#  Quintessence or Phantom : Study of scalar field dark energy models through
a general parametrization of the Hubble parameter

Nandan Roy<EMAIL_ADDRESS>Centre for Theoretical Physics & Natural
Philosophy, Mahidol University, Nakhonsawan Campus, Phayuha Khiri, Nakhonsawan
60130, Thailand  Sangita Goswami<EMAIL_ADDRESS>Department of Physics,
Visva-Bharati, Santiniketan -731235, India  Sudipta Das  sudipta.das@visva-
bharati.ac.in Department of Physics, Visva-Bharati, Santiniketan -731235,
India

###### Abstract

In this work we propose a simple general parametrization scheme of the Hubble
parameter for the scalar field dark energy models. In our approach it is
possible to incorporate both the quintessence and phantom scalar field in a
single analytical scheme and write down relevant cosmological parameters which
are independent of the nature of the scalar field. A general condition for the
phantom barrier crossing has also been obtained. To test this approach, a well
behaved parametrization of the normalized Hubble parameter has been considered
and a wide variety of observational data like CMB data, Supernovae data, BAO
data etc. has been used to constraint the various cosmological parameters. It
has been found that data prefer the present value of the equation of state of
the dark energy to be in the phantom domain. One interesting outcome of this
analysis is that although the current value of the dark energy equation of
state is phantom in nature, a phantom crossing of the EOS has taken place in
the recent past. We have also carried out the Bayesian model comparison
between $\Lambda CDM$ model and the proposed model which indicates that this
model is favored by data as compared to $\Lambda CDM$ model.

##  I Introduction

Even after two decades of its discovery, the reason behind accelerated
expansion of the universe Riess et al. (1998); Perlmutter et al. (1999);
Meszaros (2002); Arnaud et al. (2016); Ahn et al. (2012) remains a mystery.
Moreover the observational data also suggest that the onset of this
accelerated phase has taken place very recently, at around $z\sim 0.5$ Riess
et al. (2001); Amendola (2003). If we consider the theory of gravity to be
general relativity, any proposed candidate behind the accelerated expansion of
the universe referred to as “dark energy”(DE) must have a sufficient negative
pressure to counterbalance gravity and drive the accelerated expansion
Padmanabhan (2006). Cosmological constant ($\Lambda$), the most successful of
all proposals, is however troubled by the challenges coming from both
theoretical and observational sides. An alternative proposal is a dynamical DE
model. Till now a variety of such DE models have been proposed, such as
quintessence, k-essence, phantom, chaplygin gas, tachyon models, holographic
DE models and so on Sahni and Starobinsky (2006); Bamba et al. (2012);
Armendariz-Picon et al. (2001); Caldwell (2002); Carroll et al. (2003);
Kamenshchik et al. (2001); Sen (2002); Padmanabhan (2002); Copeland et al.
(2006); Amendola and Tsujikawa (2010). But the origin and nature of DE still
remains unknown despite many years of research and remains an open problem.

In the very recent years, cosmological studies have evidenced an additional
open problem, namely the $H_{0}$ tension which arises as a result of
discrepancies in the inferred value of $H_{0}$ from different types of
measurements. The CMB Planck collaboration Aghanim et al. (2020a) (including
BAO Alam et al. (2017a); Beutler et al. (2011a), BBN Alam et al. (2021) and
DES Troxel et al. (2018a); Abbott et al. (2018); Krause et al. (2017)) have
estimated the present value of the Hubble parameter to be
$H_{0}\sim(67.0-68.5)$ km/s/Mpc. On the other hand, cosmic distance ladder and
time delay measurements reported by SH0ES Riess et al. (2019a) and H0LiCOW
Wong et al. (2019) collaborations have estimated $H_{0}=(74.03\pm
1.42)$km/s/Mpc which has been obtained from the Hubble Space Telescope
observations of 70 long-period Cepheids in the Large Magellanic Cloud Riess et
al. (2019a). At the beginning the origin of this discrepancy was thought to be
from the systematic. Currently the discrepancy has moved to the order of
$\simeq 6\sigma$ forcing us to expand our thinking beyond the $\Lambda$CDM.

There have been a number of attempts to address these problems (See Normann
and Brevik (2021); Di Valentino et al. (2021a); Petronikolou et al. (2021) and
the references therein), which includes different dynamical dark energy models
with a dynamical equation of state ($w_{DE}\neq-1)$ for the dark energy (DE)
component Benisty and Staicova (2021); Karwal et al. (2021); Cedeño et al.
(2021). It has already been shown by a number of authors that a phantom-like
equation of state (EoS) of the dark energy sector can effectively speed up the
acceleration of the universe which results in a higher value of the Hubble
parameter $H_{0}$ Alestas et al. (2020); Di Valentino et al. (2021b); Vagnozzi
(2020).

Since there is no preferred theoretical model of dark energy (DE) which can
perfectly describe all the observed phenomena of the late time dynamics of the
universe, there have been attempts to construct theoretical models of DE right
from the observational data. A very useful approach in this context is
parametrization of DE models in which a functional form of a particular DE
parameter, viz, equation of state (EoS) of the dark energy, Hubble parameter
etc. is chosen. Then the evolutions of the cosmological parameters are studied
and various cosmological data sets are used to constrain these parameters.

In this work we revisited the quintessence and phantom scalar field models and
present a general scheme to write down the relevant cosmological parameters in
terms of the normalized Hubble parameter $E(z)={H(z)}/{H_{0}}$, current matter
density $\Omega_{m0}$ and the redshift $z$. The interesting aspect of this
approach is that the final expressions of the cosmological parameters do not
explicitly depend on the nature of the scalar field. We have considered a
parametrization of the Hubble parameter $H(z)$, which is one of the most
crucial and important parameter for understanding the evolution of the
universe. One can, in principle, parametrize the dark energy equation of state
(EoS) $w_{DE}$ or the dark energy density $\rho_{DE}$ as well; but the
contributions from the dark energy sector eventually enter the dynamics
through the evolution of $H(z)$. This is one of the reasons why we chose to
parametrize $H(z)$, as all other cosmological parameters are directly related
to it. Till now a number of parametric forms of Hubble parameter have been
proposed (see Banerjee et al. (2010); Pacif et al. (2017, 2020) and the
references therein) in the context of quintessence scalar field models of DE.

We have considered a general approach in which the field equations have been
written in terms of the switch parameter $\epsilon$ which can represent both
the quintessence ($\epsilon=+1$) as well as phantom scalar field models
($\epsilon=-1$). A publicly available version of the Boltzmann code CLASS Blas
et al. (2011); Lesgourgues (2011); Lesgourgues and Tram (2011) has been
amended to study the numerical evolution of the model. A detailed cosmological
data analysis has been done using the MCMC code Montepython V3.5 Brinckmann
and Lesgourgues (2018); Audren et al. (2013) using various cosmological data
sets. We have also compared our model which we name $\phi CDM$ hereafter with
the $\Lambda$ Cold Dark Matter ($\Lambda CDM$) model using the concept of
Bayes factor.

The present paper is arranged as follows. The basic mathematical formulation
and the dynamics of the scalar field have been presented in section II. This
section also deals with the parametrization of $E(z)$ and derivation of
analytical expressions for various cosmological parameters corresponding to
this parametrization. The observational constraints on the parameters of the
model and the numerical evolution of the cosmological system have been
summarized in sections III.1 and III.2 respectively. Finally we conclude by
presenting our results and findings in section IV.

##  II Scalar field dynamics

For a spatially flat, homogeneous and isotropic FRW universe filled with the
scalar field, the Einstein’s field equations and KG equation for the scalar
field are given by (with $8\pi G=c=1$)

$\displaystyle
3H^{2}=\rho_{m}+\rho_{\phi}=\rho_{m}+\frac{1}{2}\epsilon\dot{\phi^{2}}+V(\phi),$
(1) $\displaystyle
2\dot{H}+3H^{2}=-p_{\phi}=-\frac{1}{2}\epsilon\dot{\phi^{2}}+V(\phi),$ (2)
$\displaystyle\epsilon\ddot{\phi}+3\epsilon H\dot{\phi}+\frac{dV}{d\phi}=0,$
(3)

where $H=\frac{\dot{a}}{a}$ is the Hubble parameter, $\rho_{m}$ is the matter
energy density. The index $\epsilon$, which is named as the switch parameter,
characterizes the nature of the scalar field $\phi$. If $\epsilon=+1$ the
field is quintessence field and if $\epsilon=-1$ the field is phantom in
nature.

The matter energy density is represented by $\rho_{m}$, which varies with the
scale factor as $\rho_{m}=\rho_{m0}a^{-3}$ where $\rho_{m0}$ is the current
value of the matter energy density. The energy density for the scalar field
can be expressed as $\rho_{\phi}=\frac{1}{2}\epsilon\dot{\phi^{2}}+V(\phi)$
and the pressure component as
$p_{\phi}=\frac{1}{2}\epsilon\dot{\phi^{2}}-V(\phi)$. $V(\phi)$ is the
potential associated with the scalar field $\phi$.

By simple rearrangement of terms, from equations (1), (2) and (3) it is
possible to express the derivative of the Hubble parameter ($\dot{H}$) and
scalar field potential $V(\phi)$ as,

$2\dot{H}=-\frac{\rho_{m0}}{a^{3}}-\epsilon\dot{\phi^{2}}$ (4)

and

$V(\phi)=\dot{H}+3H^{2}-\frac{\rho_{m0}}{2a^{3}}$ (5)

A further rearrangement of equation (4) and use of the standard relation
$\dot{H}=\frac{1}{2}a\frac{d}{da}(H^{2})$ gives

$a\frac{d}{da}(H^{2})+\frac{\rho_{m}{}_{0}}{a^{3}}=-\epsilon\dot{\phi^{2}}.$
(6)

Again $\dot{\phi}$ can be expressed as

$\dot{\phi}=aH\left(\frac{d\phi}{da}\right).$ (7)

Using equations (6) and (7), the derivative of the scalar field $\phi$ with
respect to redshift $z$ can be expressed as

$\frac{d\phi}{dz}=\left[\frac{2E\frac{dE}{dz}-3\Omega_{m}{}_{0}(1+z)^{2}}{\epsilon
E^{2}(1+z)}\right]^{\frac{1}{2}}$ (8)

Here, we introduce the normalized Hubble parameter $E(z)=\frac{H(z)}{H_{0}}$,
where $H_{0}$ is the present value of Hubble parameter and
$\Omega_{m}{}_{0}=\frac{\rho_{m}{}_{0}}{3H_{0}^{2}}$ corresponds to the
present value of the matter density parameter. From now onwards, in the text
we would denote $E(z)$ by simply $E$ for the convenience of writing the
equations.

In a similar fashion it is possible to express the scalar field potential
given in equation (5) in terms of $E(z),\Omega_{m0},$ and $z$ as

$\frac{V(z)}{3H_{0}^{2}}=-\frac{(1+z)}{3}E\frac{dE}{dz}+E^{2}-\frac{1}{2}\Omega_{m}{}_{0}(1+z)^{3}$
(9)

One can consider $\frac{V(z)}{3H_{0}^{2}}$ as the effective potential for the
scalar field (in units of critical density $\rho_{crit}=3H_{0}^{2}$ since
$8\pi G=1$). It is interesting to note that in the expressions of $V(\phi)$ or
equivalently $V(z)$ given in equations (5) or (9), there is no explicit
dependency on the switch parameter $\epsilon$. This behavior is expected as
the nature of a scalar field depends on the kinetic term and not on the
potential term.
It is also possible to express the density parameter for the matter field
$(\Omega_{m})$, density parameter for the scalar field $(\Omega_{\phi})$, the
equation of state parameter $(w_{\phi}(z))$ and the deceleration parameter
$q(z)$ in terms of the normalized Hubble parameter $E(z)$ as

$\Omega_{m}(z)=\frac{\rho_{m}}{3H^{2}}=\frac{\Omega_{m0}(1+z)^{3}}{E^{2}},$
(10) $\Omega_{\phi}(z)=1-\Omega_{m}(z)=1-\frac{\Omega_{m0}(1+z)^{3}}{E^{2}},$
(11)
$w_{\phi}(z)=\frac{-1-\frac{2\dot{H}}{3H^{2}}}{\Omega_{\phi}}=\frac{\frac{2}{3}(1+z)E\frac{dE}{dz}-E^{2}}{E^{2}-\Omega_{m0}(1+z)^{3}},$
(12) $q(z)=-1-\frac{\dot{H}}{H^{2}}=\frac{(1+z)}{E}\frac{dE}{dz}-1.$ (13)

Similar to the potential $V(\phi)$, for the above set of equations (10) to
(13) also, it is evident that all the relevant cosmological parameters are
independent of the switch parameter $\epsilon$ and hence are identical for
quintessence as well as phantom fields. However, it deserves mention here this
independence is only apparent and the inherent dependency on $\epsilon$ comes
through the evolution of the normalized Hubble parameter $E(z)$. Since the
evolution of $E(z)$ directly depends on the nature of the scalar field, i.e,
on the value of $\epsilon$, the evolution history will obviously be different
for quintessence and phantom fields even if we consider the same form of
potential for both.

One very important aspect to note here is that in equation (8), for the term
$\frac{d\phi}{dz}$ to be real, two different conditions needs to be satisfied.
For the quintessence field $2E\frac{dE}{dz}-3\Omega_{m0}(1+z)^{2}>0$ and for
the phantom field $2E\frac{dE}{dz}-3\Omega_{m0}(1+z)^{2}<0$. More
interestingly the dark energy EoS given in equation (12) can be recast as

$w_{\phi}(z)=-1+\frac{1+z}{3{\Omega_{\phi}}E^{2}}\left(2E\frac{dE}{dz}-3\Omega_{mo}(1+z)^{2}\right),$
(14)

which is directly related to the nature of the scalar field as mentioned
above. As long as the quantity $2E\frac{dE}{dz}-3\Omega_{m0}(1+z)^{2}$ is
positive, the scalar field will behave as a quintessece field. Eventually as
the scalar field evolves with time and the quantity
$2E\frac{dE}{dz}-3\Omega_{m0}(1+z)^{2}$ becomes negative, the field will
behave as a phantom one. The phantom crossing will occur at a redshift
$z=z_{\lambda}$ for which $2E\frac{dE}{dz}-3\Omega_{m0}(1+z)^{2}=0$ and the
EoS of the scalar field will coincide with that for the cosmological constant.
A general conclusion regarding the phantom barrier crossing for the DE models
can be drawn from here. Any dark energy model for which
$2E\frac{dE}{dz}-3\Omega_{m0}(1+z)^{2}>0$ for $z>z_{\lambda}$, and
$2E\frac{dE}{dz}-3\Omega_{m0}(1+z)^{2}<0$ for $z<z_{\lambda}$ should undergo a
phantom barrier crossing.

Conventionally one can consider a particular form of a potential $V(\phi)$ and
study the evolution of the quintessence field and phantom field separately. In
our current approach one can consider a particular parametrization of $E(z)$
and reconstruct all the cosmological parameters from it. Though consideration
of a particular form of $E(z)$ is equivalent to the consideration of a
potential in view of equation (9), the advantage of our approach is that one
can study both the quintessence and phantom field in a single framework.
Moreover recent cosmological data sets can be used to constraint the
cosmological parameters together with the EoS of the dark energy which can
also help us to determine the nature of the scalar field, quintessence or
phantom.

In the present work we have considered the following parametrization of the
normalized Hubble parameter $E(z)$ as toy model, in order to reconstruct other
cosmological parameters,

$E(z)=\left[1+pz\left(b+\frac{z}{c}+\frac{z^{2}}{d}\right)\right]^{\frac{1}{2}}$
(15)

where $b$, $c$, $d$ and $p$ are nonzero real numbers. Hence the Hubble
parameter $H(z)$ can be expressed as,

$H^{2}(z)={H_{0}}^{2}\left[1+pz\left(b+\frac{z}{c}+\frac{z^{2}}{d}\right)\right].$
(16)

The functional form chosen in equation (15) (or equivalently in equation (16))
is completely phenomenological, where $H^{2}(z)$ happens to be a polynomial in
$z$. Also it can be seen from equation (15) that at $z=0$, $H(z)=H_{0}$ and
one gets back the present value of the Hubble parameter. Notice that for the
choice of $p=0$, the above equation also represents the current value of the
Hubble parameter, thus the second term in the above parametrization can be
considered as the deviation from the present value of the normalized Hubble
parameter. For non zero values of the parameter $p$, at late times when
$z<<1$, the deviation is small but for early times, when $z>1$, it can have
large deviation depending on the choice of the parameters. This behaviour is
expected for any parametrized quantity from the observational point of view.

Now for this particular choice of $E(z)$ given by (15), equations (10), (11),
(12) and (13) take the form

$\Omega_{m}(z)=\frac{\Omega_{m0}(1+z)^{3}}{\left[1+pz\left(b+\frac{z}{c}+\frac{z^{2}}{d}\right)\right]}$
(17)
$\Omega_{\phi}(z)=1-\frac{\Omega_{m0}(1+z)^{3}}{\left[1+pz\left(b+\frac{z}{c}+\frac{z^{2}}{d}\right)\right]}$
(18)
$w_{\phi}(z)=\frac{p\left(b-2bz+\frac{2z}{c}-\frac{z^{2}}{c}+\frac{3z^{2}}{d}\right)-3}{3\left[1+pz\left(b+\frac{z}{c}+\frac{z^{2}}{d}\right)-\Omega_{m}{}_{0}(1+z)^{3}\right]}$
(19)

and

$q(z)=\frac{p\left(b-bz+\frac{2z}{c}+\frac{3z^{2}}{d}+\frac{z^{3}}{d}\right)-2}{2\left[1+pz\left(b+\frac{z}{c}+\frac{z^{2}}{d}\right)\right]}$
(20)

For the sake of completeness, using equation (15) in equations (8) and (9), we
have also provided the expressions for the potential $\phi(z)$ and $V(\phi)$
for this particular choice of $E(z)$.

$\phi(z)=\int{\left[\frac{p\left(b+\frac{2z}{c}+\frac{3z^{2}}{d}\right)-3\Omega_{m0}(1+z)^{2}}{\epsilon(1+z)\left[1+pz\left(b+\frac{z}{c}+\frac{z^{2}}{d}\right)\right]}\right]^{\frac{1}{2}}}dz$
(21)

and

$\begin{split}\frac{V(z)}{3H_{0}^{2}}=-\frac{p(1+z)\left(b+\frac{2z}{c}+\frac{3z^{2}}{d}\right)}{6}+\\\
\left[(1+pz\left(b+\frac{z}{c}+\frac{z^{2}}{d}\right)\right]-\frac{1}{2}\Omega_{m0}(1+z)^{3}\end{split}$
(22)

The expression for the scalar field EoS $w_{\phi}$, given in equation (19),
can be equivalently written in a more compact form in terms of the scale
factor $(\frac{a}{a_{0}})$ as

${w}_{\phi}=\frac{3w_{0}\left(\frac{a}{a_{0}}\right)^{3}+2w_{1}\left(\frac{a}{a_{0}}\right)^{2}+w_{3}\left(\frac{a}{a_{0}}\right)}{3\left[w_{2}-w_{0}\left(\frac{a}{a_{0}}\right)^{3}-w_{1}\left(\frac{a}{a_{0}}\right)^{2}-w_{3}\left(\frac{a}{a_{0}}\right)\right]},$
(23)

where $w_{0},~{}w_{1},~{}w_{2},~{}w_{3}$ are new set of parameters which are
related to our old set of parameters $p,b,c,d$ in the following way :

$w_{0}=[p(bcd-d+c)-cd],~{}~{}w_{1}=p[2d-bcd-3c],$
$w_{3}=p(3c-d)~{}~{}{\mathrm{and}}~{}~{}w_{2}=pc-\Omega_{m0}cd~{}.$

All others cosmological parameters can also be written in terms of the new set
of parameters $w_{0},~{}w_{1},~{}w_{2},~{}w_{3}$ (please see Appendix A for
details). From equation (23), the present value of the EoS of the dark energy
$w_{\phi 0}$ can be expressed as

$w_{\phi 0}=-1+\delta(w_{0},w_{1},w_{2},w_{3}),$

where
$\delta(w_{0},w_{1},w_{3})=\frac{w_{1}-3w_{2}+2w_{3}}{3(w_{0}+w_{1}+w_{3}-w_{2})}$
can be considered as the deviation from the pure $\Lambda$CDM model at
present. Hence, depending on the values of the model parameters
($w_{0},w_{1},w_{2},w_{3}$) or equivalently ($b,c,d,p$ and $\Omega_{m0}$), the
present value of the EoS will be either in the quintessence region or in the
phantom region.

This general setup gives us the advantage to study both the quintessence and
phantom field together. It would be interesting to investigate the dynamics of
the dark energy sector considering the above mentioned parametrization of the
EOS of the dark energy, or equivalently the scalar field models, against the
current cosmological data.

##  III Numerical Investigation and Observational Constraint

Our aim here is to study the numerical evolution of our model and find the
observational constraints on the cosmological parameters by comparing it to
the observational data sets. In order to do so we have used the expression of
$w_{\phi}$ given in equation (23) and amended a public version of the CLASS
Boltzmann code to include it in the dark energy sector as a fluid. From now
onwards, we will name this scalar field model as $\phi$CDM model. It has been
shown in Roy et al. (2018); Roy and Bamba (2019) that the late time dynamics
of the scalar field as the component of the dark energy is almost independent
of the form of the potentials; it is the EoS of the scalar field which affects
the dynamics. Thus it is expected that, consideration of the EoS of the scalar
field as a fluid in the CLASS code will resemble the same dynamics as the
scalar field itself. In appendix B, we have presented a comparison between the
analytical solutions for different cosmological parameters obtained in this
work and the corresponding numerical solutions obtained from CLASS code. The
MCMC code Montepython3.5 Brinckmann and Lesgourgues (2018) has been used to
estimate the relevant cosmological parameters.

###  III.1 Observational Constraints

Cosmological data sets which we have used for the analysis are as follows:
PantheonScolnic et al. (2018), BAO (BOSS DR12 Alam et al. (2017b), 6dFGS
Beutler et al. (2011b), eBOSS DR14 (Lya) Cuceu et al. (2019) and WiggleZ Kazin
et al. (2014)). We have also included (SDSS LRG DR7 Ross et al. (2015), SDSS
LRG DR4 Tegmark et al. (2006) and WiggleZ Kazin et al. (2014)) which are the
observations related to cluster counts. A SH0ES prior together with the
compressed Planck likelihood has been imposed. The compressed Planck
likelihood has more or less the same constraining ability as the full Planck
likelihood. It has been constructed by considering only three parameters, the
baryon physical density $\omega_{b}=\Omega_{b}h^{2}$ and the two shift
parameters $\theta_{\ast}=r_{s}(z_{dec})/D_{A}(z_{dec})$ and
$\mathcal{R}=\sqrt{\Omega_{M}H^{2}_{0}}D_{A}(z_{dec})$, where $z_{dec}$ is the
redshift at decoupling and $D_{A}$ is the comoving angular diameter distance.
For more details on the compressed Planck likelihood please see Appendix A of
Arendse et al. (2020).

We have made the choice of flat priors on the base cosmological parameters as
follows: the baryon density $100\omega_{b}=[1.9,2.5]$; cold dark matter
density $\omega_{cdm}=[0.095,0.145]$; Hubble parameter
$H_{0}=[60,80]\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$.

A wide range of flat prior on the model parameters $w_{0},w_{1},w_{3}$:
$w_{0}=[0,300]$, $w_{1}=[-50,50]$ and $w_{3}=[-50,50]$ has been considered. We
have fixed the model parameter $w_{2}$ to be zero. This choice has been
considered from the point of view of the stability of the CLASS code and also
to minimize the shooting failure. The choice of $w_{2}=c(p-\Omega_{m0}~{}d)=0$
establishes a relation between $p$, $\Omega_{m0}$ and $d$ as
$\Omega_{m0}=\frac{p}{d}$, which does not put any tight constraint on other
model parameters $w_{0},~{}w_{1}$ and $w_{3}$; there will be always room for
suitable choice of parameters which can lead to a viable cosmological model.

In Fig 1, we have shown the triangular plot which shows the 2D and 1D
posterior distribution of the cosmological parameters
$100w_{b},w_{cdm},H_{0},\Omega_{m}$ and $\sigma_{8}$. A comparison has been
made with the $\Lambda CDM$ case represented by red plots in the figure. The
constraints on the various cosmological parameters for the $\Lambda CDM$ and
the $\phi CDM$ models have been enlisted in Table 1. Although there is a
slight increment in the value of $H_{0}$ for the $\phi CDM$ model, but it is
far from solving the Hubble tension. A similar result of slight increment of
the $H_{0}$ parameter from $\Lambda CDM$ value for the scalar field models has
been reported in Vázquez et al. (2021); Linares Cedeño et al. (2021). Both
these works analyze a large class of potentials for scalar field models and
reported its inability to solve the $H_{0}$ tension.

The observational constraints on the model parameters including the dark
energy EoS are shown in Fig 2. The best fit value of the EoS of the dark
energy is found to be $w_{DE}=-1.04^{+0.0204}_{-0.0166}$ which is in the
phantom region. In Fig 3, we have plotted the posterior probability
$Pr(w\vline z)$ of the dark energy EOS $w_{\phi}$ against $z$. The deeper blue
region enclosed by the solid lines represents $1\sigma(68\%)$ contour level
and the lighter blue region represents $2\sigma(95\%)$ contour level. This
plot suggests a phantom crossing of the universe around $z=2.7$ which occurs
following the conditions mentioned in section II.

In Fig 4, we have shown the posterior probability of the expansion rate of the
universe $H(z)/(1+z)$ with respect to $z$. For comparison we have plotted the
observational data from Sh0ES Riess et al. (2019b) and BAO observations Alam
et al. (2017c); Zarrouk et al. (2018); Blomqvist et al. (2019); de Sainte
Agathe et al. (2019). The blue line corresponds to the $\phi CDM$ model and
the red line corresponds to the $\Lambda CDM$ model. The deeper and lighter
blue and green regions represent the $1\sigma(68\%)$ and $2\sigma(95\%)$
contour level for the $\phi CDM$ and $\Lambda CDM$ model respectively. A
similar plot for the posterior probability of the deceleration parameter
$q(z)$ is given in Fig 5 with $1\sigma$ and $2\sigma$ confidence level. A flip
in the signature of the deceleration parameter is observed which indicates a
transition from the decelerated phase to an accelerated phase which is a must
for the unhindered structure formation of the universe.

Figure 1:  Triangular plot of 2D and 1D posterior distribution of the
cosmological parameters $100w_{1}$, $w_{cdm}$,$H_{0}$, $\Omega_{m}$ and
$\sigma_{8}$ using various dataset mentioned in section III.1. Figure 2:
Triangular plot showing observational constraints on the model parameters
$w_{0},w_{1},w_{3}$ and the EoS of the scalar field $w_{\phi 0}$. Figure 3:
This figure shows the posterior probability $Pr(w\vline z)$ of the dark energy
EOS $w_{\phi}$ against $z$. The deep blue region represents $1\sigma(68\%)$
confidence contour level and the light blue regions represent the
$2\sigma(95\%)$ confidence contour level. Figure 4:  The posterior probability
$Pr(H\vline z)$ of the expansion rate of the universe $H(z)/(1+z)$ against
$z$. The dark gray region represents $1\sigma(68\%)$ contour level and the
light gray regions represent $2\sigma(95\%)$ contour level. Figure 5:  This
figure shows the posterior probability $Pr(q\vline z)$ of the deceleration
parameter $q(z)$ against $z$. The deep blue region represents $1\sigma(68\%)$
contour level and the light blue regions represents $2\sigma(95\%)$ contour
level.

To compare the $\Lambda CDM$ model and our proposed $\phi CDM$ model, we have
used $\chi^{2}_{min}$ value difference between the models,
$\Delta\chi^{2}_{min}=\chi^{2}_{\phi}-\chi^{2}_{\Lambda}=-5$ (see Table 1).
This indicates that the proposed $\phi CDM$ model gives a better fit to the
data as compared to the $\Lambda CDM$ model. We have also computed the
Bayesian evidence for model selection to be more certain about the preference
on the model from data. Bayes factor is used to penalize the complex models
involving many parameters and hence to avoid overfitting. To compare the
models we have computed $\ln
B_{\phi\Lambda}=\ln\mathcal{Z}_{\phi}-\ln\mathcal{Z}_{\Lambda}$, where
$\mathcal{Z}$ is the Bayesian evidence. According to Jeffreys’ scale, the
strength of preference on $\phi CDM$ over $\Lambda CDM$ will depend on the
value of $\ln B_{\phi\Lambda}$. If $\ln B_{\phi\Lambda}<1$, the preference is
negative and if $\ln B_{\phi\Lambda}>1;>2.5;>5.0$ the preference is positive,
moderate and strong respectively. For more details, one can refer to Trotta
(2007). The calculation of the Bayes factor is numerically challenging; for
this, we have used the publicly available code MCEvidence Heavens et al.
(2017) which can calculate the Bayes factor directly from the MCMC chains
generated by the Monte Python. As listed in Table 1, we have obtained $\ln
B_{\phi\Lambda}=+2.005$ which indicates positive preference on the $\phi CDM$
model compared to the $\Lambda CDM$ model.

Table 1:  Best fit values of different cosmological parameters for the $\Lambda CDM$ and $\phi CDM$ models. The posteriors can be seen from the triangular plots provided in Fig 1 and Fig 2 Parameter | $\Lambda CDM$ | $\phi CDM$
---|---|---
$100~{}\omega{}_{b}$ | $2.25^{+0.013}_{-0.0129}$ | $2.24^{+0.0146}_{-0.0128}$
$\omega{}_{cdm}$ | $0.118^{+0.00069}_{-0.000705}$ | $0.119^{+0.0096}_{-0.0098}$
$H_{0}$ | $68^{+0.326}_{-0.319}$ | $68.9^{+0.573}_{-0.602}$
$\Omega_{DE}$ | $0.694^{+0.00426}_{-0.00406}$ | $0.701^{+0.00478}_{-0.00516}$
$\Omega_{m}$ | $0.306^{+0.00406}_{-0.00426}$ | $0.299^{+0.00516}_{-0.00478}$
$\sigma_{8}$ | $0.807^{+0.00255}_{-0.00254}$ | $0.821^{+0.011}_{-0.010}$
$w_{DE}$ | $-1$ | $-1.04^{+0.0204}_{-0.0166}$
$w_{0}$ | $-$ | $>96.97$
$w_{1}$ | $-$ | $<-10.03$
$w_{3}$ | $-$ | $4.25^{+1.65}_{-3.46}$
$\Delta\chi_{min}^{2}$ | $0$ | $-5$
$\ln{B_{\phi\lambda}}$ | $0$ | $+4.225$

###  III.2 Numerical Evolution

We have considered the observational constraints obtained on the parameters
$w_{0},~{}w_{1},~{}w_{3}$ in the previous section to investigate the numerical
evolution of the system. Fig 6 shows the evolution of the EoS parameter of the
scalar field component $w_{\phi}$ with respect to redshift $z$. To understand
the effect of the model parameters $w_{0},w_{1},w_{3}$ on the cosmological
parameters, we consider to vary one parameter keeping others fixed to the best
fit values obtained from the MCMC analysis. The black dashed lines in all the
plots correspond to the best fit values of the model parameters
$[w_{0},w_{1},w_{3}:207,-30.7,4.25]$. In the top panel of Fig 6, we vary
$w_{0}$ while keeping the parameters $w_{1}$ and $w_{3}$ fixed to the best-fit
value. In the middle and bottom panels, we vary $w_{1}$ and $w_{3}$
respectively keeping the other two parameters fixed. Our choice of parameters
lies well within the posteriors obtained by the MCMC analysis in the previous
section. From Fig 6 one can see that the current value of EoS of the scalar
field lies in the phantom region ($w_{\phi}<-1$) and the universe should have
undergone phantom crossing from the quintessence to phantom (see the best fit
curve, black dotted line). From the top panel it is apparent that the effect
of $w_{0}$ on the EoS parameter is not very significant at later epochs and
also the redshift at which phantom crossing occurs is insensitive to the value
of $w_{0}$. The effect of the parameter $w_{0}$ is only apparent at the early
epochs of the evolution of the universe. However, the present value of the EoS
parameter as well as the evolution of the EoS is quite sensitive to both the
parameters $w_{1},w_{3}$. This finding is in agreement with the results shown
in Fig 2 obtained by MCMC analysis. It has been observed that the evolution of
the EoS parameter will be effected only when the variation of $w_{1}$ is of
the order of 10 or more. The $w_{0},w_{1}$ parameters appear to be
unconstrained with a lower cutoff of $w_{0}>96.97$ (approximately) and a upper
cutoff $w_{1}<-5.5$ (approximately) respectively. On the other hand $w_{3}$ is
tightly constrained with allowed values in the range $4.25^{+1.65}_{-3.46}$.

Figure 6:  The evolution of the EoS ($w_{\phi}$) for the $\phi CDM$ model for
different choices of the model parameters $(w_{0},w_{1},w_{3})$ against the
redshift $z$. The offset values refer to the respective choices of the model
parameters. The solid blue line represents the $\Lambda CDM$ model and the
black dashed line represents the evolution of the EoS corresponding to the
best fit values obtained from MCMC analysis. The top panel shows the evolution
when we are varying only $w_{0}$ while keeping the parameters $w_{1},w_{3}$
fixed. The middle and bottom panels show the evolution when $w_{1}$ and
$w_{3}$ are evolving keeping the other two respective parameters fixed. The
blue horizontal line corresponds to the $\Lambda$CDM case $w_{\Lambda}=-1$.

The evolution of the energy densities of the matter $\Omega_{m}$ and the
scalar field $\Omega_{\phi}$ components has been shown in Fig 7. We have
considered the same set of parameter values for $w_{0},w_{1},w_{3}$ as in Fig
6. The blue curves represent the $\Lambda$CDM case for reference. It can be
seen from the plots that the energy densities are not very sensitive to the
model parameters $w_{0},w_{1},w_{3}$ though these parameters have significant
effect on the evolution of the EoS of dark energy.

Figure 7:  The evolution of the dark energy density parameter
($\Omega_{\phi}(z)$) (dashed line) and the matter density parameter
($\Omega_{m}(z)$) (solid line). The top, middle and bottom panels correspond
to the variation of the model parameters $w_{0},w_{1}$ and $w_{3}$
respectively keeping other two respective model parameters fixed to the best
fit values.

Figure 8:  Plot of the CMB anisotropies for the $\phi CDM$ model. The top,
middle and bottom panels correspond to the variation of the model parameters
as explained in Fig 6. The bottom panel of each figure shows the relative
difference between the $\phi CDM$ and the $\Lambda$CDM ($\Delta
D_{l}=(D_{l}-D_{l}^{\Lambda CDM})/D_{l}^{\Lambda CDM}$).

We also show in Fig 8 and Fig 9 the temperature anisotropies and the matter
power spectrum (MPS) for the $\phi CDM$ model corresponding to the same set of
model parameters as considered in Fig 6. For comparison, data from different
experiments have been plotted as references together with the numerical
solutions. For the temperature anisotropies (TT) we have used binned TT power
spectrum data from Planck 18 Aghanim et al. (2020b). For the MPS following
data sets have been used: Planck2018 CMB data Aghanim et al. (2020b), SDSS
galaxy clustering Reid et al. (2010), SDSS Ly$\alpha$ forest Abolfathi et al.
(2018) and DES cosmic shear data Troxel et al. (2018b) (for details on full
data collection, please see Chabanier et al. (2019)).

The lower panels of Fig 8 and Fig 9 represent the relative differences in
$D_{l}$ and $P(k)$ in comparison to the $\Lambda CDM$ case. It is observed
that in case of $D_{l}$, there is notable deviation from $\Lambda CDM$ case
for both the lower and higher multipoles. For $P(k)$, the deviation from the
$\Lambda CDM$ case is observed at all scales. Although the percentage
deviation is not much, but one should, in principle, take into account the
dark energy perturbations as well to obtain the complete picture. A similar
claim has been made in Ureña-López and Roy (2020) that if one does not
consider the dark energy perturbations, this can result in misleading
constraints on the cosmological parameters.

Figure 9:  Top, middle and bottom panel shows the plots of matter power
spectrum (MPS) for the variation of the model parameters as explained in
Fig.6. At the bottom panel of each figure we showed relative difference
between the $\phi CDM$ and the $\Lambda CDM$ models, ($\Delta
P(k)=(P(k)-P(k)^{\Lambda CDM})/P(k)^{\Lambda CDM}$).

##  IV Conclusion

In this work we have revisited the dynamics of the scalar field dark energy
models and proposed a general scheme which can include both the quintessence
and the phantom scalar field models. Our method is simple and straightforward
in which it is possible to express all the cosmological parameters in terms of
the redshift ($z$), the present value of the matter density parameter
($\Omega_{m0}$) and the normalized Hubble parameter ($E$). We have obtained
the expressions for various cosmological parameters and have found that the
final expressions of these cosmological parameters are independent of the
nature of the scalar field, either quintessence or phantom. A general
condition for the phantom barrier crossing of the dark energy models is also
obtained. This general condition is particularly interesting as it can help us
to check the possibility of phantom barrier crossing for a given DE model
which complies with the methodology given here without a detailed numerical
study of the model. This encourages us to consider a suitable parametrization
of the Hubble parameter as toy model and to study the model against the
current cosmological observations.

A publicly available version of the Boltzman code CLASS has been amended. The
dynamics of the scalar field has been incorporated in the CLASS code by adding
it as a fluid using our proposed parametrization. A detailed study of the
model against the recent cosmological data sets has been done using the MCMC
code Montepython. The best fit value of the EoS parameter for the scalar field
at the current epoch comes out to be less than $-1$ which indicates that at
present phantom DE models are preferred by observational data compared to
quintessence models. However a phantom barrier crossing has been observed
within $2\sigma$ confidence level for a wide range of choice of the model
parameters.

The best fit value of $H_{0}$ comes out to be $68.9^{+0.573}_{-0.602}$ which
happens to be more compatible with the PLANCK collaboration results and thus
cannot alleviate the Hubble tension. This is in agreement with the recent
results Dinda (2021) where it has been shown that CMB+BAO+SN data put stronger
constraints on $H_{0}$ and on other background cosmological parameters and
that the addition of $H_{0}$ prior from SH0ES (or from similar other local
distance observations) can not significantly pull the $H_{0}$ value towards
the corresponding SH0ES value. A comparison between $\phi CDM$ and $\Lambda
CDM$ models have been carried out using the concept of Bayes Factor and the
$\phi CDM$ model is found to have positive preference over the $\Lambda CDM$
one. We have also found that there are slight deviations in the $D_{l}$ and
$P(k)$ curves as compared to standard $\Lambda CDM$; for $D_{l}$, deviation
has been observed in both lower and higher multipoles whereas for $P(k)$ the
deviation has been observed at all scales.

We must admit that the choice of parametrization for $E(z)$ is motivated by
mathematical simplicity and is not unique. A wide variety of parametrization
can be considered which might be able to solve the $H_{0}$ tension as well. As
of now, the current choice of parametrization valid for these two scalar field
models can not alleviate the $H_{0}$ tension completely, but interesting
results may be obtained by considering an interaction between the two dark
sectors.

##  V Acknowledgement

We are grateful to Ankan Mukherjee, Luis Urena Lopez and Francisco X. Linares
Cedeño for help in the making of some plots. We acknowledge the use of the
Chalawan High Performance Computing cluster, operated and maintained by the
National Astronomical Research Institute of Thailand (NARIT). The research of
NR is supported by Mahidol University through the research project MRC-MGR
04/2565. SD would like to acknowledge IUCAA, Pune for providing support
through the associateship programme.

##  Appendix A Cosmological parameters in terms of $w_{0},w_{1},w_{2},w_{3}$

In the text we have considered two different sets of parameters $[p,b,c,d]$
and $[w_{0},w_{1},w_{2},w_{3}]$. Here we present all the cosmological
parameters provided in equations (17) to (22) in terms of the parameters
$w_{0},w_{1},w_{2}$ and $w_{3}$.

$E^{2}(z)=\frac{f_{3}}{f_{2}}$ (24)

$\begin{split}q(z)=&\frac{1}{2f_{3}}((f_{1}-w_{2})(1+z)^{3}+w_{1}(\Omega_{m0}-1)(1+z)\\\
&\quad+2w_{0}(\Omega_{m0}-1))\end{split}$ (25)

$\Omega_{m}(z)=\frac{\Omega_{m0}f_{2}(1+z)^{3}}{f_{3}}$ (26)

$\Omega_{\phi}=1-\frac{\Omega_{m0}f_{2}(1+z)^{3}}{f_{3}}$ (27)

$\begin{split}\frac{V(z)}{3H_{0}^{2}}=&\frac{1}{6f_{2}}[-(1+z)(3(f_{1}-w_{2})(1+z)^{2}\\\
&\quad-2w_{3}(\Omega_{m0}-1)(1+z)-w_{1}(\Omega_{m0}-1))\\\
&\quad+6(f_{1}-w_{2})(1+z)^{3}-6w_{3}(\Omega_{m0}-1)(1+z)^{2}\\\
&\quad-6w_{1}(\Omega_{m0}-1)(1+z)-6w_{0}(\Omega_{m0}-1)]\\\
&\quad-\frac{1}{2}\Omega_{m0}(1+z)^{3}\end{split}$ (28)

$\displaystyle\phi(z)=\int\frac{1}{\epsilon(1+z)f_{3}}\left[\left(3(f_{1}-w_{2}\right)(1+z)^{2}\right.$
$\displaystyle\left.\left.-2w_{3}\left(\Omega_{m0}-1\right)(1+z)\right.\right.$
$\displaystyle\left.\left.-w_{1}\left(\Omega_{m0}-1\right)\right)-3\Omega_{m0}f_{2}(1+z)^{2}\right]^{1/2}dz$
(29)

where

$f_{1}=(\Omega_{m0}(w_{0}+w_{1}+w_{3})),$ $f_{2}=(w_{0}+w_{1}+w_{3}-w_{2})$

and

$\begin{split}f_{3}=(f_{1}-w_{2})(1+z)^{3}-w_{3}(\Omega_{m0}-1)(1+z)^{2}\\\
-w_{1}(\Omega_{m0}-1)(1+z)-w_{0}(\Omega_{m0}-1).\end{split}$

##  Appendix B Comparison between the analytical solution and the numerical
solution

In Section III we have mentioned that the dynamics of the scalar field has
been incorporated in the CLASS code as a fluid by implementing the EoS of the
DE given in equation (23). Here we present a comparison between the numerical
solutions obtained from the CLASS code and the analytical solutions of the
cosmological parameters. In Fig 10 we have compared the expansion rate
(${H(z)}/{1+z}$) of the universe for these two solutions. The solid lines are
the numerical solutions obtained from the CLASS code and the dashed lines are
the corresponding plots for the analytical solutions which can be obtained
from equation (24). The bottom panel of Fig 10 shows the percentage difference
between the numerical and the analytical solutions. A similar comparison
between the dark energy density $(\Omega_{\phi})$ parameters for the numerical
and the analytical solutions has been provided in the upper panel of Fig 11
along with the corresponding percentage difference at the bottom panel. One
can see that for both the cases the percentage difference is less than $1\%$.
It must be mentioned here that in order to obtain the analytical solutions, we
have used the Klein Gordon (KG) equation for the scalar field whereas the
numerical solutions obtained from the CLASS code involve the integral of the
EoS of the dark energy over redshift interval. This similarity and very little
deviation justify our claim in Section III that if one considers the EoS of
the scalar field as a fluid in the CLASS code, it will resemble the same
dynamics as that of the scalar field.

Figure 10:  Comparison of the ${H(z)}/{1+z}$ for the numerical solutions
obtained from the CLASS code (represented by solid lines) and the analytical
solutions obtained from equation (24) (represented by dashed lines). In the
bottom panel we have shown the percentage difference between the numerical
solutions and the analytical solutions. Figure 11:  Comparison of
$\Omega_{\phi}(z)$ for the numerical solutions obtained from the CLASS code
(represented by solid lines) and the analytical solutions obtained from
equation (27) (represented by dashed lines). In the bottom panel we have shown
the percentage difference between the two solutions.

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