is modifed theories of gravitation, where the left-hand side of the Einstein feld equations are modifed. Scalar-tensor theories have emerged as some of the most well-established and well-studied alternatives to conservative gravity theories in the literature.
Te 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. Tis can be called the frst-ever theory of gravity, where the metric tensor represents the dynamics of space-time and the scalar feld describes the dynamics of gravity. Te 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 justifes the experiments in the Solar System domain [13]. Te gravitational constant G in this theory is to be replaced with 1/ϕ , where ϕ purely depends on the time and is coupled to gravity with a coupling parameter ω. It is evident in the literature that GR can be retrieved from the BDT if ϕ is a constant and ω ⟶ ∞ [14,15]. As and when the coupling constant ω 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][18][19]. Tere 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 infation [20]. A generalized (or modifed) version of BDT [21][22][23] supports the notion that the parameter ω depends on the scalar feld ϕ. 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][25][26][27]. Te models based on the generalized BDT have an extremely low value for the coupling parameter, unaccommodating the fndings during the implementation of the previous versions of the theory [20,[28][29][30]. A recent study in the BDT [31] has been made in the FRW models with a varying Λ-term and a dynamic deceleration parameter. Also, it has been recently justifed that for huge values of ω, the generalization of BDT can explain the Universe's accelerated expansion with the interaction between matter and a scalar feld [32].
Te anisotropy and the spatial homogeneity are the two major characterizations for the Bianchi type (BT) cosmological models. Tere are nine diferent models altogether, and these have been classifed into two classes [33]. Te 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. Tese 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 feld 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 eforts on fabricating a fne Hamiltonian system. In spite of this, these Hamiltonian forms could not be utilized to prove the collapse speculations by Lin and Wald [36,37]. Tese 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 feld 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][39][40][41].
Way back then, viscosity played an infuential 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. Te idea of perfect fuid 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 fuid models has been suggested by introducing the concept of viscosity. In particular, the bulk viscous fuids that are included in the discourse of infation are competent for explaining late-time cosmic acceleration. Te 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.
Te 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. Te concept of HDE was initially introduced by Li [76] in 2004 with respect to the holographic principle [77][78][79][80][81][82][83] to elucidate the late-time Universe's accelerated expansion. Te 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 2 Advances in Astronomy with a theory related to the boundary of that space [86] and the AdS/CFT (anti-de Sitter/Conformal feld 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 frst 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 cutof (Λ) 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 being the reduced Planck mass, and G being the Newtonian gravitational constant. Te holographic principle is considered as a central principle of quantum gravity because of its applications in various felds 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 diferent 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-ofs in the framework of the BDT, a number of authors [92][93][94][95][96][97][98][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. Tis paper is planned as, in Section 2, we explain the metric and feld equations. In Section 3, we obtain the solutions of the feld equations, along with some important properties of the Universe in Section 4. Lastly, the interpretations of our models are presented in the last section.

Metric and Field Equations
Te spatially homogeneous BT metrics II, VIII, and IX of the form, (1) have been considered, where the Eulerian angles are represented as (θ, φ, ψ), R and S are a function of t only. It represents the following: BT IX if f(θ) � sinθ and h(θ) � cosθ.
Te action of BDT in the presence of matter with Lagrangian L m in the canonical form (Jordan frame) is given by where ϕ is the Brans-Dicke scalar feld representing the inverse of Newton's constant, which is allowed to vary with space and time, R is the scalar curvature, and ω is the Brans-Dicke constant. Varying the action in (2) with respect to the metric tensor g ij and the scalar feld ϕ, the feld equations are obtained as where G ij � R ij − 1/2Rg ij is an Einstein tensor, T ij is the stress-energy tensor of the matter, and R ij is the Ricci curvature tensor. Te conservation equation is a consequence of the feld equations (3) and (4). Te energy momentum tensor for the VHDE is taken as Here, T m ij and T h ij represent matter and VHDE tensors, which are given as where ρ m is the energy density of the matter, P vhde and ρ vhde , respectively, represent the pressure and energy density of the VHDE; U i denotes the comoving velocity vector of the matter and VHDE, satisfying U i U i � and the energy conservation equation becomes Here, the over-head "dot" denotes diferentiation with respect to 't'. When δ � 0, − 1, & + 1, the feld equations (8)- (11) correspond to the BT-II, VIII, and IX Universes, respectively. Now, by using the transformation dt � R 2 SdT, the feld equations (8)- (11) can be written as Also, the energy conservation equation leads to Te conservation equation of the matter is and for VHDE, the conservation equation is Here, the over-head dash denotes diferentiation with respect to "T."

Solutions of the Field Equations
Now, the set of equations (13)-(16) forms a system of four independent equations with seven unknowns: R, S, ϕ, ρ m , ρ vhde , P vhde , and ζ. Hence, to fnd a determinate solution to these highly nonlinear diferential 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 a and scalar feld ϕ (Tripathy et al., [104]) as where m is a positive constant. We take the bulk viscosity coefcient in the following form (Ren and Meng [105]; Meng et al., [106]) where ζ 0 and ζ 1 are positive constants and H is the Hubble parameter.
Te spatial volume and average scale factor are given by From equations (27) and (21), the scalar feld ϕ is given by Te pressure of the VHDE is where Te energy density of the matter is where β 2 is the constant of integration. Te energy density of the VHDE is where Te viscosity coefcient is given by Now the metric (1) can be written as (23) can be written as

Bianchi Type-VIII
We can solve the above equation and get the deterministic solution only for n � − 1.
Tus, from the equation (36), we get Advances in Astronomy where β 2 3 � 1/2k 2 − 6 and c 2 2 � 1/2k 2 − 2 with k 2 � m + 3/3. From the equation (37), we get Te spatial volume and an average scale factor are given by a � (coshθ) From equations (40) and (21), the scalar feld ϕ is given by Te pressure of the VHDE is where Te energy density of the matter has the following expression: where β 4 is an integration constant.
Te energy density of the VHDE is obtained as where Te viscosity coefcient is given by Hence, the metric (1) takes the following form: 6 Advances in Astronomy (23) can be written as We can solve the above equation and get the deterministic solution only for n � − 1.
So, from equation (49), we get where, Te spatial volume and average scale factor are given by From equations (21) and (53), the scalar feld ϕ is given by Te pressure of the VHDE is as follows: where Te energy density of the matter is obtained as the energy density of the VHDE has the following expression: where Te viscosity coefcient is Hence, the metric (1) can be written as Tus, 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: n � 1.9, m � 0.97 for BT-II,

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Te plots of VHDE pressure (P vhde ) against redshift respectively for BT-II, VIII, and IX models have been represented in Figures 1-3for diferent values of ζ 0 & ζ 1 . Here, we observe that the behavior of P vhde for three diferent values of ζ 0 & ζ 1 is the increasing with redshift and varies in the negative region, which indicates the cosmic expansion. Moreover, as increasing the values of ζ 0 & ζ 1 , we get more acceleration of the Universe. Also, we have depicted the energy density of VHDE (ρ 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. Te bulk viscous coefcient has been plotted against redshift with various values of ζ 0 and ζ 1 in Figures 7-9 for BT-II, VIII, and IX models, respectively. Te trajectories vary in positive regions throughout the evolution of all the three models for various values of ζ 0 and ζ 1 , which indicates an accelerated expansion of the Universe.
Some of the cosmological properties of the models are discussed as follows: Te mean Hubble parameter H is given by where Hubble's parameters, which express the expansion rates of the Universe in the directions of x, y, and z, respectively. Te mean Hubble's parameter of BT-II, VIII, and IX VHDE cosmological models are, respectively, given by Te 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 A h ≠ 0, which indicates that the BT-II, VIII, and IX models are always anisotropic throughout the evolution of the Universe with respect to VHDE. Te expansion scalar (ϑ) has been defned as whose expressions for the BT-II, VIII, and IX VHDE models are, respectively, given as Te shear scalar (σ 2 ) is defned by the following equation and is followed by the expressions of σ 2 for BT-II, VIII, and IX VHDE models, respectively, as

Some Other Important Properties of the Models
Now we compute the following dynamical parameters, which are signifcant in the physical discussion of the cosmological models presented in equations (35), (48), and (61).

Deceleration Parameter (q). Te parameter is defned as
that 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 q � 0 there exists the marginal infation. Te deceleration parameters of the BT-II, VIII, and IX models are, respectively, given by

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Te behavior of the deceleration parameter(q) for BT-VIII and IX models has been depicted against the redshift (z) 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.

Jerk Parameter (j)
Te 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 fat ΛCDM model, the value of jerk becomes unity. Te 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. Te 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. 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 diferent 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 "statefnder diagnostic," defned as

Statefnder
that is based upon the derivatives of the scale factor a(t) and the deceleration parameter q. We have obtained expressions for the statefnder diagnostic pair (r, s) for the models BT-II, VIII, and IX, which are respectively given by Te interpretation of the statefnder 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 statefnder 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 statefnder parameters independent of time.

EoS Parameter (ω vhde ).
To classify the phases of the infating Universe, viz., the transition from decelerated to accelerated phases containing DE and radiation dominated eras, the EoS parameter (ω vhde ) can be broadly used, whose expression is given by ω vhde � P vhde /ρ vhde .
Accelerated phase: (i) the quintessence phase (− 1 < ω vhde < − 1/3), (ii) cosmological constant/vacuum phase (ω vhde � − 1) (iii) quintom era and phantom era (ω vhde < − 1) Te EoS Parameter for the BT-II VHDE cosmological model is given by where Advances in Astronomy 11   Advances in Astronomy 13 Te EoS parameter for the BT-VIII VHDE cosmological model is given by Te EoS parameter for the BT-IX VHDE cosmological model is given by where 14 Advances in Astronomy  (83) Figures 16-18 show the behavior of the EoS parameter (ω vhde ) taken against redshift (z) with diferent values of ζ 0 and ζ 1 for all the three models of BT-II, VIII, and IX, respectively. Here, we notice that the curves of ω vhde for BT-II and IX models for diferent values of ζ 0 and ζ 1 begin from the quintessence region (− 1 < ω vhde < − 1/3) and cross the nonrelativistic matter (ω vhde � − 1) then reaching the phantom region (ω vhde < − 1), whereas for BT-VIII, in the presence of null viscosity (i.e., ζ 0 � 0 and ζ 1 � 0), the curve of ω vhde varies in the quintessence region and as the values of ζ 0 and ζ 1 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 2018 Planck data [125], where the EoS parameter limits are as follows: where For the BT-VIII VHDE cosmological model, where Advances in Astronomy 15  16 Advances in Astronomy For the BT-IX VHDE cosmological model,

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Te evolutionary trajectories of ω vhde − ω vhde ′ plane for the BT-II, VIII, and IX models are plotted with diferent values of ζ 0 and ζ 1 in Figures 19-21, respectively and we notice that the ω vhde − ω vhde ′ plane for the BT-II and IX models vary in the freezing region (i.e., ω vhde < 0 and ω vhde ′ < 0) whereas, the ω vhde − ω vhde ′ plane of the BT-VIII model varies in thawing region (i.e., ω vhde < 0 and ω vhde ′ > 0) for small values of ζ 0 and ζ 1 . However, by increasing the values of ζ 0 and ζ 1 we get the freezing region for the BT-VIII model. Hence, the ω vhde − ω vhde ′ plane of our obtained models is in accordance with the present observational data [127,128]

Stability of the Model.
To examine the stability of any DE model, we utilize the squared speed of sound (v 2 s ). Te models with v 2 s < 0 shows instability where as models with v 2 s > 0 shows stability. Hence, the v 2 s is determined as follows [129]: where P vhde ′ and ρ vhde ′ are the diferentiation of pressure and density of VHDE w.r.t cosmic time 'T', respectively. Te squared speed of the sound for the BT-II VHDE model is given by where 18

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For the BT-VIII VHDE cosmological model, For the BT-IX VHDE cosmological model, We have plotted the squared speed of the sound (v 2 s ) versus redshift for three diferent values of ζ 0 and ζ 1 for BT-II, VIII, and IX models in Figures 22-24. Te v 2 s of BT-II and VIII models show the unstable behavior for three values of ζ 0 and ζ 1 . Whereas, v 2 s of BT-IX gives the stable to unstable behavior. Also, by increasing the values of ζ 0 and ζ 1 , i.e., increasing bulk viscosity, the models indicate the unstable behavior of the Universe.

Density Parameter (Ω vhde ). Te dimensionless density parameter of DE (Ω vhde ) is defned as
We were able to obtain the density parameter for our VHDE model by substituting the expressions for the Hubble parameter (H) and the energy density (ρ vhde ) in the above equation, and we used a graphical representation to analyze its behavior. Te density parameter of BT-II, VIII, and IX models, respectively are given by where where where Advances in Astronomy Te density parameter Ω vhde for BT-II, VIII, and IX VHDE models against redshift (z) is seen in Figures 25-27,

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Advances in Astronomy respectively. It can be seen that the trajectories of Ω vhde are decreasing against redshift (z) for all the three models and are varying in the positive region.

Om-Diagnostic (Om(z)).
To discriminate diferent phases of the Universe, Sahni et al., [130] have introduced another tool called the Om-diagnostic. It is also used to distinguish the ΛCDM for the nonminimally coupled scalar feld, quintessence model, and phantom feld through the trajectories of the curves. Te phantom DE era corresponds to the positive trajectory, whereas the negative trajectory indicates that the DE constitutes quintessence. Te omdiagnostic function is defned as Te Om-diagnostic for models which are in Equations (35), (48), and (61) are, respectively, given by Te plots of the Om-diagnostic against redshift (z) are represented for the BT-II, VIII, and IX models in Figures 28-30, respectively, which depict the quintessence behavior of the DE, as the trajectories for the three models vary in the negative region.

Interpretations of the Models
To understand the cosmological mysteries of accelerated expansion, we have constructed the feld equations of BDT for BT-II, VIII, and IX Universe in the context of VHDE. Te exact solutions of the Brans-Dicke feld equations are obtained by considering the relations between the metric potentials, the relation between scalar feld and scale factor, and by taking bulk viscosity as proposed by Ren and Meng [105] and Meng et al., [106]. Furthermore, we have discussed the evolution of the Universe by studying various    geometrical and physical parameters. We have plotted VHDE pressure, EoS parameter, EoS plane, and the squared speed of the sound by taking various values of ζ 0 & ζ 1 . Also, a comparison study has been conducted by considering some isotropic and homogeneous cosmological models. Now, we summarize the results of the obtained models as follows: For the Bianchi type-II cosmological model, we observe that the Universe shows a rapid expansion, as the VHDE pressure and energy density vary in negative and positive regions, respectively. Tis is further justifed by the bulk viscosity coefcient (ζ) as, it is varying in the positive region throughout the cosmic evolution. We have constructed plots for various parameters and have observed that the deceleration parameter (q), the jerk parameter ( j), and the statefnder plane (r − s) behave independently of time. Te Universe shows a quintom-like nature, which is interpreted with the help of the EoS parameter; and the EoS plane varies in the freezing region. Tis behaviors of the EoS parameter and the EoS plane represents the accelerated expansion of the cosmos. Te squared speed of sound (v 2 s ) varies in the negative region, depicting an unstable Universe model. Also, the density parameter (Ω vhde ) is decreasing and varying in the positive region and the Om(z) difers in the negative region, depicting the quintessence behavior of the Universe.
Te Bianchi type-VIII cosmological model has an accelerated cosmic expansion as the VHDE pressure and energy density vary in the negative and positive regions, respectively. Also, ζ is varying in the positive region, indicating an accelerated expansion of the Universe. Te deceleration parameter (q) shows a transition from an early

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Advances in Astronomy decelerating phase to a late accelerating phase, and the jerk parameter difers in a positive region approaching to one as z ⟶ 0. Moreover, the statefnder pair (r, s) starts its evolution from the quintessence and phantom regions and reaches the Λ CDM model (for r � 1, s � 0). Te model shows an unstable behavior, as v 2 s is varying in the negative region. Te plot of the EoS parameter states that the model varies in the phantom region and the EoS plane difers in the freezing region. Te density parameter is decreasing and varying in the positive region and the Om(z) varies in the negative region depicting, the quintessence behavior of the Universe.
For the Bianchi type-IX cosmological model, the plots for VHDE pressure, energy density, and viscosity coefcients indicate an accelerated cosmic expansion as they vary in negative, positive, and positive regions respectively. Te curve of deceleration parameter (q) varies in the accelerated phase, whereas the jerk parameter ( j) vary in the negative region, approaching to one in the near future as z ⟶ 0. Te squared speed of sound varies in positive to negative region, depicting the stable to an unstable behavior of the Universe. Also, the statefnder pair starts its evolution from the quintessence and phantom regions and reaches the Λ CDM model (for r � 1, s � 0). Te Universe shows the quintom region and is distinguished in the freezing region as the EoS parameter and EoS plane are varying in the negative region, respectively. Te density parameter (Ω vhde ) and Om(z) have the same behavior as the other two models. Now, it will be interesting to compare our BT-II, VIII, and IX VHDE models in BDT along with the other dark energy models in the literature with regard to the energy density of the VHDE (ρ vhde ) EoS parameter (ω vhde )r − s plane, the deceleration parameter (q), the density parameter of the VHDE (Ω vhde ) Om diagnostic, and bulk viscosity (ζ). We have considered some isotropic and homogeneous models for comparison study. Singh and Srivastava [102] have studied a fat FRW Universe flled with DM and viscous new HDE and they present four possible solutions for the model depending on the choice of the viscous term. Also, they have discussed the evolution of the cosmological quantities such as scale factor, deceleration parameter, and transition redshift to observe the efect of viscosity in the evolution. Srivastava and Singh [131] have investigated the new HDE model in modifed f(R, T) gravity theory within the framework of a fat FRW model with bulk viscous matter content and found the solution for nonviscous and viscous new HDE models. Also, they have analyzed a new HDE model with constant bulk viscosity (i.e.,ζ 1 � ζ 0 � constant) to explain the present accelerated the expansion of the Universe. Singh and Kumar [132] have examined Ricci dark energy model with bulk viscosity to observe the cosmic accelerating expansion phenomena, and they analyzed the model with deceleration parameter (q), EoS parameter (ω vhde ), bulk viscosity (ζ), and Om(z). Singh and Kaur [133] have explored a matter-dominated model with a bulk viscosity in BDT to interpret the observed cosmic accelerating expansion phenomena with fat FRW line element. Kumar and Beesham [134] have studied the concept of HDE in the frame work of BDT in the formalism of the fat FRW metric, and they have shown that the VHDE can play the role of an interacting HDE as it is able to explain the phase transition of the Universe. Rahman and Ansari [135] have studied the interacting generalized ghost polytropic gas model of DE with a specifc Hubble parameter in the spatially homogeneous and anisotropic LRS BT-II Universe in GR and also discuss the physical and geometrical properties of the Universe, which are found to be consistent with recent observations. Maurya et al., [136] have studied DE models in LRS BT-II space-time in a new perspective of time-dependent deceleration parameters in GR where various parameters of DE models are also calculated, and it is found that these are consistent with the recent observations. Naidu [137] has investigated the spatially homogeneous and totally anisotropic BT-II cosmological model flled with pressure less matter and anisotropic modifed Ricci dark energy in the presence of an attractive massive scalar feld in GR. It seems that the deceleration parameter(q) of our BT-VIII model coincides with the results of Singh and Srivastava [102], Singh and Kumar [132], Singh and Kaur [133], Kumar and Beesham [134], Rahman and Ansari [135], and Maurya et al. [136]. Also, the deceleration parameter(q) of our BT-IX model coincides with the results of Kumar and Beesham [134] and Maurya et al., [136]. Te state fnder parameters (r − s) of the BT-VIII & IX models coincide with the results of Singh and Srivastava [102], Srivastava and Singh [131], Singh and Kumar [132], Singh and Kaur [133], Kumar and Beesham [134], Rahman and Ansari [135], and Naidu [137]. Te EoS parameters of our BT-II, VIII, and IX VHDE models coincide with the results of Maurya et al., [136] and Naidu [137]. Whereas, at null viscosity, the EoS parameter of our BT-VIII model agrees with the results of Singh and Kumar [132], Singh and Kaur [133], and Rahman and Ansari [135]. Te density parameter of our BT-II, VIII, and IX VHDE models agrees with the results of Rahman and Ansari [135] and Naidu [137]. Te Om-diagnostics of our BT-II, VIII, and IX VHDE models correspond with the results of Singh and Srivastava [102], Srivastava and Singh [131], Singh and Kumar [132], and Singh and Kaur [133]. Te energy density of the VHDE of our BT-II, VIII, and IX models agree with the results of Rahman and Ansari [135] and Maurya et al., [136]. Te bulk viscosity of our models coincides with the results of Singh and Kumar [132]. Te above results lead to the conclusion that our BT-II, VIII, and IX VHDE models in BDTare in good agreement with the observational data. Also, we hope that the above investigations will help to have a deep insight into the behavior of bulk viscosity with VHDE Universes.

Data Availability
Te data used to support the fndings of this study are included within the article.