Phys. Dark Universe 32, 100804 (2021)
https://doi.org/10.1016/j.dark.2021.100804

# Late-time acceleration with a scalar field source: Observational constraints
and statefinder diagnostics

S. K. J. Pacif  0000-0003-0951-414X<EMAIL_ADDRESS>Department of
Mathematics, School of Advanced Sciences, Vellore Institute of Technology,
Vellore 632014, Tamil Nadu, India.  Simran Arora  0000-0003-0326-8945
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute of
Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.  P.K. Sahoo  0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.

###### Abstract

This article discusses a dark energy cosmological model in the standard theory
of gravity - general relativity with a broad scalar field as a source. Exact
solutions of Einstein’s field equations are derived by considering a
particular form of deceleration parameter $q$, which shows a smooth transition
from decelerated to accelerated phase in the evolution of the universe. The
external datasets such as Hubble ($H(z)$) datasets, Supernovae (SN) datasets,
and Baryonic Acoustic Oscillation (BAO) datasets are used for constraining the
model par parameters appearing in the functional form of $q$. The transition
redshift is obtained at $z_{t}=0.67_{-0.36}^{+0.26}$ for the combined data set
($H(z)+SN+BAO$), where the model shows signature-flipping and is consistent
with recent observations. Moreover, the present value of the deceleration
parameter comes out to be $q_{0}=-0.50_{-0.11}^{+0.12}$ and the jerk parameter
$j_{0}=-0.98_{-0.02}^{+0.06}$ (close to 1) for the combined datasets, which is
compatible as per Planck2018 results. The analysis also constrains the omega
value i.e., $\Omega_{m_{0}}\leq 0.269$ for the smooth evolution of the scalar
field EoS parameter. It is seen that energy density is higher for the
effective energy density of the matter field than energy density in the
presence of a scalar field. The evolution of the physical and geometrical
parameters is discussed in some details with the model parameters’ numerical
constrained values. Moreover, we have performed the state-finder analysis to
investigate the nature of dark energy.

Observational Constraints; Deceleration parameter; scalar field; Statefinder
parameters; $Om$ diagnostic

###### pacs:

04.20.-q, 04.20.Jb, 98.80.Es

##  I Introduction

Cosmological observations indicate that our universe is going through a phase
of an accelerated expansion Riess ; Perlmutter , which is also supported by
the recent SNe Ia observations SNIa , CMB observations CMB , BAO peak
experiments BAO and $H(z)$ measurements Hz . These observations also indicate
that the cosmological entity is responsible for the acceleration. In addition
to that, this entity should also create an anti-gravitational effect to push
the universe apart. The unknown force responsible for the accelerated
expansion possessing a negative pressure is generally termed ”dark
energy”(DE). According to the $\Lambda$CDM model, the best current measurement
for dark energy is 69 percentage, i.e., $\frac{3}{4}$th of the total energy in
the present-day observable universe. Since ordinary baryonic matter does not
have such an equation of state, let alone to account for such a lion-share of
the energy budget of the universe, several alternate scenarios have been
proposed and investigatedalternate .

Since dark energy is mysterious and not much idea about its nature, several
dark energy candidates have been proposed. Out of these various possible
choices to study dark energy, Einstein’s cosmological constant ($\Lambda$)
introduced in 1917, serves the best and simplest candidate, as described in
the literature. This suggests that the repulsive nature of $\Lambda$ is
responsible for accelerating the universe with the equation of state
$\omega=-1$. However, this authentic candidate suffers from some long-standing
cosmological constant problem and also the constant equation of state. As we
know, the EoS parameter is the connection between energy density and pressure
i.e. $\omega=\frac{p}{\rho}$. The EoS parameter is used to characterize the
universe’s decelerated and accelerated expansion. It categorizes different
phases of the universe as, If $\omega=\frac{1}{3}$, the model indicates the
phase dominated by radiation, while $\omega=0$ represents the phase dominated
by matter. In the current accelerated period of evolution, the quintessence
period is shown by $-1<\omega\leq 0$ and the cosmological constant
$\omega=-1$, i.e., the $\Lambda$CDM model and the phantom age by $\omega<-1$.
The recent fine-tuning problem can be minimized by considering the equation of
state as time-dependent. One such model having this property is the scalar
field, also known as quintessence Sahni . The presence of scalar fields $\phi$
is predicted by several fundamental physics theories, encouraging to study the
dynamic properties of scalar fields in cosmology. A wide range of scalar-field
dark energy models has been suggested so far. Among several, these include
quintessence, K-essence, tachyon, phantoms etc. Quintessence depends on scalar
fields’ potential energy to contribute to the acceleration of the late time
universe. The potential $V(\phi)$ has the property such that scalar fields are
rotating down potential approach a common evolutionary path. For large values
of $\phi$, the potential becomes flat, winding scalar fields to slow down,
allowing the universe to accelerate. This ordinary scalar field model is a
viable alternative that has a dynamical equation of state (EoS) wherein the
EoS parameter $\omega$ ranges in between $-1/3$ to $-1$. A route to dark
energy is also provided by Chaplygin gas AY/2001 which has a peculiar EoS and
also phantom field Bamba/2012 for which $\omega$ crosses $-1$.

Implementing SN distance measurements Riess ; Perlmutter with baryon acoustic
peak measurements in the power spectrum of cosmic microwave background(CMB)
suggests that our universe is accelerating and composed predominantly of
baryon, dark matter, and dark energy. The cosmic microwave background (CMB) is
landmark proof of the universe’s Big Bang origin. Precise CMB measurements are
essential to cosmology, as any proposed model of the universe must clarify
this radiation. Current observational data is used to constrain the $f(z)$
models and describe distance-redshift relations. Here, $f(z)$ refers to
constraining the $H(z)$ models. Nevertheless, given its role as a hypothesis,
the $\Lambda$CDM model was immensely successful in explaining most
cosmological observations. In addition to a moderately significant difference
with broad angular scale CMB observations, $\Lambda$CDM provided an almost
ideal match to measurements made by the Wilkinson Microwave Anisotropy probe
(WMAP) satellite project mission. Even in parallel with corresponding
observational data such as BAO surveys, Type Ia supernovae, and Hubble
constant direct measurements. The Baryon Acoustic Oscillations (BAO) matter
clustering provides a ‘standard ruler’ for length scale in cosmology the same
way as supernovae offer a ‘standard candle’ for astronomical observations.
There are different observational data also discussed in Valentino ; J.K.Singh
: Cosmic microwave background radiation (CMB) acts as authentication of big
bang theory, Sloan digital sky survey (SDSS), which provides a map of the
distribution of the galaxy and encodes the existing variations in the
universe, Baryon acoustic oscillations (BAO) Aghanim estimates large-scale
structures in the universe that make the dark energy more attractive, QUASARS
brings out the matter between observers and quasars, SNe Ia observations are
the instruments for measuring the cosmic distances known as standard candles.
Compilations of Hubble measurementsCasertano are regarded as cosmic
chronometers with a sample covering the redshift range of $0<z<1.97$. And the
latest type 1048 SNe Ia covers the redshift range of $0.01<z<2.26$. Also, the
luminosity distance data of 1048 type Ia supernovae from Pantheon Scolnic is
recently developed.

The current practice for finding the cosmic evolution is to develop the model
from observational data. The method of finding a viable cosmological model is
called reconstruction. Starobinsky Staro/1998 , in his pioneering work,
considered the scalar field potential, which was used as the dark energy, and
has reconstructed the cosmological model using the density perturbation data.
Later, the observational data of distance measurement from supernova has been
utilized in Huterer/1999 ; Saini/2000 . The effective equation of dark energy
through the parametrization of the quintessence scalar field and potential is
discussed in Guo/2005 . In general, two types of reconstruction available in
the literature. The first one is based on the parametric form of the DE
equation of state $\omega_{DE}$ Mukherjee/2016 and the estimation of
parameters from the observational data. The second one is a non-parametric
formulation attempting to estimate the evolution of $\omega_{DE}$ directly
from the observational data without considering any parametric form nair/2014
.

In cosmology, authors try to understand the cosmic acceleration through
analyzing the kinematic variables like the Hubble parameter ($H$), the
deceleration parameter ($q$), and the jerk parameter ($j$), where all these
parameters are derived from the derivatives of the scale factor $a$ Mamon/2018
. The kinematic approach is more advantageous as it does not depend on any
model-specific assumptions. It is described by some metric theory of gravity
and is assumed that the present Universe is isotropic and homogeneous at
cosmological scales parameter . In the literature, one can find many attempts
to constrain the current values of $H,q$, and the jerk $j$ by parametrizing
the deceleration parameter $q$ dece .

In this work, we have considered the scalar field model in flat FLRW space-
time. We come across the two field equations along with the conservation
equations in the scalar field and matter field Sudipta ; Narayan . Further,
the proposed deceleration parameter is considered so that the difficulty in
solving these equations with four unknowns $a(t)$, $\rho_{\phi}$, $p_{\phi}$
and $V(\phi)$ can be reduced. Therefore, we obtained the expressions for
$\rho_{\phi}$, $p_{\phi}$. The model parameters are constrained using the
Hubble, BAO, and SN datasets. The behavior of the equation of state parameter
$\omega=\frac{p_{\phi}}{\rho_{\phi}}$ and the deceleration parameter has been
observed showing the transition from decelerated to accelerated phase. The
various kinematic variables such as jerk, snap, and lerk are studied,
indicating the accelerated expansion. Further, the temporal evolution of dark
energy mimicked by our model has been shown by the statefinder diagnostics r-s
and r-q pairs. The two planes describe quintessence dark energy, the Chaplygin
gas model, and the $\Lambda$CDM model.

The manuscript is organized as follows: In Section II, we present the Einstein
field equations along with the scalar field. In Section III, we derive the
kinematic quantities from the second-degree parametrization of the
deceleration parameter and found the observational constraints on the model
parameters involved. The kinematic parameters of the cosmological model are
discussed in Section IV. In Section V, we present some geometrical
diagnostics. Finally, in Section VI, we present our results and conclusions.

##  II EFEs & scalar field formalism

To begin our analysis, we consider the homogeneous and isotropic flat
Friedmann-Lemaitre-Robertson-Walker (FLRW) space-time as,

$ds^{2}=dt^{2}-a(t)^{2}\left[dr^{2}+r^{2}d\Omega^{2}\right],$ (1)

where $a(t)$ is the scale factor of the universe and
$d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$.

For a general scalar field together with the cold dark matter as sources, the
Einstein field equations (EFEs) are obtained as Sudipta ; Narayan ,

$3\left(\frac{\dot{a}}{a}\right)^{2}=\rho_{m}+\frac{1}{2}\dot{\phi}^{2}+V(\phi),$
(2)

and

$2\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}=-\frac{1}{2}\dot{\phi}^{2}+V(\phi).$
(3)

Here, $\rho_{m}$ is the matter density, $\phi$ is the scalar field and
$V(\phi)$ is the scalar field potential. The overhead dot denote the
derivative of the quantity with respect to the cosmic time ‘$t$’. The units
have been chosen in such a way that $8\pi G=c=1$. The energy density and
pressure due to the field $\phi$ are,

$\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi)\text{,}$ (4)

$p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi)\text{.}$ (5)

The equation of state for the scalar field is given by,
$\omega_{\phi}=\frac{p_{\phi}}{\rho_{\phi}}$. We also consider minimal
interaction between cold dark matter and the dark energy for which the
conservation of energy and momentum yield the continuity equations for matter
and scalar fields separately as,

$\dot{\rho_{m}}+3H\rho_{m}=0\text{,}$ (6)

$\dot{\rho_{\phi}}+3H(\rho_{\phi}+p_{\phi})=0\text{,}$ (7)

where $H=$ $\frac{\dot{a}}{a}$ is the Hubble parameter. On solving Eqs. (6),
we come across the solution for the matter energy density $\rho_{m}$ as,

$\rho_{m}=\rho_{0}a^{-3}.$ (8)

With the understanding, $\rho_{eff}=\rho_{\phi}+\rho_{m}$ and
$p_{eff}=p_{\phi}+p_{m}$, from Eqs. (2), (3) and (8), we may obtain the
general expressions for the energy densities and pressure and potential
function of the scalar field as,

$\rho_{eff}=3H^{2}\text{, }\rho_{\phi}=3H^{2}-\rho_{m}\text{,}$ (9)

$p_{eff}=p_{\phi}=(2q-1)H^{2}\text{, (since }p_{m}=0\text{),}$ (10)
$V(\phi)=(2-q)H^{2}-\frac{\rho_{m}}{2}\text{,}$ (11)

where $q=-\frac{a\ddot{a}}{\dot{a}^{2}}$ is the deceleration parameter.

##  III Kinematic variables $\&$ Observational constraints

The kinematic variables play an important role in a cosmological model study
e.g. the deceleration parameter describes the behavior of the universe whether
it is ever decelerating, ever accelerating or has any transition phase or
multiple transition phases etc. Similarly, the equation of state parameter
describes the physical significance of the energy sources in the evolution of
the universe. Also, the above system of field equations need one more equation
to close the system for the complete determination of other cosmological
parameters and for their evolutionary behavior e.g. pressure, energy densities
and EoS parameter, potential function. This supplementary equation can be
assumed as a functional form of any cosmological parameter. Parameterizations
of Hubble parameter, deceleration parameter, EoS parameter etc. provide the
necessary constraint equation (see pacif2016 ). In this paper, we employ a
generalized varying deceleration parameter of the second degree introduced in
Bakry and Shafeek of the form,

$q(t)=(8\alpha^{2}-1)-12\alpha t+3t^{2},$ (12)

where $\alpha$ is an arbitrary constant and constrained from a $\chi^{2}$ test
using some external observational datasets. The corresponding Hubble parameter
reads,

$H(t)=\frac{1}{t(2\alpha-t)(4\alpha-t)}.$ (13)

Integrating the above equation, we get the explicit expression of scale factor
as,

$a(t)=\beta\frac{[t(4\alpha-t)]^{\frac{1}{8\alpha^{2}}}}{(2\alpha-t)^{\frac{1}{4\alpha^{2}}}},$
(14)

where $\beta$ is an integrating constant.
As we are interested in studying the late-time universe, we should express the
above geometrical parameters in terms of redshift $z$ related to scale factor
as $a(t)=1/(1+z)$. So, the kinematic quantities $H(z)$ and $q(z)$ are
expressed as functions of redshift as,

$H(z)=\frac{H_{0}\left((\beta(z+1))^{8\alpha^{2}}+1\right)^{3/2}}{\left(\beta^{8\alpha^{2}}+1\right)^{3/2}(z+1)^{4\alpha^{2}}},$
(15)
$q(z)=-\frac{(\beta(z+1))^{8\alpha^{2}}+\alpha^{2}\left(4-8(\beta(z+1))^{8\alpha^{2}}\right)+1}{(\beta(z+1))^{8\alpha^{2}}+1}.$
(16)

The expressions for $H(z)$ and $q(z)$ in Eqs. (15) and (16) contain two free
parameters $\alpha$ and $\beta$ (let us call them model parameters). As we can
see, the evolution depends on the values of model parameters $\alpha$ and
$\beta$, their values should be chosen properly to describe the current
evolution. So, we will constrained their values through some observational
datasets.

As we know, the modern cosmology is heavily dependent on observations and
describe the validation of any theoretical model obtained and also find
constraints on the model parameters, here in this study, we find observational
constraints on our model parameters $\alpha$ & $\beta$ using observational
Hubble datasets (OHD) containing a sample of $57$ data points, Type Ia
supernovae datasets (known as standard candles, used to measure the expansion
of Universe) containing a sample of $580$ data points (Union$2.1$ compilation
datasets) and Baryon Acoustic Oscillations (BAO) datasets (used to measure the
structure in the Universe).

###  III.1 OHD sample

A list of $57$ points of Hubble parameter data in the redshift range
$0.07\leqslant z\leqslant 2.42$ is compiled by Sharov and Vasiliev sharov (see
the Appendix in sharov ) is considered here. We also take a prior for the
present value of the Hubble constant from Planck 2018 results Hz-Plank as
$H_{0}=67.8$ $Km/s/Mpc$ to complete the data set. The mean values of the model
parameters $\alpha$ & $\beta$ are determined by minimizing the chi square
value (which is equivalent to the maximum likelihood analysis). The chi square
value is given by,

$\chi_{OHD}^{2}(p_{s})=\sum\limits_{i=1}^{28}\frac{[H_{th}(p_{s},z_{i})-H_{obs}(z_{i})]^{2}}{\sigma_{H(z_{i})}^{2}},$
(17)

where, $H_{th}$ and $H_{obs}$ respectively refers to the theoretical and
observed value of Hubble parameter $H$ and $p_{s}$ refers to the parameters of
the model to be constrained. $\sigma_{H(z_{i})}$ stands for the standard error
in the observed value of $H$.

###  III.2 Union 2.1 compilation datasets sample

For our analysis, we have used the Union $2.1$ compilation supernovae datasets
SNeIa containing $580$ points. The chi square formula for the supernovae
datasets is given by,

$\chi_{SN}^{2}(\mu_{0},p_{s})=\sum\limits_{i=1}^{580}\frac{[\mu_{th}(\mu_{0},p_{s},z_{i})-\mu_{obs}(z_{i})]^{2}}{\sigma_{\mu(z_{i})}^{2}},$
(18)

where, $\mu_{th}$ and $\mu_{obs}$ are respectively, the theoretical and
observed distance modulus with the standard error in the observed value
denoted by $\sigma_{\mu(z_{i})}$. The distance modulus $\mu(z)$ is defined by
$\mu(z)=m-M=5LogD_{l}(z)+\mu_{0},$where $m$ and $M$ are respectively, the
apparent and absolute magnitudes of a standard candle. The luminosity distance
$D_{l}(z)$ and the nuisance parameter $\mu_{0}$ are defined by
$D_{l}(z)=(1+z)H_{0}\int_{0}^{z}\frac{1}{H(z^{\ast})}dz^{\ast}$ and
$\mu_{0}=5Log\Big{(}\frac{H_{0}^{-1}}{Mpc}\Big{)}+25$ respectively. In order
to calculate luminosity distance, we have restricted the series of $H(z)$ upto
tenth term then integrate the approximate series to obtain the luminosity
distance.

###  III.3 BAO datasets sample

Baryon Acoustic Oscillation (BAO) measures the structures in the universe from
very large scales. For BAO, we have used the datasets of
$\frac{d_{A}(z_{\ast})}{D_{V}(z_{BAO})}$ gio , where $z_{\ast}$ is the photon
decoupling redshift (according to Planck 2018 results Hz-Plank ,
$z_{\ast}=1091$), the comoving angular-diameter distance
$d_{A}(z)=\int\limits_{0}^{z}\frac{dz^{{}^{\prime}}}{H(z^{{}^{\prime}})}$ and
$D_{v}(z)=\left[d_{A}(z)^{2}\frac{z}{H(z)}\right]^{\frac{1}{3}}$. The datasets
used for our analysis consisteing of six points (from surveys of SDSS(R) padn
, 6dF Galaxy survey 6df , BOSS CMASS boss and WiggleZ wig ) and given in the
following Table I.

$z_{BAO}$ | $0.106$ | $0.2$ | $0.35$ | $0.44$ | $0.6$ | $0.73$
---|---|---|---|---|---|---
$\frac{d_{A}(z_{\ast})}{D_{V}(z_{BAO})}$ | $30.95\pm 1.46$ | $17.55\pm 0.60$ | $10.11\pm 0.37$ | $8.44\pm 0.67$ | $6.69\pm 0.33$ | $5.45\pm 0.31$

Table I: Values of $\frac{d_{A}(z_{\ast})}{D_{V}(z_{BAO})}$ for different of
$z_{BAO}$.

The chi square for BAO datasets ($\chi_{BAO}^{2}$) is defined as,

$\chi_{BAO}^{2}=X_{BAO}^{T}C_{BAO}^{-1}X_{BAO}$ (19)

Where,

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

and the inverse covariance matrix is given by,

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

###  III.4 Results

With the above samples, we have found the likelihood contours for our model
parameters $\alpha$ & $\beta$ with $1$-$\sigma$, $2$-$\sigma$ and $3$-$\sigma$
errors in the $\alpha$-$\beta$ plane and shown in the following figures. We
have minimize the chi square for $OHD$ sample independently and then
combinedly as $OHD+BAO$, $OHD+SN$, $SN+BAO$ and finally $OHD+SN+BAO$ and
summarized the constrained values in Table-2.

(a) Contour plot for OHD (b) Contour plot for combined OHD+BAO (c) Contour
plot for combined OHD+SN (d) Contour plot for combined SN+BAO

(e) Contour plot for combined OHD+SN+BAO

Figure 1: Figures show the contour plots for likelihood values of the model
parameters $\alpha$ & $\beta$ with samples of OHD, SN, BAO and some combined
datasets at 1-$\sigma$, 2-$\sigma$, 3-$\sigma$ levels.

We tabulate the constrained values of the model parameters $\alpha$ & $\beta$
as follows together with the minimum chi square values. Also, we calculate the
values of deceleration parameter for different datasets at present ($z=0$)
which are obtained from Fig. 1.

Table.2: Constrained values of the model parameters with minimum chi square
values and the present values of deceleration parameter

$Datasets$ | $\alpha$ | $\beta$ | $\chi_{\min}^{2}/dof$ | $q(z=0)$ | $z_{t}$
---|---|---|---|---|---
$H(z)$ | $0.45_{-0.02}^{+0.02}$ | $1.10_{-0.11}^{+0.11}$ | $0.574$ | $-0.50_{-0.11}^{+0.11}$ | $-0.76_{-0.37}^{+0.27}$
$H(z)+BAO$ | $0.44_{-0.02}^{+0.02}$ | $1.11_{-0.11}^{+0.11}$ | $0.564$ | $-0.48_{-0.11}^{+0.13}$ | $-0.73_{-0.11}^{+0.13}$
$H(z)+SN$ | $0.45_{-0.02}^{+0.02}$ | $1.07_{-0.11}^{+0.11}$ | $0.940$ | $-0.51_{-0.15}^{+0.23}$ | $-0.74_{-0.15}^{+0.23}$
$SN+BAO$ | $0.45_{-0.01}^{+0.02}$ | $1.01_{-0.11}^{+0.11}$ | $0.966$ | $-0.57_{-0.16}^{+0.21}$ | $-0.85_{-0.60}^{+0.44}$
$H(z)+SN+BAO$ | $0.44_{-0.01}^{+0.01}$ | $1.09_{-0.11}^{+0.11}$ | $0.936$ | $-0.50_{-0.11}^{+0.12}$ | $-0.67_{-0.36}^{+0.26}$

The error bar plots of the $OHD$ sample and the Union $2.1$ compilation sample
are shown in the Fig. 2.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=216.81pt,height=180.67499pt]{6.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=216.81pt,height=180.67499pt]{7.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 2: Figures (a) and (b) are respectively the error bar plots for $57$
data points from OHD and for $580$ points from Union$2.1$ compilation
supernovae datasets. Solid red lines are presented model compared with
$\Lambda$CDM model shown in black dashed lines in both the plots.

##  IV Evolution of cosmological and cosmographic parameters

The very useful way of describing an increasing or decreasing rate of
expansion of the universe is to study the deceleration parameter. The form of
deceleration parameter considered here (see equation (16)) contain two
parameters which are constrained through some datasets, so we can now discuss
it’s evolution with the numerical values. The following plot shows the
evolution of $q$ w.r.t. redshift $z$ that explains it’s evolution in the near
past, present evolution and the signature flipping behavior for the above
constrained values of the model parameters $\alpha$ and $\beta$ (see Fig. 3).
As we know, although the negative value of q corresponds to the accelerated
period, the positive q refers to the decelerating phase. We can see from the
figure 3 that the deceleration parameter q varies with z from positive to
negative. This demonstrates a transition from early deceleration to the
universe’s present acceleration.

Figure 3: Evolution of the deceleration parameter as a function of redshift
for different constrained values of model parameters $\alpha$ & $\beta$
showing phase transition and present acceleration.

If we talk about equation of state parameter (EoS), then we see that it
reflects the relation of energy density and pressure which basically connects
to the evolution of the universe. The EoS parameter for radiation dominated
phase is illustrated by $\omega=\frac{1}{3}$ followed by the dust phase i.e
matter dominated phase with $\omega=0$. The cosmological constant is
represented by $\omega=-1$ which is also known as $\Lambda$CDM model. Also
$-1<\omega<0$, shows the quintessence phase whereas $\omega<-1$ is the phantom
stage.

Now, using equations (15) and (16) in equations (9), (10) and (11), we have
the expressions for the pressure, the energy densities and the potential
function of the scalar field can be written as,

$p_{eff}=-H_{0}^{2}\left(2\frac{(\beta(z+1))^{8\alpha^{2}}+\alpha^{2}\left(4-8(\beta(z+1))^{8\alpha^{2}}\right)+1}{(\beta(z+1))^{8\alpha^{2}}+1}+1\right)\frac{\left((\beta(z+1))^{8\alpha^{2}}+1\right)^{3}}{\left(\beta^{8\alpha^{2}}+1\right)^{3}(z+1)^{8\alpha^{2}}}\text{.}$
(20)

$\rho_{eff}=3H_{0}^{2}\frac{\left((\beta(z+1))^{8\alpha^{2}}+1\right)^{3}}{\left(\beta^{8\alpha^{2}}+1\right)^{3}(z+1)^{8\alpha^{2}}}\text{.}$
(21)

$\rho_{\phi}=3H_{0}^{2}\frac{\left((\beta(z+1))^{8\alpha^{2}}+1\right)^{3}}{\left(\beta^{8\alpha^{2}}+1\right)^{3}(z+1)^{8\alpha^{2}}}-3H_{0}^{2}\Omega_{m_{0}}(1+z)^{3}\text{.
}$ (22)

where $\Omega_{m_{0}}=\frac{\rho_{0}}{3H_{0}^{2}}$. The expression for the
potential function $V(\phi)$ of the scalar field is obtained as,

$V(\phi)=H_{0}^{2}\left(2+\frac{(\beta(z+1))^{8\alpha^{2}}+\alpha^{2}\left(4-8(\beta(z+1))^{8\alpha^{2}}\right)+1}{(\beta(z+1))^{8\alpha^{2}}+1}\right)\frac{\left((\beta(z+1))^{8\alpha^{2}}+1\right)^{3}}{\left(\beta^{8\alpha^{2}}+1\right)^{3}(z+1)^{8\alpha^{2}}}-1.5H_{0}^{2}\Omega_{m_{0}}(1+z)^{3}\text{.}$
(23)

The evolution of the effective energy density of the matter energy density,
scalar field energy density and the scalar field potential are shown in the
figure.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=216.81pt,height=180.67499pt]{8.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=216.81pt,height=180.67499pt]{9.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 4: In this figure panel (a) shows the evolution of effective energy
density, matter energy density and scalar field energy density as a function
of redshift while panel (b) shows the evolution of scalar field potential
$V(\phi)$ as function of redshift.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=216.81pt,height=180.67499pt]{Omega.pdf}\hfil\hskip
7.22743pt&\end{array}$

Figure 5: The figure shows the evolution of the density parameters for matter
($\Omega_{m}$) and the scalar field ($\Omega_{\phi}$) as functions of redshift
$z$.

Now, the effective equation of state parameter
$\omega_{eff}=\frac{p_{eff}}{\rho_{eff}}$ reads as,

$\omega_{eff}=-\frac{1}{3}\left(2\frac{(\beta(z+1))^{8\alpha^{2}}+\alpha^{2}\left(4-8(\beta(z+1))^{8\alpha^{2}}\right)+1}{(\beta(z+1))^{8\alpha^{2}}+1}+1\right).$
(24)

Moreover, with the dark energy domination, the expression for the equation of
state of scalar field or the dark energy $\omega_{\phi}$ can be written as,

$\omega_{\phi}=\frac{1}{3}\frac{\frac{\left((\beta(1+z))^{8\alpha^{2}}+1\right)^{3}}{\left(\beta^{8\alpha^{2}}+1\right)^{3}(1+z)^{8\alpha^{2}}}\left(2\frac{(\beta(z+1))^{8\alpha^{2}}+\alpha^{2}\left(4-8(\beta(z+1))^{8\alpha^{2}}\right)+1}{(\beta(z+1))^{8\alpha^{2}}+1}+1\right)}{\Omega_{m0}(1+z)^{3}-\frac{\left((\beta(1+z))^{8\alpha^{2}}+1\right)^{3}}{\left(\beta^{8\alpha^{2}}+1\right)^{3}(1+z)^{8\alpha^{2}}}}.$
(25)

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=216.81pt,height=180.67499pt]{10.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=216.81pt,height=180.67499pt]{11.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 6: In this figure, the panel (a) shows the evolution of effective EoS
parameter ($\omega^{eff}$) for the constrained values of $\alpha=0.449451$ and
$\beta=1.096408$ and the panel (b) shows the evolution of scalar field (or
dark energy) EoS parameter ($\omega_{\phi}$) as functions of redshift.

In Fig 4, the energy density $\rho_{eff}$, $\rho_{m}$ and $\rho_{\phi}$ is
showing positive behavior for according to different value of
$\Omega_{m_{0}}$. It is seen that the energy density is higher for the
effective energy density of the matter field if compared with energy density
in the presence of scalar field. Also the evolution of scalar field potential
is shown which is responsible for negative pressure.

The evolution density parameter for matter and scalar field is shown in Fig5
for different values of $\Omega_{m_{0}}$. Also, in Figs 6, the behavior of
$\omega_{eff}$ and $\omega_{\phi}$ are shown for the constrained values of
$\alpha=0.449451$ and $\beta=1.096408$ and some chosen values values of the
matter density parameter $\Omega_{m_{0}}$ as shown in the figure. This
analysis also put constrain on the $\Omega_{m_{0}}$ value which should be
$\leq 0.269$ for smooth evolution of the scalar field (or dark energy) EoS
parameter. The curve shows a negative behavior at early times and evolve
through different phases of acceleration, deceleration and finally to phantom
phase. It can also be seen that for some values of $\Omega_{m_{0}}$, the
curves shows a singularity. The transition from one phase to the other
indicates the possibility for evolution of the universe. Planck observations
are known to be considered the best approximations of cosmological effects.
Planck’s 2018 findings indicate that the Hubble constant is $H_{0}=67.8\pm
0.9$ km $s^{-1}Mpc^{-1}$ and $\Omega_{m}=0.308\pm 0.012$, but the obtained
value of $\Omega_{m_{0}}$ is 0.269. This deviation point towards a tension
with Planck results.

Moreover, the higher order derivatives of deceleration parameter $q$ such as
jerk ($j$), snap ($s$) and lerk ($l$) are important in understanding the past
and future evolution of the universe sanjay . They are represented as Pan :

$j(z)=(1+z)\frac{dq}{dz}+q(1+2q),$ (26)

$\displaystyle s(z)=-(1+z)\frac{dj}{dz}-j(2+3q),$ (27)

$\displaystyle l(z)=-(1+z)\frac{ds}{dz}-s(3+4q).$ (28)

These higher order derivatives can be useful in understanding the future
evolution of the universe owing to the fact that $q(z)$ can be strictly
constrained from observations. The jerk parameter $j$ is related to the third
time derivative of $a$ as a higher-order derivative of the scale component.
The higher-order derivatives can describe the dynamics of the universe and may
be connected to the appearance of abrupt future singularities Pan ; Dabrowski
. In the statefinder diagnostic, the jerk parameter $j$ is often used to
discriminate against various dark energy or modified gravity models. Zhai Zhai
proposed different kinds of parameterizations of $j$ as a function of the
redshift z. A vital feature of $j$ is that for the $\Lambda$CDM model, $j=1$
always. The deviation from $j=1$ enables us to constrain the departure from
the $\Lambda$CDM value.. The value of $j_{0}$ according to constrain value of
$\alpha$ and $\beta$ is $j_{0}=-0.98_{-0.02}^{+0.06}$ Mamon . The behavior of
$j(z)$ is shown in Figs. 7. It can be clearly observed that $j(z)$ increases
with redshift indicating a decelerated phase in the past and an accelerated
phase in future. Interestingly the kinematic quantity $j(z)$ is positive which
is a reminiscent of an accelerated expansion. Also note that $j(z)\neq 1$ at
$z=0$ which does not correspond to $\Lambda$CDM cosmology. As mentioned in
sanjay , this can be thought as an expansion caused purely due to
modifications of gravity.

Figure 7: The plots of jerk $j$, snap $s$ and lerk $l$ parameters  vs.
redshift $z$ for the constrained values of $\alpha=0.449451$ and
$\beta=1.096408$.

##  V Statefinder diagnostics

The statefinder pairs $\\{r,s\\}$ and $\\{r,q\\}$ are the geometrical
quantities formulated directly from the metric and are employed to identify
various dark energy model. In the literature, the $\\{r,s\\}$ and $\\{r,q\\}$
pairs are defined as Statefinder1 .

$q=-\frac{\ddot{a}}{aH^{2}}\text{, \ \ }r=\frac{\dddot{a}}{aH^{3}}\text{, \ \
}s=\frac{r-1}{3(q-\frac{1}{2})}\text{.}$ (29)

The statefinder diagnostic is an useful tool in modern day cosmology and being
used to serve the purpose of distinguishing different dark energy models
Statefinder2 . In this setup, different trajectories in $r-s$ and $r-q$ planes
define the temporal evolution for various dark energy models. In a spatially
flat FLRW background, the statefinder pair are respectively
$\\{r,s\\}=\\{1,0\\}$ and $\\{1,1\\}$ for $\Lambda$CDM and standard cold dark
matter (SCDM). In the $r-s$ and $r-q$ planes, the departure of any dark energy
model from these fixed points are analyzed. The pairs $\\{r,s\\}$ and
$\\{r,q\\}$ for our model are shown in figure below.

$\begin{array}[]{c@{\hspace{.1in}}c}\includegraphics[width=216.81pt,height=180.67499pt]{r-s.pdf}\hfil\hskip
7.22743pt&\includegraphics[width=216.81pt,height=180.67499pt]{r-q.pdf}\\\
\mbox{(}a)\hfil\hskip 7.22743pt&\mbox{(}b)\end{array}$

Figure 8: Figures (a) and (b) shows the behavior of r-s and r-q plane in red
color respectively for the constrained values of $\alpha=0.449451$ and
$\beta=1.096408$.

In Fig. 8(a) we show the temporal evolution of the dark energy model mimicked
by our model. It is observed that at early times, the model presumes values in
the range $r>1$ and $s<0$ and therefore represents a Chaplygin gas type dark
energy model. Nonetheless, the model evolves into a Quintessence type dark
energy model at some point but then quickly reverts back into CG gas at late
times. Interestingly, it can be clearly observed that throughout its temporal
evolution, the model deviates significantly from the point
$\\{r,s\\}=\\{1,0\\}$. In Fig. 8(b) we show the temporal evolution of our
model in the $\\{r,q\\}$ to get additional information regarding the
parametrization. In this diagnostic plane, the solid line in the middle
depicts the evolution of the standard $\Lambda$CDM cosmological model and also
divides the plane into two equal halves with the lower half belonging to
Quintessence dark energy models and the upper half to Chaplygin gas dark
energy models. We clearly see that the profile starts from the region $q>0$
and $r>1$ which corresponds to the SCDM universe. This is then followed by the
region $r<1$ and $q<0$ and finally approaches to towards the de-Sitter phase
with $r=1,q=-1$.

##  VI Conclusion

In this paper we have considered the scalar field with positive potential to
study the accelerated expansion of the universe. To explain the late-time
acceleration, the EoS parameter $\omega$ must be negative as this implies the
cosmological pressure is negative. Negative cosmological pressure is the
hallmark of the presence of dark energy as this is only cosmic entity which
posses an anti-gravity effect. The potential $V(\phi)$ is responsible for
negative pressure.

In order to solve the field equations, we took the assistance of a
supplementary equation since the value of $\rho_{\phi}$ and $p_{\phi}$ is
difficult to obtain. In this work, we employ a second degree parametrization
of deceleration parameter first proposed in Bakry and Shafeek . We used the
latest 57 points of $H(z)$ dataset in the redshift range $0.07\leq z\leq
2.42$, 580 points of SN data and BAO datasets to constrained the model
parameters using $\chi^{2}$ minimization technique. The values of the
parameters $\alpha\ \&\ \beta$ considered here from all the datasets given in
Table 2, the present model shows a smooth transition for the deceleration
parameter (see Fig. 3) from the deceleration ($q>0$) phase to the present
acceleration ($q<0$) phase of the Universe. It has been found that the values
of transition redshift $z_{t}$ (from decelerated phase to accelerated
expansion) and the values of present deceleration parameter $q_{0}$ depends on
the model parameters $\alpha\ \&\ \beta$. It is interesting to note that the
values of $z_{t}=0.67_{-0.36}^{+0.26}$ and $q_{0}=-0.50_{-0.11}^{+0.12}$
obtained in our model are in good agreement with the recent results as
reported in Jesus/2020 and the references therein. Further, we have studied
the other kinematical parameter like jerk parameter using the combined
datasets ($H(z)+SN+BAO$). An alternative to describing cosmological models
similar to the $\Lambda$CDM model of concordance is the jerk parameter. For
LCDM model , the value of j is 1. The deviation from $j=1$ enables us to
constrain the departure from the $\Lambda$CDM value. According to the
restricted value of $\alpha$ and $\beta$, the value of
$j_{0}=-0.98_{-0.02}^{+0.06}$ (close to 1). The jerk parameter increasing with
respect to redshift indicating the accelerated phase in the future Universe.
Also, the evolution of $\omega_{eff}$ and $\omega_{\phi}$ is shown for the
constrained $\alpha=0.449451$ and $\beta=1.096408$ values and some of the
chosen $\Omega_{m_{0}}$ parameter density values. The transition from positive
to negative values indicates the decelerated to accelerated phase of the
universe. This study also constrains the value of $\Omega_{m_{0}}$ which
should be $\leq 0.269$ for the smooth evolution of the EoS parameter scalar
field. It can also be shown that the curves display a singularity for specific
values of $\Omega_{m_{0}}$. It is understood that Planck observations are
regarded as the best approximations of cosmological results. Planck’s 2018
results suggest that $H_{0}=67.8\pm 0.9$ km $s^{-1}Mpc^{-1}$ and
$\Omega_{m}=0.308\pm 0.012$ are the Hubble constants, but the obtained value
of $\Omega_{m_{0}}$ is 0.269 which results in tension with Planck
approximations.

Furthermore, to understand the parametrization from a cosmological point of
view, we also diagnose it geometrically using $\\{r,s\\}$, $\\{r,q\\}$ planes
and $Om(z)$ parameter. We observe that at early times, the model represents a
Chaplygin gas type dark energy model and later evolves into a Quintessence
type dark energy model at some point but then quickly reverts back into CG gas
at late times. Interestingly, the model deviates significantly from the point
$\\{r,s\\}=\\{1,0\\}$ and therefore do not coincide with $\Lambda$CDM
cosmology throughout the cosmic aeon.

###### Acknowledgements.

S. A. acknowledges CSIR, Govt. of India, New Delhi, for awarding Junior
Research Fellowship. PKS acknowledges CSIR, New Delhi, India for financial
support to carry out the Research project [No.03(1454)/19/EMR-II
Dt.02/08/2019]. We are very much grateful to the honorable referee and to the
editor for the illuminating suggestions that have significantly improved our
work in terms of research quality, and presentation.

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