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Bianchi Type V Universe with QVDP in f (R, T) Theory of Gravity 
 

Vandana Soni1 & Sudha Agrawal2 
1Department of Mathematics, AKS University, Satna-485001, (M.P.), India 

sonivandana06@gmail.com 
2Department of Mathematics, AKS University, -485001, (M.P.), India 

Corresponding Author: sudha.agrawal10@gmail.com 

 

Article History: 

Received: 04-01-2025 

Revised: 25-01-2025 

Accepted: 21-02-2025 

Abstract:  

Throughout the report, The Bianchi cosmological model is contemplating within a 

context of f (R, T) Gravity. Under the recently proposed the deceleration parameter in 

quadratic form, the solutions to the reconditioned field equations have been found by 

Bakery and Shafeek (2019). Big explosion uniqueness at infinite time t = 0 and shred 

with respect to t = 1 are expose to view by the model. Studies have been accompanied 

on the development of the universe. Physical and dynamical properties and a graphical 

depiction has also been provided. Typical parameter of field equations is determined 

Also we discussed the jerk parameter of the proposed model. 

 

Key words: f (R, T) theory of gravity, B-V galactic model, Deceleration Parameter, 

State finder parameter. 

 

I. INTRODUCTION 

According to the   discussion on many observations by many researchers, the cause of the 

accelerated universe is dark energy is most responsible candidate. sinister energy to be 

contemplate as a substance with under pressure which influence the current cosmological 

energy movement. By various observations, it becomes necessary to envision and concede the 

nature of force field [1-4]. Among the most broadly considered claimant of dark energy is the 

empty space of the cosmological constant Λ. But in cosmology, it brawls with remarkable 

problems among them like cosmic coincidence and fine tuning [5-7]. Some other features of 

dark energy are Chaplygin gas (a dark fluid which might describe the past decelerating matter 

dominated era and at present time it provides an accelerating expansion of the universe.) [8], 

tachyons (a hypothetical particle that travels faster than the speed of light.) [9], Quintessence 

(bottom distillation) [10], phantom (wraith) [11], k-essence [12]. 

By notify the angular part of the EH action, make alteration to gravity theories can be acquired.  

These theories have recently leap up to the top of the list of options for clarification the unsolved 

problem of the accelerating phase of the universe enlargement. The simplest and most common 

converting of GR in this situation is f(R) gravity. The f(R) gravity theory was first dispensed in 

Buchdah,1970 [13], and it was afterwards utilized to discover an isotropic, Non-Singular de-

Sitter type Cosmological Solutions [14]. The masses authors have inquired various aspects of 

𝑓(𝑅) gravity for different type cosmological models, [15-18]. Moreover, it is signified that the 

f(R) gravity is congenial with recent observations [19-22]. 

Nevertheless, modern cosmology now addresses the anisotropy problem in inclusion to the Dark 

Energy problem. Span the latest factual data supports the prospect of an anisotropic aspects in 

mailto:sonivandana06@gmail.com
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previous periods of the cosmos, the FRW models deduce the nature of the modern universe as 

spatially uniform and isotropic [23]. This observation is understood to mean that the universe's 

initial anisotropy is currently changing to an isotropy [24-25]. Therefore, the homogeneous and 

anisotropic Bianchi type models can be quite useful in examining the universe's early time 

anisotropic behaviour. 

In this context, several theories of modified gravity were later developed.  Otherwise, f (R, T) 

theory is a further imprecise one than the f (R). Geometry and matter are integrated in the 

gravitational action to enlarge f (𝑅, 𝑇) theory [26]. Gravitational Lagrangian in this alter gravity 

theory requires an unknown function of the scalar curvature and the stress energy-momentum 

tensor trace. 

 

Astronomers who have researched contemporary cosmological issues like as dark energy and 

anisotropy have a tendency to look at modified theories involving Bianchi type models. 

Formerly ten years f (R, T) gravity assumption increased attention, among other things. In this 

context, a large number of studies exist. 

 

In f (R, T) gravity, Adhav stumbled on errorless solutions to the field equations with view to 

LRS Bianchi type-I space [27]. By embracing a linearly substituting deceleration parameter and 

taking into account numerous functional configurations of 𝑓 (𝑅, 𝑇) Sahoo and Sivakumar 

obtained the require solutions for LRS Bianchi type-I cosmological models in 𝑓 (𝑅, 𝑇) theory 

of gravity [28]. The wordsmith concluded that there is no way to avoid the Big Rip scenario. 

Using the presumption of the constant deceleration parameter for Bianchi type-V space-time in 

the f(R,T) theory of gravity and the variant law of the Hubble parameter, Shamir set up two 

exact solutions that correlate with  singular and non-singular models [29].Using a linear 

deceleration parameter with a negative slope, Chaubey et al. analysed the broad class of Bianchi 

cosmological models in f(𝑅, 𝑇) gravity with dark energy as standard and modified Chaplygin 

gas and bulk viscosity for three types of average scale factor [30]. 

 

FLRW cosmological model with orthogonal deceleration parameter in f (R, T) gravity has been 

scrutinize by Bishi et al. [31]. Moreover, using a unique form of Hubble's parameter, Bishi, et 

al. range over the LRS Bianchi type-I cosmological model in f (R, T) gravity [32]. For LRS 

Bianchi type-I model, Tiwari et al. derived the suspension to the field equations of f (𝑅, 𝑇) 

gravity by assuming that the deceleration parameter is a linear function of the Hubble parameter 

and indicated that the model starts with an initial singular phase and transitions from an early 

deceleration phase to a late-time acceleration phase [33]. Mamon and Das have examined the 

field equations of 

f (𝑅, 𝑇) gravity for the parametric remodelling model of the deceleration parameter [34]. 

 

Existed come across that the obtained model of the Universe restrain an initial singularity, a finite 

life full lenght, and a big rip at the end. Akarsu and Dereli explored cosmological models in which 

the cosmos evinces quintom like behavior and bring to a conclusion in a large rip with a linearly 

diverse deceleration value [35]. The big-rip theory advance that the universe expands before 



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shrinking and moving back to its initial state at the time of the big bang. Bakry and Shafeek 

glanced into the universe model with the time-dependent deceleration parameter of the second 

degree and came to this inference [36]. Tiwari et al. inquire into the time-dependent deceleration 

parameter in the framework of f (R, T) gravity and concluded that the cosmos has become 

distended periodically. i.e., the universe expansion at embarks on with a slowing then facilitate 

super exponentially [37]. According to Tiwari and Sofuoglu's compulsion investigation of 

quadratic changing deceleration parameter in the context of f (R, T) gravity, universe set about 

with a big bang and ends with a enormous rip [38]. Kumar and Arora have studied Bi-Quadratic 

Varying Deceleration Parameter to Study the Cosmological Model [39]. Mishra et al. have 

provided a comprehensive analysis of the evolution of geometrical and physical parameters, 

complete with the numerically constrained values.[40]. 

 

In this paper, we have evolved anisotropic cosmological model in f(𝑅, 𝑇) gravity theory to acquire 

the exact solution of the alter field equations in the activity of the newly offered condition of  [36] 

quadratic form of deceleration parameter and requiring the functional form of f(𝑅, 𝑇) = f1(R) +

 f2(T), where f1(R) = 𝜆 𝑅 and f2(T) = 𝜆 𝑇. The tangible major of the considered deceleration 

parameter is presented as; 

The cosmological model starts with a big bang at t = 0 and ends at a big rip at t=2𝛽 

i.e., The universe ‘begins’ with q = (8 𝛽2 − 1) and ends with q = (-4 𝛽2 − 1). 

 

II. BASIC FORMALISM OF F (R, T) GRAVITY THEORY: 

 

In f (R, T) gravity, the field equations are obtained from a dissimilarity, Hilbert-Einstein type, 

principle. The action for the specified f (R, T) gravity is, 

S =   ∫  (
1

16π
f(R, T) + sm)√−𝑔 𝑑𝑥4                                                              

(1) 

where f (R, T) is an arbitrary function of the Ricci tensor R and energy momentum tensor 

T. 

Three obvious requirements of the operational from of f (R, T) are standardly considered, 

𝑓(𝑅, 𝑇) =  {

𝑅 + 2𝑓(𝑇)

𝑓1(𝑅) + 𝑓2(𝑇)

𝑓1(𝑅) + 𝑓2(𝑅)𝑓3(𝑇)
                          

(2) 

 

Assuming as a choice of f (R, T) = f1(R) + f2(T), where f1(R) = 𝜆R and f2(T) = 𝜆T, where 𝜆 is 

free parameter. 

Therefore, Equation (2) can be written as, 

𝑓1
′(𝑅)𝑅𝑖𝑗 − 

1

2
 𝑓1(𝑅)𝑔𝑖𝑗 + (𝑔𝑖𝑗 ∇𝑖∇𝑖 - ∇𝑖∇𝑗)  𝑓1 (𝑅) = k𝑇𝑖𝑗 − 𝑓2

′(𝑇) 𝑇𝑖𝑗 −𝑓2
′(𝑇)Θ𝑖𝑗 + 

1

2
𝑓2(T)𝑔𝑖𝑗                                                                                                                                           (3) 



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Where the prime denotes differentiation with respect to the arguments. where  Θ𝑖𝑗 is expressed 

as, 

Θ𝑖𝑗 = 𝑔𝑙𝑚 δTlm

δ𝑔𝑖𝑗                              

(4) 

where Tij is energy momentum tensor with perfect fluid and defined as, 

  𝑇𝑖𝑗 = (𝜌 + 𝑝) u𝑖u𝑗 − 𝑝𝑔𝑖𝑗              

(5) 

where 𝑢𝑖 = (0,0,0,1) is four velocity vector satisfying 𝑢𝑖𝑢𝑗  = 1              

(6) 

p and 𝜌 are pressure and energy density of the matter respectively. 

 

Assuming that the matter content consists of a perfect fluid and defined as, 

Θ𝑖𝑗 = −2 𝑇𝑖𝑗 − 𝑝gij                     

(7) 

 

The field equation takes the form, 

     λ𝑅𝑖𝑗 − 
1

2
 λ(𝑅 + 𝑇)𝑔𝑖𝑗 + (𝑔𝑖𝑗  ∇𝑖∇𝑖 - ∇𝑖∇𝑗) λ   = k𝑇𝑖𝑗 −  λ 𝑇𝑖𝑗  + λ [ 2𝑇𝑖𝑗 +

1

2
𝑔𝑖𝑗 )]                       

(8) 

Generally, the field equations also pivot on the whole time the tensor 𝜃𝑖𝑗, on the physical nature of 

the matter 

field. Hence in the case of f (R, T) gravity depending on the nature of the matter source, we prevailed 

several theoretical models corresponding to each choice of f (R, T). Out of three explicit 

specifications 

of the functional form of f (R, T). 

Now, we have 

     𝑅𝑖𝑗 − 
1

2
𝑅𝑔𝑖𝑗 − (𝑝 +

1

2
 𝑇)𝑔𝑖𝑗   = (

k+λ

λ
)  𝑇𝑖𝑗                          

(9) 

 

Recollecting Einstein equations inclusive of cosmological constant, 

𝐺𝑖𝑗 −  Λ𝑔𝑖𝑗 = − k𝑇𝑖𝑗           

(10) 

Pick out k=1. 

We choose a negative small value for the arbitrary λ, so that we have the same sign of the RHS. 

of (9) and (10), we keep this choice of λ throughout. (p + 
1

2
 T) can be regarded now as a 

cosmological constant term t. Hence, we can write, 

Λ(T) = p +
1

2
𝑇           

(11) 

Hence Eq. (11) reduces to , 



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Λ(T) =
1

2
(ρ −  p)                                                                         

(12) 

 

III. GRAVITATIONAL FIELD EQUATION 

 

We consider a homogeneous Bianchi type–V metric, which is given by, 

 

𝑑𝑠2= 𝑑𝑡2 − 𝑎1
2𝑑𝑥2 − 𝑒2𝑚𝑥[𝑎2

2𝑑𝑦2 + 𝑎3
2𝑑𝑧2]                                                                                   (13) 

where, A, B and C are functions of cosmic time t only. 

Now assuming commoving coordinate system, the field Equation (9) with the help of (5) and 

(6) for the metric (12) can be written as, 
𝑎2̇𝑎3̇

𝑎2𝑎3
+ 

𝑎3̈

𝑎3
+

𝑎2̈

𝑎2
−

𝑚

𝑎1
2 = (

8π+λ

λ
)p - Λ                   

(14) 

𝑎1̇𝑎3̇

𝑎1𝑎3
+

𝑎1̈

𝑎1
 +

𝑎3̈

𝑎3
−

𝑚

𝑎1
2 = (

8π+λ

λ
)p - Λ                      

(15) 

𝑎1̇𝑎2̇

𝑎1𝑎2
+

𝑎1̈

  𝑎1
+

𝑎2̈

𝑎2
−

𝑚

𝑎1
2 = (

8π+λ

λ
)p – Λ                            

(16) 

𝑎1̇

𝑎1

𝑎2̇

𝑎2
+

𝑎2𝑎3̇
̇

𝑎2𝑎3
+

𝑎1̇𝑎3̇

𝑎1𝑎3
   −

 3𝑚

𝑎1
2 = - (

8π+λ

λ
) 𝜌- Λ              

(17) 

2
𝑎1̇

𝑎1
 −

𝑎2̇

𝑎2
−

𝑎3̇

𝑎3
  = 0                                                                                          

(18) 

From equations (14)-(16) together with equation (17), we get – 
𝑎1̇

𝑎1
 = 

ȧ

a
                                                                               

(19) 
𝑎2̇

𝑎2
 = 

ȧ

a
 – 

𝑘

 a3                                          

(20) 
𝑎3̇

𝑎3
 = 

ȧ

a
 +  

𝑘

 a3                  

(21) 

where 𝑘 is constant. 

The spatial volume and the scale factor of the Metric (13) are defined   by, 𝑉 = 𝑎3 = 𝑎1𝑎2𝑎3     

                                                             (22) 

The physical quantities of observational interest in cosmology are as; 

The mean generalized Hubble parameter becomes, 



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H = 
ȧ

a
   = 

1

3
(
𝑎1̇

𝑎1
+

𝑎2̇

𝑎2
+

𝑎3̇

𝑎3
 )                                                                                               

(23) 

 

The energy density, pressure and cosmological term for this model are demonstrates as, 

 

𝜌 =
2𝜆2

[(8π+3λ)2−λ2
[�̇� −

3(8π+λ )

λ
𝐻2 +

(8π+2λ )k2 

λ𝑎6 +
(24π+4λ )λ

𝑎2
]                         

(24) 

 

p = 
2λ2

[(8π+3λ)2 −λ2]
[
(16𝜋+3λ)

λ
𝐻 ̇ + 

3(8π+λ)

λ
𝐻2 +

(8π+2λ)𝑘2

λ𝑎6 −
8𝜋λ

𝑎2
]                            

(25) 

and 

Ʌ =  
λ2

[(8π+3λ)2 −λ2]
[
(16𝜋+2λ)

λ
𝐻 ̇ −

6(8𝜋+λ)

λ 
𝐻2 +

(32𝜋+4λ)

𝑎2 λ]                         

(26) 

 

In sequence to find the exact solution of field equations following [36], we assume the quadratic 

deceleration parameter q which is given by, 

 

q = 
−𝑎�̈�

�̇�2  = 3𝑡2-12𝛽𝑡 + (8 𝛽2 − 1)            

(27) 

 

where  𝛽 is positive constant. At t=0, q = (8 𝛽2 − 1),which is always positive for all  𝛽 > 0  i.e. 

Given form of the deceleration parameter in (27) concludes the constant form of deceleration 

parameter which is discussed by [41,42]. Remoter this was extended in the quadratic form of 

time. At the cyclic term, this model starts with the big-bang singularity and late time in a big 

rip. 

 

The equation for the deceleration parameter is given by, 

 

q = 
−𝑎�̈�

�̇�2  = −
�̇�

𝐻2 − 1          

(28) 

 

With the help of equation (27) and (28) Hubble parameter can be found as, 

 

H= 
1

𝑡(𝑡−2𝛽)(𝑡−4𝛽)
                                   

(29) 

 



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we can see from equation (29), that the Hubble parameter H is always positive in the range  

 0 ≤ 𝑡 ≤  4𝛽. Also we can observe that the Hubble parameter diverges at t=0, t = 2𝛽 and t =4𝛽 

 

Integrating equation (29), we obtain the scale factor as, 

 

a = 
[𝑡(4𝛽−𝑡)]

1

8𝛽2

[(2𝛽−𝑡)]

1

4𝛽2

                                                                                    

(30) 

 

which gives the declaration of the scale factor as function of cosmic time. It is remarked that   

the scale factor a= 0 for t=0 and at t =4𝛽 and scale factor diverges at t= 2𝛽, which manifest that 

the universe starts with big-bang at t=0 and ends with big crunch at t=2 𝛽. 

 

From equations (19)-(21) together with equation (31) & (32), we get – 
𝑎1̇

𝑎1
 = 

1

𝑡(2𝛽−𝑡)(4𝛽−𝑡)
                                                                            

(31) 

𝑎2̇

𝑎2
 = 

1

𝑡(2𝛽−𝑡)(4𝛽−𝑡)
 – 

[(2𝛽−𝑡)]

3

4𝛽2
𝑘

 [𝑡(4𝛽−𝑡)]

3

8𝛽2

                               

(32) 

𝑎3̇

𝑎3
 = 

1

𝑡(2𝛽−𝑡)(4𝛽−𝑡)
 +  

[(2𝛽−𝑡)]

3

4𝛽2
𝑘

 [𝑡(4𝛽−𝑡)]

3

8𝛽2

                                                           

(33) 

which gives, 

𝑎1 =   [
𝑡(4𝛽−𝑡)

(2𝛽−𝑡)2
]

1

8𝛽2
                                                                                           (34) 

𝑎2 = [
𝑡(4𝛽−𝑡)

(2𝛽−𝑡)2
]

1

8𝛽2
   − ∫

[(2𝛽−𝑡)]

3

4𝛽2
𝑘

 [𝑡(4𝛽−𝑡)]

3

8𝛽2

 dt                                                            (35) 

𝑎3 = [
𝑡(4𝛽−𝑡)

(2𝛽−𝑡)2
]

1

8𝛽2
   +  ∫

[(2𝛽−𝑡)]

3

4𝛽2
𝑘

 [𝑡(4𝛽−𝑡)]

3

8𝛽2

 dt                   

(36) 

 

 

Relation between scale factor and redshift is given as, 

 
𝑎 

𝑎0
 = 

1

1+𝑧
                                                                                                                                         (37) 

 

From equation (30) and (37) we have, 



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[
𝑡(4𝛽−𝑡)

(2𝛽−𝑡)2
]

1

8𝛽2
=  

1

1+𝑧
                                                            (38) 

 

To find the value of time in the term of red shift, we have taken the choice of 𝛽 . 

So, we assume,  
1

4𝛽2 = 1 ⟹   𝛽 = 0.5, which can help to express t as, 

 

t = 1± 
1+𝑍

√𝑍(𝑍+2)+2
                                                                         (39) 

 

The Hubble Parameter in the terms of redshift (z) conceivably expressed as, 

 

H = 
Ho

[1± 
1+𝑧

√𝑧(𝑧+2)+2
 ]

3

−6𝛽[1± 
1+𝑧

√𝑧(𝑧+2)+2
 ]

2

+8 𝛽2[1± 
1+𝑧

√𝑧(𝑧+2)+2
  ]

 

 

concerning red shift, Deceleration parameter can be stated as, 

 

q (z) =  
𝑧2+2𝑧−1

𝑧2+2𝑧+2
                                                                                                                          (40) 

 

as the consequence (26), (27), (28), (31) & (32), the energy density, pressure and cosmological 

term of model are express as, 

 

𝜌 =
2𝜆2

[(8π+3λ)2−λ2
[
3𝜆𝑡2−12𝛽𝜆𝑡 + 8 𝜆𝛽2−(24π+3λ )

𝜆𝑡2(2𝛽−𝑡)2(4𝛽−𝑡)2
+

(8π+2λ )k2 [(2𝛽−𝑡)]

3

2𝛽2

λ𝑡

3

4𝛽2(4𝛽−𝑡)]

3

4𝛽2

+
(24π+4λ )[(2𝛽−𝑡)]

1

2𝛽2
λ

𝑡

1

4𝛽2(4𝛽−𝑡)]

1

4𝛽2

]

            

                             (41) 

 

 

p = 
2𝜆2

[(8π+3λ)2−λ2]
[
(16π+3λ )3𝑡2−12𝛽𝑡+ 24 𝛽2(8π+λ )

λ𝑡2(2𝛽−𝑡)2(4𝛽−𝑡)2
+

(8π+2λ )k2 [(2𝛽−𝑡)]

3

2𝛽2

λ𝑡

3

4𝛽2(4𝛽−𝑡)

3

4𝛽2

−
(8πλ )[(2𝛽−𝑡)]

1

2𝛽2

𝑡

1

4𝛽2(4𝛽−𝑡)

1

4𝛽2

]       

                                 

(42) 

 

Ʌ =  
λ2(8π+λ)

[(8π+3λ)2 −λ2]
[
(6𝑡2−24𝛽𝑡 + 16𝛽2+3)

λ𝑡2(2𝛽−𝑡)2(4𝛽−𝑡)2
+ 

4[(2𝛽−𝑡)]

1

2𝛽2

𝑡(4𝛽−𝑡)]

1

4𝛽2

]           

(43) 

 

The Equalization of state parameter (p = 𝜔𝜌)is given by, 

 



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𝜔= 

[
(16π+3λ )3𝑡2−12𝛽𝑡+ 24 𝛽2(8π+λ )

λ𝑡2(2𝛽−𝑡)2(4𝛽−𝑡)2
+

(8π+2λ )k2 [(2𝛽−𝑡)]

3

2𝛽2

λ𝑡

3

4𝛽2
(4𝛽−𝑡)

3

4𝛽2

−
(8πλ )[(2𝛽−𝑡)]

1

2𝛽2

𝑡

1

4𝛽2
(4𝛽−𝑡)

1

4𝛽2

]

[
3𝜆𝑡2−12𝛽𝜆𝑡 + 8 𝜆𝛽2−(24π+3λ )

𝜆𝑡2(2𝛽−𝑡)2(4𝛽−𝑡)2
+

(8π+2λ )k2 [(2𝛽−𝑡)]

3

2𝛽2

λ𝑡

3

4𝛽2
(4𝛽−𝑡)]

3

4𝛽2

+
(24π+4λ )λ[(2𝛽−𝑡)]

1

2𝛽2

𝑡

1

4𝛽2
(4𝛽−𝑡)]

1

4𝛽2

]

                

                 

(44) 

identification of restriction (𝜔), as regards red shift as, 

 

𝜔(𝑧)= 

[
 
 
 
 
 
 
 
 
 
 
 
 

(16π + 3λ )3 [1 ± 
1 + 𝑍

√𝑍(𝑍 + 2) + 2
  ]

2

− 12𝛽 [1 ± 
1 + 𝑍

√𝑍(𝑍 + 2) + 2
  ]  +  24 𝛽2(8π + λ )

λ [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ]

2

([1 ±  
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ] − 2𝛽)

2

([1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ] − 4𝛽)

2 +

(8π + 2λ )k2  (2𝛽 − [1 ± 
1 + 𝑍

√𝑍(𝑍 + 2) + 2
  ])

3
2𝛽2

λ [1 ±  
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ]

3
4𝛽2

(4𝛽 − [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ])

3
4𝛽2

−     

8πλ (2𝛽 − [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ])

1
2𝛽2

[1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ]

1
4𝛽2

(4𝛽 − [1 ±  
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ])

1
4𝛽2

]
 
 
 
 
 
 
 
 
 
 
 
 

 

 

[
 
 
 
 
 
 
 
 
 
 
 
 

3𝜆 [1 ±  
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ]

2

− 12𝛽𝜆 [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ]  +  8 𝜆𝛽2 − (24π + 3λ )

𝜆 [1 ±  
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ]

2

([1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ] − 2𝛽)

2

([1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ] − 4𝛽)

2 +

(8π + 2λ )k2  (2𝛽 − [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ])

3
2𝛽2

λ [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ]

3
4𝛽2

(4𝛽 − [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ])

3
4𝛽2

 +

(24π + 4λ )λ(2𝛽 − [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ])

1
2𝛽2

[1 ±  
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ]

1
4𝛽2

(4𝛽 − [1 ±  
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ])

1
4𝛽2

]
 
 
 
 
 
 
 
 
 
 
 
 

 

 

(45) 

 

 

The Density parameter, 

Ω(t) = 
𝜌

3𝐻2 

Ω(t)   =
2𝜆2𝑡2(2𝛽−𝑡)2(4𝛽−𝑡)2

3[(8π+3λ)2−λ2
[
3𝜆𝑡2−12𝛽𝜆𝑡 + 8 𝜆𝛽2−(24π+3λ )

𝜆𝑡2(2𝛽−𝑡)2(4𝛽−𝑡)2
+

             
[(2𝛽−𝑡)]

1

2𝛽2
{(8π+2λ )k2 [(2𝛽−𝑡)]

1

𝛽2
−(24π+4λ )}

λ𝑡

3

4𝛽2(4𝛽−𝑡)]

3

4𝛽2

]                  

(46) 

 

 

 

 



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Density parameter in the terms of red shift, 

 

 

Ω(z) = 

2𝜆2 [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ]

2

([1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ] − 2𝛽)

2

([1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ] − 4𝛽)

2

3[(8π + 3λ)2 − λ2
 

[
 
 
 
 
 
 

3𝜆 [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ]

2

− 12𝛽𝜆 [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ]  +  8 𝜆𝛽2 − (24π + 3λ )

𝜆 [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ]

2

([1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ] − 2𝛽)

2

([1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ] − 4𝛽)

2

+

(2𝛽 − [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ])

1
2𝛽2

{(8π + 2λ )k2  [(2𝛽 − [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ])]

1
𝛽2

− (24π + 4λ )

λ [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ]

3
4𝛽2

(4𝛽 − [1 ± 
1 + 𝑧

√𝑧(𝑧 + 2) + 2
  ])

3
4𝛽2

]
 
 
 
 
 
 

 

 

 
Figure.1 average scale factor (a) against cosmic time(t) 

 



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Figure.2 Deceleration parameter against cosmic time(t) and redshift (z) 

 

 

 
Figure 3 Hubble Parameter (H) averse to cosmic time(t) 

 

 
Figure 4. Density parameter and pressure against cosmic time(t) 

 

 

 



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STATE- FINDER PARAMETER 

The state finder parameter is decisive in order to study the cosmological models. The pair {r, 

s} is useful for distinguishing the properties of dark Energy. The r − s plane advisable the 

different qualitative behaviour for different cosmological models. The state appropriator 

diagnostic along with SNAP (Super Novae Acceleration Probe) inspection may distinguish 

different dark energy models. The state finder pair {r, s} is defined as, 

 

r =
𝑎

𝑎𝐻3 together with s = 
𝑟−1

3(𝑞−0.5)
 

Here, the associated with parameters have the accustomed denotation talk throughout the paper. 

The state finder parameters for the discussed model are obtained as, 

r =
𝑎

𝑎𝐻3  
    =   

𝑧2+2𝑧−1

𝑧2+2𝑧+2
   [2 (

𝑧2+2𝑧−1

𝑧2+2𝑧+2
) +  1] +   

6(1+z)2

(𝑧2+2𝑧+2)2
 

s = 
−18 (1+z)2 (2𝑧2+4𝑧+1)

(𝑧2+2𝑧+2)2
 + 3𝑧 

 (𝑧2+2𝑧−1) (𝑧+2) (5𝑧2+10𝑧+1)

(𝑧2+2𝑧+2)3
 

IV.   RESULT AND DISCUSSION 

In The figure1, opposed to time we have taken the interval [0, 2] for cosmic time, while the 

roughly is notable in interval [0,1] only. Noteworthy this, the creativity from the value of scale 

factor i.e., scale factor is negative in the domain [1,2]. So that this region is insignificant from 

the observed value. 

Figure 2, represents behaviour of the deceleration parameter hostile to cosmic time and red shift 

separately from the figures for equation (29), it can be observed that at  𝛽 = 1/2  , the 

deceleration parameter starts at q = 1 at initial time t = 0 and ends at q= -2 at t =1. So, 

observations show that deceleration parameter be sited in the interval [-2,1] for the time interval 

[0,1]. In the course of the time interval of t ∈ [0,0.18], q is positive., q lies between [1, 0.0172] 

which advisable the decelerating phase. i.e. (q > 0). At t = 0.19, q is negative, which indicates 

the accelerating phase. i.e. (q < 0). In the interval t ∈ [1, 1.57] present model forecasting the 

super exponential phase i.e., (q < -1). The qualitative behaviour of q(z) is consistent with 

deceleration parameter discussed by Farook [43]. 

Figure 3 represents The Hubble parameter against cosmic time which shows that, H diverges at 

t = 0 and t = 2 𝛽, i.e., t = 1. Further, one can say that the big rip moment occurs at t = 1. 

 

Figure 4, characterized the variation of the energy density and pressure of model for the flat, 

open and closed universe [44]. It is strong matches out that in both the models, for flat, open 

and closed universes, the energy density has singularities at t = 0 and t = 1, is non-negative (𝜌 

≥ 0).  The energy density in all flat, open and closed universes has a big rip moment at t = 1 due 

to the divergent essence of the scale factor and energy density of the fluid at that time. 

 

At the initial point t = 0, the pressure has the maximum value. The pressure is positive valued 

in the interval [0, 0.08] and negative valued in the interval [0.09, 1]. for the closed universe of 

the model. 



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