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

Authors: M. Vijaya Santhi; T. Chinnappalanaidu; S. Srivani Madhu; Daba Meshesha Gusu
Journal: Advances in Astronomy
 (2022)

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## 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 asR=Sn and the relation between the scalar field ϕ and the scale factor a as ϕ=am. 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 ωvhde, the density parameter Ωvhde, Om-diagnostic, squared speed of sound vs2, EoS plane ωvhde−ωvhde′, and statefinder plane r−s 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.


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## Body

## 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 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 [17–19]. 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 ω depends on the scalar field 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 [24–27]. 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 Λ-term and a dynamic deceleration parameter. Also, it has been recently justified 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 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 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 ρΛ=3C2MPl/L2, with C being a constant, MPl=1/8πG being the reduced Planck mass, and G 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 Section2, 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,(1)ds2=dt2−R2dθ2+f2θdφ2−S2dψ+hθdφ2,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 II iffθ=1 and hθ=θ;BT VIII iffθ=coshθ and hθ=sinhθ;BT IX iffθ=sinθ and hθ=cosθ.The action of BDT in the presence of matter with LagrangianLm in the canonical form (Jordan frame) is given by(2)S=∫d4x−g−ϕℛ+ωϕ□μϕ□μϕ+Lm,where ϕ is the Brans–Dicke scalar field 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 gij and the scalar field ϕ, the field equations are obtained as(3)Gij=−8πϕ−1Tij−ωϕ−2ϕ,iϕ,j−12gijϕ,κϕ,κ−ϕ−1ϕi;j−gijϕ;κ,κ,(4)ϕ;κ,κ=8π3+2ω−1T,where Gij=ℛij−1/2ℛgij is an Einstein tensor, Tij is the stress-energy tensor of the matter, and ℛij is the Ricci curvature tensor.The conservation equation(5)T;jij=0,is a consequence of the field equations (3) and (4).The energy momentum tensor for the VHDE is taken as(6)Tij=Tijm+Tijh.Here,Tijm and Tijh represent matter and VHDE tensors, which are given as(7)Tijh=P¯vhde+ρvhdeUihUjh−P¯vhdegij,Tijm=ρmUimUjm,where ρm is the energy density of the matter, P¯vhde and ρvhde, respectively, represent the pressure and energy density of the VHDE; Ui denotes the comoving velocity vector of the matter and VHDE, satisfying UiUi=1. The VHDE pressure satisfies the relation P¯vhde=Pvhde−3ζH, where ζ is the bulk viscosity coefficient. Equation of state parameters for VHDE and HDE, respectively, are defined as ωvhde=Pvhde/ρvhde and ωhde=Phde/ρhde. 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 (8)R¨R+S¨S+R˙S˙RS+S24R4+ωϕ˙22ϕ2+ϕ˙ϕR˙R+S˙S+ϕ¨ϕ=−8πPvhde−3ζHϕ,(9)2R¨R+R˙2R2+δR2−3S24R4+ωϕ˙22ϕ2+2R˙ϕ˙Rϕ+ϕ¨ϕ=−8πPvhde−3ζHϕ,(10)R˙2R2+δR2+2R˙S˙RS−S24R4−ωϕ˙22ϕ2+ϕ˙ϕ2R˙R+S˙S=8πρm+ρvhdeϕ,(11)&S˙S+2R˙Rϕ˙+ϕ¨=8πρm+ρvhde−3Pvhde−3ζH3+2ω,and the energy conservation equation becomes(12)ρ˙m+ρ˙vhde+S˙S+2R˙Rρm+ρvhde+Pvhde−3ζH=0.Here, the over-head “dot” denotes differentiation with respect to ‘t‘. When δ=0,−1,&+1, the field equations (8)–(11) correspond to the BT-II, VIII, and IX Universes, respectively. Now, by using the transformation dt=R2SdT, the field equations (8)–(11) can be written as(13)R'′R+S'′S−2R'2R2−S'2S2−2R′S′RS+S44+ωϕ'22ϕ2+ϕ'′ϕ−R′ϕ′Rϕ=−8πPvhde−3ζHR4S2ϕ,(14)2R'′R−2R′S′RS−3R'2R2+δR2S2−3S44+ωϕ'22ϕ2−ϕ′S′ϕS+ϕ'′ϕ=−8πPvhde−3ζHR4S2ϕ,(15)R2R2+δR2S2+2R′S′RS−S44−ωϕ'22ϕ2+ϕ′ϕS′S+2R′R=8πρm+ρvhdeR4S2ϕ,(16)ϕ″R4S2=8πρm+ρvhde−3Pvhde−3ζHϕ.Also, the energy conservation equation leads to(17)ρ′m+ρ′vhde+S′S+2R′Rρm+ρvhde+Pvhde−3ζH=0.The conservation equation of the matter is(18)ρ′m+S′S+2R′Rρm=0,and for VHDE, the conservation equation is(19)ρ′vhde+S′S+2R′Rρvhde+Pvhde−3ζH=0.Here, the over-head dash denotes differentiation with respect to “T.”


## 3. 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, ρm, ρvhde, Pvhde, and ζ. 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(20)R=Sn,wheren > 1.We consider the relation between scale factora and scalar field ϕ (Tripathy et al., [104]) as(21)ϕ=am,wherem is a positive constant.We take the bulk viscosity coefficient in the following form (Ren and Meng [105]; Meng et al., [106])(22)ζ=ζ0+ζ1H,where ζ0 and ζ1 are positive constants and H is the Hubble parameter.Now, using conditions (20) and (21) in field equations (13) and (14), we get(23)S″S+m2n+1−33S'2S2=δS2n+2−S41−n.


### 3.1. Bianchi Type- IIδ=0 Cosmological Model

Ifδ=0, equation (23) can be written as(24)S″S+m2n+1−33S′2S2=−S41−n.On solving equation (24), we get(25)S=2β1−2γ1T−1/2,R=2β1−2γ1T−n/2.where, γ1=2/n−16+2k11/2, with k1=m2n+1−3/3, whereas β1 is an integration constant and n>1.The spatial volume and average scale factor are given by(26)V=R2Sfθ=2β1−2Tγ1−2n−1/2,(27)a=R2Sfθ1/3=2β1−2Tγ1−2n−1/6.From equations (27) and (21), the scalar field ϕ is given by(28)ϕ=2β1−2γ1T−m2n+1/6.The pressure of the VHDE is(29)Pvhde=Υ1576πTγ1−β12,where(30)Υ1=643+n+12mTγ1−β13γ12n+12m2β1−2Tγ1−2n−1m+12n−12/6−144n2γ12Tγ1−β122β1−2Tγ1−2n−1m+12n−6/6+8Tγ1−β1−278+n+122ωm2+−3n−32m−27n22+9nγ122β1−2Tγ1−2n−1m+12n/6−576Tζ0−ζ1n3−ζ16γ1−β1ζ0πγ1n+12,The energy density of the matter is(31)ρm=β22β1−2γ1T2n+1/2,where β2 is the constant of integration.The energy density of the VHDE is(32)ρvhde=Υ2576πTγ1−β1,where(33)Υ2=9+8ωn+122m2−48n+122m−36n2−72nγ122β1−2Tγ1−2n−1m+12n/6−576πβ22β1−2Tγ1n+1/2Tγ1−β1.The viscosity coefficient is given by(34)ζ=ζ0+ζ1γ12n+132β1−2γ1T.Now the metric (1) can be written as(35)ds2=dt2−2β1−2γ1−ndθ2+dφ2−2β1−2γ1−1dψ+θdφ2.


### 3.2. Bianchi Type-VIIIδ=−1 Cosmological Model

Ifδ=−1, equation (23) can be written as(36)S″S+m2n+1−33S'2S2=−S2n+2−S41−n.We can solve the above equation and get the deterministic solution only forn=−1.Thus, from the equation (36), we get(37)S′2=β32S6+γ22S2,where β32=1/2k2−6 and γ22=1/2k2−2 with k2=m+3/3.From the equation (37), we get(38)S=γ2β3cosech2γ2T1/2,R=γ2β3cosech2γ2T−1/2.The spatial volume and an average scale factor are given by(39)V=coshθγ2β3cosech2γ2T−1/2,(40)a=coshθ13γ2β3cosech2γ2T−1/6.From equations (40) and (21), the scalar field ϕ is given by(41)ϕ=γ2β3cosech2γ2T−m/6coshθm/3.The pressure of the VHDE is(42)Pvhde=Υ3−144πβ33sinh32γ2T,where(43)Υ3=coshm3θγ2β3cosech2γ2T−m6−18+18+ω+2m2+6mγ22β32cosh22γ2T+18+−12m−72γ22β32−27γ222,−144β33cosh2γ2Tsinh2γ2Tζ1γ23+cosh22γ2Tζ0−ζ0cosh2γ2Tπγ2.The energy density of the matter has the following expression:(44)ρm=β4γ2β3cosech2γ2T1/2,where β4 is an integration constant.The energy density of the VHDE is obtained as(45)ρvhde=Υ4288πβ33sinh32γ2T,where(46)Υ4=−2γ2coshm/3θ18+m2ω−6m+18γ22β32cosh22γ2T−18β32+9γ222γ2β3cosech2γ2T−m/6,−288β4γ2β3cosech2γ2T1/2sinh2γ2Tπβ33cosh22γ2T−1.The viscosity coefficient is given by(47)ζ=ζ0+ζ1γ2coth2γ2T3.Hence, the metric (1) takes the following form:(48)ds2=dt2−γ2β3cosech2γ2T−1dθ2+coshθdφ2−γ2β3cosech2γ2Tdψ+sinhθdφ2.


### 3.3. Bianchi Type-IXδ=1 Cosmological Model

Ifδ=1, equation (23) can be written as(49)S″S+m2n+1−33S′2S2=S2n+2−S41−n.We can solve the above equation and get the deterministic solution only forn=−1.So, from equation (49), we get(50)S′2=γ32S2−β52S6,where, β52=1/6−2k3 and γ32=1/2−2k3 with k3=m+3/3.From equation (49), we get(51)S=γ3β5sech2γ3T1/2,R=γ3β5sech2γ3T−1/2.The spatial volume and average scale factor are given by(52)V=sinθγ3β5sech2γ3T−1/2,(53)a=sinθ1/3γ3β5sech2γ3T−1/6.From equations (21) and (53), the scalar field ϕ is given by(54)ϕ=γ3β5sech2γ3T−m/6sinθm/3.The pressure of the VHDE is as follows:(55)Pvhde=Υ5−144πβ53cosh32γ3T,where(56)Υ5=−18+18+ω+2m2+6mγ32β52cosh22γ3T−γ32272+−54+ω+2m2−6mβ52γ32sinm/3θγ3β5sech2γ3T−m/6,−48cosh2γ3Tπβ53cosh22γ3Tζ1γ3+3sinh2γ3Tζ0cosh2γ3T−ζ1γ3γ3.The energy density of the matter is obtained as(57)ρm=β6γ3β5sech2γ3T1/2,the energy density of the VHDE has the following expression:(58)ρvhde=Υ6288πβ53cosh32γ3T,where(59)Υ6=−2sinm/3θγ3−18+m2ω−6m+18γ32β52cosh22γ3T−γ32−92+m2ω−6m+18β52γ3γ3β5sech2γ3T−m/6,−288β6γ3β5sech2γ3T1/2πβ53cosh32γ3T.The viscosity coefficient is(60)ζ=ζ0+ζ1γ3tanh2γ3T3.Hence, the metric (1) can be written as(61)ds2=dt2−γ3β5sech2γ3T−1dθ2+sinθdφ2−γ3β5sech2γ3Tdψ+cosθdφ2.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: n=1.9, m=0.97 for BT-II, m=0.011 for BT-VIII, m=3.999 for BT-IX, γ1=−0.099, β1=0.145, β2=−1045, ω=958899, ζ1=9.15,018.3, ζ0=0.35,02, γ2=0.099, β3=0.045, β4=−9995, γ3=0.095, β5=0.011, β6=−999, a0=1.5, H0=85. The plots of VHDE pressure (Pvhde) against redshift respectively for BT-II, VIII, and IX models have been represented in Figures 1–3for different values of ζ0&ζ1. Here, we observe that the behavior of Pvhde for three different values of ζ0&ζ1 is the increasing with redshift and varies in the negative region, which indicates the cosmic expansion. Moreover, as increasing the values of ζ0&ζ1, we get more acceleration of the Universe. Also, we have depicted the energy density of VHDE ρvhde 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 ζ0 and ζ1 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 ζ0 and ζ1, which indicates an accelerated expansion of the Universe.Figure 1
Pressure of VHDE (Pvhde) versus redshift (z) for BT-II.Figure 2
Pressure of VHDE (Pvhde) versus redshift for BT-VIII.Figure 3
The pressure of VHDE(Pvhde) versus redshift (z) for BT-IX.Figure 4
Energy density of VHDE (ρvhde) versus redshift (z) for BT-II.Figure 5
Energy density of VHDE (ρvhde) versus redshift (z) for BT-VIII.Figure 6
The energy density of VHDE (ρvhde) versus redshift (z) for BT-IX.Figure 7
Bulk viscosity (ζ) versus redshift (z) for BT-II.Figure 8
Bulk viscosity (ζ) versus redshift (z) for BT-VIII.Figure 9
Bulk viscosity (ζ) versus redshift (z) for BT-IX.Some of the cosmological properties of the models are discussed as follows:The mean Hubble parameterH is given by(62)H=H1+H2+H33,whereH1=H2=R˙/R,H3=S˙/S are directional Hubble’s parameters, which express the expansion rates of the Universe in the directions of x, y, and z, respectively. The mean Hubble’s parameter of BT-II, VIII, and IX VHDE cosmological models are, respectively, given by(63)H=γ12n+132β1−2γ1T,H=γ2coth2γ2T3,H=γ3tanh2γ3T3.The anisotropic parameter of the BT-II VHDE model is given by(64)Ah=13∑i=13Hi−HH2=2n−122n+12,and for the BT-VIII and IX VHDE cosmological models, we get(65)Ah=8.From equations (64) and (65), we can observe that Ah≠0, 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 (ϑ) has been defined as(66)ϑ=2R˙R+S˙S,whose expressions for the BT-II, VIII, and IX VHDE models are, respectively, given as(67)ϑ=γ12n+12β1−2γ1T,ϑ=γ2coth2γ2T,&ϑ=γ3tanh2γ3T.The shear scalar (σ2) is defined by the following equation and is followed by the expressions of σ2 for BT–II, VIII, and IX VHDE models, respectively, as(68)σ2=12σijσij=12∑i=13Hi2−ϑ23,(69)σ2=γ122β1−2γ1T−2n−123,(70)σ2=4γ22coth22γ2T3,(71)&σ2=4γ32tanh22γ3T3.


## 3.1. Bianchi Type- IIδ=0 Cosmological Model

Ifδ=0, equation (23) can be written as(24)S″S+m2n+1−33S′2S2=−S41−n.On solving equation (24), we get(25)S=2β1−2γ1T−1/2,R=2β1−2γ1T−n/2.where, γ1=2/n−16+2k11/2, with k1=m2n+1−3/3, whereas β1 is an integration constant and n>1.The spatial volume and average scale factor are given by(26)V=R2Sfθ=2β1−2Tγ1−2n−1/2,(27)a=R2Sfθ1/3=2β1−2Tγ1−2n−1/6.From equations (27) and (21), the scalar field ϕ is given by(28)ϕ=2β1−2γ1T−m2n+1/6.The pressure of the VHDE is(29)Pvhde=Υ1576πTγ1−β12,where(30)Υ1=643+n+12mTγ1−β13γ12n+12m2β1−2Tγ1−2n−1m+12n−12/6−144n2γ12Tγ1−β122β1−2Tγ1−2n−1m+12n−6/6+8Tγ1−β1−278+n+122ωm2+−3n−32m−27n22+9nγ122β1−2Tγ1−2n−1m+12n/6−576Tζ0−ζ1n3−ζ16γ1−β1ζ0πγ1n+12,The energy density of the matter is(31)ρm=β22β1−2γ1T2n+1/2,where β2 is the constant of integration.The energy density of the VHDE is(32)ρvhde=Υ2576πTγ1−β1,where(33)Υ2=9+8ωn+122m2−48n+122m−36n2−72nγ122β1−2Tγ1−2n−1m+12n/6−576πβ22β1−2Tγ1n+1/2Tγ1−β1.The viscosity coefficient is given by(34)ζ=ζ0+ζ1γ12n+132β1−2γ1T.Now the metric (1) can be written as(35)ds2=dt2−2β1−2γ1−ndθ2+dφ2−2β1−2γ1−1dψ+θdφ2.


## 3.2. Bianchi Type-VIIIδ=−1 Cosmological Model

Ifδ=−1, equation (23) can be written as(36)S″S+m2n+1−33S'2S2=−S2n+2−S41−n.We can solve the above equation and get the deterministic solution only forn=−1.Thus, from the equation (36), we get(37)S′2=β32S6+γ22S2,where β32=1/2k2−6 and γ22=1/2k2−2 with k2=m+3/3.From the equation (37), we get(38)S=γ2β3cosech2γ2T1/2,R=γ2β3cosech2γ2T−1/2.The spatial volume and an average scale factor are given by(39)V=coshθγ2β3cosech2γ2T−1/2,(40)a=coshθ13γ2β3cosech2γ2T−1/6.From equations (40) and (21), the scalar field ϕ is given by(41)ϕ=γ2β3cosech2γ2T−m/6coshθm/3.The pressure of the VHDE is(42)Pvhde=Υ3−144πβ33sinh32γ2T,where(43)Υ3=coshm3θγ2β3cosech2γ2T−m6−18+18+ω+2m2+6mγ22β32cosh22γ2T+18+−12m−72γ22β32−27γ222,−144β33cosh2γ2Tsinh2γ2Tζ1γ23+cosh22γ2Tζ0−ζ0cosh2γ2Tπγ2.The energy density of the matter has the following expression:(44)ρm=β4γ2β3cosech2γ2T1/2,where β4 is an integration constant.The energy density of the VHDE is obtained as(45)ρvhde=Υ4288πβ33sinh32γ2T,where(46)Υ4=−2γ2coshm/3θ18+m2ω−6m+18γ22β32cosh22γ2T−18β32+9γ222γ2β3cosech2γ2T−m/6,−288β4γ2β3cosech2γ2T1/2sinh2γ2Tπβ33cosh22γ2T−1.The viscosity coefficient is given by(47)ζ=ζ0+ζ1γ2coth2γ2T3.Hence, the metric (1) takes the following form:(48)ds2=dt2−γ2β3cosech2γ2T−1dθ2+coshθdφ2−γ2β3cosech2γ2Tdψ+sinhθdφ2.


## 3.3. Bianchi Type-IXδ=1 Cosmological Model

Ifδ=1, equation (23) can be written as(49)S″S+m2n+1−33S′2S2=S2n+2−S41−n.We can solve the above equation and get the deterministic solution only forn=−1.So, from equation (49), we get(50)S′2=γ32S2−β52S6,where, β52=1/6−2k3 and γ32=1/2−2k3 with k3=m+3/3.From equation (49), we get(51)S=γ3β5sech2γ3T1/2,R=γ3β5sech2γ3T−1/2.The spatial volume and average scale factor are given by(52)V=sinθγ3β5sech2γ3T−1/2,(53)a=sinθ1/3γ3β5sech2γ3T−1/6.From equations (21) and (53), the scalar field ϕ is given by(54)ϕ=γ3β5sech2γ3T−m/6sinθm/3.The pressure of the VHDE is as follows:(55)Pvhde=Υ5−144πβ53cosh32γ3T,where(56)Υ5=−18+18+ω+2m2+6mγ32β52cosh22γ3T−γ32272+−54+ω+2m2−6mβ52γ32sinm/3θγ3β5sech2γ3T−m/6,−48cosh2γ3Tπβ53cosh22γ3Tζ1γ3+3sinh2γ3Tζ0cosh2γ3T−ζ1γ3γ3.The energy density of the matter is obtained as(57)ρm=β6γ3β5sech2γ3T1/2,the energy density of the VHDE has the following expression:(58)ρvhde=Υ6288πβ53cosh32γ3T,where(59)Υ6=−2sinm/3θγ3−18+m2ω−6m+18γ32β52cosh22γ3T−γ32−92+m2ω−6m+18β52γ3γ3β5sech2γ3T−m/6,−288β6γ3β5sech2γ3T1/2πβ53cosh32γ3T.The viscosity coefficient is(60)ζ=ζ0+ζ1γ3tanh2γ3T3.Hence, the metric (1) can be written as(61)ds2=dt2−γ3β5sech2γ3T−1dθ2+sinθdφ2−γ3β5sech2γ3Tdψ+cosθdφ2.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: n=1.9, m=0.97 for BT-II, m=0.011 for BT-VIII, m=3.999 for BT-IX, γ1=−0.099, β1=0.145, β2=−1045, ω=958899, ζ1=9.15,018.3, ζ0=0.35,02, γ2=0.099, β3=0.045, β4=−9995, γ3=0.095, β5=0.011, β6=−999, a0=1.5, H0=85. The plots of VHDE pressure (Pvhde) against redshift respectively for BT-II, VIII, and IX models have been represented in Figures 1–3for different values of ζ0&ζ1. Here, we observe that the behavior of Pvhde for three different values of ζ0&ζ1 is the increasing with redshift and varies in the negative region, which indicates the cosmic expansion. Moreover, as increasing the values of ζ0&ζ1, we get more acceleration of the Universe. Also, we have depicted the energy density of VHDE ρvhde 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 ζ0 and ζ1 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 ζ0 and ζ1, which indicates an accelerated expansion of the Universe.Figure 1
Pressure of VHDE (Pvhde) versus redshift (z) for BT-II.Figure 2
Pressure of VHDE (Pvhde) versus redshift for BT-VIII.Figure 3
The pressure of VHDE(Pvhde) versus redshift (z) for BT-IX.Figure 4
Energy density of VHDE (ρvhde) versus redshift (z) for BT-II.Figure 5
Energy density of VHDE (ρvhde) versus redshift (z) for BT-VIII.Figure 6
The energy density of VHDE (ρvhde) versus redshift (z) for BT-IX.Figure 7
Bulk viscosity (ζ) versus redshift (z) for BT-II.Figure 8
Bulk viscosity (ζ) versus redshift (z) for BT-VIII.Figure 9
Bulk viscosity (ζ) versus redshift (z) for BT-IX.Some of the cosmological properties of the models are discussed as follows:The mean Hubble parameterH is given by(62)H=H1+H2+H33,whereH1=H2=R˙/R,H3=S˙/S are directional Hubble’s parameters, which express the expansion rates of the Universe in the directions of x, y, and z, respectively. The mean Hubble’s parameter of BT-II, VIII, and IX VHDE cosmological models are, respectively, given by(63)H=γ12n+132β1−2γ1T,H=γ2coth2γ2T3,H=γ3tanh2γ3T3.The anisotropic parameter of the BT-II VHDE model is given by(64)Ah=13∑i=13Hi−HH2=2n−122n+12,and for the BT-VIII and IX VHDE cosmological models, we get(65)Ah=8.From equations (64) and (65), we can observe that Ah≠0, 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 (ϑ) has been defined as(66)ϑ=2R˙R+S˙S,whose expressions for the BT-II, VIII, and IX VHDE models are, respectively, given as(67)ϑ=γ12n+12β1−2γ1T,ϑ=γ2coth2γ2T,&ϑ=γ3tanh2γ3T.The shear scalar (σ2) is defined by the following equation and is followed by the expressions of σ2 for BT–II, VIII, and IX VHDE models, respectively, as(68)σ2=12σijσij=12∑i=13Hi2−ϑ23,(69)σ2=γ122β1−2γ1T−2n−123,(70)σ2=4γ22coth22γ2T3,(71)&σ2=4γ32tanh22γ3T3.


## 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 (q)

The parameter is defined as(72)q=−aa¨a˙2,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 inflation. The deceleration parameters of the BT-II, VIII, and IX models are, respectively, given by(73)q=−62n+1−1,q=6sech22γ2T−1,q=−6cosech22γ3T−1.The 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.Figure 10
Deceleration parameter (q) versus redshift (z) for BT-VIII.Figure 11
Deceleration parameter (q) versus redshift (z) for BT-IX.


### 4.2. Jerk Parameter (j)

The cosmic jerk,(74)j=1aH3d3adt3,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(75)j=4n+40n+912n+12,j=54sech22γ2T+1,j=1−54cosech22γ3T.Figures12 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.Figure 12
Jerk (j) versus redshift (z) for BT-VIII.Figure 13
Jerk (j) versus redshift (z) for BT-IX.


### 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 as(76)r=a⃛aH3&s=r−13q−1/2,that 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(77)r=2n+72n+132n+12&s=−42n+1,r=54sech22γ2T+1&s=−12cosh22γ2T−4,&r=1−54cosech22γ3T&s=12cosh22γ3T+3.The interpretation of the statefinder pair from Figures14 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.Figure 14
Statefinder plane (r − s) for BT-VIII.Figure 15
Statefinder plane (r − s) for BT-IX.


### 4.4. EoS Parameterωvhde

To classify the phases of the inflating Universe, viz., the transition from decelerated to accelerated phases containing DE and radiation dominated eras, the EoS parameterωvhde can be broadly used, whose expression is given by ωvhde=Pvhde/ρvhde.It categorizes various epochs as follows:Decelerated phase:(i)
stiff fluid (ωvhde=1),(ii)
the radiation dominated phase (0<ωvhde<1/3) and(iii)
dust fluid phase or cold dark matter (ωvhde=0).Accelerated phase:(i)
the quintessence phase (−1<ωvhde<−1/3),(ii)
cosmological constant/vacuum phase (ωvhde=−1)(iii)
quintom era and phantom era (ωvhde<−1)The EoS Parameter for the BT-II VHDE cosmological model is given by(78)ωvhde=Υ7Υ8,where(79)Υ7=1576−643+n+12mTγ1−β13γ12mn+12−2Tγ1+2β1−2n−1m+6n−126,+144γ12n2Tγ1−β12−2Tγ1+2β1−2n−1m+6n−66,−8Tγ1−β1−278+n+122ωm2+−3n−32m−27n22+9nγ12−2Tγ1+2β1−2n−1m+6n6,+576−2Tγ1+2β1−nπγ1Tζ0−ζ1n3−ζ16γ1−ζ0β1n+12,Υ8=Tγ1−β1−164+−172n+122ωm2+112n+122m+n216+n8γ12−2Tγ1+2β1−2n−1m+6n6,+−2Tγ1+2β11/2πβ2Tγ1−β1.The EoS parameter for the BT-VIII VHDE cosmological model is given by(80)ωvhde=Υ9Υ10,where(81)Υ9=8coshm/3θ−18+18+ω+2m2+6mγ22β32cosh4Tγ2−−18+18+ω+2m2+6mγ22β32cosh2γ2T+14−272+−54+ω+2m2−6mβ32γ22γ2β3cosech2γ2T−m/6−96cosh2γ2T−12πγ2cosh2γ2Tsinhγ2Tζ1+3cosh3γ2Tζ0−γ22ζ1sinhγ2T−3coshγ2Tζ0coshγ2Tβ33γ2,Υ10=1152β4cosh3γ2Tsinhγ2Tπ2β33coshγ2T−1coshγ2T+1γ2β3cosechγ2Tsechγ2T1/2+8coshm/3θ18+m2ω−6m+18γ22β32cosh4γ2T−18+m2ω−6m+18γ22β32cosh2γ2T+1492+m2ω−6m+18β32γ22γ2β3cosech2γ2T−m/6γ2.The EoS parameter for the BT-IX VHDE cosmological model is given by(82)ωvhde=Υ11Υ12,where(83)Υ11=8sinm3θ−18+18+ω+2m2+6mγ32β52cosh4γ3T−−18+18+ω+2m2+6mγ32β52cosh2γ3T+−278+3m+18β52γ32−9β522γ3β5sech2γ3T−m/6−96coshγ3Tγ3cosh3γ3Tζ1+3cosh2γ3Tsinhγ3Tζ0−γ3ζ1coshγ3T−3ζ02sinhγ3Tπβ53cosh2γ3T−12γ3,Υ12=2304πβ53β6cosh2γ3T−123γ3β52cosh2γ3T−11/2+8γ3sinm/3θγ3β5sech2γ3T−m/6−18+m2ω−6m+18γ32β52cosh4γ3T−−18+m2ω−6m+18γ32β52cosh2γ3T+9γ328−9β522.Figures16–18 show the behavior of the EoS parameter ωvhde taken against redshift (z) with different values of ζ0 and ζ1 for all the three models of BT-II, VIII, and IX, respectively. Here, we notice that the curves of ωvhde for BT-II and IX models for different values of ζ0 and ζ1 begin from the quintessence region (−1<ωvhde<−1/3) and cross the nonrelativistic matter (ωvhde=−1) then reaching the phantom region (ωvhde<−1), whereas for BT-VIII, in the presence of null viscosity (i.e.,ζ0=0and ζ1=0), the curve of ωvhde varies in the quintessence region and as the values of ζ0 and ζ1 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:(84)ωvhde=−1.56−0.48+0.60Planck+TT+loE,−1.58−0.41+0.52Planck+TT,EE+loE,−1.57−0.40+0.50Planck+TT,TE,EE+loE+lensing,−1.04−0.10+0.10Planck+TT,TE,EE+loE+lensing+BAO.Figure 16
EoS parameterωvhde versus redshift (z) for BT-II.Figure 17
EoS parameterωvhdeωvhde versus redshift (z) for BT-VIII.Figure 18
EoS parameter versus redshift (z) for BT-IX.


### 4.5.ωvhde−ωvhde′ plane

Cadwell and Linder [126] have suggested ωvhde−ωvhde′ plane (where ' signifies differentiation w.r.t ln 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 ωvhde and ωvhde′, the plane describes two distinct areas. The plane is described as the thawing zone for ωvhde′>0 when ωvhde<0 and the freezing region for ωvhde′<0 when ωvhde<0.For the BT-II VHDE cosmological model,(85)ω′=Υ13Υ14,where(86)ϒ13=136π83+n+12mn+12m−3n+92Tγ1−β15β2n+12γ12m−2Tγ1+2β12n−1m+6n−15/6−18n2n+12m−3n+92Tγ1−β14β2γ12−2Tγ1+2β1−2n−1m+6n−9/6+12n+12γ1n+122γ21ωm2−6n+122γ12m−9γ12n22−9γ12n+98n+12−γ1ζ1n3+Tζ0−ζ16γ1−ζ0β1m+2n2ζ1γ1+−6Tζ0+2ζ1γ1+6ζ0β1n+γ1ζ12−2Tγ1+2β1−2mn−m/6+Tγ1−β12β2−216−n2ζ1γ13+Tζ0−ζ1γ1−ζ0β1n+3Tζ02−5ζ112γ1−3Tζ0β12πn+12γ1−2Tγ1+2β1−n−1/2+n+12m−3n+92Tγ1−β1−2Tγ1+2β1−2n−1m+6n−3/6n+122γ21ωm2−3n+12γ21m−278−27γ21n22+9γ21n,Υ14=Tγ1−β1−172n+122γ12ωm2+112n+122γ12m+γ12n216+γ12n8−64−2Tγ1+2β1−2n−1m+6n/6+−2Tγ1+2β11/2πβ2Tγ1−β12n+12..For the BT-VIII VHDE cosmological model,(87)ω′vhde=Υ15Υ16,where(88)Y16=γ2β3cosech2γ2T1/2β4cosh3γ2Tsinhγ2Tπ2β33cosh2γ2T−1γ2β3cosechγ2Tsechγ2T1/2+coshm/3θ14418+m2ω−6m+18γ22β32cosh4γ2T−18+m2ω−6m+18γ22β32cosh2γ2T+1492+m2ω−6m+18β32γ22γ2β3cosech2γ2T−m/6γ22cosh2γ2T,ϒ15=11442883coshm/3θπζ1m−618+m2ω−6m+18γ22β32γ2cosh10γ2T+3ζ0m−6sinhγ2T18+m2ω−6m+18γ22β32cosh9γ2T−5ζ12m−618+m2ω−6m+18γ22β32γ2cosh8γ2T−6ζ0m−6sinhγ2T18+m2ω−6m+18γ22β32cosh7γ2T+5ζ12m−6m2ω−6m+18γ22+81m5−3785β32+9γ2220m−18γ2cosh6γ2T+9ζ02sinhγ2Tm−7m2ω−6m+18γ22+15m−72β32+3γ224m−18cosh5γ2T−5ζ14m−6m2ω−6m+18γ22+63m5−545β32+27γ2220m−18γ2cos4γ2T−3ζ02sinhγ2Tm−9m2ω−6m+18γ22+9mβ32+9γ224m−18cosh3γ2T+5ζ116m−6m2ω−6m+18γ22+36m5+2165β3227γ2210m−14γ2cosh2γ2T+3ζ016m−12sinhγ2T92+m2ω−6m+18β32γ22coshγ2T−ζ132m−692+m2ω−6m+18β32γ23β3γ2β3cosech2γ2T−m/6+cosh2γ2T−12cosh2m/3θγ2β3cosech2γ2T−m/3sinhγ2Tγ22m+6m2ω−6m+18γ22+54−3ω−3m2+18mβ32+272+27+3ω2+34m2−9m2γ22coshγ2Tγ2β3cosechγ2Tsechγ2T1/2+β4πcoshm/3θcosh2γ2T−12−18+18+ω+2m2+6mγ22m−3β32cosh4γ2T−−18+18+ω+2m2+6mγ22m−3β32cosh2γ3T+14−272+−54+ω+2m2−6mβ32m−15γ22sinhγ2Tγ2β3coseh2γ2T−m/6−288coshγ2T+1γ2cosh6γ2Tζ1+3cosh5γ2Tsinhγ2Tζ0−3γ22cosh4γ2Tζ1−3cosh3γ2Tsinhγ2Tζ0−γ2ζ14cosh2γ2T−34coshγ2Tsinhγ2Tζ0+3ζ1γ28πcoshγ2T−1β33coshγ2Tcoshγ2Tsinh2γ2Tβ32γ22.For the BT-IX VHDE cosmological model,(89)ωvhde′=Υ17Υ18,where(90)ϒ17=−36γ3β52cosh2γ3T−833−18+m2ω−6m+18γ32β52ς0m−6cosh10γ3T+ς1−18+m2ω−6m+18γ32γ3β52m−6sinhγ3Tcosh9γ3T−152−18+m2ω−6m+18γ32β52ς0m−6cosh8γ3T−2ς1−18+m2ω−6m+18γ32γ3β52m−6sinhγ3Tcosh7γ3T+6ς0m−214m2ω−6m+18γ32−81m4+135β52+9γ3216m−18cosh6γ3T+m−16m2−6m+18γ32−45m2+189β52+9γ328m−18ς1γ3sinhγ3Tcosh5γ3T−32m2ω−6m+18m−32γ32−63m2+270β52+27γ38m−18ς0cosh4γ3T−9ς08γ3sinhγ3Tγ3−2β5γ3+2β5m−18cosh3γ3T+94m2ω−6m+18γ32−3m+45β52+3γ324m−21ς0cosh2γ3T−278ς1γ3sinhγ3Tγ3−2β5γ3+2β5coshγ3T+81ς0γ3232−81β52ς08πβ5sinm/3θγ3β5sech2γ3T−m/6+γ3β5sech2γ3T−m/3m2ω−6m+18γ32+3m−18m+6β52+27+−27+−3ω2−34m2+9m2γ32sin2m/3θcoshγ3Tcosh2γ3T−12γ32sinhγ3Tγ3β52cosh2γ3T−11/2−4πcosh2γ3T−12β6coshγ3T−18+18+ω+2m2+6mγ32m−3β52cosh4γ3T−−18+18+ω+2m2+6mγ32m−3β52cosh2γ3T+−216+3ω+9m2−9mγ32−9m2+272β52−27γ328m−15sinm/3θsinhγ3Tγ3β5cosh2γ3T−m/6−288γ3cosh5γ3Tsinhγ3Tς1+3ς0cosh6γ3T−γ3cosh3γ3Tsinhγ3Tς1−9ς02cosh4γ3T+γ3coshγ3Tsinhγ3Tς1+3ς0cosh2γ3T−3ς04πcosh2γ3T−12β52,Υ18=sin2m/3θ−18+m2ω−6m+18γ32β52cosh4γ3T−−18+m2ω−6m+18γ32β52cosh2γ3T+9γ328−9β5222γ3γ3β5sech2γ3T−m/3+41472π2cosh2γ3T−125β55β66γ3β52cosh2γ3T−11/2+288πcosh2γ3T−12−18+m2ω−6m+18γ32β52cosh4γ3T−−18+m2ω−6m+18γ32β52cosh2γ3T+9γ328−9β522γ3β5sech2γ3T−m/6γ3β52sinm/3θβ6sinh2γ3T.The evolutionary trajectories ofωvhde−ωvhde′ plane for the BT–II, VIII, and IX models are plotted with different values of ζ0 and ζ1 in Figures 19–21, respectively and we notice that the ωvhde−ωvhde′ plane for the BT-II and IX models vary in the freezing region (i.e.,ωvhde<0 and ωvhde′<0) whereas, the ωvhde−ωvhde′ plane of the BT-VIII model varies in thawing region (i.e.,ωvhde<0 and ωvhde′>0) for small values of ζ0 and ζ1. However, by increasing the values of ζ0 and ζ1 we get the freezing region for the BT-VIII model. Hence, the ωvhde−ωvhde′ plane of our obtained models is in accordance with the present observational data [127, 128] as follows:(91)ωvhde=−1.17−0.12+0.13,ωvhde′=0.85−0.49+0.50WMAP+eCAMB+BAO+H0+SNe,ωvhde=−1.13−0.25+0.24,ωvhde′<1.32Planck+WP+BAO,ωvhde=−1.34−0.18+0.18,ωvhde′=0.85±0.7WMAP+eCAMB+BAO+H0.Figure 19
EoS planeωvhde−ωvhde′ for BT-II.Figure 20
EoS planeωvhde−ωvhde′ for BT-VIII.Figure 21
EoS planeωvhde−ωvhde′ for BT-IX.


### 4.6. Stability of the Model

To examine the stability of any DE model, we utilize the squared speed of sound (vs2). The models with vs2<0 shows instability where as models with vs2>0 shows stability. Hence, the vs2 is determined as follows [129]:(92)vs2=Pvhde′ρvhde′,where Pvhde′ and ρvhde′ are the differentiation of pressure and density of VHDE w.r.t cosmic time ‘T’, respectively.The squared speed of the sound for the BT-II VHDE model is given by(93)vs2=Υ19Υ20,where(94)Υ19=−154Tγ1−β13n+12m−6n+3γ12n+12m3+n+12m−2Tγ1+2β1−2n−1m+6n−12/6,−9n24Tγ1−β12n+12m−6n+3γ12−2Tγ1+2β1−2n−1m+6n−6/6+18Tγ1−β1n+12m−6n+3,−278+n+122ωm2+−3n−32m−27n22+9nγ12−2Tγ1+2β1−2n−1m+6n/6,−27−2Tγ1+2β1−nπγ1n+12Tζ0−2nζ13−ζ13γ1−ζ0β1−2Tγ1+2β11/2,Υ20=Tγ1−β1−1432n+12m−6n+398+n+122ωm2−6n+122m−9n22−9nγ12,−2Tγ1+2β1−2m+6n−m+3/6+Tγ1−β12πβ2n+12.For the BT-VIII VHDE cosmological model,(95)vs2=Υ21Υ22,where(96)Υ21=4coshm/3θcosh2γ2T−12−18+18+ω+2m2+6mγ22m−6β32cosh4γ2T−−18+18+ω+2m2+6mγ22m−6β32cosh2γ2T+14m−18γ22−272+−54+ω+2m2−6mβ32γ2β3cosech2γ2T−m/6+288γ2cosh2γ2Tsinhγ2Tζ1+32cosh3γ2Tζ0−γ22sinhγ2Tζ1−32coshγ2Tζ0πcoshγ2Tβ33γ2β3cosech2γ2T1/2,Υ22=cosh2γ2Tcoshm/3θγ2β3cosech2γ2T−m/62m−618+m2ω−6m+18γ22β32cosh4γ2T−m−618+m2ω−6m+18γ22β32cosh2γ2T+14m−1892+m2ω−6m+18γ32γ2γ2β3cosechγ2Tsechγ2T1/2−864πβ32β4cosh2γ2Tcosh2γ2T−1.For the BT-IX VHDE cosmological model,(97)vs2=Υ23Υ24,where(98)Υ23=2γ3β5sech2γ3T1/2coshγ3Tsinhγ3Tsinm3θ−18+18+ω+2m2+6mγ32β52m−6cosh4γ3T−−18+18+ω+2m2+6mγ32β52m−6cosh2γ3T+−270+3ω+9m2−18mγ32+27−9m2β52−27γ328m−18γ3β5sech2γ3T−m/6−288cosh2γ3T−12πγ3coshγ3Tsinhγ3Tζ1+3cosh2γ3Tζ02−3ζ04β52,Υ24=sinh2γ3Tγ3β5cosh2γ3T−m/6sinm/3θ−18+m2ω−6m+18γ32β52m−6cosh4γ3T−−18+m2ω−6m+18γ32β52m−6cosh2γ3T+3m2ω−18m+54γ32+27−9m2β52+9γ328m−18γ3β52cosh2γ3T−11/2−432cosh2γ3T−122πβ52β6.We have plotted the squared speed of the soundvs2 versus redshift for three different values of ζ0and ζ1 for BT-II, VIII, and IX models in Figures 22–24. The vs2 of BT-II and VIII models show the unstable behavior for three values of ζ0 and ζ1. Whereas, vs2 of BT-IX gives the stable to unstable behavior. Also, by increasing the values of ζ0and ζ1, i.e., increasing bulk viscosity, the models indicate the unstable behavior of the Universe.Figure 22
Squared speed of sound (v2) versus redshift (z) for BT-II.Figure 23
Squared speed of soundvs2 versus redshift (z) for BT-VIII.Figure 24
Squared speed of soundvs2. versus redshift (z) for BT-IX.


### 4.7. Density Parameter (Ωvhde)

The dimensionless density parameter of DE (Ωvhde) is defined as(99)Ωvhde=ρvhde3H2.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 ρvhde in the above equation, and we used a graphical representation to analyze its behavior. The density parameter of BT-II, VIII, and IX models, respectively are given by(100)Ωvhde=Υ256γ122n+12π,where(101)Υ25=−72πβ22β1−2Tγ1n+1/2Tγ1−β1+2β1−2Tγ1m−2n−1+12n/6γ12ωn+122m2−6n+122mγ12+98+−9n22−9nγ12Tγ1−β1,(102)Ωvhde=Υ2696γ22πβ33sinh2γ2Tcosh22γ2T,where(103)Υ26=−2γ2coshθm/3γ2β3cosech2γ2T−m/618+m2ω−6m+18γ22β32cosh22γ2T−18β32+9γ222−288β4γ2β3cosech2γ2T1/2sinh32γ2Tπβ33cosh22γ2T−1,(104)Ωvhde=Υ2796γ32cosh2γ3Tπβ53sinh22γ3T,where(105)Υ27=−2−18+m2ω−6m+18γ32β52cosh22γ3T−γ32−92+m2ω−6m+18β52γ3γ3β5sech2γ3T−m/6−288β6γ3β5sech2γ3T1/2πβ53cosh32γ3T.The density parameterΩvhde for BT–II, VIII, and IX VHDE models against redshift (z) is seen in Figures 25–27, respectively. It can be seen that the trajectories of Ωvhde are decreasing against redshift (z) for all the three models and are varying in the positive region.Figure 25
Density parameter of VHDEΩvhde) versus redshift (z) for BT-II.Figure 26
Density parameter of VHDEΩvhde). versus redshift (z) for BT-VIII.Figure 27
Density parameter of VHDEΩvhde) versus redshift (z) for BT-IX.


### 4.8. Om-Diagnostic (Om(z))

To discriminate different 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 field, quintessence model, and phantom field through the trajectories of the curves. The phantom DE era corresponds to the positive trajectory, whereas the negative trajectory indicates that the DE constitutes quintessence. The om-diagnostic function is defined as(106)Omz=H2z−H02H02z3−1.The Om-diagnostic for models which are in Equations (35), (48), and (61) are, respectively, given by(107)Omz=γ12n+1−3H021+z/a06/2n+13H021+z/a06/2n+1z3−1,Omz=γ22/91+β3/γ2cosh2θ1+z/a062−H02H02z3−1,Omz=γ32/91+β5/γ3sin2θ1+z/a062−H02H02z3−1.The 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.Figure 28
Om-diagnostics (Om(z)) versus. redshift (z) for BT-II.Figure 29
Om-diagnostics (Om(z)) versus redshift (z) for BT-VIII.Figure 30
Om-diagnostics (Om(z)) versus. redshift (z) for BT-IX.


## 4.1. Deceleration Parameter (q)

The parameter is defined as(72)q=−aa¨a˙2,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 inflation. The deceleration parameters of the BT-II, VIII, and IX models are, respectively, given by(73)q=−62n+1−1,q=6sech22γ2T−1,q=−6cosech22γ3T−1.The 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.Figure 10
Deceleration parameter (q) versus redshift (z) for BT-VIII.Figure 11
Deceleration parameter (q) versus redshift (z) for BT-IX.


## 4.2. Jerk Parameter (j)

The cosmic jerk,(74)j=1aH3d3adt3,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(75)j=4n+40n+912n+12,j=54sech22γ2T+1,j=1−54cosech22γ3T.Figures12 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.Figure 12
Jerk (j) versus redshift (z) for BT-VIII.Figure 13
Jerk (j) versus redshift (z) for BT-IX.


## 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 as(76)r=a⃛aH3&s=r−13q−1/2,that 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(77)r=2n+72n+132n+12&s=−42n+1,r=54sech22γ2T+1&s=−12cosh22γ2T−4,&r=1−54cosech22γ3T&s=12cosh22γ3T+3.The interpretation of the statefinder pair from Figures14 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.Figure 14
Statefinder plane (r − s) for BT-VIII.Figure 15
Statefinder plane (r − s) for BT-IX.


## 4.4. EoS Parameterωvhde

To classify the phases of the inflating Universe, viz., the transition from decelerated to accelerated phases containing DE and radiation dominated eras, the EoS parameterωvhde can be broadly used, whose expression is given by ωvhde=Pvhde/ρvhde.It categorizes various epochs as follows:Decelerated phase:(i)
stiff fluid (ωvhde=1),(ii)
the radiation dominated phase (0<ωvhde<1/3) and(iii)
dust fluid phase or cold dark matter (ωvhde=0).Accelerated phase:(i)
the quintessence phase (−1<ωvhde<−1/3),(ii)
cosmological constant/vacuum phase (ωvhde=−1)(iii)
quintom era and phantom era (ωvhde<−1)The EoS Parameter for the BT-II VHDE cosmological model is given by(78)ωvhde=Υ7Υ8,where(79)Υ7=1576−643+n+12mTγ1−β13γ12mn+12−2Tγ1+2β1−2n−1m+6n−126,+144γ12n2Tγ1−β12−2Tγ1+2β1−2n−1m+6n−66,−8Tγ1−β1−278+n+122ωm2+−3n−32m−27n22+9nγ12−2Tγ1+2β1−2n−1m+6n6,+576−2Tγ1+2β1−nπγ1Tζ0−ζ1n3−ζ16γ1−ζ0β1n+12,Υ8=Tγ1−β1−164+−172n+122ωm2+112n+122m+n216+n8γ12−2Tγ1+2β1−2n−1m+6n6,+−2Tγ1+2β11/2πβ2Tγ1−β1.The EoS parameter for the BT-VIII VHDE cosmological model is given by(80)ωvhde=Υ9Υ10,where(81)Υ9=8coshm/3θ−18+18+ω+2m2+6mγ22β32cosh4Tγ2−−18+18+ω+2m2+6mγ22β32cosh2γ2T+14−272+−54+ω+2m2−6mβ32γ22γ2β3cosech2γ2T−m/6−96cosh2γ2T−12πγ2cosh2γ2Tsinhγ2Tζ1+3cosh3γ2Tζ0−γ22ζ1sinhγ2T−3coshγ2Tζ0coshγ2Tβ33γ2,Υ10=1152β4cosh3γ2Tsinhγ2Tπ2β33coshγ2T−1coshγ2T+1γ2β3cosechγ2Tsechγ2T1/2+8coshm/3θ18+m2ω−6m+18γ22β32cosh4γ2T−18+m2ω−6m+18γ22β32cosh2γ2T+1492+m2ω−6m+18β32γ22γ2β3cosech2γ2T−m/6γ2.The EoS parameter for the BT-IX VHDE cosmological model is given by(82)ωvhde=Υ11Υ12,where(83)Υ11=8sinm3θ−18+18+ω+2m2+6mγ32β52cosh4γ3T−−18+18+ω+2m2+6mγ32β52cosh2γ3T+−278+3m+18β52γ32−9β522γ3β5sech2γ3T−m/6−96coshγ3Tγ3cosh3γ3Tζ1+3cosh2γ3Tsinhγ3Tζ0−γ3ζ1coshγ3T−3ζ02sinhγ3Tπβ53cosh2γ3T−12γ3,Υ12=2304πβ53β6cosh2γ3T−123γ3β52cosh2γ3T−11/2+8γ3sinm/3θγ3β5sech2γ3T−m/6−18+m2ω−6m+18γ32β52cosh4γ3T−−18+m2ω−6m+18γ32β52cosh2γ3T+9γ328−9β522.Figures16–18 show the behavior of the EoS parameter ωvhde taken against redshift (z) with different values of ζ0 and ζ1 for all the three models of BT-II, VIII, and IX, respectively. Here, we notice that the curves of ωvhde for BT-II and IX models for different values of ζ0 and ζ1 begin from the quintessence region (−1<ωvhde<−1/3) and cross the nonrelativistic matter (ωvhde=−1) then reaching the phantom region (ωvhde<−1), whereas for BT-VIII, in the presence of null viscosity (i.e.,ζ0=0and ζ1=0), the curve of ωvhde varies in the quintessence region and as the values of ζ0 and ζ1 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:(84)ωvhde=−1.56−0.48+0.60Planck+TT+loE,−1.58−0.41+0.52Planck+TT,EE+loE,−1.57−0.40+0.50Planck+TT,TE,EE+loE+lensing,−1.04−0.10+0.10Planck+TT,TE,EE+loE+lensing+BAO.Figure 16
EoS parameterωvhde versus redshift (z) for BT-II.Figure 17
EoS parameterωvhdeωvhde versus redshift (z) for BT-VIII.Figure 18
EoS parameter versus redshift (z) for BT-IX.


## 4.5.ωvhde−ωvhde′ plane

Cadwell and Linder [126] have suggested ωvhde−ωvhde′ plane (where ' signifies differentiation w.r.t ln 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 ωvhde and ωvhde′, the plane describes two distinct areas. The plane is described as the thawing zone for ωvhde′>0 when ωvhde<0 and the freezing region for ωvhde′<0 when ωvhde<0.For the BT-II VHDE cosmological model,(85)ω′=Υ13Υ14,where(86)ϒ13=136π83+n+12mn+12m−3n+92Tγ1−β15β2n+12γ12m−2Tγ1+2β12n−1m+6n−15/6−18n2n+12m−3n+92Tγ1−β14β2γ12−2Tγ1+2β1−2n−1m+6n−9/6+12n+12γ1n+122γ21ωm2−6n+122γ12m−9γ12n22−9γ12n+98n+12−γ1ζ1n3+Tζ0−ζ16γ1−ζ0β1m+2n2ζ1γ1+−6Tζ0+2ζ1γ1+6ζ0β1n+γ1ζ12−2Tγ1+2β1−2mn−m/6+Tγ1−β12β2−216−n2ζ1γ13+Tζ0−ζ1γ1−ζ0β1n+3Tζ02−5ζ112γ1−3Tζ0β12πn+12γ1−2Tγ1+2β1−n−1/2+n+12m−3n+92Tγ1−β1−2Tγ1+2β1−2n−1m+6n−3/6n+122γ21ωm2−3n+12γ21m−278−27γ21n22+9γ21n,Υ14=Tγ1−β1−172n+122γ12ωm2+112n+122γ12m+γ12n216+γ12n8−64−2Tγ1+2β1−2n−1m+6n/6+−2Tγ1+2β11/2πβ2Tγ1−β12n+12..For the BT-VIII VHDE cosmological model,(87)ω′vhde=Υ15Υ16,where(88)Y16=γ2β3cosech2γ2T1/2β4cosh3γ2Tsinhγ2Tπ2β33cosh2γ2T−1γ2β3cosechγ2Tsechγ2T1/2+coshm/3θ14418+m2ω−6m+18γ22β32cosh4γ2T−18+m2ω−6m+18γ22β32cosh2γ2T+1492+m2ω−6m+18β32γ22γ2β3cosech2γ2T−m/6γ22cosh2γ2T,ϒ15=11442883coshm/3θπζ1m−618+m2ω−6m+18γ22β32γ2cosh10γ2T+3ζ0m−6sinhγ2T18+m2ω−6m+18γ22β32cosh9γ2T−5ζ12m−618+m2ω−6m+18γ22β32γ2cosh8γ2T−6ζ0m−6sinhγ2T18+m2ω−6m+18γ22β32cosh7γ2T+5ζ12m−6m2ω−6m+18γ22+81m5−3785β32+9γ2220m−18γ2cosh6γ2T+9ζ02sinhγ2Tm−7m2ω−6m+18γ22+15m−72β32+3γ224m−18cosh5γ2T−5ζ14m−6m2ω−6m+18γ22+63m5−545β32+27γ2220m−18γ2cos4γ2T−3ζ02sinhγ2Tm−9m2ω−6m+18γ22+9mβ32+9γ224m−18cosh3γ2T+5ζ116m−6m2ω−6m+18γ22+36m5+2165β3227γ2210m−14γ2cosh2γ2T+3ζ016m−12sinhγ2T92+m2ω−6m+18β32γ22coshγ2T−ζ132m−692+m2ω−6m+18β32γ23β3γ2β3cosech2γ2T−m/6+cosh2γ2T−12cosh2m/3θγ2β3cosech2γ2T−m/3sinhγ2Tγ22m+6m2ω−6m+18γ22+54−3ω−3m2+18mβ32+272+27+3ω2+34m2−9m2γ22coshγ2Tγ2β3cosechγ2Tsechγ2T1/2+β4πcoshm/3θcosh2γ2T−12−18+18+ω+2m2+6mγ22m−3β32cosh4γ2T−−18+18+ω+2m2+6mγ22m−3β32cosh2γ3T+14−272+−54+ω+2m2−6mβ32m−15γ22sinhγ2Tγ2β3coseh2γ2T−m/6−288coshγ2T+1γ2cosh6γ2Tζ1+3cosh5γ2Tsinhγ2Tζ0−3γ22cosh4γ2Tζ1−3cosh3γ2Tsinhγ2Tζ0−γ2ζ14cosh2γ2T−34coshγ2Tsinhγ2Tζ0+3ζ1γ28πcoshγ2T−1β33coshγ2Tcoshγ2Tsinh2γ2Tβ32γ22.For the BT-IX VHDE cosmological model,(89)ωvhde′=Υ17Υ18,where(90)ϒ17=−36γ3β52cosh2γ3T−833−18+m2ω−6m+18γ32β52ς0m−6cosh10γ3T+ς1−18+m2ω−6m+18γ32γ3β52m−6sinhγ3Tcosh9γ3T−152−18+m2ω−6m+18γ32β52ς0m−6cosh8γ3T−2ς1−18+m2ω−6m+18γ32γ3β52m−6sinhγ3Tcosh7γ3T+6ς0m−214m2ω−6m+18γ32−81m4+135β52+9γ3216m−18cosh6γ3T+m−16m2−6m+18γ32−45m2+189β52+9γ328m−18ς1γ3sinhγ3Tcosh5γ3T−32m2ω−6m+18m−32γ32−63m2+270β52+27γ38m−18ς0cosh4γ3T−9ς08γ3sinhγ3Tγ3−2β5γ3+2β5m−18cosh3γ3T+94m2ω−6m+18γ32−3m+45β52+3γ324m−21ς0cosh2γ3T−278ς1γ3sinhγ3Tγ3−2β5γ3+2β5coshγ3T+81ς0γ3232−81β52ς08πβ5sinm/3θγ3β5sech2γ3T−m/6+γ3β5sech2γ3T−m/3m2ω−6m+18γ32+3m−18m+6β52+27+−27+−3ω2−34m2+9m2γ32sin2m/3θcoshγ3Tcosh2γ3T−12γ32sinhγ3Tγ3β52cosh2γ3T−11/2−4πcosh2γ3T−12β6coshγ3T−18+18+ω+2m2+6mγ32m−3β52cosh4γ3T−−18+18+ω+2m2+6mγ32m−3β52cosh2γ3T+−216+3ω+9m2−9mγ32−9m2+272β52−27γ328m−15sinm/3θsinhγ3Tγ3β5cosh2γ3T−m/6−288γ3cosh5γ3Tsinhγ3Tς1+3ς0cosh6γ3T−γ3cosh3γ3Tsinhγ3Tς1−9ς02cosh4γ3T+γ3coshγ3Tsinhγ3Tς1+3ς0cosh2γ3T−3ς04πcosh2γ3T−12β52,Υ18=sin2m/3θ−18+m2ω−6m+18γ32β52cosh4γ3T−−18+m2ω−6m+18γ32β52cosh2γ3T+9γ328−9β5222γ3γ3β5sech2γ3T−m/3+41472π2cosh2γ3T−125β55β66γ3β52cosh2γ3T−11/2+288πcosh2γ3T−12−18+m2ω−6m+18γ32β52cosh4γ3T−−18+m2ω−6m+18γ32β52cosh2γ3T+9γ328−9β522γ3β5sech2γ3T−m/6γ3β52sinm/3θβ6sinh2γ3T.The evolutionary trajectories ofωvhde−ωvhde′ plane for the BT–II, VIII, and IX models are plotted with different values of ζ0 and ζ1 in Figures 19–21, respectively and we notice that the ωvhde−ωvhde′ plane for the BT-II and IX models vary in the freezing region (i.e.,ωvhde<0 and ωvhde′<0) whereas, the ωvhde−ωvhde′ plane of the BT-VIII model varies in thawing region (i.e.,ωvhde<0 and ωvhde′>0) for small values of ζ0 and ζ1. However, by increasing the values of ζ0 and ζ1 we get the freezing region for the BT-VIII model. Hence, the ωvhde−ωvhde′ plane of our obtained models is in accordance with the present observational data [127, 128] as follows:(91)ωvhde=−1.17−0.12+0.13,ωvhde′=0.85−0.49+0.50WMAP+eCAMB+BAO+H0+SNe,ωvhde=−1.13−0.25+0.24,ωvhde′<1.32Planck+WP+BAO,ωvhde=−1.34−0.18+0.18,ωvhde′=0.85±0.7WMAP+eCAMB+BAO+H0.Figure 19
EoS planeωvhde−ωvhde′ for BT-II.Figure 20
EoS planeωvhde−ωvhde′ for BT-VIII.Figure 21
EoS planeωvhde−ωvhde′ for BT-IX.


## 4.6. Stability of the Model

To examine the stability of any DE model, we utilize the squared speed of sound (vs2). The models with vs2<0 shows instability where as models with vs2>0 shows stability. Hence, the vs2 is determined as follows [129]:(92)vs2=Pvhde′ρvhde′,where Pvhde′ and ρvhde′ are the differentiation of pressure and density of VHDE w.r.t cosmic time ‘T’, respectively.The squared speed of the sound for the BT-II VHDE model is given by(93)vs2=Υ19Υ20,where(94)Υ19=−154Tγ1−β13n+12m−6n+3γ12n+12m3+n+12m−2Tγ1+2β1−2n−1m+6n−12/6,−9n24Tγ1−β12n+12m−6n+3γ12−2Tγ1+2β1−2n−1m+6n−6/6+18Tγ1−β1n+12m−6n+3,−278+n+122ωm2+−3n−32m−27n22+9nγ12−2Tγ1+2β1−2n−1m+6n/6,−27−2Tγ1+2β1−nπγ1n+12Tζ0−2nζ13−ζ13γ1−ζ0β1−2Tγ1+2β11/2,Υ20=Tγ1−β1−1432n+12m−6n+398+n+122ωm2−6n+122m−9n22−9nγ12,−2Tγ1+2β1−2m+6n−m+3/6+Tγ1−β12πβ2n+12.For the BT-VIII VHDE cosmological model,(95)vs2=Υ21Υ22,where(96)Υ21=4coshm/3θcosh2γ2T−12−18+18+ω+2m2+6mγ22m−6β32cosh4γ2T−−18+18+ω+2m2+6mγ22m−6β32cosh2γ2T+14m−18γ22−272+−54+ω+2m2−6mβ32γ2β3cosech2γ2T−m/6+288γ2cosh2γ2Tsinhγ2Tζ1+32cosh3γ2Tζ0−γ22sinhγ2Tζ1−32coshγ2Tζ0πcoshγ2Tβ33γ2β3cosech2γ2T1/2,Υ22=cosh2γ2Tcoshm/3θγ2β3cosech2γ2T−m/62m−618+m2ω−6m+18γ22β32cosh4γ2T−m−618+m2ω−6m+18γ22β32cosh2γ2T+14m−1892+m2ω−6m+18γ32γ2γ2β3cosechγ2Tsechγ2T1/2−864πβ32β4cosh2γ2Tcosh2γ2T−1.For the BT-IX VHDE cosmological model,(97)vs2=Υ23Υ24,where(98)Υ23=2γ3β5sech2γ3T1/2coshγ3Tsinhγ3Tsinm3θ−18+18+ω+2m2+6mγ32β52m−6cosh4γ3T−−18+18+ω+2m2+6mγ32β52m−6cosh2γ3T+−270+3ω+9m2−18mγ32+27−9m2β52−27γ328m−18γ3β5sech2γ3T−m/6−288cosh2γ3T−12πγ3coshγ3Tsinhγ3Tζ1+3cosh2γ3Tζ02−3ζ04β52,Υ24=sinh2γ3Tγ3β5cosh2γ3T−m/6sinm/3θ−18+m2ω−6m+18γ32β52m−6cosh4γ3T−−18+m2ω−6m+18γ32β52m−6cosh2γ3T+3m2ω−18m+54γ32+27−9m2β52+9γ328m−18γ3β52cosh2γ3T−11/2−432cosh2γ3T−122πβ52β6.We have plotted the squared speed of the soundvs2 versus redshift for three different values of ζ0and ζ1 for BT-II, VIII, and IX models in Figures 22–24. The vs2 of BT-II and VIII models show the unstable behavior for three values of ζ0 and ζ1. Whereas, vs2 of BT-IX gives the stable to unstable behavior. Also, by increasing the values of ζ0and ζ1, i.e., increasing bulk viscosity, the models indicate the unstable behavior of the Universe.Figure 22
Squared speed of sound (v2) versus redshift (z) for BT-II.Figure 23
Squared speed of soundvs2 versus redshift (z) for BT-VIII.Figure 24
Squared speed of soundvs2. versus redshift (z) for BT-IX.


## 4.7. Density Parameter (Ωvhde)

The dimensionless density parameter of DE (Ωvhde) is defined as(99)Ωvhde=ρvhde3H2.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 ρvhde in the above equation, and we used a graphical representation to analyze its behavior. The density parameter of BT-II, VIII, and IX models, respectively are given by(100)Ωvhde=Υ256γ122n+12π,where(101)Υ25=−72πβ22β1−2Tγ1n+1/2Tγ1−β1+2β1−2Tγ1m−2n−1+12n/6γ12ωn+122m2−6n+122mγ12+98+−9n22−9nγ12Tγ1−β1,(102)Ωvhde=Υ2696γ22πβ33sinh2γ2Tcosh22γ2T,where(103)Υ26=−2γ2coshθm/3γ2β3cosech2γ2T−m/618+m2ω−6m+18γ22β32cosh22γ2T−18β32+9γ222−288β4γ2β3cosech2γ2T1/2sinh32γ2Tπβ33cosh22γ2T−1,(104)Ωvhde=Υ2796γ32cosh2γ3Tπβ53sinh22γ3T,where(105)Υ27=−2−18+m2ω−6m+18γ32β52cosh22γ3T−γ32−92+m2ω−6m+18β52γ3γ3β5sech2γ3T−m/6−288β6γ3β5sech2γ3T1/2πβ53cosh32γ3T.The density parameterΩvhde for BT–II, VIII, and IX VHDE models against redshift (z) is seen in Figures 25–27, respectively. It can be seen that the trajectories of Ωvhde are decreasing against redshift (z) for all the three models and are varying in the positive region.Figure 25
Density parameter of VHDEΩvhde) versus redshift (z) for BT-II.Figure 26
Density parameter of VHDEΩvhde). versus redshift (z) for BT-VIII.Figure 27
Density parameter of VHDEΩvhde) versus redshift (z) for BT-IX.


## 4.8. Om-Diagnostic (Om(z))

To discriminate different 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 field, quintessence model, and phantom field through the trajectories of the curves. The phantom DE era corresponds to the positive trajectory, whereas the negative trajectory indicates that the DE constitutes quintessence. The om-diagnostic function is defined as(106)Omz=H2z−H02H02z3−1.The Om-diagnostic for models which are in Equations (35), (48), and (61) are, respectively, given by(107)Omz=γ12n+1−3H021+z/a06/2n+13H021+z/a06/2n+1z3−1,Omz=γ22/91+β3/γ2cosh2θ1+z/a062−H02H02z3−1,Omz=γ32/91+β5/γ3sin2θ1+z/a062−H02H02z3−1.The 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.Figure 28
Om-diagnostics (Om(z)) versus. redshift (z) for BT-II.Figure 29
Om-diagnostics (Om(z)) versus redshift (z) for BT-VIII.Figure 30
Om-diagnostics (Om(z)) versus. redshift (z) for BT-IX.


## 5. Interpretations of the Models

To understand the cosmological mysteries of accelerated expansion, we have constructed the field equations of BDT for BT-II, VIII, and IX Universe in the context of VHDE. The exact solutions of the Brans–Dicke field equations are obtained by considering the relations between the metric potentials, the relation between scalar field 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 ζ0&ζ1. 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. This is further justified by the bulk viscosity coefficientζ 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 statefinder plane r−s behave independently of time. The 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. This behaviors of the EoS parameter and the EoS plane represents the accelerated expansion of the cosmos. The squared speed of sound vs2 varies in the negative region, depicting an unstable Universe model. Also, the density parameter Ωvhde is decreasing and varying in the positive region and the Om(z) differs in the negative region, depicting the quintessence behavior of the Universe.The 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. The deceleration parameter q shows a transition from an early decelerating phase to a late accelerating phase, and the jerk parameter differs in a positive region approaching to one as z⟶0. Moreover, the statefinder pair r,s starts its evolution from the quintessence and phantom regions and reaches the Λ CDM model (for r = 1, s = 0). The model shows an unstable behavior, as vs2 is varying in the negative region. The plot of the EoS parameter states that the model varies in the phantom region and the EoS plane differs in the freezing region. The 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 coefficients indicate an accelerated cosmic expansion as they vary in negative, positive, and positive regions respectively. The curve of deceleration parameterq 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. The squared speed of sound varies in positive to negative region, depicting the stable to an unstable behavior of the Universe. Also, the statefinder pair starts its evolution from the quintessence and phantom regions and reaches the Λ CDM model (for r = 1, s = 0). The 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. The density parameter Ωvhde 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 VHDEρvhde EoS parameter ωvhder−s plane, the deceleration parameter q, the density parameter of the VHDE Ωvhde Om diagnostic, and bulk viscosity ζ. We have considered some isotropic and homogeneous models for comparison study. Singh and Srivastava [102] have studied a flat FRW Universe filled 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 effect of viscosity in the evolution. Srivastava and Singh [131] have investigated the new HDE model in modified fR,T gravity theory within the framework of a flat 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.,ζ1=ζ0= 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 ωvhde, 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 flat FRW line element. Kumar and Beesham [134] have studied the concept of HDE in the frame work of BDT in the formalism of the flat 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 specific 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 filled with pressure less matter and anisotropic modified Ricci dark energy in the presence of an attractive massive scalar field 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]. The state finder 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]. The 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]. The density parameter of our BT-II, VIII, and IX VHDE models agrees with the results of Rahman and Ansari [135] and Naidu [137]. The 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]. The 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]. The bulk viscosity of our models coincides with the results of Singh and Kumar [132]. The above results lead to the conclusion that our BT-II, VIII, and IX VHDE models in BDT are 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.



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*Source: 5364541-2022-11-02.xml*