# Bianchi type-I Universe with Cosmological constant and periodic varying
deceleration parameter.

N. Hulke  G. P. Singh  Department of Mathematics, Visvesvaraya National
Institute of Technology, Nagpur-440010, India<EMAIL_ADDRESS><EMAIL_ADDRESS>Binaya K. Bishi  Department of Mathematics, Lovely
Professional University, Phagwara, Punjab-144411, India.
<EMAIL_ADDRESS>author

(Day Month Year; Day Month Year)

###### Abstract

This manuscript investigate the dark energy Bianchi type-I cosmological models
in presence of generalized Chaplygin gas, variable gravitational and
cosmological constants. In this manuscript, exact solutions of Einstein field
equations are obtained under the assumption of time periodic varying
deceleration parameter. The physical and dynamical behaviors of the models
have been discussed with the help of graphical representations. Also we have
discussed the stability and physical acceptability of the obtained solutions.

###### keywords:

$f(R,T)$ gravity; Bianchi type-I space-time; periodic varying deceleration
parameter

PACS numbers:98.80.-K, 04.50.Kd, 98.80.Es

##  1 Introduction

Cosmological and astronomical data such as the type Ia supernovae (SNe-Ia) [1,
2] cosmic microwave background (CMB) [3], large scale structure (LSS) [4],
Wilkinson Microwave Anisotropy Probe (WMAP) [5, 6, 7], Sloan Digital Sky
Survey (SDSS) [8] and the Plank 2015 [9] reveal that we live in some 13.8
billion years old Universe which experiences a speedy expansion stage that
have been originated with a bang from a phase of very high density and
temperature. For a long time, it was assumed that either the Universe expands
eternally or the inward pull of gravity gradually slows down the expansion of
the Universe and would ultimately came to a halt after which Universe start to
contract into a big crunch. However, it was at the end of twentieth century,
the unexpected discovery that the Universe might be expanding with an
acceleration surprised the cosmologist as the idea of cosmic acceleration was
against the standard predictions of decelerating expansion caused by gravity.
Latter on it was observed that the expansion of the Universe is accelerated
which is dominated by some exotic stuff called as “Dark energy”(DE) with
highly negative pressure which is the cause of acceleration of the Universe.
The successive disclosure in this direction gave more and more documentation
for a flat, dark energy dominated accelerating Universe. However, by modifying
the theory of gravity one can find the alternative way to explain the
acceleration of the expanding Universe.

The Quantum field theory (QFT) and general theory of relativity (GTR) pointed
out cosmological constant $\Lambda$ [10, 11, 12] introduced by Einstein as the
most feasible candidate of dark energy. The cosmological constant is the
simplest form of dark energy and though the model with cosmological constant
is known as $\Lambda$CDM model. It faces some serious problematic issues [13,
14] such as the fine tuning (a typical small value), cosmic coincidence
problems (although the Universe is in accelerated phase of expansion, why the
dark matter and the dark energy are of the same order?). Chaplygin gas (CG)
[15] with EoS $p=-\frac{B}{\rho}$ [16] where $\rho$ and $p$ are energy density
and pressure respectively, is one of the most expected candidate of dark
energy (DE). Among the various class of dark energy models Chaplygin gas is a
simple characterisation in order to understand the cosmic acceleration and has
attracted large group of researchers in the field of cosmology. Chaplygin gas
is also a fascinating subject of holography [17] and string theory [18].
Chaplygin gas depicts a transition to the present cosmic acceleration from a
decelerated cosmic expansion and conceivably submit a deformation of
$\Lambda$CDM model. Chaplygin gas model fails the tests connected with
structure formation and observed strong oscillations of matter power spectrum
[19]. To overcome this failure of Chaplygin gas model the generalised
Chaplygin gas with EoS $p=-\frac{B}{\rho^{\alpha}}$ with
$0\leqslant\alpha\leqslant 1$ has been proposed. From the holographic point of
view also the generalised Chaplygin gas is interesting. Further the
generalised Chaplygin gas (GCG) model is modified as Modified Chaplygin gas
(MGCG) model with EoS $p=\mu\rho-\frac{B}{\rho^{\alpha}}$, where $\mu$ is a
positive constant [20]. This modification of GCG to MGCG has been done because
the inferences from GCG models are almost similar to the CDM models [21].

Several authors have studied the evolution of universe with time varying
deceleration parameter in various therious of gravity. Shen and Zhao [22] have
given the model of oscillating universe with quintom matter in the framework
of $FRW$ line element. Shen [23] has investigated the Barber’s second self-
creation theory of gravitation with varying cosmological constant $\Lambda$
and field equations were solved using time periodic varying deceleration
parameter. Actas and Aygun [24], Sahoo et.al. [25], Ahmed and Alamri [26] and
Bharadwaj and Rana [27] have used the time varrying deceleration parameter to
study the evolution of universe in $f(R,T)$ gravity. In view of the above
research, we have analysed the Bianchi type-I Universe with periodic varying
deceleration parameter in presence of generalized Chaplygin gas.

##  2 Field equations

Let us consider the gravitational field of matter distribution represented by
a Bianchi type I space-time as

$ds^{2}=dt^{2}-\left(R_{1}^{2}(t)dx^{2}+R_{2}^{2}(t)dy^{2}+R_{3}^{2}(t)dz^{2}\right).$
(1)

Einstein’s field equations with gravitational and cosmological constant for
perfect fluid distribution are given as

$R_{ij}-\frac{1}{2}Rg_{ij}=-8\pi GT_{ij}+\Lambda g_{ij},$ (2)

where $T_{ij}$ is the energy momentum tensor, which is given as

$T_{ij}=(\rho+p)u_{i}u_{j}-pg_{ij}.$ (3)

Here $\rho$ is the energy density, $p$ represents perfect fluid pressure and
$u^{i}$ is the four velocity vector.

The Einstein’s field equation (2) for the space- time metric (1) yields
following equations

$\frac{\ddot{R_{2}}}{R_{2}}+\frac{\ddot{R_{3}}}{R_{3}}+\frac{\dot{R_{2}}}{R_{2}}\frac{\dot{R_{3}}}{R_{3}}=-8\pi
Gp+\Lambda.$ (4)
$\frac{\ddot{R_{1}}}{R_{1}}+\frac{\ddot{R_{3}}}{R_{3}}+\frac{\dot{R_{1}}}{R_{1}}\frac{\dot{R_{3}}}{R_{3}}=-8\pi
Gp+\Lambda.$ (5)
$\frac{\ddot{R_{1}}}{R_{1}}+\frac{\ddot{R_{2}}}{R_{2}}+\frac{\dot{R_{1}}}{R_{1}}\frac{\dot{R_{2}}}{R_{2}}=-8\pi
Gp+\Lambda.$ (6)
$\frac{\dot{R_{1}}}{R_{1}}\frac{\dot{R_{2}}}{R_{2}}+\frac{\dot{R_{2}}}{R_{2}}\frac{\dot{R_{3}}}{R_{3}}+\frac{\dot{R_{3}}}{R_{3}}\frac{\dot{R_{1}}}{R_{1}}=8\pi
G\rho+\Lambda.$ (7)

By a combination of equations (4)-(7) one can easily obtain

$\dot{\rho}+(\rho+p)\left(\frac{\dot{R_{1}}}{R_{1}}+\frac{\dot{R_{2}}}{R_{2}}+\frac{\dot{R_{3}}}{R_{3}}\right)+\rho\frac{\dot{G}}{G}+\frac{\dot{\Lambda}}{8\pi
G}=0.$ (8)

The energy momentum conservation equation $(T_{;j}^{ij})$ suggests

$\dot{\rho}+(\rho+p)\left(\frac{\dot{R_{1}}}{R_{1}}+\frac{\dot{R_{2}}}{R_{2}}+\frac{\dot{R_{3}}}{R_{3}}\right)=0.$
(9)

From equations (8) and (9) we have

$\rho\frac{\dot{G}}{G}+\frac{\dot{\Lambda}}{8\pi G}=0.$ (10)

Further, equations (4)-(6) yield the solutions

$R_{i}=c_{i}Re^{\frac{k_{i}}{3}\int\frac{1}{R^{3}}dt},$ (11)

Here $k_{i}$ and $c_{i}$ $(i=1,2,3)$ are constants obeying the relation
$\sum_{i=1}^{3}k_{i}=0$ and $\prod_{i=1}^{3}c_{i}=1$. The above results
suggest that the metric potential can be explicitly expresses in terms of
scale factor represented by Bianchi type I space-time.

##  3 Cosmological Solutions

It can be easily seen that we have four equations (4)-(7) with six unknowns
$R_{1}$, $R_{2}$, $R_{3}$, $\rho$, $G$ and $\Lambda$. Hence to solve the
system of equations completely we need two additional physically plausible
relations among these variables. To obtain the cosmological solution we
considered the deceleration parameter of the form

$q=m\cos(nt)-1$ (12)

where $m$ and $n$ are positive constant quantity. This type of deceleration
parameter is known as time periodic varying deceleration parameter (TPVDP).
The deceleration parameter play a crucial role in determining the nature of
the constructed models of our universe i.e. decelerating or accelerating in
nature. According to the value/ranges of $q$ the universe exhibits the
expansion in the following ways

  * •

$q>0$ : Decelerating expansion

  * •

$q=0$ : Expansion with constant rate

  * •

$-1<q<0$ : Accelerating power law expansion

  * •

$q=-1$ : Exponential expansion/de Sitter expansion

  * •

$q<-1$ : Super exponential expansion

From the considered form of $q$ in equation (12), the deceleration parameter
shows periodic nature due to the presence of $\cos(nt)$. The deceleration
parameter lies in the interval $-(m+1)\leq q\leq m-1$. Here we observed that
(i) for $m=0$, the deceleration parameter $q$ is equal to $-1$ and the
universe exhibits exponential expansion/de Sitter expansion. (ii) for
$m\in(0,1)$, the deceleration parameter $q$ becomes negative and leads to
accelerated expansion in a periodic way. (iii) for $m=1$, $q$ lies in the
interval $[-2,0]$. This shows that, the universe evolves from expansion with
constant rate to super exponential expansion in a periodic way followed by
accelerating power law expansion to de Sitter expansion. (iv) $m>1$, phase
transition takes place from decelerating phase to accelerating phase in a
periodic way and the universe start with a decelerating expansion and evolves
to super exponential expansion. The present observational limit of the
considered deceleration parameter $q_{0}=-0.53^{+0.17}_{-0.13}$ suggests the
following values of $m$ and $n$, which is given in Table (3). In order to
obtain the Hubble parameter from equation (12), we used the relation among
Hubble parameter and $q$ as $q=\frac{d}{dt}(\frac{1}{H})-1$, which leads to

$H=\frac{n}{m\sin(nt)+nc_{4}},\;c_{4}\,\text{is the constant of integration}$
(13)

The value of the integration constant $c_{4}$ does not affect the qualitative
behaviour of the Hubble parameter but it effect the scale factor. Thus we have
classified our cosmological models in three different cases as per $c_{4}=0$,
$c_{4}>0$ and $c_{4}<0,\>i.e.\ c_{4}=-c_{5},\;c_{5}>0$.

Computational range of the parameter $m$ for fixed values of $n$, obtained
from the present value of the deceleration parameter.  n Interval of $m$ 0.01
$0.34\leq m\leq 0.64$ 0.02 $0.35\leq m\leq 0.66$ 0.03 $0.37\leq m\leq 0.69$
0.04 $0.39\leq m\leq 0.74$ 0.05 $0.43\leq m\leq 0.82$ 0.06 $0.49\leq m\leq
0.94$ 0.07 $0.59\leq m\leq 1.11$ 0.08 $0.74\leq m\leq 1.39$ 0.09 $1.02\leq
m\leq 1.93$ 0.10 $1.70\leq m\leq 3.20$

Figure 1: Profile of deceleration parameter $q$ against cosmic time for
$0.34\leq m\leq 0.64$ and $n=0.01$.

Figure 2: Profile of deceleration parameter $q$ against cosmic time for
$1.70\leq m\leq 3.20$ and $n=0.10$.

In order to analyse the behaviour of the deceleration parameter $q$, we have
represented graphically in Figures 2 and 2 for different values of $m$ and
$n$. From the figures, it is noticed that (i) for $n=0.01$ and $0.34\leq m\leq
0.64$ our models are accelerating in nature i.e $q<0$ and (ii) for $n=0.10$
and $1.70\leq m\leq 3.20$ our models show phase transition i.e decelerating
phase to accelerating phase. In this situation $q$ take values from positive
to negative. We are mainly focus to investigate the phase transition scenario
so in all our discussed models as a representative case we have considered the
value of $m$ in the range $1.70\leq m\leq 3.20$ and $n=0.10$.

###  3.1 Case-I: $c_{4}=0$

In this case the Hubble parameter in equation (13) leads to

$H=\frac{n}{m\sin(nt)}$ (14)

We know the relationship among Hubble parameter and scale factor as
$H=\frac{\dot{R}}{R}$, which along with (14) leads to the scale factor of the
form

$R=c_{6}\left[\frac{1-\cos(nt)}{\sin(nt)}\right]^{\frac{1}{m}}=c_{6}\left[\tan\left(\frac{nt}{2}\right)\right]^{\frac{1}{m}},\>\>c_{6}\;\text{is
the constant of integration}.$ (15)

Further, equations (11) and (15) yields the following metric potentials as

$R_{i}=c_{i1}\left[\tan\left(\frac{nt}{2}\right)\right]^{\frac{1}{m}}exp\left[{c_{i2}\int\left[\tan\left(\frac{nt}{2}\right)\right]^{-\frac{3}{m}}dt}\right],$
(16)

where $c_{i1}=c_{i}c_{6}$ and $c_{i2}=\frac{k_{i}}{3c_{6}^{3}}$ for $i=1,2,3$.
It is known that ordinary matter fields available from standard model of
particle physics in general relativity, fails to account the present
observations. Therefore modifications of the matter sector of the
Einstein–Hilbert action with exotic matter are considered in the literature.
Chaplygin gas (CG) is considered to be one such candidate for dark energy so
in this case we have considered the equation of state for an exotic background
fluid, the generalized Chaplygin gas, described by equation of state

$p=-\frac{A}{\rho^{\alpha}},\;A>0\>\text{and}\>0\leq\alpha\leq 1$ (17)

The expression for the energy density can be obtained from equations (9),(14)
and (17) as

$\rho=\left[A+\frac{c_{7}}{R^{3(1+\alpha)}}\right]^{\frac{1}{1+\alpha}}=\left[A+\rho_{0}\left(\sin(nt)\right)^{-\frac{3(1+\alpha)}{m}}\left(1+\cos(nt)\right)^{\frac{3(1+\alpha)}{m}}\right]^{\frac{1}{1+\alpha}}$
(18)

where $\rho_{0}=c_{7}c_{6}^{-3(1+\alpha)}$ and $c_{7}$ is the constant of
integration. The directional Hubble parameter $H_{i}$ are given by

$H_{i}=\frac{\dot{R}_{i}}{R_{i}}=\frac{3c_{6}^{3}n+\left(\sin(nt)\right)^{\frac{m-3}{m}}\left(1+\cos(nt)\right)^{\frac{3}{n}}k_{i}m}{3c_{6}^{3}m\sin(nt)},\;i=1,2,3$
(19)

Eqs. (4),(7) , (16) and (17)-(18), we can obtain the expression for
Gravitational constant as

$G=\frac{1}{8\pi(\rho+p)}\left[\frac{k_{4}\left(\sin(nt)\right)^{-\frac{6}{m}}(\cos(nt)-1)\left(\cos(nt)+1\right)^{\frac{m+6}{m}}-18c_{6}^{6}n^{2}\cos(nt)}{9c_{6}^{6}m(\cos^{2}(nt)-1)}\right]$
(20)

where $k_{4}=m(k_{1}k_{2}+k_{1}k_{3}-k_{2}^{2}-k_{3}^{2})$. With the help of
equations (4) and (7), one can obtain the expression for cosmological constant
as

$\Lambda=\frac{1}{\rho+p}\bigg{[}\frac{\begin{multlined}m^{2}\left(\sin(nt)\right)^{-\frac{6}{m}}(\cos(nt)-1)\left(\cos(nt)+1\right)^{\frac{m+6}{m}}(k_{5}\rho+k_{6}p)\\\
-27c_{6}^{6}n^{2}(\rho+p)+18c_{6}^{6}n^{2}m\cos(nt)\rho\end{multlined}m^{2}\left(\sin(nt)\right)^{-\frac{6}{m}}(\cos(nt)-1)\left(\cos(nt)+1\right)^{\frac{m+6}{m}}(k_{5}\rho+k_{6}p)\\\
-27c_{6}^{6}n^{2}(\rho+p)+18c_{6}^{6}n^{2}m\cos(nt)\rho}{9c_{6}^{6}m^{2}(\cos^{2}(nt)-1)}\bigg{]}$
(21)

where $k_{5}=k_{2}k_{3}+k_{2}^{2}+k_{3}^{2}$ and
$k_{6}=k_{1}k_{2}+k_{1}k3+k_{2}k_{3}$.
The physical quantities of the observational interest are expansion scalar
($\Theta$), shear scalar ($\sigma^{2}$) and the anisotropic parameter
($A_{m}$), which are defined as follows:

$\Theta=3H$ (22)
$\sigma^{2}=\frac{1}{2}\left[H_{x}^{2}+H_{y}^{2}+H_{z}^{2}\right]-\frac{\Theta^{2}}{6}$
(23) $A_{m}=\frac{1}{3}\sum_{i=1}^{3}\left(\frac{H_{i}-H}{H}\right)^{2}$ (24)

In this case, the physical quantities are obtained as

$\Theta=\frac{3n}{m\sin(nt)}$ (25)
$\sigma^{2}=\frac{(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})\left[1+\cos(nt)\right]^{\frac{6}{m}}\left[\sin(nt)\right]^{\frac{-6+2m}{m}}}{18c_{6}^{6}(1-\cos^{2}(nt))}=\frac{k_{1}^{2}+k_{2}^{2}+k_{3}^{2}}{18c_{6}^{6}}\left[\frac{1+\cos(nt)}{\sin(nt)}\right]^{\frac{6}{m}}$
(26)
$A_{m}=\frac{(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})m^{2}\left[1+\cos(nt)\right]^{\frac{6}{m}}\left[\sin(nt)\right]^{\frac{-6+2m}{m}}}{27c_{6}^{6}n^{2}}$
(27)

In this case the state finder parameters are defined and expressed as

$r=\frac{\dddot{R}}{RH^{3}}=m^{2}\cos^{2}(nt)-3m\cos(nt)+1+m^{2}$ (28)
$s=\frac{r-1}{3(q-0.5)}=\frac{2m(m\cos^{2}(nt)-3\cos(nt)+m)}{3(2m\cos(nt)-3)}$
(29)

The relationship among $r$ and $s$ is given by

$r=1+\frac{9}{2}s^{2}\pm\frac{3}{2}s\sqrt{9+9s^{2}-4m^{2}}$ (30)

In terms of the deceleration parameter the state finder parameters are given
as

$r=-1-q+q^{2}+m^{2}$ (31) $s=\frac{2(-2-q+q^{2}+m^{2})}{3(-1+2q)}$ (32)

The profile of scale factor $R$, energy density $\rho$, pressure $p$,
cosmological constant $\Lambda$ and gravitational constant $G$ against time is
presented in the Figure 3 to Figure 7 respectively for fix $n$ and different
values of $m$ with suitable choice of arbitrary constant involve in the
expressions of the physical quantities. The scale factor is increasing with
the evolvement of time in the provided range of time, which can be seen from
the Figure 3. However, the qualitative behaviour is similar to $\tan$ function
due to the presence of $\tan\left(\frac{nt}{2}\right)$ in equation (15) and
$R\rightarrow\infty$ at $t=\frac{(2n_{1}+1)\pi}{n},\;n_{1}\in\textbf{Z}$.

{r,s}=(1,0) corresponding to the different values of $m$ and $q$.  $q$ $m$
{r,s} pair -1.00 0.000 (1.000,0) -0.80 0.748 (1.0,2.846726e-017)$\approx$(1,0)
-0.60 1.020 (1.0,0) -0.40 1.200 (1.0,-8.223874e-017)$\approx$ (1,0) -0.20
1.327 (1.0,1.057355e-016)$\approx$ (1,0) 0.00 1.414
(1.0,-2.960595e-016)$\approx$ (1,0) 0.20 1.470 (1.0,0) 0.40 1.497 (1.0,0) 0.60
1.497 (1.0,0) 0.80 1.470 (1.0,0) 1.00 1.414 (1.0,2.960595e-016)$\approx$ (1,0)

Figure 3: Profile of Scale factor parameter $R$ against cosmic time for
different $m$ and $n$.

Figure 4: Profile of energy density $\rho$ against cosmic time for different
$m$, $n=0.1$, $\alpha=0.5$, $\rho_{0}=1$ and $A=1$.

Figure 5: Profile of pressure $p$ against cosmic time for different $m$,
$n=0.1$, $\alpha=0.5$, $\rho_{0}=1$ and $A=1$.

Figure 6: Profile of cosmological constant $\Lambda$ against cosmic time for
different $m$, $n=0.1$, $\alpha=0.5$, and $A=1$.

Figure 7: Profile of Gravitational constant $G$ against cosmic time for
different $m$, $n=0.1$, $\alpha=0.5$ and $A=1$.

The energy density,pressure,cosmological constant and gravitational constant
are periodic in nature, which is noticed from the Figure 5 to Figure 7
respectively due to the presence of $\cos(nt)$ and $\sin(nt)$ terms in the
expressions of these physical quantities. Here also we noted that
$\rho,p,\Lambda,G\rightarrow\infty$ at $t=\frac{n_{2}\pi}{n},\;\forall
n_{2}\in\textbf{Z}$. The energy density is positive where as pressure is
negative for different values of $m$ with evolvement of time. In this case,
the qualitative behavior of energy density follow the pattern of higher energy
density value to lower energy density value (approaching to zero) to higher
energy density value. This process will continue due to the periodic nature of
the terms $\cos(nt)$ and $\sin(nt)$ involve in the expression of energy
density (18). Again it is also pointed out that, cosmological constant is
positive for some values of $m$ and positive to negative values for some $m$
(See Figure 7). Gravitational constant takes values from negative to positive
values and positive to negative values in a periodic way for different values
of $m$ (See Figure 7). Further, it is noticed that the expansion scalar, shear
scalar and anisotropy parameter have also singularity at
$t=\frac{n_{2}\pi}{n},\;\forall n_{2}\in\textbf{Z}$ and they behaves
periodically.

###  3.2 Case-II:$c_{4}>0$

In this case the Hubble parameter in equation (13) takes the form

$H=\frac{n}{m\sin(nt)+nc_{4}}$ (33)

We know the relationship among Hubble parameter and scale factor as
$H=\frac{\dot{R}}{R}$, which along with (33) leads to the scale factor of the
form

$R=c_{8}e^{\frac{2}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)},\>\>c_{8}\;\text{is
the constant of integration}.$ (34)

Further, equations (11) and (34) yields the following metric potentials as

$R_{i}=c_{i3}e^{\frac{2\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}}exp\left[c_{i4}\int
e^{-\frac{6\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}}dt\right],\;i=1,2,3$
(35)

where $c_{i3}=c_{i}c_{8}$ and $c_{i4}=\frac{k_{i}}{3c_{8}^{3}}\>(i=1,2,3)$.
The directional Hubble parameters are expressed as

$H_{i}=\frac{\left[3nc_{8}^{3}e^{\frac{6\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}}+nc_{4}k_{i}+mk_{i}\sin(nt)\right]e^{-\frac{6\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}}}{3c_{8}^{3}(nc_{4}+m\sin(nt))}$
(36)

The expression for the energy density can be obtained from equations (9) and
(34) as

$\rho=\left[A+\frac{c_{7}}{R^{3(1+\alpha)}}\right]^{\frac{1}{1+\alpha}}=\left[A+\rho_{1}e^{-\frac{6(1+\alpha)\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}}\right]^{\frac{1}{1+\alpha}}$
(37)

where $\rho_{1}=c_{7}c_{8}^{-3(1+\alpha)}$. The expression for gravitational
and cosmological constant are obtained as

$G=\frac{\begin{multlined}-9n^{2}mc_{8}^{6}\left(\cos^{2}\left(\frac{nt}{2}\right)-\frac{1}{2}\right)e^{\frac{12\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}}+\bigg{(}m^{2}\cos^{4}\left(\frac{nt}{2}\right)\\\
-m^{2}\cos^{2}\left(\frac{nt}{2}\right)-nmc_{4}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{4}^{2}}{4}\end{multlined}-9n^{2}mc_{8}^{6}\left(\cos^{2}\left(\frac{nt}{2}\right)-\frac{1}{2}\right)e^{\frac{12\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}}+\bigg{(}m^{2}\cos^{4}\left(\frac{nt}{2}\right)\\\
-m^{2}\cos^{2}\left(\frac{nt}{2}\right)-nmc_{4}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{4}^{2}}{4}\bigg{)}k_{7}}{72\pi
c_{8}^{6}(\rho+p)\left(\begin{multlined}m^{2}\cos^{4}\left(\frac{nt}{2}\right)-m^{2}\cos^{2}\left(\frac{nt}{2}\right)\\\
-nmc_{4}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{4}^{2}}{4}\end{multlined}m^{2}\cos^{4}\left(\frac{nt}{2}\right)-m^{2}\cos^{2}\left(\frac{nt}{2}\right)\\\
-nmc_{4}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{4}^{2}}{4}\right)e^{\frac{12\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}}}$
(38)

where $k_{7}=\frac{k_{4}}{m}$.

$\Lambda=\frac{\begin{multlined}9n^{2}c_{8}^{6}\left(\rho
m\left(\cos^{2}\left(\frac{nt}{2}\right)-\frac{1}{2}\right)-\frac{3}{4}(\rho+p)\right)e^{\frac{12\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}}\\\
+\bigg{(}m^{2}\cos^{4}\left(\frac{nt}{2}\right)-m^{2}\cos^{2}\left(\frac{nt}{2}\right)-nmc_{4}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{4}^{2}}{4}\bigg{)}(k_{5}\rho+k_{6}p)\end{multlined}9n^{2}c_{8}^{6}\left(\rho
m\left(\cos^{2}\left(\frac{nt}{2}\right)-\frac{1}{2}\right)-\frac{3}{4}(\rho+p)\right)e^{\frac{12\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}}\\\
+\bigg{(}m^{2}\cos^{4}\left(\frac{nt}{2}\right)-m^{2}\cos^{2}\left(\frac{nt}{2}\right)-nmc_{4}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{4}^{2}}{4}\bigg{)}(k_{5}\rho+k_{6}p)}{9c_{8}^{6}(\rho+p)\left(\begin{multlined}m^{2}\cos^{4}\left(\frac{nt}{2}\right)-m^{2}\cos^{2}\left(\frac{nt}{2}\right)\\\
-nmc_{4}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{4}^{2}}{4}\end{multlined}m^{2}\cos^{4}\left(\frac{nt}{2}\right)-m^{2}\cos^{2}\left(\frac{nt}{2}\right)\\\
-nmc_{4}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{4}^{2}}{4}\right)e^{\frac{12\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}}}$
(39)

In this case, the physical quantities are obtained as

$\Theta=\frac{3n}{m\sin(nt)+nc_{4}}$ (40)
$\sigma^{2}=\frac{(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})}{18c_{8}^{6}}e^{-\frac{12\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}}$
(41)
$A_{m}=-\frac{(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})(m^{2}\cos^{2}(nt)-2nmc_{4}\sin(nt)-m^{2}-n^{2}c_{4}^{2})}{27n^{2}c_{8}^{6}}e^{-\frac{12\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}}$
(42)

In this case the state finder parameters are defined and expressed as

$r=\frac{\dddot{R}}{RH^{3}}=-\frac{(m\sin(nt)+nc_{4})^{3}\bigg{(}\begin{multlined}1+3m+2m^{2}+nmc_{4}\sin(nt)\\\
+4m^{2}\cos^{4}\left(\frac{nt}{2}\right)-(4m^{2}+6m)\cos^{2}\left(\frac{nt}{2}\right)\bigg{)}\end{multlined}1+3m+2m^{2}+nmc_{4}\sin(nt)\\\
+4m^{2}\cos^{4}\left(\frac{nt}{2}\right)-(4m^{2}+6m)\cos^{2}\left(\frac{nt}{2}\right)\bigg{)}}{\begin{multlined}\left(12nm^{2}c_{4}+4m^{3}\sin(nt)\right)\cos^{4}\left(\frac{nt}{2}\right)-\left(12nm^{2}c_{4}+4m^{3}\sin(nt)\right)\cos^{2}\left(\frac{nt}{2}\right)\\\
-3n^{2}mc_{4}^{2}\sin(nt)-n^{3}c_{4}^{3}\end{multlined}\left(12nm^{2}c_{4}+4m^{3}\sin(nt)\right)\cos^{4}\left(\frac{nt}{2}\right)-\left(12nm^{2}c_{4}+4m^{3}\sin(nt)\right)\cos^{2}\left(\frac{nt}{2}\right)\\\
-3n^{2}mc_{4}^{2}\sin(nt)-n^{3}c_{4}^{3}}$ (43) $\displaystyle s$
$\displaystyle=$
$\displaystyle\frac{r-1}{3(q-0.5)}=-\frac{2}{3(2m\cos(nt)-3)}$ (48)
$\displaystyle\times$
$\displaystyle\left[\frac{(m\sin(nt)+nc_{4})^{3}\bigg{(}\begin{multlined}1+3m+2m^{2}+nmc_{4}\sin(nt)\\\
+4m^{2}\cos^{4}\left(\frac{nt}{2}\right)-(4m^{2}+6m)\cos^{2}\left(\frac{nt}{2}\right)\bigg{)}\end{multlined}1+3m+2m^{2}+nmc_{4}\sin(nt)\\\
+4m^{2}\cos^{4}\left(\frac{nt}{2}\right)-(4m^{2}+6m)\cos^{2}\left(\frac{nt}{2}\right)\bigg{)}}{\begin{multlined}\left(12nm^{2}c_{4}+4m^{3}\sin(nt)\right)\cos^{4}\left(\frac{nt}{2}\right)-\left(12nm^{2}c_{4}+4m^{3}\sin(nt)\right)\cos^{2}\left(\frac{nt}{2}\right)\\\
-3n^{2}mc_{4}^{2}\sin(nt)-n^{3}c_{4}^{3}\end{multlined}\left(12nm^{2}c_{4}+4m^{3}\sin(nt)\right)\cos^{4}\left(\frac{nt}{2}\right)-\left(12nm^{2}c_{4}+4m^{3}\sin(nt)\right)\cos^{2}\left(\frac{nt}{2}\right)\\\
-3n^{2}mc_{4}^{2}\sin(nt)-n^{3}c_{4}^{3}}+1\right]$

In terms of the deceleration parameter the state finder parameters are given
as

$r=\frac{\left(\sqrt{m^{2}-1-2q-q^{2}}+nc_{4}\right)^{3}\left(m^{2}-1-q+q^{2}+nc_{4}\sqrt{m^{2}-1-2q-q^{2}}\right)}{\left(m^{2}-1-2q-q^{2}+3n^{2}c_{4}^{2}\right)\sqrt{m^{2}-1-2q-q^{2}}+3nc_{4}\left(m^{2}-1-2q-q^{2}+\frac{n^{2}c_{4}^{2}}{3}\right)}$
(49) $\displaystyle s$ $\displaystyle=$
$\displaystyle\frac{2\left(\sqrt{m^{2}-1-2q-q^{2}}+nc_{4}\right)^{3}\left(m^{2}-1-q+q^{2}+nc_{4}\sqrt{m^{2}-1-2q-q^{2}}\right)}{3(2q-1)\left(\left(m^{2}-1-2q-q^{2}+3n^{2}c_{4}^{2}\right)\sqrt{m^{2}-1-2q-q^{2}}+3nc_{4}\left(m^{2}-1-2q-q^{2}+\frac{n^{2}c_{4}^{2}}{3}\right)\right)}$
(50) $\displaystyle-$ $\displaystyle\frac{2}{3(2q-1)}$

Figure 8 indicates the profile of scale factor against the cosmic time. Here
we noticed that, the scale factor is increasing with respect to cosmic time in
a periodic way.

Figure 10 and Figure 10 are representing the profile of pressure and energy
density. As per the evolution of time, energy density and pressure are
positive and negative respectively in a periodic way for different values of
m.

Figure 12 and Figure 12 shows the qualitative behaviour of gravitational
constant and cosmological constant with respect to cosmic time. We noticed
that, both behaves in a periodic way with time and also $\Lambda>0$ and $G<0$
with the evolution of cosmic time. $\Lambda>0$ and $G<0$ are free from the
initial singularity.

Figure 14 and Figure 14 shows the qualitative behaviour of shear scalar and
anisotropic parameter respect to cosmic time. Both the parameters behave in a
periodic way with time and positive with the evolution of cosmic time.

Figure 8: Profile of scale factor $R$ against cosmic time for different $m$

Figure 9: Profile of pressure $p$ against cosmic time for $1.70\leq m\leq
3.20$ and $n=0.10$.

Figure 10: Profile of energy density $\rho$ against cosmic time for $1.70\leq
m\leq 3.20$ and $n=0.10$.

Figure 11: Profile of Gravitational constant $G$ against cosmic time for
$1.70\leq m\leq 3.20$,$c_{7}=c_{8}=A=1$,$\alpha=0.5$,$c_{4}=55$ and $n=0.10$.

Figure 12: Profile of Cosmological constant $\Lambda$ against cosmic time for
$1.70\leq m\leq 3.20$,$c_{7}=c_{8}=A=1$,$\alpha=0.5$,$c_{4}=55$ and $n=0.10$.

Figure 13: Profile of shear scalar $\sigma^{2}$ against cosmic time for
$1.70\leq m\leq 3.20$,$c_{7}=c_{8}=A=1$,$\alpha=0.5$,$c_{4}=55$ and $n=0.10$.

Figure 14: Profile of anisotropic parameter $A_{m}$ against cosmic time for
$1.70\leq m\leq 3.20$,$c_{7}=c_{8}=A=1$,$\alpha=0.5$,$c_{4}=55$ and $n=0.10$.

$\\{r,s\\}$ pair for different model parameters with fixed $q$ $q$ $n$ $m$
$c_{4}$ $r$ $s$ -0.5 0.02 0.5358059184 250 1.000000001$\approx$ 1
-3.33333333$\times 10^{-10}\approx 0$ -0.5 0.04 0.5097105689 250
0.9999999951$\approx$ 1 1.633333333$\times 10^{-9}\approx 0$ -0.5 0.06
0.5043861337 250 1.000000008$\approx$ 1 -2.666666667$\times 10^{-9}\approx 0$
-0.5 0.08 0.8416493515 10 1.000000001 $\approx$ 1 -3.333333333$\times
10^{-10}\approx 0$

###  3.3 Case 3: $c_{4}<0\;\text{i.e.\ }c_{4}=-c_{5},\;c_{5}>0$

In this case the Hubble parameter in (13) takes the form

$H=\frac{n}{m\sin(nt)-nc_{5}}$ (51)

We know the relationship among Hubble parameter and scale factor as
$H=\frac{\dot{R}}{R}$, which along with (51) leads to the scale factor of the
form

$R=c_{9}e^{-\frac{2}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\arctan\left(\frac{nc_{5}\tan\left(\frac{nt}{2}\right)-m}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)},\>\>c_{9}\;\text{is
the constant of integration}.$ (52)

Further, equations ((11)) and ((52)) yields the following metric potentials as

$R_{i}=c_{i5}e^{-\frac{2\arctan\left(\frac{nc_{5}\tan\left(\frac{nt}{2}\right)-m}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}exp\left[c_{i6}\int
e^{\frac{6\arctan\left(\frac{nc_{5}\tan\left(\frac{nt}{2}\right)-m}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}dt\right]$
(53)

where $c_{i5}=c_{i}c_{9}$ and $c_{i6}=\frac{k_{i}}{3c_{9}^{3}}\>(i=1,2,3)$.
The directional Hubble parameters are expressed as

$H_{i}=\frac{\left[3nc_{9}^{3}e^{-\frac{6\arctan\left(\frac{nc_{5}\tan\left(\frac{nt}{2}\right)-m}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}-nc_{5}k_{i}+mk_{i}\sin(nt)\right]e^{\frac{6\arctan\left(\frac{nc_{5}\tan\left(\frac{nt}{2}\right)-m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}}{3c_{9}^{3}(m\sin(nt)-nc_{5})}$
(54)

The expression for the energy density can be obtained from equations (9) and
(21) as

$\rho=\left[A+\frac{c_{7}}{R^{3(1+\alpha)}}\right]^{\frac{1}{1+\alpha}}=\left[A+\rho_{2}e^{\frac{6(1+\alpha)\arctan\left(\frac{nc_{5}\tan\left(\frac{nt}{2}\right)-m}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}\right]^{\frac{1}{1+\alpha}}$
(55)

where $\rho_{2}=c_{7}c_{9}^{-3(1+\alpha)}$.

$G=\frac{\begin{multlined}-9n^{2}mc_{9}^{6}\left(\cos^{2}\left(\frac{nt}{2}\right)-\frac{1}{2}\right)e^{\frac{12\arctan\left(\frac{m-nc_{5}\tan\left(\frac{nt}{2}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}+\bigg{(}m^{2}\cos^{4}\left(\frac{nt}{2}\right)\\\
-m^{2}\cos^{2}\left(\frac{nt}{2}\right)+nmc_{5}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{5}^{2}}{4}\end{multlined}-9n^{2}mc_{9}^{6}\left(\cos^{2}\left(\frac{nt}{2}\right)-\frac{1}{2}\right)e^{\frac{12\arctan\left(\frac{m-nc_{5}\tan\left(\frac{nt}{2}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}+\bigg{(}m^{2}\cos^{4}\left(\frac{nt}{2}\right)\\\
-m^{2}\cos^{2}\left(\frac{nt}{2}\right)+nmc_{5}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{5}^{2}}{4}\bigg{)}k_{7}}{72\pi
c_{9}^{6}(\rho+p)\left(\begin{multlined}m^{2}\cos^{4}\left(\frac{nt}{2}\right)-m^{2}\cos^{2}\left(\frac{nt}{2}\right)\\\
+nmc_{5}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{5}^{2}}{4}\end{multlined}m^{2}\cos^{4}\left(\frac{nt}{2}\right)-m^{2}\cos^{2}\left(\frac{nt}{2}\right)\\\
+nmc_{5}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{5}^{2}}{4}\right)e^{\frac{12\arctan\left(\frac{m-nc_{5}\tan\left(\frac{nt}{2}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}}$
(56)

where $k_{7}=\frac{k_{4}}{m}$.

$\Lambda=\frac{\begin{multlined}9n^{2}c_{9}^{6}\left(\rho
m\left(\cos^{2}\left(\frac{nt}{2}\right)-\frac{1}{2}\right)-\frac{3}{4}(\rho+p)\right)e^{\frac{12\arctan\left(\frac{m-nc_{5}\tan\left(\frac{nt}{2}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}\\\
+\bigg{(}m^{2}\cos^{4}\left(\frac{nt}{2}\right)-m^{2}\cos^{2}\left(\frac{nt}{2}\right)+nmc_{5}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{5}^{2}}{4}\bigg{)}(k_{5}\rho+k_{6}p)\end{multlined}9n^{2}c_{9}^{6}\left(\rho
m\left(\cos^{2}\left(\frac{nt}{2}\right)-\frac{1}{2}\right)-\frac{3}{4}(\rho+p)\right)e^{\frac{12\arctan\left(\frac{m-nc_{5}\tan\left(\frac{nt}{2}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}\\\
+\bigg{(}m^{2}\cos^{4}\left(\frac{nt}{2}\right)-m^{2}\cos^{2}\left(\frac{nt}{2}\right)+nmc_{5}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{5}^{2}}{4}\bigg{)}(k_{5}\rho+k_{6}p)}{9c_{9}^{6}(\rho+p)\left(\begin{multlined}m^{2}\cos^{4}\left(\frac{nt}{2}\right)-m^{2}\cos^{2}\left(\frac{nt}{2}\right)\\\
+nmc_{5}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{5}^{2}}{4}\end{multlined}m^{2}\cos^{4}\left(\frac{nt}{2}\right)-m^{2}\cos^{2}\left(\frac{nt}{2}\right)\\\
+nmc_{5}\sin\left(\frac{nt}{2}\right)\cos\left(\frac{nt}{2}\right)-\frac{n^{2}c_{5}^{2}}{4}\right)e^{\frac{12\arctan\left(\frac{m-nc_{5}\tan\left(\frac{nt}{2}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}}$
(57)

In this case, the physical quantities are obtained as

$\Theta=\frac{3n}{m\sin(nt)-nc_{5}}$ (58)
$\sigma^{2}=\frac{(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})}{18c_{9}^{6}}e^{-\frac{12\arctan\left(\frac{m-nc_{5}\tan\left(\frac{nt}{2}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}$
(59)
$A_{m}=\frac{\begin{multlined}27n^{2}m^{2}c_{9}^{6}\left(-2\tan\left(\frac{nt}{2}\right)+\tan^{2}\left(\frac{nt}{2}\right)\sin(nt)+\sin(nt)\right)^{2}e^{-\frac{12\arctan\left(\frac{nc_{5}\tan\left(\frac{nt}{2}\right)-m}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}\\\
+4\left(-\frac{nc_{5}}{2}-\frac{nc_{5}\tan\left(\frac{nt}{2}\right)}{2}+m\tan\left(\frac{nt}{2}\right)\right)^{2}(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})(m\sin(nt)-nc_{5})^{2}e^{\frac{12\arctan\left(\frac{nc_{5}\tan\left(\frac{nt}{2}\right)-m}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}\end{multlined}27n^{2}m^{2}c_{9}^{6}\left(-2\tan\left(\frac{nt}{2}\right)+\tan^{2}\left(\frac{nt}{2}\right)\sin(nt)+\sin(nt)\right)^{2}e^{-\frac{12\arctan\left(\frac{nc_{5}\tan\left(\frac{nt}{2}\right)-m}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}\\\
+4\left(-\frac{nc_{5}}{2}-\frac{nc_{5}\tan\left(\frac{nt}{2}\right)}{2}+m\tan\left(\frac{nt}{2}\right)\right)^{2}(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})(m\sin(nt)-nc_{5})^{2}e^{\frac{12\arctan\left(\frac{nc_{5}\tan\left(\frac{nt}{2}\right)-m}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}}{27n^{2}c_{9}^{6}\left(-nc_{5}-nc_{5}\tan^{2}\left(\frac{nt}{2}\right)+2m\tan\left(\frac{nt}{2}\right)\right)^{2}}$
(60)

In this case the state finder parameters are defined and expressed as

$r=\frac{\dddot{R}}{RH^{3}}=\frac{(m\sin(nt)-nc_{5})^{3}\left(\begin{multlined}-4m^{2}\cos^{4}\left(\frac{nt}{2}\right)+(6m+4m^{2})\cos^{2}\left(\frac{nt}{2}\right)\\\
+nmc_{5}\sin(nt)-3m-2m^{2}-1\end{multlined}-4m^{2}\cos^{4}\left(\frac{nt}{2}\right)+(6m+4m^{2})\cos^{2}\left(\frac{nt}{2}\right)\\\
+nmc_{5}\sin(nt)-3m-2m^{2}-1\right)}{4m^{2}(m\sin(nt)-3nc_{5})\cos^{2}\left(\frac{nt}{2}\right)\left(\cos^{2}\left(\frac{nt}{2}\right)-1\right)+n^{3}c_{5}^{3}-3mn^{2}c_{5}^{2}\sin(nt)}$
(61) $\displaystyle s$ $\displaystyle=$
$\displaystyle\frac{r-1}{3(q-0.5)}=\frac{2}{3(2m\cos(nt)-3)}$ (64)
$\displaystyle\times$
$\displaystyle\left[\frac{(m\sin(nt)-nc_{5})^{3}\left(\begin{multlined}-4m^{2}\cos^{4}\left(\frac{nt}{2}\right)+(6m+4m^{2})\cos^{2}\left(\frac{nt}{2}\right)\\\
+nmc_{5}\sin(nt)-3m-2m^{2}-1\end{multlined}-4m^{2}\cos^{4}\left(\frac{nt}{2}\right)+(6m+4m^{2})\cos^{2}\left(\frac{nt}{2}\right)\\\
+nmc_{5}\sin(nt)-3m-2m^{2}-1\right)}{4m^{2}(m\sin(nt)-3nc_{5})\cos^{2}\left(\frac{nt}{2}\right)\left(\cos^{2}\left(\frac{nt}{2}\right)-1\right)+n^{3}c_{5}^{3}-3mn^{2}c_{5}^{2}\sin(nt)}-1\right]$

In terms of the deceleration parameter the state finder parameters are given
as

$r=\frac{\left(\sqrt{m^{2}-1-2q-q^{2}}-nc_{5}\right)^{3}\left(m^{2}-1-q+q^{2}-nc_{5}\sqrt{m^{2}-1-2q-q^{2}}\right)}{\left(m^{2}-1-2q-q^{2}+3n^{2}c_{5}^{2}\right)\sqrt{m^{2}-1-2q-q^{2}}-3nc_{5}\left(m^{2}-1-2q-q^{2}+\frac{n^{2}c_{5}^{2}}{3}\right)}$
(65) $\displaystyle s$ $\displaystyle=$ $\displaystyle\frac{2}{3(2q-1)}\times$
$\displaystyle\left[\frac{\left(\sqrt{m^{2}-1-2q-q^{2}}-nc_{5}\right)^{3}\left(m^{2}-1-q+q^{2}-nc_{5}\sqrt{m^{2}-1-2q-q^{2}}\right)}{\left(m^{2}-1-2q-q^{2}+3n^{2}c_{5}^{2}\right)\sqrt{m^{2}-1-2q-q^{2}}-3nc_{5}\left(m^{2}-1-2q-q^{2}+\frac{n^{2}c_{5}^{2}}{3}\right)}-1\right]$

Figure 16 and Figure 16 indicates the profile of Hubble parameter against
cosmic time for different values of $c_{4}$. Similar qualitative behaviour as
that of case-II is observed for physical parameters likes energy
density[Figure 18], pressure[Figure 18], gravitational constant [Figure 20],
cosmological constant [Figure 20], shear scalar [Figure 22] and anisotropic
parameter [Figure 22].

Further, we have analyzed whether these derived models are approaching
$\Lambda$CDM model or not for the computational range of $n$ and $m$ (Table
3). From equations (31),(32),(49), (50), (65) and (3.3), we have evaluated the
$\\{r,s\\}$ pair for the different model parameters, which are presented in
Table 3.1 to Table 3.3 respectively for three cases. Here we have noticed that
model discussed in case-I approaches to $\Lambda$CDM model for different value
of $q$ and $m$. At current value of $q=-\frac{1}{2}$, the model in case-I
approaches to $\Lambda$CDM model for $m=\frac{\sqrt{5}}{2}$. The models in
case-II and Case-III, the $\\{r,s\\}$ pair depends on the model parameters
$n$, $m$, $c_{4}$,$c_{5}$ and $q$. At current value of $q=-\frac{1}{2}$, the
model in case-II approaches to $\Lambda$CDM model whereas model in Case-III
does not approaches to $\Lambda$CDM model in view of the values of $n$ and $m$
presented in Table 3.

Figure 15: Profile of Hubble parameter $H$ against cosmic time for $m=2.2$,
$n=0.1$ and different $c_{4}$.

Figure 16: Profile of Hubble parameter $H$ against cosmic time for $m=0.5$,
$n=0.01$ and different $c_{4}$.

Figure 17: Profile of energy density $\rho$ against cosmic time for $1.70\leq
m\leq 3.20$ and $n=0.10$.

Figure 18: Profile of pressure $p$ against cosmic time for $1.70\leq m\leq
3.20$ and $n=0.10$.

Figure 19: Profile of Gravitational constant $G$ against cosmic time for
$1.70\leq m\leq 3.20$,$c_{7}=c_{8}=A=1$,$\alpha=0.5$,$c_{4}=55$ and $n=0.10$.

Figure 20: Profile of Cosmological constant $\Lambda$ against cosmic time for
$1.70\leq m\leq 3.20$,$c_{7}=c_{8}=A=1$,$\alpha=0.5$,$c_{4}=55$ and $n=0.10$.

Figure 21: Profile of shear scalar $\sigma^{2}$ against cosmic time for
$1.70\leq m\leq 3.20$,$c_{7}=c_{8}=A=1$,$\alpha=0.5$,$c_{4}=55$ and $n=0.10$.

Figure 22: Profile of anisotropic parameter $A_{m}$ against cosmic time for
$1.70\leq m\leq 3.20$,$c_{7}=c_{8}=A=1$,$\alpha=0.5$,$c_{4}=55$ and $n=0.10$.

$\\{r,s\\}$ pair for different model parameters with fixed $q$ $q$ $n$ $m$
$c_{5}$ $r$ $s$ -0.5 0.02 1.127031913 1 1.000000001$\approx 1$
-3.333333333$\times 10^{-10}\approx 0$ -0.5 0.04 1.136137315 1 1.020403999
-0.006801333000 -0.5 0.06 1.144350162 1 1.041217996 -0.01373933200 -0.5 0.08
1.154670505 1 1.062447982 -0.01373933200

##  4 Stability and physical acceptability of the solutions

###  4.1 The squared sound speed

Here we introduced the adiabatic squared sound speed for the stability of our
system. It is one of the important quantity in cosmology. The squared of sound
speed, for any fluid is defined as

$c_{s}^{2}=\frac{\partial p}{\partial\rho},$ (67)

here $c_{s}^{2}$ has three possibilities i.e $c_{s}^{2}<0$ or $c_{s}^{2}=0$ or
$c_{s}^{2}>0$. The sign of $c_{s}^{2}$ is very important to investigate as it
leads to the instability of the cosmological models through which one can
reject or accept the constructed cosmological models. The case when,
$c_{s}^{2}<0$, leads to classical instability of the cosmological models due
to the uncontrolled grow of the energy density perturbation. The case when
$c_{s}^{2}>0$ may leads to the issue of occurrence of casuality. As a matter
of fact, it is usually considered as $c_{s}\leq 1$ and the bound on
$c_{s}^{2}$ is $0\leq c_{s}^{2}\leq 1$. In addition to that, the complementary
bound $c_{s}>1$ is used as a condition for rejecting the theories. The details
regarding $c_{s}^{2}$ can be found in [22].

In our present study, $c_{s}^{2}$ is obtained as

$c_{s}^{2}=\frac{\partial
p}{\partial\rho}=\frac{A\alpha}{\rho^{1+\alpha}}=\begin{cases}\frac{A\alpha}{A+\rho_{0}\left(\sin(nt)\right)^{-\frac{3(1+\alpha)}{m}}\left(1+\cos(nt)\right)^{\frac{3(1+\alpha)}{m}}},&\text{for
}c_{4}=0\\\\[20.0pt]
\frac{A\alpha}{A+\rho_{1}e^{-\frac{6(1+\alpha)\arctan\left(\frac{nc_{4}\tan\left(\frac{nt}{2}\right)+m}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{4}^{2}-m^{2}}}}},&\text{for}\;c_{4}>0\\\\[20.0pt]
\frac{A\alpha}{A+\rho_{2}e^{\frac{6(1+\alpha)\arctan\left(\frac{nc_{5}\tan\left(\frac{nt}{2}\right)-m}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}\right)}{\sqrt{n^{2}c_{5}^{2}-m^{2}}}}},&\text{for}\;c_{4}<0\end{cases}$
(68)

In the Figure 23, we have presented the profile of squared of sound speed
against time for different values of $m$. It is observed from the Figure 23
that, $c_{s}^{2}$ is periodic in nature and $0\leq c_{s}^{2}<0.6$ in all the
cases. Thus in account of the prescribe range of $c_{s}^{2}$, the solution
presented here are stable.

Figure 23: Profile of squared of sound speed against time for three different
cases: $c_{4}=0$ (Left),$c_{4}>0$ (Right) and $c_{4}<0$ (Middle).

###  4.2 Energy conditions

The role of energy conditions (ECs) can not be ignored in defining the
cosmological evolution. The energy conditions are contractions of time like or
null vector field with respect to the Einstein’s tensor and energy momentum
tensor coming from Einstein’s field equations. Energy conditions can be
imposed in order to investigate the constraints to the free parameters
involved in the cosmological models. For example the evolution of acceleration
or deceleration of cosmic fluid and the emergence of Big Rip singularities,
can be related to the constraints imposed by the energy conditions. In our
study,we have investigated the four ECs namely

  * •

Null energy condition (NEC): $\rho+p\geq 0$

  * •

Weak energy condition (WEC): $\rho\geq 0$ and $\rho+p\geq 0$

  * •

Strong energy condition (SEC): $\rho+p\geq 0$ and $\rho+3p\geq 0$

  * •

Dominant energy condition (DEC): $\rho\geq 0$ and $\rho\pm p\geq 0$

It is important to note that,the above ECs are formulated from the
Raychaudhuri equation. We have used the above ECs to investigate the energy
conditions in this discussed gravity. Cosmological model should satisfy WEC
and SEC and violets SEC for late time accelerated expansion of the universe.
For each of the three cases, Figures (24), (25) and (26) represents the
profile of WEC, DEC and SEC respectively versus cosmic time. From these
figures it can seen that profile of energy conditions follows periodic
variation for each of the cases. Considered model satisfies WEC and DEC
whereas SEC is satisfied and violated periodically throughout the evolution of
the universe.

Figure 24: Profile of energy conditions (WEC(Left), DEC (Middle), SEC (Right))
against time for case-I.

Figure 25: Profile of energy conditions (WEC(Left), DEC (Middle), SEC (Right))
against time for case-II.

Figure 26: Profile of energy conditions (WEC(Left), DEC (Middle), SEC (Right))
against time for case-III.

##  5 Outlook

In this work we have considered Bianchi type-I cosmological models with
generalised Chaplygin gas EoS represented by equation (17) and cosmological
solutions are obtained by using time periodic varying deceleration parameter
represented by equation (12). By fixing the value of constant parameter $n$,
we have computed the range of parameter $m$ (see Table-3 for present
observational value (range) of deceleration parameter. Figures 2 and 2 shows
the variation of deceleration parameter with respect to cosmic time for
$n=0.01$ and $n=0.10$ and corresponding range of $m$. For $n=0.01$
deceleration parameter is negative throughout the evolution of the universe
whereas for $n=0.10$ it shows transition from decelerated phase to accelerated
phase periodically throughout the evolution of the universe. Hubble parameter
represented by (13) is computed by using it’s relation with deceleration
parameter. The constant of integration $c_{4}$ effects the scale factor so we
have discussed three different cases for the considered model depending on the
positive negative and neutral nature of $c_{4}$. In each of the cases we have
calculated metric potentials, energy density ($\rho$), pressure ($p$)
directional Hubble parameters, Gravitational constant ($G$), cosmological
constant ($\Lambda$), physical quantities of the observational interest (such
as expansion scalar ($\Theta$), shear scalar ($\sigma^{2}$) and the anisotropy
parameter ($A_{m}$) and state finder parameters ($r$ and $s$). In this work we
have considered the time periodic deceleration parameter and EoS of
generalised Chaplygin gas as exotic fluid background in the framework of
Bianchi type-I space-time. The conclusion of our study are as follows

  * •

Almost all the parameters which we have discussed in our study shows periodic
behaviour which is because of the choice of deceleration parameter.

  * •

For each of the cases energy $\rho$ is positive and $p$ is negative throughout
the evolution of the universe and this negative pressure guarantees the late
time expansion of the universe.

  * •

Models discussed in Case-I and Case-II, tending to $\Lambda$CDM model whereas
model in Case-III fails for the value of $n$ and $m$ provided in Table 3. The
expressions obtained for expansion scalar, shear scalar, and anisotropic
parameter bears singularities in Case-I whereas these are free from
singularities in Case-II and Case-III.

  * •

The square sound speed satisfies the bounds $0\leq c_{s}^{2}\leq 1$ for each
of the cases. So the considered model is stable and physically acceptable.

  * •

For each of the cases considered model satisfies WEC and DEC where as it
violate SEC periodically throughout the evolution of the universe. So from the
results of energy conditions one can conclude that the violation of SEC may
leads to the accelerating Universe.

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