# Dark energy models from a parametrization of $H$: A comprehensive analysis
and observational constraints

S. K. J. Pacif<EMAIL_ADDRESS>Department of Mathematics, School of
Advanced Sciences, Vellore Institute of Technology, Vellore 632014, Tamil
Nadu, India

###### Abstract

The presented paper is a comprehensive analysis of two dark energy (DE)
cosmological models wherein exact solutions of the Einstein field equations
(EFEs) are obtained in a model-independent way (or by cosmological
parametrization). A simple parametrization of Hubble parameter ($H$) is
considered for the purpose in the flat Friedmann-Lemaitre-Robertson-Walker
(FLRW) background. The parametrization of $H$ covers some known models for
some specific values of the model parameters involved. Two models are of
special interest which show the behavior of cosmological phase transition from
deceleration in the past to acceleration at late-times. The model parameters
are constrained with $57$ points of Hubble datasets together with the $580$
points of Union $2.1$ compilation supernovae datasets and baryonic acoustic
oscillation (BAO) datasets. With the constrained values of the model
parameters, both the models are analyzed and compared with the standard
$\Lambda$CDM model and showing nice fit to the datasets. Two different
candidates of DE is considered, cosmological constant $\Lambda$ and a general
scalar field $\phi$ and their dynamics are discussed on the geometrical base
built. The geometrical and physical interpretations of the two models in
consideration are discussed in details and the evolution of various
cosmological parameters are shown graphically. The age of the Universe in both
models are also calculated. Various cosmological parametrization schemes used
in the past few decades to find exact solutions of the EFEs are also
summarized at the end which can serve as a unified reference for the readers.

Parametrization; cosmological parameters; observational constraints.

##  I Introduction

Late-time cosmic acceleration is an essential constituent of precision
cosmology at present. The idea of cosmic acceleration was first evidenced by
the observations of high redshift supernova of type Ia perlmutter ; riess .
The idea of cosmic acceleration was later accepted quickly by the scientific
community largely because of the independent observations with different
methodology adopted by the supernova search teams lead by Perlmutter and Riess
and also the CMB and the large scale structure data were providing substantial
evidence for a cosmological constant, indirectly bernadis ; hanany ;
netterfield . Later on some robust analysis and precise observations
strengthen the idea of cosmic acceleration and a flat Universe consistent with
$\Omega_{\Lambda}=1-\Omega_{m}=0.75$ mould ; spergel ; komastu ; essence .
What causing the accelerating expansion is still a mystery and we are mostly
in dark in this context. However, the theoretical predictions and advanced
surveys in observational point of view indicating the presence of a weird form
of energy in the Universe with high negative pressure with increasing density.
The mysterious energy is named as dark energy turner  as it’s nature,
characteristics is speculative only without any laboratory tests. Also, the
candidature of dark energy is a debatable topic at present cosmological
studies. Moreover, the age crisis in the standard model need cosmic
acceleration globular . Although, the modification of gravity theory at infra
red scale attracted attention to explain the late-time acceleration without
invoking any extra source term capozilo ; dvali , but the theory of dark
energy became quite popular copeland-sami ; bamba-de .

Very recently, gravitational wave detection and the picture of black hole
shadow strenghten the Einstein’s general theory of relativity and any
modifications in the Einstein’s theory (specifically to the geometry part) is
not worth appreciated. However, Einstein himself was not convinced with the
matter distribution in the Universe i.e. the right hand side of his field
equations (representing matter sector) is considered to made up of low grade
wood while the geometry part is of solid marble (representing the space-time).
Any extra source term such as Einstein’s cosmological constant (representing
energy density of vacuum) could be added into the energy momentum tensor and
serve as a candidate of dark energy. The most favoured candidate of dark
energy is the well known cosmological constant $\Lambda$. Also, $\Lambda$CDM
models have the best fit with many observational datasets. However, with this
significant $\Lambda$, due to its non dynamical and the long standing fine
tuning problem, researchers thought beyond it for a better candidate of dark
energy. So, scalar field models were discussed after the cosmological constant
for which $\Lambda$ could also be generated from particle creation effect
sahni . The dynamically evolving scalar field models have been utilized for
the purpose are quintessence zlatev ; brax ; barreiro , K-essence mukhanov1 ;
chiba ; steinhardt1 , phantom caldwell  and tachyonic field sen1 ; garousi1 ;
bergshoeff . The exotic fluid is also serve the purpose to explain the cosmic
acceleration phenomenology that considered an equation of state producing
large negative pressure e.g. Chapligyn gas equation of state gorini ,
Polytropic gas equation of state chavanis  etc.

Soon after the formulation of EFEs, theoreticians worked on finding exact
solutions. The first exact solution of the EFEs is the Schwarzschild exterior
solution schwar  wherein the prefect fluid equation of state was considered as
a suplementary condition. Despite of the high non linearity of the EFEs,
various exact solutions are obtained for static and spherically symmetric
metrics. Einstein’s static solution static , de-Sitter solution de-Sitter ,
Tolman’s solutions Tolman , Adler’s solutions Adlar , Buchdahl’s solution
Buchd , Vaidya and Tikekar solution Vaidya , Durgapal’s solutions Durgapal ,
Knutsen’s solutions Knutsen  and many more well-known solutions of EFEs are
obtained which are summarized in the literature Kramer  and also discussed in
Negi . Milne’s model milne , steady state model Steady  are some different
models proposed. All those phenomenological cosmological models explain the
Universe theoretically very well. However, observations play a major role in
modern cosmology which validate or discard a model. Now, numerical
computations are also playing big role in modern cosmology and estimating
cosmological parameters and also parameters of a model. In this study, an
important discussion is given on a technique of finding exact solution of EFEs
known as model independent way study. Moreover, two models are discussed and
analyzed comprehensively with current trendz in theoretical comology.

The paper is organized as follows. The first section is an introduction to
present cosmological scenario. The second section describes the Einstein’s
field equations in general relativity in presence of dark energy. The third
section is a motivation to the idea of model independent way or the
cosmological parametrization study to obtain exact solutions to EFEs. A simple
parametrization of Hubble parameter is considered in the light of
cosmographical study in the fourth section. In the fifth section,
observational constraints have been found for the model parameters involved in
the functional form of $H$ for the two models obtained. The sixth section is
devoted to the geometrical dynamics and analysis of some important
cosmological parameters describing the geometrical behavior of the Universe
for both the models. In the seventh section, two candidates of dark energy is
explored, cosmological constant and a general scalar field and the physical
parameters such as energy density, density parameter, potential of scalar
field and equation of state parameter are discussed for both the models under
considerations. In the eighth section, the age of the Universe for the
obtained models are calculated. The final section summarizes the physical
insights of the results obtained and concluded. A brief summary of the various
parametrization schemes used in the past few decades are given in the
appendix.

##  II EFEs in presence of Dark Energy

The nature of dark energy and its candidature is a mystery and it is a matter
of speculation to express it as a source term into the Einstein field
equations. However, DE is speculated to be homogeneous permeating all over the
space for which the energy momentum tensor can be represented in the form of a
perfect fluid as

$T_{ij}^{DE}=(\rho_{DE}+p_{DE})U_{i}U_{j}+p_{DE}~{}g_{ij},$ (1)

with its equation of state in the form $p_{DE}=\omega_{DE}\rho_{DE}$, where
$\omega_{DE}$ is the equation of state (EoS) parameter and is a function of
time in general satisfying the inequality $\omega_{DE}<0$. There is hot debate
going on for a suitable value of $\omega_{DE}$ and the analysis of some
observational data shows that its value lies in the range
$-1.61<\omega_{DE}<-0.78$ spergel ; knop ; tegmark . But recent analysis of
datasets provide more tighter constraints on $\omega_{DE}$ w-de1 ; w-de2 . The
different values of $\omega_{DE}$ in certain ranges gives rise to different
candidates and can broadly be be classified as follows. For

$\blacktriangleright$ $\omega_{DE}=-1$, the case is for the cosmological
constant;

$\blacktriangleright$ $\omega_{DE}=constant\neq-1$, the case is for cosmic
strings, domain walls, etc.;

$\blacktriangleright$ $\omega_{DE}\neq constant$, the cases for scalar fields
(quintessence, k-essence etc.), braneworlds, Dirac-Born-Infeld(DBI) action,
Chaplygin gas etc.;

$\blacktriangleright$ $\omega_{DE}<-1$, the case is for phantom models.

For a broader list of dark energy models see copeland-sami ; bamba-de  (and
refs. therein). There are interesting cases in each of them with some problems
though. For example, Cosmological constant $\Lambda$ is the most consistent
model for dark energy explaining observations but is plagued with fine tuning
problem. Similarly, the phantom models are interesting where the weak energy
condition ($\rho>0,~{}\rho+p>0$) is violated with the feature of finite time
singularity caldwell .

In general relativity, dark energy can be introduced by supplementing the
energy momentum tensor $T_{ij}^{DE}$ into the Einstein field equations
$G_{ij}=-8\pi GT_{ij}$ together with the matter source $T_{ij}^{M}$ as a
perfect fluid,

$T_{ij}^{tot}=T_{ij}^{m}+T_{ij}^{de}=(\rho_{tot}+p_{tot})U_{i}U_{j}+p_{tot}~{}g_{ij},$
(2)

with $\rho_{tot}=\sum\rho+\rho_{de}$ and $p_{tot}=\sum p+p_{de}$ denoting the
total energy densities and total pressure due to all types of matter (baryonic
matter, dark matter and radiation) and dark energy respectively. $U_{i}$ is
the usual four velocity vector and $g_{ij}$ is the metric tensor. Now, the
modified Einstein Field Equations for a flat FLRW metric

$ds^{2}=-dt^{2}+a^{2}(t)\left[dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right]\text{, }$ (3)

where $a(t)$ is the scale factor of the Universe, can be written as

$M_{pl}^{-2}\rho_{tot}=3\left(\frac{\dot{a}}{a}\right)^{2}=3H^{2},$ (4)

$M_{pl}^{-2}p_{tot}=-2\frac{\ddot{a}}{a}-\left(\frac{\dot{a}}{a}\right)^{2}=(2q-1)H^{2}.$
(5)

The conservation of energy-momentum (or from Eqs. (4) and (5) yields

$\dot{\rho}_{tot}+3(p_{tot}+\rho_{tot})\frac{\dot{a}}{a}=0.$ (6)

The continuity equation (6) play significant role in the evolution as it deals
with the matter and its interaction. In current cosmology, two kinds of dark
energy models generally discussed; interacting models of dark energy
(considering the interaction between cold dark matter and dark energy) zimdahl
; bertolemi ; banerjee  and non-interacting models of dark energy where all
the matters allowed to evolve separately ellis ; sahni-2 ; saini ; simon . Up
to date, there are no known interaction other than gravity between the matter
and dark energy. The present study refers to non interacting models only. The
system of equations are non linear ordinary differential equations and is
difficult to find exact solutions. There are tremendous efforts to find both
the exact and numerical solutions to EFEs in the past. In the next section,
the solution techniques of the above system of equations will be discussed
elaborately.

##  III Cosmological parametrization

The above system of equations (4), (5) and (6) possesses only two independent
equations with five unknowns $a$, $\rho$, $p$, $\rho_{de}$, $p_{de}$ (or
$\omega_{de}$). Due to the homogeneous distribution of matter in the Universe
at large scale, it is customary to consider the barotropic equation of state
$p=\omega\rho$, $\omega\in\left[0,1\right]$. The equation of state describes
different types of matter source in the Universe depending upon the discrete
or dynamical values of equation of state parameter $\omega$ that includes
baryonic matter ($\omega=0$), dark matter ($\omega=0$), radiation
($\omega=1/3$), stiff matter ($\omega=1$), etc. This additional equation
provides the third constraint equation. Another constraint equation can be the
consideration of equation of state of dark energy ($\omega_{de}=$ constant or
a function of time $t$ or function of scale factor $a$ or function of redshift
$z$) - best known as parametrization of dark energy equation of state. These
four equations can explain the cosmological dynamics of the Universe where all
the geometrical parameters (Hubble parameter $H$, deceleration parameter $q$,
jerk parameter $j$, etc.) or physical parameters (densities $\rho$,
$\rho_{de}$, pressures $p$, $p_{de}$, EoS parameter $\omega_{de}$, density
parameter $\Omega_{i}$, etc.) are expressed as functions of either scale
factor $a$ or the redshift $z$ ($=\frac{a_{0}}{a}-1$, $a_{0}$ being the
present value of scale factor generally normalized to $a_{0}=1$). But, there
is still one more equation short to close the system for the complete
determination of the system; the time evolution of scale factor $a$ is yet to
be determined. In literature, there are various schemes of parametrization of
the scale factor and it’s higher order derivatives ($H$, $q$, $j$ etc.)
providing the complete solution of the EFEs i.e. the explicit forms of
cosmological parameters as a function of cosmic time $t$.

In fact, a critical analysis of the solution techniques of EFEs in general
relativity theory or in modified theories has two aspects; one is the
parametrization of geometrical parameters $a$, $H$, $q$, $j$ giving the time
dependent functions of all the cosmological parameters; another is the
parametrization of the physical parameters $\rho$, $p$, $\rho_{de}$, $p_{de}$
(or $\omega_{de}$) giving the scale factor dependence or redshift dependence
of all the cosmological parameters. See the appendix for a broad list of
different schemes of parametrization of geometrical parameters and physical
parameters and also some phenomenological ansatzs used in the past few decades
to find the exact solutions of Einstein field equations. If we, observe
carefully, we can say that the first kind of parametrization schemes (of
geometrical parameters) are considered to find exact solutions that discusses
the expansion dynamics of the Universe and provides the time evolution of the
physical parameters $\rho$, $p$, $\rho_{de}$, $p_{de}$ (or $\omega_{de}$).
This method is generally known as model independent way study of cosmological
models or the cosmological parametrization chuna ; edvard ; pacif2016 . The
method do not affect the background theory anyway and provide solutions to the
EFEs explicitly and also has an advantage of reconstructing the cosmic history
of the Universe explaining some phenomena of the Universe. Also, this method
provides the simplest way to resolve some of the problems of standard model
e.g. the initial singularity problem, cosmological constant problem, etc. and
also explain the late-time acceleration conundrum, theoretically. While the
second kind of parametrization (of physical parameters) are generally
considered to discuss all the physical aspects (thermodynamics, structure
formation, nucleosynthesis etc.) of the Universe. However, both the schemes of
parametrization are adhoc choices or some phenomenological ansatzs (e.g.
$\Lambda$-varying cosmologies). All parametrization schemes (see appendix-1)
contain some arbitrary constants, referred to as model parameters which are
constrained through any observational datasets.

The purpose here is to obtain an exact solution of the Einstein field
equations in standard general relativity theory with a simple parametrization
of the Hubble parameter $H$ and discuss the reconstructed cosmic evolution.

##  IV Parametrization of $H$ & the models

The cosmographic analysis provide clues to study the evolution of the
observable Universe in a model independent way in terms of the kinematic
variables capozilo-lakoz . Moreover, analysis of cosmographic parameters helps
in studying the dark energy without any assumption of any particular
cosmological model except only the cosmological principle. In the standard
approximation the scale factor can be expanded in Taylors series around the
present time $t_{0}$ (which is the current age of the Universe also) and is
the simple strategy adopted in cosmographical analysis. Here and afterwards a
suffix $0$ denotes the value of the parameter at present time $t_{0}$. The
Taylor’s series expansion can be written as:

$\displaystyle a^{(n)}$ $\displaystyle=$ $\displaystyle
1+H_{0}(t-t_{0})-\frac{1}{2!}q_{0}H_{0}^{2}(t-t_{0})^{2}+\frac{1}{3!}j_{0}H_{0}^{3}(t-t_{0})^{3}$
(7)
$\displaystyle+\frac{1}{4!}s_{0}H_{0}^{4}(t-t_{0})^{4}+\frac{1}{5!}l_{0}H_{0}^{5}(t-t_{0})^{5}+.....$

where $H(t)=\frac{1}{a}\frac{da}{dt}$ is the Hubble parameter measuring
velocity,
$q(t)=-\frac{1}{a}\frac{d^{2}a}{dt^{2}}\left[\frac{1}{a}\frac{da}{dt}\right]^{-2}$
is the deceleration parameter measuring acceleration,
$j(t)=\frac{1}{a}\frac{d^{3}a}{dt^{3}}\left[\frac{1}{a}\frac{da}{dt}\right]^{-3}$
jerk parameter measuring jerk,
$s(t)=\frac{1}{a}\frac{d^{4}a}{dt^{4}}\left[\frac{1}{a}\frac{da}{dt}\right]^{-4}$
is the snap parameter and
$l(t)=\frac{1}{a}\frac{d^{5}a}{dt^{5}}\left[\frac{1}{a}\frac{da}{dt}\right]^{-5}$
is the lerk parameter. All of these parameters play significant roles in the
cosmographic analysis of the Universe (specifically the $H$, $q$ and $j$) and
distinguish various dark energy models.

Motivated by the above discussion, in this paper, a simple parametrization of
the Hubble parameter ($H$) is considered as an explicit function of cosmic
time ‘$t$’ in the form pacif2016

$H(t)=\frac{k_{2}t^{m}}{\left(t^{n}+k_{1}\right)^{p}}$ (8)

where $k_{1},$ $k_{2}\neq 0,$ $m,$ $n,$ $p$ are real constants (or model
parameters). $k_{1}$ and $k_{2}$ both have the dimensions of time. Some
specific values of the parameters $m,$ $n,$ $p$ suggest some distinguished
models which are elaborated by Pacif et al pacif2016 . It is easy to see that,
the single parametrization (8) generalizes several known models e.g.
$\Lambda$CDM model, power law model, hybrid expansion model, bouncing model,
linearly varying deceleration parameter model and some more. Out of the twelve
models deduced for some integral or non integral values of $m$, $n$, $p$ in
the functional form of HP in (8), two models (with $m=-1$, $p=1$, $n=1$ and
with $m=-1$, $p=1$, $n=2$) show the possibility of describing the phenomena of
cosmological phase transition for negative $k_{1}$ & $k_{2}$ and is described
as in the following Table-1.

Table-1: The models
---
Models | $H(t)$ | $a(t)$ | $q(t)$
M1 | $\frac{k_{2}}{t\left(k_{1}-t\right)}$ | $\beta\left(\frac{t}{k_{1}-t}\right)^{\frac{k_{2}}{k_{1}}}$ | $-1+\frac{k_{1}}{k_{2}}-\frac{2}{k_{2}}t$
M2 | $\frac{k_{2}}{t\left(k_{1}-t^{2}\right)}$ | $\beta\left(\frac{t^{2}}{k_{1}-t^{2}}\right)^{\frac{k_{2}}{2k_{1}}}$ | $-1+\frac{k_{1}}{k_{2}}-\frac{3}{k_{2}}t^{2}$

where, $\beta$ is an integrating constant which also play an important role in
the evolution. Pacif et al. obtained solutions for these two models in a
scalar field background and also found the observational constraints on model
parameters with $28$ points of $H(z)$ datasets. The present paper is an
extension of the same study for these two models M1 and M2 wherein much deeper
analysis have been done.

One can see, for both the models M1 and M2, the Hubble parameter and scale
factor both diverge in finite time and show a big rip singularity in near
future at $t=t_{s}=k_{1}$ for model M1 and at $t=t_{s}=\sqrt{k_{1}}$ for model
M2. The phase transition occurs at time $t_{tr}=\frac{k_{1}-k_{2}}{2}$ for
model M1 and at time $t_{tr}=\sqrt{\frac{k_{1}-k_{2}}{3}}$ for model M2 and
suggest that $k_{1}$ must be greater than $k_{2}$. With some suitable choice
of model parameters $k_{1}$, $k_{2}$ and $\beta$, rough sketches for the time-
evolution of scale factor (SF) and the Hubble parameter (HP) are made and are
shown graphically in the figures FIG. 1 and FIG. 2 respectively showing that
$a(t)$ diverges in finite time and the $H(t)$ becomes asymptotic showing big
rip in near future.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=166.2212pt]{aM1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=166.2212pt]{aM2.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 1:  Figures (a) and (b) respectively show rough sketches of the
evolotion of the scale factor w.r.t. cosmic time ‘$t$’ for both models M1 and
M2 with some arbitrary values of the model parameters $k_{1}$, $k_{2}$,
$\beta$.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=166.2212pt]{HM1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=166.2212pt]{HM2.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 2:  Figures (a) and (b) respectively show rough sketches of the
evolotion of the Hubble parameter w.r.t. cosmic time ‘$t$’ for both models M1
and M2 with some arbitrary values of the model parameters $k_{1}$, $k_{2}$,
$\beta$.

In order to check the consistency of the theoretical models obtained here with
the observations, some available datasets are used in the next section. The
model parameters are constrained through these datasets.

##  V Observational constraints

Three datasets are considered here for our analysis namely Hubble datasets
($Hz$), Type Ia supernovae datasets ($SN$) and Baryon Acoustic Oscillations
datasets ($BAO$). The detailed datasets and the method used are explained
below.

In the study of late-time Universe and the observational studies, it is
convenient to express all the cosmological parameters as functions of redshift
$z$. As the cosmological parameters here are functions of cosmic time $t$, the
time-redshift relationship must be established. The $t$-$z$ relations are
obtained as:

$t(z)=k_{1}\left[1+\left\\{\beta\left(1+z\right)\right\\}^{\frac{k_{1}}{k_{2}}}\right]^{-1},$
(9)

for model M1 and

$t(z)=\sqrt{k_{1}}\left[1+\left\\{\beta\left(1+z\right)\right\\}^{2\frac{k_{1}}{k_{2}}}\right]^{-\frac{1}{2}}$
(10)

for model M2. The above expressions (9) and (10) contain three parameters
$\beta$, $k_{1}$ and $k_{2}$ but actually two model parameters are sufficient
to describe these models by taking $\frac{k_{1}}{k_{2}}=\alpha$ which is also
beneficial for further analysis and numerical computations for which the
expressions for the Hubble parameter for both the models M1 and M2 are written
in terms of redshift $z$ as follows:

$H(z)=H_{0}\left(1+\beta^{\alpha}\right)^{-2}(1+z)^{-\alpha}\left[1+\left\\{\beta\left(1+z\right)\right\\}^{\alpha}\right]^{2},$
(11)

for model M1 and

$H(z)=H_{0}\left(1+\beta^{2\alpha}\right)^{-\frac{3}{2}}(1+z)^{-2\alpha}\left[1+\left\\{\beta\left(1+z\right)\right\\}^{2\alpha}\right]^{\frac{3}{2}}$
(12)

for model M2. The different datasets are described below.

###  V.1 H(z) datasets

It is well known that the Hubble parameter ($H=\frac{\dot{a}}{a}$) directly
probes the expansion history of the Universe where $\dot{a}$ is the rate of
change of the scale factor $a$ of the Universe. Hubble parameter is also
related to the differential redshift as, $H(z)$
$=-\frac{1}{1+z}\frac{dz}{dt}$, where $dz$ is obtained from the spectroscopic
surveys and so a measurement of $dt$ provides the Hubble parameter which will
be independent of the model. In fact, two methods are generally used to
measure the Hubble parameter values $H(z)$ at certain redshift and are
extraction of $H(z)$ from line-of-sight BAO data and differential age method
H1 -H19  estimating $H(z)$. Here, in this paper, an updated list of $57$ data
points are used as listed in Table-2 out of which $31$ data points measured
with DA method and $26$ data points are obtained with BAO and other methods in
the redshift range $0.07\leqslant z\leqslant 2.42$ sharov . Moreover, the
value of $H_{0}$ is taken as prior for our analysis as $H_{0}=67.8$ $Km/s/Mpc$
H0-value . The chi square function to determine the mean values of the model
parameters $\alpha$ & $\beta$ (which is equivalent to the maximum likelihood
analysis) is,

$\chi_{H}^{2}(\alpha,\beta)=\sum\limits_{i=1}^{28}\frac{[H_{th}(z_{i},\alpha,\beta)-H_{obs}(z_{i})]^{2}}{\sigma_{H(z_{i})}^{2}},$
(13)

where, $H_{th}$ is the theoretical, $H_{obs}$ is the observed value and
$\sigma_{H(z_{i})}$ is the standard error in the observed value of the Hubble
parameter $H$. The $57$ points of Hubble parameter values $H(z)$ with errors
$\sigma_{H}$ from differential age ($31$ points) method and BAO and other
($26$ points) methods are tabulated in Table-2 with references.

Table-2: 57 points of $H(z)$ datasets
---
31 points from DA method | 26 points from BAO & other method
$z$ | $H(z)$ | $\sigma_{H}$ | Ref. | $z$ | $H(z)$ | $\sigma_{H}$ | Ref. | $z$ | $H(z)$ | $\sigma_{H}$ | Ref. | $z$ | $H(z)$ | $\sigma_{H}$ | Ref.
$0.070$ | $69$ | $19.6$ | H1  | $0.4783$ | $80$ | $99$ | H5  | $0.24$ | $79.69$ | $2.99$ | H8  | $0.52$ | $94.35$ | $2.64$ | H10
$0.90$ | $69$ | $12$ | H2  | $0.480$ | $97$ | $62$ | H1  | $0.30$ | $81.7$ | $6.22$ | H9  | $0.56$ | $93.34$ | $2.3$ | H10
$0.120$ | $68.6$ | $26.2$ | H1  | $0.593$ | $104$ | $13$ | H3  | $0.31$ | $78.18$ | $4.74$ | H10  | $0.57$ | $87.6$ | $7.8$ | H14
$0.170$ | $83$ | $8$ | H2  | $0.6797$ | $92$ | $8$ | H3  | $0.34$ | $83.8$ | $3.66$ | H8  | $0.57$ | $96.8$ | $3.4$ | H15
$0.1791$ | $75$ | $4$ | H3  | $0.7812$ | $105$ | $12$ | H3  | $0.35$ | $82.7$ | $9.1$ | H11  | $0.59$ | $98.48$ | $3.18$ | H10
$0.1993$ | $75$ | $5$ | H3  | $0.8754$ | $125$ | $17$ | H3  | $0.36$ | $79.94$ | $3.38$ | H10  | $0.60$ | $87.9$ | $6.1$ | H13
$0.200$ | $72.9$ | $29.6$ | H4  | $0.880$ | $90$ | $40$ | H1  | $0.38$ | $81.5$ | $1.9$ | H12  | $0.61$ | $97.3$ | $2.1$ | H12
$0.270$ | $77$ | $14$ | H2  | $0.900$ | $117$ | $23$ | H2  | $0.40$ | $82.04$ | $2.03$ | H10  | $0.64$ | $98.82$ | $2.98$ | H10
$0.280$ | $88.8$ | $36.6$ | H4  | $1.037$ | $154$ | $20$ | H3  | $0.43$ | $86.45$ | $3.97$ | H8  | $0.73$ | $97.3$ | $7.0$ | H13
$0.3519$ | $83$ | $14$ | H3  | $1.300$ | $168$ | $17$ | H2  | $0.44$ | $82.6$ | $7.8$ | H13  | $2.30$ | $224$ | $8.6$ | H16
$0.3802$ | $83$ | $13.5$ | H5  | $1.363$ | $160$ | $33.6$ | H7  | $0.44$ | $84.81$ | $1.83$ | H10  | $2.33$ | $224$ | $8$ | H17
$0.400$ | $95$ | $17$ | H2  | $1.430$ | $177$ | $18$ | H2  | $0.48$ | $87.79$ | $2.03$ | H10  | $2.34$ | $222$ | $8.5$ | H18
$0.4004$ | $77$ | $10.2$ | H5  | $1.530$ | $140$ | $14$ | H2  | $0.51$ | $90.4$ | $1.9$ | H12  | $2.36$ | $226$ | $9.3$ | H19
$0.4247$ | $87.1$ | $11.2$ | H5  | $1.750$ | $202$ | $40$ | H2  |  |  |  |  |  |  |  |
$0.4497$ | $92.8$ | $12.9$ | H5  | $1.965$ | $186.5$ | $50.4$ | H7  |  |  |  |  |  |  |  |
$0.470$ | $89$ | $34$ | H6  |  |  |  |  |  |  |  |  |  |  |  |

###  V.2 SN Ia datasets

The first indication for the accelerating expansion of the Universe is due to
observations of supernovae of type Ia. Since then, several new SN Ia datasets
have been published. In this analysis, the Union $2.1$ compilation supernovae
datasets is considered containing $580$ points from SNeIa  which provides the
estimated values of the distance moduli $\mu_{i}=\mu_{i}^{obs}$ for a
particular redshift $z_{i}$ in the interval $0<z_{i}\leq 1.41$. The model
parameters of the models are to be fitted with, comparing the observed
$\mu_{i}^{obs}$ value to the theoretical $\mu_{i}^{th}$ value of the distance
moduli which are the logarithms
$\mu_{i}^{th}=\mu(D_{L})=m-M=5\log_{10}(D_{L})+\mu_{0}$, where $m$ and $M$ are
the apparent and absolute magnitudes and
$\mu_{0}=5\log\left(H_{0}^{-1}/Mpc\right)+25$ is the nuisance parameter that
has been marginalized. The luminosity distance is defined by,

$\displaystyle D_{l}(z)$ $\displaystyle=$
$\displaystyle\frac{c(1+z)}{H_{0}}S_{k}\left(H_{0}\int_{0}^{z}\frac{1}{H(z^{\ast})}dz^{\ast}\right),$
$\displaystyle\text{where }S_{k}(x)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{c}\sinh(x\sqrt{\Omega_{k}})/\Omega_{k}\text{,
}\Omega_{k}>0\\\ x\text{, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
}\Omega_{k}=0\\\ \sin
x\sqrt{\left|\Omega_{k}\right|})/\left|\Omega_{k}\right|\text{,
}\Omega_{k}<0\end{array}\right..$

Here, $\Omega_{k}=0$ (flat space-time). For our cosmological models M1 and M2
with theoretical value $H(z)$ which are depending on the model parameters
$\alpha$ & $\beta$, the distance $D_{L}(z)$ is calculated and the
corresponding chi square function measuring differences between the SN Ia
observational data and values predicted by the models is given by,

$\chi_{SN}^{2}(\mu_{0},\alpha,\beta)=\sum\limits_{i=1}^{580}\frac{[\mu^{th}(\mu_{0},z_{i},\alpha,\beta)-\mu^{obs}(z_{i})]^{2}}{\sigma_{\mu(z_{i})}^{2}},$
(15)

$\sigma_{\mu(z_{i})}^{2}$ is the standard error in the observed value.
Following perivolar  after marginalizing $\mu_{0}$, the chi square function is
written as,

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\chi_{SN}^{2}(\alpha,\beta)=A(\alpha,\beta)-[B(\alpha,\beta)]^{2}/C(\alpha,\beta)$

where

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
A(\alpha,\beta)=\sum\limits_{i=1}^{580}\frac{[\mu^{th}(\mu_{0}=0,z_{i},\alpha,\beta)-\mu^{obs}(z_{i})]^{2}}{\sigma_{\mu(z_{i})}^{2}},$

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
B(\alpha,\beta)=\sum\limits_{i=1}^{580}\frac{[\mu^{th}(\mu_{0}=0,z_{i},\alpha,\beta)-\mu^{obs}(z_{i})]^{2}}{\sigma_{\mu(z_{i})}^{2}},$

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
C(\alpha,\beta)=\sum\limits_{i=1}^{580}\frac{1}{\sigma_{\mu(z_{i})}^{2}}$.

###  V.3 BAO datasets

Baryonic acoustic oscillations is an analysis dealing with the early Universe.
It is known that the early Universe filled with baryons, photons and dark
matter. Moreover, baryons and photons together act as single fluid (coupled
tightly through the Thompson scattering) and can not collapse under gravity
rather oscillate due to the large pressure of photons. These oscillations are
termed a Baryonic acoustic oscillations (BAO). The characteristic scale of BAO
is governed by the sound horizon $r_{s}$ at the photon decoupling epoch
$z_{\ast}$ is given as,

$r_{s}(z_{\ast})=\frac{c}{\sqrt{3}}\int_{0}^{\frac{1}{1+z_{\ast}}}\frac{da}{a^{2}H(a)\sqrt{1+(3\Omega_{0b}/4\Omega_{0\gamma})a}},$

where $\Omega_{0b}$ stands for the baryon density and $\Omega_{0\gamma}$
stands for the photon density at present time.

The BAO sound horizon scale is also used to derive the angular diameter
distance $D_{A}$ and the Hubble expansion rate $H$ as a function of $z$. If
$\triangle\theta$ be the measured angular separation of the BAO feature in the
2 point correlation function of the galaxy distribution on the sky and the
$\triangle z$ be the measured redshift separation of the BAO feature in the 2
point correlation function along the line of sight then,

$\triangle\theta=\frac{r_{s}}{d_{A}(z)}$ where
$d_{A}(z)=\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})}$ and $\triangle
z=H(z)r_{s}$.

In this work, BAO datasets of $d_{A}(z_{\ast})/D_{V}(z_{BAO})$ from the
references BAO1 ; BAO2 ; BAO3 ; BAO4 ; BAO5 ; BAO6  is considered where the
photon decoupling redshift is $z_{\ast}\approx 1091$ and $d_{A}(z)$ is the co-
moving angular diameter distance and
$D_{V}(z)=\left(d_{A}(z)^{2}z/H(z)\right)^{1/3}$ is the dilation scale. The
data used for this analysis is given in the Table-3

Table-3: Values of $d_{A}(z_{\ast})/D_{V}(z_{BAO})$ for distinct values of
$z_{BAO}$
---
$z_{BAO}$ | $0.106$ | $0.2$ | $0.35$ | $0.44$ | $0.6$ | $0.73$
$\frac{d_{A}(z_{\ast})}{D_{V}(z_{BAO})}$ | $30.95\pm 1.46$ | $17.55\pm 0.60$ | $10.11\pm 0.37$ | $8.44\pm 0.67$ | $6.69\pm 0.33$ | $5.45\pm 0.31$

The chi square function for BAO is given by BAO6

$\chi_{BAO}^{2}=X^{T}C^{-1}X\,,$ (16)

where

$X=\left(\begin{array}[]{c}\frac{d_{A}(z_{\star})}{D_{V}(0.106)}-30.95\\\
\frac{d_{A}(z_{\star})}{D_{V}(0.2)}-17.55\\\
\frac{d_{A}(z_{\star})}{D_{V}(0.35)}-10.11\\\
\frac{d_{A}(z_{\star})}{D_{V}(0.44)}-8.44\\\
\frac{d_{A}(z_{\star})}{D_{V}(0.6)}-6.69\\\
\frac{d_{A}(z_{\star})}{D_{V}(0.73)}-5.45\end{array}\right)\,,$

and $C^{-1}$ is the inverse covariance matrix defined in BAO6 .

$C^{-1}=\left(\begin{array}[]{cccccc}0.48435&-0.101383&-0.164945&-0.0305703&-0.097874&-0.106738\\\
-0.101383&3.2882&-2.45497&-0.0787898&-0.252254&-0.2751\\\
-0.164945&-2.454987&9.55916&-0.128187&-0.410404&-0.447574\\\
-0.0305703&-0.0787898&-0.128187&2.78728&-2.75632&1.16437\\\
-0.097874&-0.252254&-0.410404&-2.75632&14.9245&-7.32441\\\
-0.106738&-0.2751&-0.447574&1.16437&-7.32441&14.5022\end{array}\right)\,$

With the above samples of Hubble ($Hz$), supernovae of type Ia ($SN$) and
baryon acoustic oscillations ($BAO$) datasets, the chi square functions (13),
(15) and (16) are minimized to get the average values of the model parameters
$\alpha$ & $\beta$. The maximum likelihood contours for the model parameters
$\alpha$ & $\beta$ are shown in the following figures FIG. 3, FIG. 4, FIG. 5,
and FIG. 6 for independent $Hz$ datasets and combined $Hz+SN$, $SN+BAO$ and
$Hz+SN+BAO$ datasets respectively with $1$-$\sigma$, $2$-$\sigma$ and
$3$-$\sigma$ error contours in the $\alpha$-$\beta$ plane.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=166.2212pt]{cont-
hz-M1-Red.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=166.2212pt]{cont-
hz-M2-Red.pdf}\\\ \mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 3:  Figures (a) and (b) are contour plots for Hubble datasets ($Hz$)
for models M1 and M2 respectvely.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=166.2212pt]{cont-
hz+sn-M1-Yellow.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=166.2212pt]{cont-
hz+sn-M2-Yellow.pdf}\\\ \mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 4:  Figures (a) and (b) are contour plots for combined $Hz+SN$ datasets
for models M1 and M2 respectvely.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=166.2212pt]{cont-
sn+bao-M1-Green.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=166.2212pt]{cont-
sn+bao-M2-Green.pdf}\\\ \mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 5:  Figures (a) and (b) are contour plots for combibed $SN+BAO$
datasets for models M1 and M2 respectvely.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=166.2212pt]{cont-
hz+sn+bao-M1-Blue.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=166.2212pt]{cont-
hz+sn+bao-M2-Blue.pdf}\\\ \mbox{(}a)\hfil\hskip
7.22743pt&\mbox{(}b)\end{array}$

Figure 6:  Figures (a) and (b) are contour plots for combined $Hz+SN+BAO$
datasets for models M1 and M2 respectvely.

The average mean values (constrained values) of the model parameters and the
minimum chi square values are tabulated in Table-4 for independent $Hz$
datasets and combined $Hz+SN$, $SN+BAO$ and $Hz+SN+BAO$ datasets.

Table-4: Constrained values of model parameters and chi square values
---
Datasets | Models | $\alpha$ | $\beta$ | $\chi_{\min}^{2}$ | $\chi^{2}/dof$
$H(z)$ |  | M1
---
M2
| $1.58064$
---
$1.31611$
| $1.48729$
---
$1.56124$
| $31.329529$
---
$29.972660$
| $0.56962$
---
$0.54495$
$H(z)+SN$ |  | M1
---
M2
| $1.60094$
---
$1.32551$
| $1.44572$
---
$1.53587$
| $596.49325$
---
$595.02853$
| $0.93935$
---
$0.93705$
$SN+BAO$ |  | M1
---
M2
| $1.61116$
---
$1.45677$
| $1.36647$
---
$1.36451$
| $564.45777$
---
$566.44641$
| $0.96653$
---
$0.96994$
$H(z)+SN+BAO$ |  | M1
---
M2
| $1.59173$
---
$1.42829$
| $1.45678$
---
$1.40637$
| $599.07805$
---
$614.40132$
| $0.93459$
---
$0.95850$

The error bar plots of the $57$ points of $Hz$ and $580$ points of Union $2.1$
compilation datasets are plotted and shown in the following figures FIG. 7 and
FIG. 8, using the constrained values of the model parameters as in Table-4 for
both the models M1 and M2.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=166.2212pt]{Error-
HP-M1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=166.2212pt]{Error-
HP-M2.pdf}\\\ \mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 7:  Figures (a) and (b) are the error bar plots for $57$ data points
from Hubble datasets together with the models M1 and M2 shown in solid red
lines respectively. The dashed lines in both the figures are $\Lambda$CDM
model shown for comparision.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=166.2212pt]{Error-
SN-M1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=166.2212pt]{Error-
SN-M2.pdf}\\\ \mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 8:  Figures (a) and (b) are the error bar plots for Union $2.1$
compilation supernovae datasets together with the models M1 and M2 shown in
solid red lines respectively. The dashed lines in both the figures are
$\Lambda$CDM model shown for comparision.

##  VI Geometrical Dynamics of the models

###  VI.1 Deceleration parameter & Phase transition

The expressions for the deceleration parameter can be written in terms of
redshift $z$ as:

$q(z)=-1+\alpha-2\alpha\left[1+\left\\{\beta\left(1+z\right)\right\\}^{\alpha}\right]^{-1}$
(17)

for model M1 and

$q(z)=-1+\alpha-3\alpha\left[1+\left\\{\beta\left(1+z\right)\right\\}^{2\alpha}\right]^{-1}$
(18)

for model M2. The behavior of $q$ is shown in the following FIG. 9 and the
important values assumed by the deceleration parameter $q$ in the course of
evolution are tabulated in the following Table-5 for different sets of
$\alpha$ & $\beta$ as obtained.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=166.2212pt]{DP-M1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=166.2212pt]{DP-M2.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 9:  Figures (a) and (b) show the evolution of deceleration parameter
from past $(z=4)$ to far future with a phase transition for models M1 and M2
respectively.

The initial value of the deceleration parameter ($q_{i}$ as
$z\longrightarrow\infty$), the present value of the deceleration parameter
($q_{0}$ as $z\longrightarrow 0$) and the far future value of the deceleration
parameter ($q_{f}$ as $z\longrightarrow-1$) are calculated together with the
phase transition redshift ($z_{tr}$ for which $q=0$, the redshift at which the
Universe transited from decelerating expansion to accelerating one) using the
constrained (numerical) values of the model parameters $\alpha$ & $\beta$ (see
Table-4) for both the models M1 and model M2 in the following Table-5 and
Table-6 respectively.

Table-5: Values of $q$ at different epochs & phase transition redshift for
model M1
---
redshift | formula | $Hz$ | $Hz+SN$ | $SN+BAO$ | $Hz+SN+BAO$
$z\longrightarrow\infty$ ($q_{i}$) | $q_{i}=-1+\alpha$ | $0.58064$ | $0.60094$ | $0.61116$ | $0.59173$
$z\longrightarrow 0$ ($q_{0}$) | $q_{0}=-1+\alpha-\frac{2\alpha}{1+\beta^{\alpha}}$ | $-0.51977$ | $-0.54087$ | $-0.60308$ | $-0.53714$
$z\longrightarrow-1$ ($q_{f}$) | $q_{f}=-1-\alpha$ | $-2.58064$ | $-2.60094$ | $-2.61116$ | $-2.59173$
$z_{tr}$ ($q=0$) | $z_{tr}=-1+\frac{1}{\beta}\left(\frac{\alpha+1}{\alpha-1}\right)^{\frac{1}{\alpha}}$ | $0.72763$ | $0.72730$ | $0.80236$ | $0.73622$
Table-6: Values of $q$ at different epochs & phase transition redshift for
model M2
---
redshift | formula | $Hz$ | $Hz+SN$ | $SN+BAO$ | $Hz+SN+BAO$
$z\longrightarrow\infty$ ($q_{i}$) | $q_{i}=-1+\alpha$ | $0.31611$ | $0.32551$ | $0.45677$ | $0.42829$
$z\longrightarrow 0$ ($q_{0}$) | $q_{0}=-1+\alpha-\frac{3\alpha}{1+\beta^{2\alpha}}$ | $-0.61721$ | $-0.63987$ | $-0.80152$ | $-0.74601$
$z\longrightarrow-1$ ($q_{f}$) | $q_{f}=-1-2\alpha$ | $-3.63222$ | $-3.65102$ | $-3.91354$ | $-3.85658$
$z_{tr}$ ($q=0$) | $z_{tr}=-1+\frac{1}{\beta}\left(\frac{2\alpha+1}{\alpha-1}\right)^{\frac{1}{2\alpha}}$ | $0.61941$ | $0.62055$ | $0.53182$ | $0.53471$

From the above Table-5 and Table-6, it is observed that in both the models M1
and M2, the Universe begins smoothly with deceleration (means no inflationary
phase in these models) and transit to accelerating phase at around
$z_{tr}\approx 0.72$ in model M1 for all numerical constrained values of model
parameters $\alpha$ & $\beta$ while in model M2, the phase transition occurs
at $z_{tr}\approx 0.62$ $Hz$ and $Hz+SN$ constrained values of $\alpha$ &
$\beta$ and $z_{tr}\approx 0.53$ $SN+BAO$ and $Hz+SN+BAO$ constrained values
of $\alpha$ & $\beta$. The present values of the deceleration parameter
$q_{0}$ found in both models can be seen in the above tables consistent with
predicted values. In the future, the Universe enter into super acceleration
phase ($q<-1$) for both the models M1 and M2 and finally attains maximum
values $q_{f}<-2.58$ in model M1 and $q_{f}<-3.61$ in model M2 for all the
values of $\alpha$ & $\beta$.

###  VI.2 Statefinder diagnostics

Statefinder diagnostics stfnd1 ; stfnd2 ; stfnd3 ; stfnd4  is a technique
generally used to distinguish various dark energy models and compare their
behavior using the higher order derivatives of the scale factor. The
parameters are $s$ & $r$ and calculated using the relations:

$r=\frac{\dddot{a}}{aH^{3}}\text{, \ \
}s=\frac{r-1}{3(q-\frac{1}{2})}\text{.}$ (19)

The statefinder diagnostics pairs are constructed as $\\{s,r\\}$ and
$\\{q,r\\}$ wherein different trajectories in the $s$-$r$ and $q$-$r$ planes
are plotted to see the temporal evolutions of different dark energy models.
The fixed points in this contexts are generally considered as
$\\{s,r\\}=\\{0,1\\}$ for $\Lambda$CDM model and $\\{s,r\\}=\\{1,1\\}$ for
SCDM (standard cold dark matter) model in FLRW background and the departures
of any dark energy model from these fixed points are analyzed. The other
diagnostic pair is $\left\\{q,r\right\\}$ and the fixed points considered are
$\left\\{q,r\right\\}=\left\\{-1,1\right\\}$ for $\Lambda$CDM model and
$\left\\{q,r\right\\}=\left\\{0.5,1\right\\}$ for SCDM model. The statefinder
parameters for the considered model M1 are calculated as,

$r(z)=1+\alpha(2\alpha-3)+\frac{6\alpha}{1+\left\\{\beta(1+z)\right\\}^{\alpha}}\left[1-\alpha+\frac{\alpha}{1+\left\\{\beta(1+z)\right\\}^{\alpha}}\right]$
(20)

$s(z)=\frac{2\alpha}{3}-\frac{\alpha}{1+\left\\{\beta(1+z)\right\\}^{\alpha}}+\frac{\alpha(3+2\alpha)}{3\left[-3-2\alpha+(2\alpha-3)\left\\{\beta(1+z)\right\\}^{\alpha}\right]}$
(21)

and for model M2

$r(z)=1-3\alpha+2\alpha^{2}+\frac{12\alpha^{2}}{\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{2}}+\frac{3\alpha(3-2\alpha)}{1+\left\\{\beta(1+z)\right\\}^{2\alpha}}$
(22)

$s(z)=\frac{2}{3}\left[\alpha-\frac{2\alpha}{1+\left\\{\beta(1+z)\right\\}^{2\alpha}}+\frac{\alpha(3+4\alpha)}{\left[-3-4\alpha+(2\alpha-3)\left\\{\beta(1+z)\right\\}^{2\alpha}\right]}\right]$
(23)

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=166.2212pt]{srM1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=166.2212pt]{srM2.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 10:  Figures (a) and (b) are the $s$-$r$ plots for model M1 and M2
respectively showing the different trajectories of the models.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=166.2212pt]{qrM1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=166.2212pt]{qrM2.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 11:  Figures (a) and (b) are $q$-$r$ plots for models M1 and M2
respectively showing different trajectories of the models.

In the above figure FIG. 10, one can see the diverge evolutions of the model
M1 and model M2 in the $s$-$r$ and $q$-$r$ planes. Both the models showing
distinctive features as compared to the other standard models. One can observe
that at early times, model M1 presumes values in the range $r>1$ and $s<0$
representing Chaplygin gas type DE model and evolutes to quintessence region
and again reverts to Chapligyn gas region at late times by crossing the
intermediate $\Lambda$CDM fixed point $\left\\{0,1\right\\}$ during evolution.
But, the model M2 is different and evolutes from quintessence region in the
past and goes to Chaplygin gas region intermediating the $\Lambda$CDM fixed
point $\left\\{0,1\right\\}$ during it’s evolution for all cases. The figure
FIG. 11 depicts the temporal evolution of the models M1 and M2 in the
$\\{q,r\\}$ plane providing additional information about the models M1 and M2
wherein the dashed lines describe the evolution of the $\Lambda$CDM model
below which quintessence region and the upper one is Chaplygin gas region are
shown. The evolution of model M1 and M2 are clearly observed. Both the models
M1 and M2 deviates from de Sitter point ($-1,1$).

###  VI.3 $Om$ diagnostic

$Om$ diagnostic is another tool introduced in omdig1 ; omdig2 ; omdig3 ;
omdig4 , using the Hubble parameter and serving the purpose of providing a
null test of the $\Lambda$CDM model. Like, statefinder diagnostic, $Om$
diagnostic is also an effective method to discriminate various DE models from
$\Lambda$CDM model according to the slope variation of $Om(z)$. Positive slope
of diagnostic implies a Quintessence nature ($\omega>-1$), Negative slope of
diagnostic implies a Phantom nature ($\omega<-1$) and Constant slope with
respect to redshift tells the nature of dark energy coincide with that of the
cosmological constant ($\omega=-1$).

The $Om(z)$ for a flat Universe is defined as:

$Om\left(z\right)=\frac{\left(\frac{H(z)}{H_{0}}\right)^{2}-1}{\left(1+z\right)^{3}-1}\text{,{}}$
(24)

which can also be represented as
$Om\left(z\right)=\Omega_{m0}+\left(1-\Omega_{m0}\right)\frac{(1+z)^{3(1+\omega)}-1}{(1+z)^{3}-1}$.
For a constant EoS parameter $\omega$ imply $Om\left(z\right)=\Omega_{m0}$ and
different values of $Om\left(z\right)$ suggest whether the model is a
$\Lambda$CDM model or quintessence or phantom models. For the models of
consideration here, the expressions for $Om(z)$ for models M1 and M2 are
obtained as,

$Om\left(z\right)=\frac{\frac{\left[1+\left\\{\beta\left(1+z\right)\right\\}^{\alpha}\right]^{4}}{\left(1+\beta^{\alpha}\right)^{4}(1+z)^{2\alpha}}-1}{\left(1+z\right)^{3}-1}\text{{}}$
(25)
$Om\left(z\right)=\frac{\frac{\left[1+\left\\{\beta\left(1+z\right)\right\\}^{2\alpha}\right]^{3}}{\left(1+\beta^{2\alpha}\right)^{3}(1+z)^{4\alpha}}-1}{\left(1+z\right)^{3}-1}\text{{}}$
(26)

The slope variation of $Om(z)$ vs. $z$ are shown in the following figure FIG.
12 for models M1 and M2. For both the models M1 and M2 and for all values of
$\alpha$ & $\beta$, the $Om\left(z\right)$ values is less than $\Omega_{m0}$
in the redshift range $z>0$ showing the models are in quintessence region in
the past and for the redshift range $z<0$, $Om\left(z\right)$ values decreases
sharply and becomes negative implying the both the models enter into phantom
region.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=158.99377pt]{OmM1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=158.99377pt]{OmM2.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 12:  Figures (a) and (b) show the plots for $Om(z)$ vs. $z$ for models
M1 and M2 respectively.

###  VI.4 Jerk, Snap and Lerk parameters

Likewise, the Hubble and the deceleration parameters, the other cosmographic
parameters, jerk, snap and lerk parameters also play significant roles in
analyzing a cosmological model. The cosmic jerk $j(z=0)\simeq 1$ signify a
cosmic acceleration. The evolution of jerk parameter for the models M1 and M2
are shown in FIG. 13 showing that for all numerical constrained values of
model parameters $\alpha$ & $\beta$, $j_{0}\in(1.1,1.4)$ for model M1 and
$j_{0}\in(0.6,1.2)$ for model M2. The increasing values of jerk, snap and lerk
parameters in the future ($z<0$) showing the deviation from the $\Lambda$CDM
model which can also be interpreted from the statefinder diagrams FIG. 9 and
FIG. 10. Similary, the evolution of snap and lerk parameters are shown in FIG.
14 and FIG. 15 respectively for both the models M1 and M2. From the figures,
it can be seen that for all values of model parameters $\alpha$ & $\beta$,
$s_{0}\in(1.2,2.2)$ for model M1 and $s_{0}\in(-1.5,0-0.7)$ for model M2 and
$l_{0}\in(6,9)$ for model M1 and $l_{0}\in(5,13)$ for model M2. These values
of $j_{0}$, $s_{0}$, $l_{0}$ are in good agreement with the expected values.
One can also interpret that the model M2 has better fit to the observational
datasets as compared to model M1 which can also be seen from FIG. 14 and FIG.
15.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=158.99377pt]{jerk-M1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=158.99377pt]{jerk-M2.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 13:  Figures (a) and (b) show the plots for jerk parameter $j(z)$ vs.
$z$ for models M1 and M2 respectively.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=158.99377pt]{snap-M1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=158.99377pt]{snap-M2.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 14:  Figures (a) and (b) show the plots for snap parameter $s(z)$ vs.
$z$ for models M1 and M2 respectively.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=158.99377pt]{lerk-M1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=158.99377pt]{lerk-M2.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 15:  Figures (a) and (b) show the plots for lerk parameter $l(z)$ vs.
$z$ for models M1 and M2 respectively.

##  VII Physical Dynamics of the models

The geometrical part of the Einstein field equations is discussed elaborately
and now the physical interpretations can be discussed for the obtained models
once the matter content of the Universe is specified. In the introduction, it
is mentioned that the candidate of dark energy is still unknown and it is a
matter of speculation only to choose any candidate described in literature.
However, the most discussed candidate and having best fit with some
observations is the Einstein’s cosmological constant. So, in the following,
the cosmological constant will be considered as a candidate of dark energy for
further analysis.

So, let us consider the two fluid Universe, cold dark matter and dark energy
only, since the radiation contribution at present is negligible. The matter
pressure is $p=p_{m}=0$ for cold dark matter and for dark energy the equation
of state is $p_{DE}=\omega_{DE}\rho_{DE}$. In the following, the physical
behavior of the matter and dark energy densities and pressures are found out
and their evolutions are shown graphically.

###  VII.1 Cosmological constant

When the candidate of dark energy is the cosmological constant implying
$\rho_{DE}=\rho_{\Lambda}=M_{pl}^{-2}\Lambda$ and for which the equation of
state parameter $\omega_{DE}$ reduces to $-1$. Solving equations (4) and (5),
it is easy to obtain the explicit expressions for the matter energy density
and the energy density of cosmological constant as,

$\displaystyle\frac{\rho_{m}}{M_{pl}^{2}H_{0}^{2}}$ $\displaystyle=$
$\displaystyle\frac{2\alpha\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]^{4}-4\alpha\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]^{3}}{\left(1+\beta^{\alpha}\right)^{4}\left(1+z\right)^{2\alpha}}\text{,}$
(27) $\displaystyle\frac{\rho_{\Lambda}}{M_{pl}^{2}H_{0}^{2}}$
$\displaystyle=$
$\displaystyle\frac{\Lambda}{H_{0}^{2}}=\frac{(3-2\alpha)\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]^{4}+4\alpha\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]^{3}}{\left(1+\beta^{\alpha}\right)^{4}\left(1+z\right)^{2\alpha}}\text{.}$
(28)

for model M1 and

$\displaystyle\frac{\rho_{m}}{M_{pl}^{2}H_{0}^{2}}$ $\displaystyle=$
$\displaystyle\frac{2\alpha\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{3}-6\alpha\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{2}}{\left(1+\beta^{2\alpha}\right)^{3}\left(1+z\right)^{4\alpha}}\text{,}$
(29) $\displaystyle\frac{\rho_{\Lambda}}{M_{pl}^{2}H_{0}^{2}}$
$\displaystyle=$
$\displaystyle\frac{\Lambda}{H_{0}^{2}}=\frac{(3-2\alpha)\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{3}+6\alpha\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{2}}{\left(1+\beta^{2\alpha}\right)^{3}\left(1+z\right)^{4\alpha}}\text{.}$
(30)

for model M2. The evolution of these physical parameters are shown in the FIG.
16 and FIG. 17.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=158.99377pt]{CC-
rho-M1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=158.99377pt]{CC-
rho-M2.pdf}\\\ \mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 16:  Figures (a) and (b) show the evolution of the matter energy
densities ($\rho_{m}$) for models M1 and M2 respectively.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=158.99377pt]{CC-
lam-M1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=158.99377pt]{CC-
lam-M2.pdf}\\\ \mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 17:  Figures (a) and (b) show the evolution of the energy densities of
the cosmological constant ($\rho_{\Lambda}$) for models M1 and M2
respectively.

The density parameters for matter
$\left(\Omega_{m}=\frac{\rho_{m}}{3Mpl^{2}H^{2}}\right)$ and density parameter
for cosmological constant
$\left(\Omega_{\Lambda}=\frac{\Lambda}{3H^{2}}\right)$ can also be computed
for both the models M1 and M2 as,

$\Omega_{m}=\frac{2\alpha}{3}-\frac{4\alpha}{3\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]}\text{,
}\Omega_{\Lambda}=1-\frac{2\alpha}{3}+\frac{4\alpha}{3\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]}$
(31)

for model M1 and

$\Omega_{m}=\frac{2\alpha}{3}-\frac{2\alpha}{\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]}\text{,
}\Omega_{\Lambda}=1-\frac{2\alpha}{3}+\frac{2\alpha}{\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]}$
(32)

for model M2. One can see from the above expressions that the sum total of the
density parameters with these components is equal to $1$. The evolution of the
density parameters are shown in FIG. 18.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=158.99377pt]{CC-
omega-M1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=158.99377pt]{CC-
omega-M2.pdf}\\\ \mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 18:  Figures (a) and (b) show the evolution of the density parameters
for matter ($\Omega_{m}$) and cosmological constant ($\Omega_{\Lambda}$) for
models M1 and M2 respectively.

###  VII.2 Scalar field

Since the equation of state for cosmological constant is non dynamical and
observations reveal it’s dynamical characteristics, other candidates such as a
general scalar field came into picture for a suitable candidate of dark
energy. For an ordinary scalar field $\phi$ for the action can be represented
as,

$S=\int
d^{4}x\sqrt{-g}\left\\{\frac{M_{p}^{2}}{2}R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)+L_{Matter}\right\\}\text{.}$
(33)

The term $V(\phi)$ is the potential function for the scalar field $\phi$. In
the considered FLRW background the energy density $\rho_{\phi}$ will take the
form $\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V\left(\phi\right)$ and pressure
$p_{\phi}$ will take the form
$p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V\left(\phi\right)$. For a two component
Universe, scalar field and cold dark matter with minimal interaction between
them (i.e. they conserve separately giving $\rho=ca^{-3}=c(1+z)^{3}$, $c$ is a
constant of integration), then the solutions obtained from Eqs. (4) and (5)
are,

$\frac{V(\phi)}{M_{pl}^{2}H_{0}^{2}}=\frac{(3-\alpha)\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]^{4}+2\alpha\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]^{3}}{\left(1+\beta^{\alpha}\right)^{4}\left(1+z\right)^{2\alpha}}-\frac{c}{2M_{pl}^{2}H_{0}^{2}}(1+z)^{3}$
(34)

$\frac{\rho_{\phi}}{M_{pl}^{2}H_{0}^{2}}=\frac{3\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]^{4}}{\left(1+\beta^{\alpha}\right)^{4}\left(1+z\right)^{2\alpha}}-\frac{c}{M_{pl}^{2}H_{0}^{2}}(1+z)^{3}$
(35)

and the expression for the scalar field $\phi(z)$ can be calculated by
integrating,

$\frac{\phi-\phi_{0}}{\sqrt{2}M_{pl}}=-\int\left[\frac{\alpha\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]^{4}-2\alpha\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]^{3}}{\left(1+\beta^{\alpha}\right)^{4}\left(1+z\right)^{2\alpha}}-\frac{c}{2M_{pl}^{2}H_{0}^{2}}(1+z)^{3}\right]^{\frac{1}{2}}\frac{(1+\beta^{\alpha})^{2}(1+z)^{\alpha-1}}{\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]^{2}}dz$
(36)

for model M1. Here, $\phi_{0}$ is an integrating constant. Similarly, the
potential and energy densities for model M2 are obtained as,

$\frac{V(\phi)}{M_{pl}^{2}H_{0}^{2}}=\frac{(3-\alpha)\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{3}+3\alpha\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{2}}{\left(1+\beta^{2\alpha}\right)^{3}\left(1+z\right)^{4\alpha}}-\frac{c}{2M_{pl}^{2}H_{0}^{2}}(1+z)^{3}$
(37)

$\frac{\rho_{\phi}}{M_{pl}^{2}H_{0}^{2}}=\frac{3\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{3}}{\left(1+\beta^{2\alpha}\right)^{3}\left(1+z\right)^{4\alpha}}-\frac{c}{M_{pl}^{2}H_{0}^{2}}(1+z)^{3}$
(38)

and the expression for the scalar field $\phi(z)$ can be calculated by
integrating,

$\frac{\phi-\phi_{0}}{\sqrt{2}M_{pl}}=-\int\left[\frac{\alpha\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{3}-3\alpha\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{2}}{\left(1+\beta^{2\alpha}\right)^{3}\left(1+z\right)^{4\alpha}}-\frac{c}{2M_{pl}^{2}H_{0}^{2}}(1+z)^{3}\right]^{\frac{1}{2}}\frac{(1+\beta^{2\alpha})^{\frac{3}{2}}(1+z)^{2\alpha-1}}{\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{\frac{3}{2}}}dz\text{.}$
(39)

The density parameters for matter
$\left(\Omega_{m}=\frac{\rho_{m}}{3M^{2}plH^{2}}\right)$ and density parameter
for the scalar field
$\left(\Omega_{\phi}=\frac{\rho_{\phi}}{3M^{2}plH^{2}}=\frac{\frac{1}{2}\dot{\phi}^{2}+V\left(\phi\right)}{3M^{2}plH^{2}}\right)$
can be computed for both the models M1 and M2 as,

$\Omega_{\phi}=1-\Omega_{m}\text{\thinspace,
}\Omega_{m}=\frac{c(1+z)^{3}\left(1+\beta^{\alpha}\right)^{4}\left(1+z\right)^{2\alpha}}{3M_{pl}^{2}H_{0}^{2}\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]^{4}}$
(40)

for model M1 and

$\Omega_{\phi}=1-\Omega_{m}\text{\thinspace,
}\Omega_{m}=\frac{c(1+z)^{3}\left(1+\beta^{2\alpha}\right)^{3}\left(1+z\right)^{4\alpha}}{3M_{pl}^{2}H_{0}^{2}\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{3}}$
(41)

for model M2. From equations (40) and (41), one obtains
$\Omega_{m0}=\frac{c}{3M_{pl}^{2}H_{0}^{2}}\Longrightarrow
c=\Omega_{m0}3M_{pl}^{2}H_{0}^{2}$. The equations of state parameter
($\omega_{\phi}=\frac{p_{\phi}}{\rho_{\phi}}$) are given by,

$\omega_{\phi}^{eff}=\frac{1}{3}\frac{(2\alpha-3)\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]^{4}-4\alpha\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]^{3}}{\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]^{4}-\Omega_{m0}\left(1+\beta^{\alpha}\right)^{4}\left(1+z\right)^{2\alpha+3}}$
(42)
$\omega_{\phi}^{eff}=\frac{1}{3}\frac{(2\alpha-3)\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{3}-6\alpha\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{2}}{\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{3}-\Omega_{m0}\left(1+\beta^{2\alpha}\right)^{3}\left(1+z\right)^{4\alpha+3}}$
(43)

The evolution of the Scalar field energy density, scalar field potential and
the density parameters are shown in FIG. 19, FIG. 20 and FIG. 21 respectively
for models M1 and M2.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=158.99377pt]{rho-
phi-M1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=158.99377pt]{rho-
phi-M2.pdf}\\\ \mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 19:  Figures (a) and (b) show the evolution of the scalar field energy
density ($\rho_{\phi}$) for models M1 and M2 respectively.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=158.99377pt]{V-phi-M1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=158.99377pt]{V-phi-M2.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 20:  Figures (a) and (b) show the evolution of the scalar field
potential $V({\phi)}\sim z$ for models M1 and M2 respectively.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=158.99377pt]{Density-M1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=158.99377pt]{Density-M2.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 21:  Figures (a) and (b) show the evolution of the density parameters
$\Omega_{\phi}$ & $\Omega_{m}$ w.r.t. redshift $z$ for models M1 and M2
respectively.

The evolution of the equation of state parameter ($\omega_{\phi}(z)$) vs.
redshift $z$ is plotted by neglecting the matter contribution and shown in the
FIG. 22 for models M1 and M2.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=195.12877pt,height=158.99377pt]{Eos-
Scalar-M1.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=195.12877pt,height=158.99377pt]{Eos-
Scalar-M2.pdf}\\\ \mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 22:  Figures (a) and (b) show the evolution of equation of state
parameter vs. redshift ($\omega_{\phi}(z)\sim z$) for models M1 and M2
respectively.

##  VIII Age of the Universe

The calculation of the age of the Universe is associated to the values of the
cosmological parameters, specifically the Hubble parameter. In general, using
the Friedmann equation one can obtain the relation as
$t_{0}=\frac{1}{H_{0}}F(\Omega_{x})$, $x=$ radiation, matter, dark energy,
neutrino etc. The functional $F$ contributes a fraction and largely the term
$1/H_{0}$ in the age calculation e.g. for $H_{0}=69$ $km/s/Mpc$, one obtains
$1/H_{0}\approx$ $14.5$ $Gyr$ (Giga years) and the factor $F=0.956$ for
$\Lambda$CDM model with
$\left(\Omega_{m},\Omega_{\Lambda}\right)=\left(0.3086,0.6914\right)$ giving
pretty good estimate of $t_{0}$ and $F=0.666$ for Einstein-de-Sitter model
with $\left(\Omega_{m},\Omega_{\Lambda}\right)=\left(1,0\right)$ giving much
smaller value of $t_{0}$. So, the introduction of cosmological constant is
significant as matter-only Universe was not enough to explain the globular
clusters in the Milky Way which appeared to be older than the age of the
Universe calculated then. According to the Planck2015 results age of the
universe is estimated to be $13.799\pm 0.021$ $Gyr$ with $H_{0}=67.74\pm 0.46$
within $68\%$ confidence limits for $\Lambda$CDM model constrained by combined
CMB power spectra, Planck polarization data, CMB lensing reconstruction and
external data of BAO, JLA (Joint light curve analysis) and Hubble datasets.

Here, the present work is a model independent study wherein the geometrical
parameter $H$ is parametrized for which the calculation of the age is
unaffected by the matter content and solely depend on the functional form of
the Hubble parameter $H(t)$. We have already established the $t$-$z$
relationships for models M1 and M2 which can be rewritten as,

$t(z)=\frac{\left(1+\beta^{\alpha}\right)^{2}}{\alpha\beta^{\alpha}\left[1+\left\\{\beta(1+z)\right\\}^{\alpha}\right]}\frac{1}{H_{0}}\text{
and
}t(z)=\frac{\left(1+\beta^{2\alpha}\right)^{\frac{3}{2}}}{\alpha\beta^{2\alpha}\left[1+\left\\{\beta(1+z)\right\\}^{2\alpha}\right]^{\frac{1}{2}}}\frac{1}{H_{0}}$

respectively. By considering the present value of the Hubble parameter,
$H_{0}=67.8$ $Km/Sec/Mpc$, the terms multiplied to $1/H_{0}$ are calculated
and for both the models are greater than $0.96$ for all constrained values of
$\alpha$ & $\beta$ and give pretty good estimate for the present age of the
Universe and is larger than the standard model. The age calculation is
tabulated in the following Table-7 for all the constrained numerical values of
$\alpha$ & $\beta$ for both the models M1 and M2.

Table-7
---
Models | M1 | M2
Datasets | $(\alpha,\beta)$ | Factor | Age (in $Gyr$) | $(\alpha,\beta)$ | Factor | Age (in $Gyr$)
$Hz$ | $(1.58064,1.48729)$ | $0.97046$ | $14.0068$ | $(1.31611,1.56124)$ | $0.99502$ | $14.3613$
$Hz+SN$ | $(1.60094,1.44572)$ | $0.97084$ | $14.0123$ | $(1.32551,1.53587)$ | $0.99630$ | $14.3797$
$SN+BAO$ | $(1.61116,1.36647)$ | $0.99597$ | $14.3751$ | $(1.45677,1.36451)$ | $0.96400$ | $13.9136$
$Hz+SN+BAO$ | $(1.59173,1.45678)$ | $0.97343$ | $14.0496$ | $(1.42829,1.40637)$ | $0.96445$ | $13.9201$

##  IX Results and Conclusion

To summarize the results, the philosophy behind writing this present paper is
to discuss the phenomenology of cosmological parametrization to obtain exact
solutions of Einstein field equations. As an exemplification, a simple
parametrization of Hubble parameter is considered with some model parameters
which reduce to some known models (see pacif2016 ) for some specific values of
the model parameters involved. Two models discussed here in details and both
the models M1 and M2 exhibit a phase transition from deceleration to
acceleration. Also, both the models diverges in finite time and show big rip
singularity. For consistency of the models obtained here, some observational
datasets namely, $H(z)$ datasets with updated $57$ data points, Supernovae
datasets from union 2.1 compilation datasets containing $580$ data points and
BAO datasets with $6$ data points are considered and compared with the
standard $\Lambda$CDM model. Both the models M1 and M2 contain two model
parameters $\alpha$ & $\beta$ which are constrained through these datasets and
some numerical values are obtained in pairs with independent $Hz$, combined
$Hz+SN$, $SN+BAO$ and $Hz+SN+BAO$ datasets which are then used for further
analysis for geometrical and physical interpretations of the models. The
present values of the deceleration parameters obtained for these constrained
values of model parameters $\alpha$ & $\beta$ are calculated which are
tabulated in Table-5 and Table-6 together with the phase transition redshifts
and are in certain standard estimated range. In the future the Universe in
both the models enters into super acceleration phases and diverges in finite
times. The other geometrical parameters such as jerk, snap and lerk parameters
are also discussed and their evolutions are shown graphically. The statefinder
diagnostics and $om$ diagnostics are also presented to compare the obtained
models with the standard $\Lambda$CDM model and the models behavior are shown
in plots compared with the standard $\Lambda$CDM model and SCDM model. After
the brief cosmographic analysis, the physical interpretation of the models are
discussed by considering the cosmological constant and scalar field as
candidates of dark energy. The matter energy density ($\rho_{m}$), energy
density of ($\rho_{\Lambda}$), the density parameters $\Omega_{\Lambda}$ and
$\Omega_{m}$ are also calculated and their dynamical behavior w.r.t. redshift
$z$ are shown graphically for both the models M1 and M2 using the numerical
constrained values of the model parameters $\alpha$ & $\beta$. Similarly, the
evolution of energy density ($\rho_{\phi}$), the potential ($V(\phi)$) of the
scalar field ($\phi$), density parameters $\Omega_{\Lambda}$ and $\Omega_{m}$
and also the equation of state parameter $\omega_{\phi}$ of the scalar field
are shown graphically for both the models M1 and M2. The geometrical and
physical analysis for both the models M1 and M2 interpret that both models M1
and M2 have the quintessence behavior in the past and phantom like behavior in
the future. Finally, the age of the Universe for both the models M1 and M2 are
calculated for the constrained numerical values of model parameters $\alpha$ &
$\beta$. It is found that the age found for both the models are greater than
the standard model and consistent with the age constraints of $\Lambda$CDM
model.

The conclusion is that the model M2 which is a quadratic varying deceleration
parameter model has better fit to the observational datasets (see FIG. 7 and
FIG. 8) and shows better approximation to the present cosmological scenario on
geometrical as well as physical grounds as compared to the model M1 which is a
linearly varying deceleration parameter model. The presented study is an
example of doing a comprehensive analysis of any cosmological model that
describe a simple methodology of finding exact solution of the Einstein field
equations, comparing to the observations and estimating model parameters from
the observational datasets. A brief list of various schemes of parametrization
of different geometrical and physical parameters used in the past few decades
to obtain the exact solutions of EFEs are also summarized here which will help
the readers for their studies in cosmological modelling.

##  X Appendix

A brief list of various parametrization schemes of parametrization of
geometrical and physical parameters used in the past few decades to find exact
solutions of Einstein Field Equations is given below.

###  X.1 PARAMETRIZATIONS OF GEOMETRICAL PARAMETERS

Scale factor $a(t)$

Given below a list of different expansion laws of the scale factor those have
been extensively studied in different contexts.

$a(t)=constant$ static  (Static model)

$a(t)=ct$ milne ; melia  (Milne model or Linear expansion)

$a(t)\sim\exp(H_{0}t)$ ellismadsen  ($\Lambda$CDM model or Exponential
expansion)

$a(t)\sim\exp\left[-\alpha t\ln\left(\frac{t}{t_{0}}\right)+\beta t\right]$
a-zhou  (Inflationary model)

$a(t)\sim\exp\left[-\alpha t-\beta t^{n}\right]$ a-zhou  (Inflationary model)

$a(t)\sim\left[\exp(\alpha t)-\beta\exp(-\alpha t)\right]^{n}$ a-zhou
(Inflationary model)

$a(t)\sim\exp\left(\frac{t}{M}\right)\left[1+\cos\left(\frac{\varsigma(t)}{N}\right)\right]$
qssc  (quasi steady state cosmology, Cyclic Universe)

$a(t)\sim t^{\alpha}$ plc-lohiya  (Power law Cosmology)

$a(t)\sim t^{n}\exp(\alpha t)$ hsf  (Hybrid expansion)

$a(t)\sim\exp\left[n(\log t)^{m}\right]$ logamediate  (Logamediate expansion)

$a(t)\sim\cosh\alpha t$ ellismadsen  (Hyperbolic expansion)

$a(t)\sim\left(\sinh\alpha t\right)^{\frac{1}{n}}$ ritika  (Hyperbolic
expansion)

$a(t)\sim\left(\frac{t}{t_{s}-t}\right)^{n}$ odintsov1  (Singular model)

$a(t)\sim t^{n}\exp\left[\alpha(t_{s}-t)\right]$ odintsov1  (Singular model)

$a(t)\sim\exp\left(\alpha\frac{t^{2}}{t_{\ast}^{2}}\right)$ bouncing
(Bouncing Model)

$a(t)\sim\exp\left(\frac{\beta}{\alpha+1}(t-t_{s})^{\alpha+1}\right)$ bouncing
(Bouncing Model)

$a(t)\sim\left(\frac{3}{2}\rho_{cr}t^{2}+1\right)^{\frac{1}{3}}$ bouncing
(Bouncing Model)

$a(t)\sim\left(\frac{t_{s}-t}{t_{\ast}}\right)$ bouncing  (Bouncing Model)

$a(t)\sim\sin^{2}\left(\alpha\frac{t}{t_{\ast}}\right)$ bouncing  (Bouncing
Model)

Hubble parameter $H(t)$ or $H(a)$

$H(a)=Da^{-m}$ H-berman1983

$H(a)=e^{\frac{1-\gamma a^{2}}{\alpha a}}$ H-banerjee2010

$H(a)=\alpha(1+a^{-n})$ H-JPSINGH

$H(t)=\frac{m}{\alpha t+\beta}$ H-Pacif1

$H(t)=\frac{16\alpha t}{15\left[1+(8\alpha t^{2})/5\right]}$ H-pks1

$H(t)=m+\frac{n}{t}$ H-pks2

$H(t)=\frac{\alpha t_{R}}{t(t_{R}-t)}$ H-Cannata

$H(t)=\frac{\alpha}{3}\left(t+T_{0}\right)^{3}-\beta\left(t+T_{0}\right)+\gamma$
H-nojiri2006

$H(t)=\alpha e^{\lambda t}$ H-odintsov2012

$H(t)=\alpha+\beta(t_{s}-t)^{n}$ H-odintsov2012

$H(t)=\alpha-\beta e^{-nt}$ H-odin2015

$H(t)=f_{1}(t)+f_{2}(t)(t_{s}-t)^{n}$ H-odin2014

$H(t)=\frac{\beta t^{m}}{\left(t^{n}+\alpha\right)^{p}}$ pacif2016

$H(t)=n\alpha\tanh(m-nt)+\beta$ H-ashutosh

$H(t)=\alpha\tanh\left(\frac{t}{t_{0}}\right)$ bamba-de

$H(z)=\left[\alpha+\left(1-\alpha\right)\left(1+z\right)^{n}\right]^{\frac{3}{2n}}$
H-abdulla

Deceleration parameter $q(t)$ or $q(a)$, $q(z)$

$q(t)=m-1$ q-berman1998

$q(t)=-\alpha t+m-1$ q-akarsu2012

$q(t)=\alpha\cos(\beta t)-1$ q-pks1

$q(t)=-\frac{\alpha t}{1+t}$ q-pks2

$q(t)=-\frac{\alpha(1-t)}{1+t}$ q-pks2

$q(t)=-\frac{\alpha}{t^{2}}+\beta-1$ q-ASSRP

$q(t)=(8n^{2}-1)-12nt+3t^{2}$ q-bakry

$q(a)=-1-\frac{\alpha a^{\alpha}}{1+a^{\alpha}}$ q-banerjee2005

$q(z)=q_{0}+q_{1}z$ q-riess

$q(z)=q_{0}+q_{1}z(1+z)^{-1}$ q-santos

$q(z)=q_{0}+q_{1}z(1+z)(1+z^{2})^{-1}$ q-sdas

$q(z)=\frac{1}{2}+q_{1}(1+z)^{-2}$ q-nair

$q(z)=q_{0}+q_{1}[1+\ln(1+z)]^{-1}$ q-Xu2008

$q(z)=\frac{1}{2}+(q_{1}z+q_{2})(1+z)^{-2}$ q-gong2006

$q(z)=-1+\frac{3}{2}\left(\frac{\left(1+z\right)^{q_{2}}}{q_{1}+(1+z)^{q_{2}}}\right)$
q-campo

$q(z)=-\frac{1}{4}\left[3q_{1}+1-3(q_{1}+1)\left(\frac{q_{1}e^{q_{2}\left(1+z\right)}-e^{-q_{2}\left(1+z\right)}}{q_{1}e^{q_{2}\left(1+z\right)}+e^{-q_{2}\left(1+z\right)}}\right)\right]$
q-pavon

$q(z)=-\frac{1}{4}+\frac{3}{4}\left(\frac{q_{1}e^{q_{2}\frac{z}{\sqrt{1+z}}}-e^{-q_{2}\frac{z}{\sqrt{1+z}}}}{q_{1}e^{q_{2}\frac{z}{\sqrt{1+z}}}+e^{-q_{2}\frac{z}{\sqrt{1+z}}}}\right)$
q-pavon

$q(z)=q_{f}+\frac{q_{i}-q_{f}}{1-\frac{q_{i}}{q_{f}}\left(\frac{1+z_{t}}{1+z}\right)^{\frac{1}{\tau}}}$
q-ishida

$q(z)=q_{0}-q_{1}\left(\frac{(1+z)^{-\alpha}-1}{\alpha}\right)$ q-abdulla1

$q(z)=q_{0}+q_{1}\left[\frac{\ln(\alpha+z)}{1+z}-\beta\right]$ q-abdulla2

$q(z)=q_{0}-(q_{0}-q_{1})(1+z)\exp\left[z_{c}^{2}-(z+z_{c})^{2}\right]$
q-garza

Jerk parameter $j(z)$

$j(z)=-1+j_{1}\frac{f(z)}{E^{2}(z)},$ where $f(z)=z$, $\frac{z}{1+z}$,
$\frac{z}{(1+z)^{2}}$, $\log(1+z)$ and $E(z)=\frac{H(z)}{H_{0}}$jerk-para

$j(z)=-1+j_{1}\frac{f(z)}{h^{2}(z)},$ where $f(z)=1$, $1+z$, $(1+z)^{2}$,
$(1+z)^{-1}$ and $h(z)=\frac{H(z)}{H_{0}}$jerk-para2

###  X.2 PARAMETRIZATIONS OF PHYSICAL PARAMETERS

Pressure $p\mathbf{(}\rho\mathbf{)}$, $p(z)$

The matter content in the Universe is not properly known but it can be
categorized with its equations of states $p=p(\rho)$. Following is a list of
some cosmic fluid considerations with their EoS. Also, some dark energy
pressure parametrization are listed.

$p(\rho)=w\rho$ (Perfect fluid EoS)

$p{\small(\rho)}=w\rho-f(H)$ viscous1  (Viscous fluid EoS)

$p{\small(\rho)}=w\rho+k\rho^{1+\frac{1}{n}}$ poly0  (Polytropic gas EoS)

$p{\small(\rho)}=\frac{8w\rho}{3-\rho}-3\rho^{2}$ vanderwall  (Vanderwaal gas
EoS)

$p{\small(\rho)}=-(w+1)\frac{\rho^{2}}{\rho_{P}}+w\rho+(w+1)\rho_{\Lambda}$
quad  (EoS in quadratic form)

$p{\small(\rho)}=-\frac{B}{\rho}$CG1  (Chaplygin gas EoS)

$p{\small(\rho)}=-\frac{B}{\rho^{\alpha}}$ GCG1  (Generalized Chaplygin gas
EoS)

$p{\small(\rho)}=A\rho-\frac{B}{\rho^{\alpha}}$ MCG1  (Modified Chaplygin gas
EoS)

$p{\small(\rho)}=A\rho-\frac{B(a)}{\rho^{\alpha}}$ VMCG  (Variable modified
Chaplygin gas EoS)

$p{\small(\rho)}=A(a)\rho-\frac{B(a)}{\rho^{\alpha}}$ NVMCG  (New variable
modified Chaplygin gas EoS)

$p{\small(\rho)}=-\rho-\rho^{\alpha}$ NOJI-EOS  (DE EoS)

$p(z)=\alpha+\beta z$ eos-zhang  (DE EoS)

$p(z)=\alpha+\beta\frac{z}{1+z}$ eos-zhang  (DE EoS)

$p(z)=\alpha+\beta\left(z+\frac{z}{1+z}\right)$ eos-wang  (DE EoS)

$p(z)=\alpha+\beta\ln(1+z)$ eos-jun  (DE EoS)

Equation of state parameter $w(z)$

$w(z)=w_{0}+w_{1}z$ w-LIN2  (Linear parametrization)

$w(z)=w_{0}+w_{1}\frac{z}{\left(1+z\right)^{2}}$ w-JBP  (JBP parametrization)

$w(z)=w_{0}+w_{1}\frac{z}{\left(1+z\right)^{n}}$ w-nCPL  (Generalized JBP
parametrization)

$w(z)=w_{0}+w_{1}\frac{z}{1+z}$ w-CPL1  (CPL parametrization)

$w(z)=w_{0}+w_{1}\left(\frac{z}{1+z}\right)^{n}$ w-nCPL  (Generalized CPL
parametrization)

$w(z)=w_{0}+w_{1}\frac{z}{\sqrt{1+z^{2}}}$ w-sqrt  (Square-root
parametrization)

$w(z)=w_{0}+w_{1}\sin(z)$ w-sin  (Sine parametrization)

$w(z)=w_{0}+w_{1}\ln(1+z)$ w-LOG  (Logarithmic parametrization)

$w(z)=w_{0}+w_{1}\ln\left({\small 1+\frac{z}{1+z}}\right)$ w-feng
(Logarithmic parametrization)

$w(z)=w_{0}+w_{1}\frac{z(1+z)}{1+z^{2}}$ w-BA  (BA parametrization)

$w(z)=w_{0}+w_{1}\left(\frac{\ln(2+z)}{1+z}-\ln 2\right)$ (MZ parametrization)

$w(z)=w_{0}+w_{1}\left(\frac{\sin(1+z)}{1+z}-\sin 1\right)$ w-MZ  (MZ
parametrization)

$w(z)=w_{0}+w_{1}\frac{z}{1+z^{2}}$ (FSLL parametrization)

$w(z)=w_{0}+w_{1}\frac{z^{2}}{1+z^{2}}$ w-FSSL  (FSLL parametrization)

$w(z)=-1+\frac{1+z}{3}\frac{\alpha+2\beta(1+z)}{\gamma+2\alpha(1+z)+\beta(1+z)^{2}}$
w-ASSS  (ASSS parametrization)

$w(z)=\frac{1+\left(\frac{1+z}{1+z_{s}}\right)^{\alpha}}{w_{0}+w_{1}\left(\frac{1+z}{1+z_{s}}\right)^{\alpha}}$
w-hansted  (Hannestad Mortsell parametrization)

$w(z)=-1+\alpha(1+z)+\beta(1+z)^{2}$ w-weller  (Polynomial parametrization)

$w(z)=-1+\alpha\left[1+f(z)\right]+\beta\left[1+f(z)\right]^{2}$ w-sendra
(Generalized Polynomial parametrization)

$w(z)=w_{0}+z\left(\frac{dw}{dz}\right)_{0}$ w-cooray

$w(z)=\frac{-2(1+z)d_{c}^{{}^{\prime\prime}}-3d_{c}^{{}^{\prime}}}{3\left[d_{c}^{{}^{\prime}}-\Omega_{M}(1+z)^{3}\left(d_{c}^{{}^{\prime}}\right)^{3}\right]}$
where $d_{c}^{{}^{\prime}}=\int\limits_{0}^{z}\frac{H_{0}dz}{H(z)}$ w-hai

$w_{x}(a)=w_{0}\exp(a-1)$ w-yang

$w_{x}(a)=w_{0}a(1-\log a)$ w-yang

$w_{x}(a)=w_{0}a\exp(1-a)$ w-yang

$w_{x}(a)=w_{0}a(1+\sin(1-a))$ w-yang

$w_{x}(a)=w_{0}a(1+\arcsin(1-a))$ w-yang

$w_{de}(z)=w_{0}+w_{1}q$ w-elizalde

$w_{de}(z)=w_{0}+w_{1}q(1+z)^{\alpha}$ w-elizalde

$w_{de}(z)=\frac{w_{0}}{\left[1+b\ln(1+z)\right]^{2}}$ w-wetterich

$w_{x}(z)=w_{0}+b\left\\{1-\cos\left[\ln(1+z)\right]\right\\}$ w-supriya

$w_{x}(z)=w_{0}+b\sin\left[\ln(1+z)\right]$ w-supriya

$w_{x}(z)=w_{0}+b\left[\frac{\sin(1+z)}{1+z}-\sin 1\right]$ w-supriya

$w_{x}(z)=w_{0}+b\left(\frac{z}{1+z}\right)\cos(1+z)$ w-supriya

$w(z)=w_{0}+w_{a}\left[\frac{\ln(2+z)}{1+z}-\ln 2\right]$ w-mazhang

$w(z)=w_{0}+w_{a}\left[\frac{\ln(\alpha+1+z)}{\alpha+z}-\frac{\ln(\alpha+1)}{\alpha}\right]$
w-sello

Energy density $\rho$

$\rho=\rho_{c}$ OT2 , ABDEL

$\rho\sim\theta^{2}$ SKJP1

$\rho=\frac{A}{a^{4}}\sqrt{{\small a}^{2}{\small+b}}$ SKJP3

$\left({\small\rho+3p}\right)a^{3}=A$ RGVgrg

$\rho+p=\rho_{c}$ ASRGVprd

$\rho_{de}(z)=\rho_{de}(0)\left[1+\alpha\left(\frac{z}{1+z}\right)^{n}\right]$
rho-abdulla

$\rho_{de}(z)=\frac{1}{\rho_{\phi}}\left(\frac{d\rho_{\phi}}{d\phi}\right)=-\frac{\alpha
a}{\left(\beta+a\right)^{2}}$ rho-sdas

$\rho_{de}(z)=\alpha H(z)$ rho-rezaei

$\rho_{de}(z)=\alpha H(z)+\beta H^{2}(z)$ rho-rezaei

$\rho_{de}(z)=\frac{3}{\kappa^{2}}\left[\alpha+\beta H^{2}(z)\right]$ rho-
rezaei

$\rho_{de}(z)=\frac{3}{\kappa^{2}}\left[\alpha+\frac{2}{3}\beta\dot{H}(z)\right]$
rho-rezaei

$\rho_{de}(z)=\frac{3}{\kappa^{2}}\left[\alpha
H^{2}(z)+\frac{2}{3}\beta\dot{H}(z)\right]$ rho-rezaei

$\rho_{de}(z)=\frac{3}{\kappa^{2}}\left[\alpha+\beta
H^{2}(z)+\frac{2}{3}\gamma\dot{H}(z)\right]$ rho-rezaei

$\rho_{de}(z)=\rho_{\phi 0}\left(1+z\right)^{\alpha}e^{\beta z}$ rho-abdulla2

Cosmological constant ($\Lambda$)

In order to resolve the long standing cosmological constant problem, authors
have considered some variation laws for the cosmological constant in the past
forty years, commonly known as “$\Lambda$-varying cosmologies” or “Decaying
vacuum cosmologies”. Later the idea was adopted to explain the accelerated
expansion of the Universe considering varying $\Lambda$. Following is list of
such decay laws of $\Lambda$.

$\Lambda\sim a^{-n}$ RGVcqg1

$\Lambda\sim H^{n}$ COOPER

$\Lambda\sim\rho$ RGVcqg1

$\Lambda\sim t^{n}$ COOPER

$\Lambda\sim q^{n}$ COOPER

$\Lambda\sim e^{-\beta a}$ SGRAJ

$\Lambda=\Lambda(T)$ LINDE  $T$ is Temperature

$\Lambda\sim C+e^{-\beta t}$ var-lam3

$\Lambda=3\beta H^{2}+\alpha a^{-2}$ CARVALHO

$\Lambda=\beta\frac{\ddot{a}}{a}$ SRAY

$\Lambda=3\beta H^{2}+\alpha\frac{\ddot{a}}{a}$ ArbabLAM2

$\frac{d\Lambda}{dt}\sim\beta\Lambda-\Lambda^{2}$ var-lammoffat

Scalar field Potentials $V(\phi)$

$V(\phi)=V_{0}\phi^{n}$ copeland-sami  (Power law)

$V(\phi)=V_{0}\exp\left[-\frac{\alpha\phi}{M_{pl}}\right]$ copeland-sami
(exponential)

$V(\phi)=\frac{V_{0}}{\cosh\left[\phi/\phi_{0}\right]}$ copeland-sami

$V(\phi)=V_{0}\left[\cosh\left(\alpha\phi/M_{pl}\right)\right]^{-\beta}$
(hyperbolic) copeland-sami

$V(\phi)=\frac{\alpha}{\phi^{n}}$ (Inverse power law) v-inverse

$V(\phi)=\frac{V_{0}}{1+\beta\exp(-\alpha\kappa\phi)}$ (Woods-Saxon potential)
v-wood

$V(\phi)=\alpha c^{2}\left[\tanh\frac{\phi}{\sqrt{6}\alpha}\right]^{2}$
($\alpha$-attractor) v-shah

$V(\phi)=V_{0}(1+\phi^{\alpha})^{2}$ v-shah2

$V(\phi)=V_{0}\exp(\alpha\phi^{2})$ v-shah2

$V(\phi)=\frac{1}{4}(\phi^{2}-1)^{2}$ v-arefe

Note: All the parametrization listed above contain some arbitrary constants
such as $\alpha$, $\beta$, $\gamma$, $m$, $n$, $p$, $q_{0}$, $q_{1}$, $q_{2}$,
$w_{0}$, $w_{1}$, $A$, $B$ are model parameters which are generally
constrained through observational datasets or through any analytical methods
and also some arbitrary functions $f_{1}(t)$, $f_{2}(t)$. $t_{s}$ denote the
bouncing time or future singularity time and $t_{\ast}$ some arbitrary time.

## References

  * (1) S. Perlmutter et al., Astrophys. J. 517, 565 (1999).
  * (2) A. Riess et al., Astrophys. J. 117, 707 (1999).
  * (3) de Bernardis et al., Nature 404, 955 (2000).
  * (4) S. Hanany et al., Astrophy. J. Lett. 545, L5 (2000).
  * (5) C. Netterfield et al., Astrophy. J. Lett. 474, 47 (1997).
  * (6) J. R. Mould et al., Astrophy. J. 529, 786 (2000).
  * (7) D. Spergal et al., Astrophys. J. Suppl. 148, 175 (2003).
  * (8) E. Komatsu et al., Astrophys. J. Suppl. 192, 18 (2011).
  * (9) W. M. Wood-Vasey, Astrophys. J. 666, 694 (2007).
  * (10) M. S. Turner and D. Huterer, J. Phys. Soc. Jap. 76, 111015 (2007).
  * (11) B. Chaboyer, Phys. Rept. 307, 23 (1998).
  * (12) S. Capozziello and L. Z. Fang, Int. J. Mod. Phys. D 11, 483 (2002).
  * (13) G. R. Dvali et al., Phys. Lett. B 485, 208 (2000).
  * (14) E. J. Copeland, M. Sami, S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006).
  * (15) K. Bamba et al., Astrophys. Space Sci. 342, 155 (2012).
  * (16) V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 9 (2000) 373.
  * (17) I. Zlatev, L.M. Wang and P. J. Steinhardt, Phys. Rev. Lett. 82 (1999) 896.
  * (18) P. Brax and J. Martin, Phys. Rev. D 61 (2000) 103502.
  * (19) T. Barreiro, E.J. Copeland and N.J. Nunes, Phys. Rev. D 61 (2000) 127301.
  * (20) C. Armendariz-Picon, T. Damour, and V. Mukhanov, Phys. Lett. B 458 (1999) 219.
  * (21) T. Chiba, T. Okabe and M. Yamaguchi, Phys. Rev. D 62 (2000) 023511.
  * (22) C. Armendariz-Picon, V. Mukhanov and P.J. Steinhardt, Phys. Rev. Lett. 85 (2000) 4438.
  * (23) R.R. Caldwell, Phys. Lett. B 545, 23 (2003).
  * (24) A. Sen, J. High Energy Phys., 0204 048 (2002).
  * (25) M.R. Garousi, Nucl. Phys. B 584 284 (2000).
  * (26) E.A. Bergshoeff et al., J. High Energy Phys. 5 009 (2000).
  * (27) V. Gorini et al., Phys. Rev. D 67, 063509 (2003).
  * (28) P. H. Chavanis, Eur. Phys. J. Plus 129, 38 (2012).
  * (29) K. Schwarzschild, Sitzungsb. der König. Preuss. Akad. der Wissen. 7, 189 (1916).
  * (30) H. A. Lorentz, A. Einstein, H. Minkowski, H. Weyl, The Principle of Relativity (New York: Metheun & Co.) 175 (1923).
  * (31) W. de Sitter, Mon. Not. R. Astron. Soc. 76, 699 (1916); 77, 155 (1916).
  * (32) R. C. Tolman, Phys. Rev. 55, 364 (1939).
  * (33) R. J. Adlar, J. Math. Phys. 15, 727 (1974).
  * (34) H. A. Buchdahl, Astrophys. J. 147, 310 (1967).
  * (35) P. C. Vaidya and R. Tikekar, J. Astrophys. & Astron. 3, 325 (1982).
  * (36) M. C. Durgapal, J. Phys. A. Math. Gen. 15, 2637 (1982).
  * (37) H. Knutsen, Astrophys. & Space Science 140, 385 (1988).
  * (38) D. Kramer et al., “Exact solutions of Einstein’s equations”, Cambridge: CUP, (1980).
  * (39) P. S. Negi, Int. J. Theor. Phys. 45, 1684 (2006).
  * (40) Edward A. Milne, Relativity, Gravitation and World Structure, (Oxford University Press) (1935).
  * (41) H. Bondi and T. Gold, Mon. Not. R. Astron. Soc. 108, 252 (1948); F. Hoyle, Mon. Not. R. Astron. Soc. 108, 372 (1948).
  * (42) R. A. Knop et al., Astrophys. J. 598,102 (2003).
  * (43) M. Tegmark et al., Astrophys. J. 606 702 (2004).
  * (44) T. M. C. Abbott et al., Astrophys. J. Lett., 872, L30 (2019).
  * (45) Jing-Fei Zhang et al., Physics Letters B 799, 135064 (2019).
  * (46) W. Zimdahl and D. Pavon, D., Gen. Relativ. Gravit. 36, 1483 (2004).
  * (47) O. Bertolami and D. Pavon, D., Phys. Rev. D 61, 064007 (2000).
  * (48) N. Banerjee and D. Pavon, Phys. Rev. D 63, 043504 (2001).
  * (49) G. F. R. Ellis and M. Madsen, M., Class. Quantum Gravity 8, 667 (1991).
  * (50) 43\. Sahni, V.: Lect. Notes Phys. 653, 141 (2004).
  * (51) T. D. Saini et al., Phys. Rev. Lett. 85, 1162 (2000).
  * (52) J. Simon et al., Phys. Rev. D 71, 123001 (2005).
  * (53) J. V. Cunha J. V. J. A. S. Lima, Mon. Not. R. Astr. Soc. 390, 210 (2008).
  * (54) E. M̈ortsell and C. Clarkson, J. Cosm. Astropar. Phys., 2009, 044 (2009).
  * (55) S. K. J. Pacif et al., Int. J. Geom. Meth. Mod. Phys., 14(7), 1750111 (2017).
  * (56) S. Capozziello, R. Lazkoz, and V. Salzano, Phys. Rev. D 84, 124061 (2011).
  * (57) D. Stern et al., J. Cosmol. Astropart. Phys., 02, 008 (2010).
  * (58) J. Simon, L. Verde, R. Jimenez, Phys. Rev. D, 71, 123001 (2005).
  * (59) M. Moresco et al., J. Cosmol. Astropart. Phys., 08, 006 (2012).
  * (60) C. Zhang et al., Research in Astron. and Astrop., 14, 1221 (2014).
  * (61) M. Moresco et al., J. Cosmol. Astropart. Phys., 05, 014 (2016).
  * (62) A.L. Ratsimbazafy et al., Mon. Not. Roy. Astron. Soc., 467, 3239 (2017).
  * (63) M. Moresco, Mon. Not. Roy. Astron. Soc.: Letters. , 450, L16 (2015).
  * (64) E. Gaztañaga, A. Cabre, L. Hui, Mon. Not. Roy. Astron. Soc., 399, 1663 (2009).
  * (65) A. Oka et al., Mon. Not. Roy. Astron. Soc., 439, 2515 (2014).
  * (66) Y. Wang et al., Mon. Not. Roy. Astron. Soc. 469, 3762 (2017).
  * (67) C. H. Chuang, Y. Wang, Mon. Not. Roy. Astron. Soc., 435, 255 (2013).
  * (68) S. Alam et al., Mon. Not. Roy. Astron. Soc., 470, 2617 (2017).
  * (69) C. Blake et al., Mon. Not. Roy. Astron. Soc., 425, 405 (2012).
  * (70) C. H. Chuang et al. , Mon. Not. Roy. Astron. Soc., 433, 3559 (2013).
  * (71) L. Anderson et al., Mon. Not. Roy. Astron. Soc. , 441, 24 (2014).
  * (72) N. G. Busca et al., Astron. Astrop., 552, A96 (2013).
  * (73) J. E. Bautista et al. Astron. Astrophys., 603, A12 (2017).
  * (74) T. Delubac et al., Astron. Astrophys. , 574, A59 (2015).
  * (75) A. Font-Ribera et al., J. Cosmol. Astropart. Phys., 05, 027 (2014).
  * (76) G. S. Sharov, V.O. Vasiliev, Mathematical Modelling and Geometry 6, 1 (2018).
  * (77) E. Macaulay et al., Mon. Not. R. Astron. Soc. 486(2), 2184 (2019).
  * (78) N. Suzuki et al., Astrophys. J. 746, 85 (2012).
  * (79) S. Nesseris and L. Perivolaropoulos, Phys. Rev. D 70, 043531 (2004).
  * (80) C. Blake et al., Mon. Not. Roy. Astron. Soc. 418, 1707 (2011).
  * (81) W. J. Percival et al., Mon. Not. Roy. Astron. Soc. 401, 2148 (2010).
  * (82) F. Beutler et al., Mon. Not. Roy. Astron. Soc. 416, 3017 (2011).
  * (83) N. Jarosik et al., Astrophys. J. Suppl. 192, 14 (2011).
  * (84) D. J. Eisenstein et al., Astrophys. J. 633, 560 (2005).
  * (85) R. Giostri et al., J. Cosm. Astropart. Phys. 1203, 027 (2012).
  * (86) V. Sahni, et al. JETP Lett. 77, 201 (2003).
  * (87) U. Alam, et al., Mon. Not. R. Astron. Soc. 344, 1057 (2003).
  * (88) M. Sami et al., Phys. Rev. D 86, 103532 (2012)
  * (89) R. Myrzakulov,et al., J. Cosm. Astrop. Phys. 1310, 047 (2013).
  * (90) V. Sahni, A. Shaifeloo, A. A. Starobinsky, Phys. Rev. D 78, 103502 (2008).
  * (91) C. Zunckel, etal., Phys. Rev. Lett. 101,181301 (2008).
  * (92) M. Shahalam, et al., Mon. Not. Roy. Astron. Soc. 448, 2948 (2015)
  * (93) A. Agarwal et al., Int. J. Mod. Phys. D 28, 1950083 (2019).
  * (94) F. Melia, Astronom. J., 144, 110 (2012).
  * (95) G. F. R. Ellis and M. S. Madsen, Class. Quantum Grav. 8, 667 (1991).
  * (96) X. Zhou, Chinese Phys. B 18, 3115 (2009).
  * (97) F. Hoyle, G. Burbidge, & J. V. Narlikar, Astrophys. J., 410(2), 437 (1993).
  * (98) D. Lohiya and M. Sethi, Class. Quantum Grav. 16, 1545 (1999).
  * (99) Ozgur Akarsu et al., J. Cosmol. Astropart. Phys. 01, 22 (2014).
  * (100) J. D. Barrow, N. J. Nunes, Phys. Rev. D 76, 043501 (2007).
  * (101) Ritika Nagpal et al., Annals of Phys. 405, 234 (2019).
  * (102) S. D. Odintsov and V. K. Oikonomou, Phys.Rev. D 92(2), 024016 (2015).
  * (103) Alvaro de la Cruz-Dombriz et al., Phys. Rev. D 97, 104040 (2018).
  * (104) M. S. Berman, Nuovo Cimento B, Series 74B, 182 (1983)
  * (105) N Banerjee, S. Das and K. Ganguly, Pramana 74(3), 481 (2010).
  * (106) J. P. Singh, Astrophys. Space Sci. 318, 103 (2008).
  * (107) S. K. J. Pacif and B. Mishra, Res. Astron. Astrophys. 15(12), 2141 (2015).
  * (108) S. Bhattacharjee and P. K. Sahoo, Phys. Dark Univ. 28, 100537 (2020).
  * (109) P. H. R. S. Moraes et al., Adv. Astron. 2019, 8574798 (2019).
  * (110) F. Cannata and A. Y. Kamenshchik, Int. J. Mod. Phys. D 20, 121 (2010).
  * (111) S. Noniri and S. D. Odintsov, Gen. Relativ. Grav. 38(8), 1285 (2006).
  * (112) Paul H. Frampton et al., Phys. Lett. B 708, 204 (2012).
  * (113) S. Nojiri et al., J. Cosmol. Astropart. Phys. 1509, 044 (2015).
  * (114) K. Bamba et al., Phys. Lett. B 732, 349 (2014).
  * (115) A. Singh, Astrophys. Space Sci. 365, 54 (2020).
  * (116) Abdulla Al Mamon, Int. J. Mod. Phys. D 26(11), 1750136 (2017).
  * (117) M. S. Berman and F. M. Gomide, Gen Relativ. Grav. 20, 191 (1998).
  * (118) O. Akarsu and T. Dereli, Int. J. Theoret. Phys. 51, 612 (2012).
  * (119) P. K. Sahoo et al., Mod. Phys. Lett. A, 33(33), 1850193 (2018).
  * (120) P. K. Sahoo et al., New Astron. 60, 80 (2018).
  * (121) Abdussattar and S. R. Prajapati, Astrophys. Space Sci. 331, 657 (2011).
  * (122) M. A. Bakry, A. T. Shafeek, Astrophys. Space Sci. 364,135 (2019).
  * (123) N. Banerjee and S. Das, Gen. Relativ. Grav. 37, 1695 (2005).
  * (124) A. G. Riess et al., Astrophy. J. 607, 665 (2004).
  * (125) B. Santos, J. C. Carvalho, J. S. Alcaniz, Astropart. Phys. 35, 17 (2011).
  * (126) Abdulla Al Mamon and Sudipta Das, Int. J. Mod. Phys. D 25(03), 1650032 (2016).
  * (127) R. Nair et al., J. Cosmol. Astropart. Phys. 01, 018 (2012).
  * (128) L. Xu and H. Liu, Mod. Phys. Lett. A 23, 1939 (2008).
  * (129) Y. G. Gong and A. Wang, Phys. Rev. D 73, 083506 (2006).
  * (130) J. V. Cunha and J. A. S. Lima, Mon. Not. R. Astron. Soc. 390, 210 (2008).
  * (131) S. del Campo et al., Phys. Rev. D 86, 083509 (2012).
  * (132) D. Pavon et al., Proc. MG13 Meeting Gen. Relativ., Stockholm Univ., Sweden (2012).
  * (133) E. E. O. Ishida et al., Astroparticle Phys. 28, 7 (2007).
  * (134) Abdulla Al Mamon, Mod. Phys. Lett. A 33(10), 1850056 (2018).
  * (135) Abdulla Al Mamon and S. Das, Eur. Phys. J. C 77(7), 495 (2017).
  * (136) Jaime Roman-Garza et al., Eur. Phys. J. C 79, 890 (2019).
  * (137) Z. X. Zhai et al., Physics Letters B 727, 8 (2013).
  * (138) A. Mukherjee and N. Banerjee, Phys. Rev. D 93, 043002 (2016).
  * (139) C. Ekart, Phys. Rev. 58, 919 (1940).
  * (140) K. Karami, S. Ghaari, and J. Fehri, Eur. Phys. J. C 64(1), 85 (2009).
  * (141) G. M. Kremer, Phys. Rev. D 68 123507 (2003).
  * (142) P. H. Chavanis, Universe 1, 357 (2015).
  * (143) A. Yu. Kamenshchik, U. Moschella and V. Pasquier, Phys. Lett. B 511, 265 (2001).
  * (144) M. C. Bento, O. Bertolami and A. A. Sen, Phys. Rev. D 66, 043507 (2002).
  * (145) H. B. Benaoum, arxiv:hep-th/0205140v1 (2002).
  * (146) U. Debnath, Astrophys. Space Sci. 312, 295 (2007).
  * (147) W. Chakraborty, U. Debnath, Gravit. and Cosmol. 16, 223 (2010).
  * (148) S. Nojiri and S. D. Odintsov, Phys. Rev. D 70, 103522 (2004).
  * (149) Q. Zhang et al., Eur. Phys. J. C 75(7), 300 (2015).
  * (150) D. Wang et al., Eur. Phys. J. C 77(4), 263 (2017).
  * (151) Jun-Chao Wang, Xin-He Meng, Comm. Theor. Phys., 70(12), 67 (2018).
  * (152) J. Weller and A. Albrecht, Phys. Rev. D 65, 103512 (2002).
  * (153) H. K. Jassal, J. S. Bagla, and T. Padmanabhan, Mon. Not. R. Astron. Soc. Letters 356(1), L11 (2005).
  * (154) Dao-Jun Liu et al., Mon. Not. R. Astron. Soc. 388, 275 (2008).
  * (155) M. Chevallier and D. Polarski, Int. J. Mod. Phys. D 10, 213 (2001).
  * (156) G. Pantazis, S. Nesseris and L. Perivolaropoulos, Phys. Rev. D 93, 103503 (2016).
  * (157) R. Lazkoz, V. Salzano, and I. Sendra, Phys. Lett. B 694, 198 (2010).
  * (158) G. Efstathiou, Mon. Not. R. Astron. Soc. 310, 842 (1999).
  * (159) L. Feng and T. Lu, J. Cosmol. Astropart. Phys. 11, 034 (2011).
  * (160) E. M. Barboza, Jr. and J. S. Alcaniz, J. Cosmol. Astropart. Phys. 02, 042 (2012).
  * (161) J. -Z. Ma and X. Zhang, Phys. Lett. B 699, 233 (2011).
  * (162) C. -J. Feng et al., J. Cosmol. Astropart. Phys. 09, 023 (2012).
  * (163) U. Alam et al., Mon. Not. R. Astron. Soc. 354, 275 (2004).
  * (164) S. Hannestad, E. Mrtsell, J. Cosm. Astropart. Phys., 0409, 001 (2004).
  * (165) J. Weller, A. Albrecht, Phys.Rev. D 65, 103512 (2002).
  * (166) I. Sendra and R. Lazkoz, Mon. Not. R. Astron. Soc., 422(1), 776 (2012).
  * (167) A. R. Cooray, D. Huterer, Astrophys. J. 513, L95 (1999).
  * (168) Hai-Nan Lin, Xin Li, Li Tang, Chinese Physics C 43, 075101 (2019).
  * (169) W. Yang et al., Phys. Rev. D 99, 043543 (2019).
  * (170) E. Elizalde et al., Int. J. Mod. Phys. D 28(01), 1950019 (2019).
  * (171) C. Wetterich, Physics Letters B 594(1-2), 17 (2004).
  * (172) S. Pan et al., Phys. Rev. D 98, 063510 (2018).
  * (173) J. Z. Ma and X. Zhang, Phys. Lett. B 699, 233 (2011).
  * (174) S. Sello, arXiv:1308.0449 [astro-ph.CO]
  * (175) M. Ozer and M.O. Taha, Nucl. Phys. B 287, 776 (1987).
  * (176) A-M. M. Abdel-Rahman, Gen. Relativ. Gravit. 22, 655 (1990).
  * (177) S. K. J. Pacif and Abdussattar, Eur. Phys. J. Plus 129, 244 (2014).
  * (178) Abdussattar and S. R. Prajapati, Chin. Phys. Lett. Vol. 28(2), 029803 (2011).
  * (179) R. G. Vishwakarma, Gen. Relativ. Grav. 33(11), 1973 (2001).
  * (180) R. G. Vishwakarma, Abdussattar and A. Beesham, Phys. Rev. D 60, 063507 (1999).
  * (181) Abdulla Al Mamon, Mod. Phys. Lett. A 33(20), 1850113 (2018).
  * (182) S. Das et al., Res. Astron. Astrophys. 18(11), 131 (2018).
  * (183) M. Rezaei et al., Phys. Rev. D 100, 023539 (2019).
  * (184) Abdulla Al Mamon, K. Bamba and S. Das, Eur. Phys. J. C 77(1), 29 (2017).
  * (185) R. G. Vishwakarma, Class. Quantum Grav. 18, 1159 (2001).
  * (186) J. M. Overduin and F. I. Cooperstock, Phys. Rev. D 58, 043506 (1998).
  * (187) S. G. Rajeev, Phys. Lett. B 125, 144 (1983).
  * (188) A. D. Linde, JETP Lett. 19, 183 (1974).
  * (189) A. Beesham, Phys. Rev. D 48, 3539 (1993).
  * (190) J. C. Carvalho, J A S Lima and I Waga, Phys. Rev. D 46, 2404 (1992).
  * (191) S. Ray et al., Int. J. Theor. Phys. 48(9), 2499 (2009).
  * (192) Arbab I Arbab, Grav. Cosmol. 8, 227 (2002).
  * (193) J. W. Moffat, Los Alamos report astro-ph/9608202 (1996).
  * (194) P.J.E. Peebles, B. Ratra, Astrophys. Phys. J. 325, L17 (1988).
  * (195) V.K. Oikonomou, N.Th. Chatzarakis, Annals of Phys. 411, 167999 (2019).
  * (196) M. Shahalam, M. Sami, Anzhong Wang, Phys. Rev. D 98, 043524 (2018).
  * (197) M. Shahalam et al., Gen. Relativ. Gravit. 51, 125 (2019).
  * (198) I. Ya. Aref’eva, AIP Conf. Proc. 826, 301 (2006).