# Roles of Modified Chaplygin-Jacobi and Chaplygin-Abel Gases in FRW Universe

Ujjal Debnath
Department of Mathematics, Indian Institute of Engineering
Science and Technology, Shibpur, Howrah-711 103, India.
<EMAIL_ADDRESS>
###### Abstract

We have considered flat Friedmann-Robertson-Walker (FRW) model of the universe
and reviewed the modified Chaplygin gas as the fluid source. Associated with
the scalar field model, we have determined the Hubble parameter as a
generating function in terms of the scalar field. Instead of hyperbolic
function, we have taken Jacobi elliptic function and Abel function in the
generating function and obtained modified Chaplygin-Jacobi gas (MCJG) and
modified Chaplygin-Abel gas (MCAG) equation of states, respectively. Next, we
have assumed that the universe filled in dark matter, radiation, and dark
energy. The sources of dark energy candidates are assumed as MCJG and MCAG. We
have constrained the model parameters by recent observational data analysis.
Using $\chi^{2}$ minimum test (maximum likelihood estimation), we have
determined the best fit values of the model parameters by OHD+CMB+BAO+SNIa
joint data analysis. To examine the viability of the MCJG and MCAG models, we
have determined the values of the deviations of information criteria like
$\triangle$AIC, $\triangle$BIC and $\triangle$DIC. The evolutions of
cosmological and cosmographical parameters (like equation of state,
deceleration, jerk, snap, lerk, statefinder, Om diagnostic) have been studied
for our best fit values of model parameters. To check the classical stability
of the models, we have examined the values of square speed of sound
$v_{s}^{2}$ in the interval $(0,1)$ for expansion of the universe.

Keywords: Dark energy, Chaplygin gas, Data analysis, Parameters

###### Contents

  1. 1 Introduction
  2. 2 Review of Modified Chaplygin Gas in FRW Model
  3. 3 Modified Chaplygin-Jacobi Gas
  4. 4 Modified Chaplygin-Abel Gas
  5. 5 Observational Data Analysis
    1. 5.1 OHD
    2. 5.2 CMB Data
    3. 5.3 BAO Data
    4. 5.4 SNIa Data
    5. 5.5 Information Criteria
    6. 5.6 Results
  6. 6 Evolutions of Cosmological and Cosmographical Parameters
  7. 7 Discussions and Conclusions

##  1 Introduction

From Supernova type Ia [1, 2, 3] observation, it has well-established that our
universe is undergoing acceleration phase. This is also confirmed by other
observations like Baryon Acoustic Oscillation (BAO) [4, 5], Cosmic Microwave
Background (CMB) [6], WMAP [7, 8] and Planck [9] observations. There is some
unknown matter which possesses sufficient negative pressure (repulsive force),
which derives the universe to the acceleration, dubbed as dark energy (DE).
According to the most simple and successful candidate of dark energy is the
cosmological constant $\Lambda$ (or vacuum energy) and its density consist of
$\sim$ 70% of total energy density in the universe [10, 11, 12, 13]. The
concordance $\Lambda$ cosmology is consistent with the cosmological
observations, but it cannot solve the fine-tuning, and the cosmic coincidence
problems [11, 12, 14]. So other candidates of dark energy obey time-evolving
energy density have been proposed to solve the above cosmological problems
[15, 16, 17, 18]. The quintessence (or scalar field) [18, 19, 20, 21, 22, 23,
24, 25] is another most accepted candidate of DE. There are other scalar field
models which are also the candidates of DE models [26, 27, 28, 29, 30, 31, 32,
33]. On the other hand, modified theories of gravity may be the alternatives
to the DE to derive the acceleration of the universe [34, 35, 36, 37, 38, 39,
40, 41].

Another alternative to DE is an exotic type of fluid - the so-called pure
Chaplygin gas. The pure Chaplygin gas derives to the present-day acceleration
of the universe, which can be treated as a unification of dark matter and DE
[42, 44, 43]. Then it has been generalized to the generalized Chaplygin gas,
which also unifies dark matter and DE [45, 46, 47]. Further, it has been
modified to the modified Chaplygin gas (MCG), which propagates between
radiation and $\Lambda$CDM stages [48, 49]. Other extensions of Chaplygin gas
are variable Chaplygin gas [50], variable modified Chaplygin gas [51], viscous
Chaplygin gas [52] etc., which are also the candidates of DE models that unify
dark matter and DE. Due the Supernova, CMBR, WMAP observations, the parameters
of the generalized Chaplygin gas have been constrained [46, 53, 54, 55]. Also,
the MCG parameters have been constrained by CMBR, WMAP data [56, 57]. On the
other hand, the MCG parameters have been constrained in modified gravity
theories [58, 59, 60, 61, 62, 63].

Villanueva [64] has obtained generalized Chaplygin-Jacobi gas through Hubble
parameter associated to the generalized Chaplygin scalar field and using
Jacobi’s elliptic function [65]. The relevant inflationary quantities have
also been obtained [66]. Motivated by this work, we obtain modified Chaplygin-
Jacobi gas and modified Chaplygin-Abel gas equation of states. Then we study
the observational data analysis and constrains the model parameters. We check
the compatibility of the models with observational data employing AIC, BIC,
and DIC model selection tools. Using the best-fitting values of the models’
free parameters, we study the evolutions of cosmological and cosmographical
parameters and check the classical stability of the models. The paper is
structured as follows. In section 2, we briefly write the basic equations for
MCG. In sections 3 and 4, we obtain modified Chaplygin-Jacobi gas and modified
Chaplygin-Abel gas equation of states through the Hubble parameter. In section
5, we write the data analysis tools in our constructed models. Then we study
the OHD+CMB+BAO+SNIa joint data analysis and the information criteria for our
models. Section 4 deals with the evolutions of cosmological and cosmographical
parameters using best-fitting values of the model parameters. Finally, in
section 7, we discuss the results of the whole work.

##  2 Review of Modified Chaplygin Gas in FRW Model

The line element for homogeneous and isotropic universe in spatially flat
Friedmann-Robertson-Walker (FRW) model is (choosing speed of light $c=1$)

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

where $a(t)$ is a scale factor depends on time $t$. The Einstein’s field
equations are given by

$3H^{2}=8\pi G\rho$ (2)

and

$\dot{H}=-4\pi G(\rho+p)$ (3)

where $\rho$ and $p$ are the energy density and pressure respectively,
$H=\dot{a}/a$ denotes the time dependent Hubble parameter and $G$ is the
Newtonian gravitational constant. The energy conservation equation is given by

$\dot{\rho}+3H(\rho+p)=0$ (4)

We consider the fluid source is modified Chaplygin gas (MCG), which obeys the
equation of state [48, 49]

$p=A\rho-\frac{B}{\rho^{\alpha}}$ (5)

where $A$, $B(>0)$ and $\alpha\in(0,1)$ are constants. We have the solution of
$\rho$ as [48, 49]

$\rho=\left[\frac{B}{1+A}+\frac{C}{a^{3(1+A)(1+\alpha)}}\right]^{\frac{1}{1+\alpha}}$
(6)

where $C$ is an integration constant.

From field-theoretic point of view, we may describe this MCG model by
introducing scalar field $\phi$ and corresponding self-interacting potential
$V(\phi)$, which have the effective Lagrangian

${\cal L_{\phi}}=\frac{1}{2}\dot{\phi}^{2}-V(\phi)$ (7)

The analogous energy density $\rho_{\phi}$ and pressure $p_{\phi}$ for the
scalar field are the following [49]:

$\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi)=\rho$ (8)

and

$p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi)=A\rho-\frac{B}{\rho^{\alpha}}$ (9)

From these equations with the help of field equation (2), we get the solution
[49]:

$\phi=\phi_{c}+\frac{1}{\sqrt{6\pi
G(1+A)}(1+\alpha)}~{}Sinh^{-1}\left\\{\sqrt{\frac{C(1+A)}{B}}\frac{1}{a^{\frac{3}{2}(1+\alpha)(1+A)}}\right\\}$
(10)

where $\phi_{c}$ is constant. Using equations (2), (6) and (10), we obtain the
generating function for MCG

$H(\phi)=H_{c}~{}Sech^{-\frac{1}{1+\alpha}}(\Phi)$ (11)

where $\Phi=\sqrt{6\pi G(1+A)}(1+\alpha)(\phi-\phi_{c})$ and
$H_{c}=\sqrt{\frac{8\pi
G}{3}}\left(\frac{B}{1+A}\right)^{\frac{1}{2(1+\alpha)}}$ is the value of the
Hubble parameter when $\phi=\phi_{c}$. Now from equation (3), we get

$p=-\rho+\frac{1}{(4\pi G)^{2}}\left(\frac{dH}{d\phi}\right)^{2}$ (12)

##  3 Modified Chaplygin-Jacobi Gas

We know that hyperbolic function is a particular case of elliptic functions
and Jacobi elliptic functions are a set of basic elliptic functions. In 2015,
Villanueva [64] has replaced the cosine hyperbolic function in the Hubble
parameter (obtained by generalized Chaplygin gas) by Jacobi’s elliptic
function and derived the equation of state, termed as generalized Chaplygin-
Jacobi gas. So replacing the hyperbolic function in the generating function
(11) by the Jacobi elliptic cosine function $cn(\Phi)$, the generating
function can be written as

$H(\phi)=H_{c}~{}cn^{-\frac{1}{1+\alpha}}(\Phi)$ (13)

where $cn(\Phi)\equiv cn(\Phi,k)$ and $k\in[0,1]$ is the elliptic modulus. For
$k=1$, the above equation (13) reduces to the equation (11). Using the
relations
$cn^{2}(\Phi)+sn^{2}(\Phi)=1,~{}cn^{2}(\Phi)+(1-k)sn^{2}(\Phi)=dn^{2}(\Phi),~{}cn^{\prime}(\Phi)=-sn(\Phi)~{}dn(\Phi)$
(the notations are defined in [64]) and taking the derivative of (13), we
obtain

$\left(\frac{dH}{d\phi}\right)^{2}=6\pi
G(1+A)H_{c}^{2}cn^{-\frac{2(2+\alpha)}{1+\alpha}}(\Phi)(1-cn^{2}(\Phi))((1-k)+k~{}cn^{2}(\Phi))$
(14)

Also using equations (8), (9), (12), (13) and (14) we can obtain the potential
function

$V(\phi)=\frac{1}{2}\left(\frac{B}{1+A}\right)^{\frac{1}{2(1+\alpha)}}~{}cn^{-\frac{2}{2+\alpha}}(\Phi)[2+(1-cn^{-2}(\Phi))((1-k)+k~{}cn^{2}(\Phi))]$
(15)

Using equations (2), (12), (13) and (14), we finally obtain the relation
between pressure and density as in the form

$p=[(2k-1)(1+A)-1]\rho-\frac{kB}{\rho^{\alpha}}+\frac{(1-k)(1+A)^{2}}{B}~{}\rho^{2+\alpha}$
(16)

which may be called “Modified Chaplygin-Jacobi Gas” (MCJG) equation of state.
For $A=0$, the MCJG equation of state can be reduced to the generalized
Chaplygin-Jacobi gas equation of state [64]. Putting the expression of $p$
from equation (16) into (4), we obtain

$\rho^{1+\alpha}=\frac{B}{1+A}\left[\frac{a^{3(1+\alpha)(1+A)}+kD}{a^{3(1+\alpha)(1+A)}-(1-k)D}\right]$
(17)

where $D$ is a positive constant. For large value of scale factor $a(t)$, we
have
$\rho\simeq\left(\frac{B}{1+A}\right)^{\frac{1}{1+\alpha}}~{}\text{and}~{}p\simeq-\left(\frac{B}{1+A}\right)^{\frac{1}{1+\alpha}}=-\rho$
which corresponds to the $\Lambda$CDM model with the cosmological constant
$=\left(\frac{B}{1+A}\right)^{\frac{1}{1+\alpha}}$. But on the other hand, in
the phase of the universe where $x=a^{3(1+\alpha)(1+A)}-(1-k)D\simeq 0$, we
have
$\rho\simeq\left(\frac{B}{1+A}~{}\frac{D}{x}\right)^{\frac{1}{1+\alpha}}$.
This corresponds to a phase of the universe which follows polytropic equation
of state $p=\frac{(1-k)(1+A)}{B}\rho^{\alpha+2}$. The above expression of
$\rho$ can be written in the following form

$\rho^{1+\alpha}=\frac{B}{1+A}\left[\frac{A_{s}+(1-A_{s})(1+z)^{3(1+\alpha)(1+A)}}{kA_{s}-(1-k)(1-A_{s})(1+z)^{3(1+\alpha)(1+A)}}\right]$
(18)

where $A_{s}=\frac{1}{1+kD}$ satisfying $1-k<A_{s}<1$. So the present value of
the energy density is
$\rho_{CJ0}^{1+\alpha}=\frac{B}{(1+A)[kA_{s}-(1-k)(1-A_{s})]}$. The equation
of state (EoS) parameter is obtained in the form

$\displaystyle
W=\frac{p}{\rho}=[(2k-1)(1+A)-1]-k(1+A)\left[\frac{kA_{s}-(1-k)(1-A_{s})(1+z)^{3(1+\alpha)(1+A)}}{A_{s}+(1-A_{s})(1+z)^{3(1+\alpha)(1+A)}}\right]$
$\displaystyle+(1-k)(1+A)\left[\frac{A_{s}+(1-A_{s})(1+z)^{3(1+\alpha)(1+A)}}{kA_{s}-(1-k)(1-A_{s})(1+z)^{3(1+\alpha)(1+A)}}\right]$
(19)

##  4 Modified Chaplygin-Abel Gas

Abel elliptic functions are a special kind of elliptic function. If we replace
the hyperbolic function in the generating function (11) by the Abel elliptic
function $F(\Phi)$, the generating function (11) in terms of Abel elliptic
function can be written as

$H(\phi)=H_{c}~{}F^{-\frac{1}{1+\alpha}}(\Phi)$ (20)

Here $F(\Phi)=\sqrt{1+e^{2}\varphi^{2}(\Phi)}$ where
$\varphi(\Phi)\equiv\varphi(\Phi,c,e)$ is also Abel elliptic function (here
$c,e\in\mathbb{R}$). Using the relation
$\varphi^{\prime}(\Phi)=\sqrt{(1-c^{2}\varphi^{2}(\Phi))(1+e^{2}\varphi^{2}(\Phi))}$
and taking the derivative of (20), we obtain

$\left(\frac{dH}{d\phi}\right)^{2}=6\pi
G(1+A)H_{c}^{2}F^{-\frac{2(2+\alpha)}{1+\alpha}}(\Phi)(F^{2}(\Phi)-1)((c^{2}+e^{2})-c^{2}F^{2}(\Phi))$
(21)

Also using equations (8), (9), (12), (20) and (21) we can obtain the potential
function

$V(\phi)=\frac{1}{2}\left(\frac{B}{1+A}\right)^{\frac{1}{2(1+\alpha)}}~{}F^{-\frac{2}{2+\alpha}}(\Phi)[2+(F^{-2}(\Phi)-1)((c^{2}+e^{2})-c^{2}F^{2}(\Phi))]$
(22)

Using equations (2), (12), (20) and (21), we finally obtain the relation
between pressure and density as in the form

$p=[(e^{2}+2c^{2})(1+A)-1]\rho-\frac{c^{2}B}{\rho^{\alpha}}-\frac{(e^{2}+c^{2})(1+A)^{2}}{B}~{}\rho^{2+\alpha}$
(23)

which may be called “Modified Chaplygin-Abel Gas” (MCAG) equation of state.
For $A=0$, the MCAG equation of state may be called the generalized Chaplygin-
Abel gas equation of state. Putting the expression of $p$ from equation (23)
into (4), we obtain

$\rho^{1+\alpha}=\frac{B}{1+A}\left[\frac{a^{3e^{2}(1+\alpha)(1+A)}+c^{2}K}{a^{3e^{2}(1+\alpha)(1+A)}+(e^{2}+c^{2})K}\right]$
(24)

where $K$ is a positive constant. For large value of scale factor $a(t)$, we
obtain $p\simeq-\rho$ which corresponds to the $\Lambda$CDM model with the
cosmological constant $=\left(\frac{B}{1+A}\right)^{\frac{1}{1+\alpha}}$. For
small value of scale factor $a(t)$, we also obtain $p\simeq-\rho$, which
corresponds to the inflationary stage of the universe. So the MCAG propagates
between inflation and $\Lambda$CDM stages. The above expression of $\rho$ can
be written in the form

$\rho^{1+\alpha}=\frac{c^{2}B}{1+A}\left[\frac{B_{s}+(1-B_{s})(1+z)^{3e^{2}(1+\alpha)(1+A)}}{c^{2}B_{s}+(e^{2}+c^{2})(1-B_{s})(1+z)^{3e^{2}(1+\alpha)(1+A)}}\right]$
(25)

where $B_{s}=\frac{1}{1+c^{2}K}$ satisfying $0<B_{s}<1$. So the present value
of the energy density is
$\rho_{CA0}^{1+\alpha}=\frac{c^{2}B}{(1+A)[c^{2}B_{s}+(e^{2}+c^{2})(1-B_{s})]}$.
The equation of state (EoS) parameter is obtained in the form

$\displaystyle
W=\frac{p}{\rho}=[(e^{2}+2c^{2})(1+A)-1]-(1+A)\left[\frac{c^{2}B_{s}+(e^{2}+c^{2})(1-B_{s})(1+z)^{3e^{2}(1+\alpha)(1+A)}}{B_{s}+(1-B_{s})(1+z)^{3e^{2}(1+\alpha)(1+A)}}\right]$
$\displaystyle-c^{2}(e^{2}+c^{2})(1+A)\left[\frac{B_{s}+(1-B_{s})(1+z)^{3e^{2}(1+\alpha)(1+A)}}{c^{2}B_{s}+(e^{2}+c^{2})(1-B_{s})(1+z)^{3e^{2}(1+\alpha)(1+A)}}\right]$
(26)

##  5 Observational Data Analysis

Now, we assume that the universe is composed of radiation, dark matter (DM),
and dark energy (DE). So in the Einstein field equations (2), (3) and in the
conservation equation (4), we can consider $\rho=\rho_{r}+\rho_{m}+\rho_{d}$
and $p=p_{r}+p_{m}+p_{d}$. Here suffices $r,~{}m,~{}d$ denote the radiation,
dark matter, and dark energy, respectively. If we assume that there is no
interaction between radiation, DM, and DE, then their conservation equations
will be

$\dot{\rho}_{r}+3H(\rho_{r}+p_{r})=0,$ (27)
$\dot{\rho}_{m}+3H(\rho_{m}+p_{m})=0$ (28)

and

$\dot{\rho}_{d}+3H(\rho_{d}+p_{d})=0$ (29)

For radiation, the EoS is $p_{r}=\frac{1}{3}\rho_{r}$, so from equation (27)
we get the solution $\rho_{r}=\rho_{r0}(1+z)^{4}$. Also since the DM has
negligible pressure (i.e., $p_{m}\approx 0$), so from equation (28) we obtain
the solution $\rho_{m}=\rho_{m0}(1+z)^{3}$. Here $\rho_{r0}$ and $\rho_{m0}$
represent the present values of radiation density and DM density,
respectively. Next we assume that the DE of the universe is in the form of
MCJG or MCAG.

$\bullet{}$ MCJG : If the DE of the universe obeys the MCJG equation of state
(16), then using the equation (18), the equation (2) reduces to the following
equation for $E(z)$ as

$\displaystyle
E^{2}(z)=\Omega_{r0}(1+z)^{4}+\Omega_{m0}(1+z)^{3}+\Omega_{CJ0}[kA_{s}-(1-k)(1-A_{s})]^{\frac{1}{1+\alpha}}$
$\displaystyle\times\left[\frac{A_{s}+(1-A_{s})(1+z)^{3(1+\alpha)(1+A)}}{kA_{s}-(1-k)(1-A_{s})(1+z)^{3(1+\alpha)(1+A)}}\right]^{\frac{1}{1+\alpha}}$
(30)

where we have defined the normalized Hubble parameter
$E(z)=\frac{H(z)}{H_{0}}$ and the dimensionless density parameters
$\Omega_{r0}=\frac{8\pi G\rho_{r0}}{3H_{0}^{2}}$, $\Omega_{m0}=\frac{8\pi
G\rho_{m0}}{3H_{0}^{2}}$ and $\Omega_{CJ0}=\frac{8\pi
G\rho_{{}_{CJ0}}}{3H_{0}^{2}}$. Putting $z=0$ in the above equation, we obtain
$\Omega_{r0}+\Omega_{m0}+\Omega_{CJ0}=1$.

$\bullet{}$ MCAG : If the DE of the universe obeys the MCAG equation of state
(23), then using the equation (25), the equation (2) reduces to the following
equation for $E(z)$ as

$\displaystyle
E^{2}(z)=\Omega_{r0}(1+z)^{4}+\Omega_{m0}(1+z)^{3}+\Omega_{CA0}[c^{2}B_{s}+(e^{2}+c^{2})(1-B_{s})]^{\frac{1}{1+\alpha}}$
$\displaystyle\times\left[\frac{B_{s}+(1-B_{s})(1+z)^{3e^{2}(1+\alpha)(1+A)}}{c^{2}B_{s}+(e^{2}+c^{2})(1-B_{s})(1+z)^{3e^{2}(1+\alpha)(1+A)}}\right]^{\frac{1}{1+\alpha}}$
(31)

where the dimensionless density parameter $\Omega_{CA0}=\frac{8\pi
G\rho_{{}_{CA0}}}{3H_{0}^{2}}$. Putting $z=0$ in the above equation, we obtain
$\Omega_{r0}+\Omega_{m0}+\Omega_{CA0}=1$.

In this section, we’ll study the observational data analysis tools for fitting
the theoretical MCJG and MCAG models in flat FRW universe. To determine the
best fit values of the unknown parameters of the models, we use the observed
Hubble data (OHD), cosmic microwave background (CMB) data, baryonic acoustic
oscillations (BAO) data and type Ia supernova (SNIa) data survey. For this
purpose, we use minimum values of $\chi^{2}$ with maximum likelihood analysis
using different confidence levels (like $1\sigma,~{}2\sigma$ and $3\sigma$)
due to OHD+CMB+BAO+SNIa observational data.

###  5.1 OHD

Due to latest observed Hubble data (OHD) sets compilation [67, 68, 69] with 57
data points in the region $0.07\lesssim z\lesssim 2.42$ [70], the $\chi^{2}$
value (as a sum of standard normal distribution) can be written as

${\chi}^{2}_{OHD}=\sum\frac{(H(z)-H_{obs}(z))^{2}}{\sigma^{2}_{obs}(z)}$ (32)

where $H(z)$ and $H_{obs}(z)$ respectively represent the theoretical and
observed values of the Hubble parameter. Also $\sigma_{obs}(z)$ refers to the
standard error in the observed value of $H$. Due to Planck 2015 results [9],
we take a prior for the present value of the Hubble constant
$H_{0}=67.8$km/s/MPc.

###  5.2 CMB Data

To constraints the dark energy, we use the Cosmic Microwave Background (CMB)
anisotropy spectrum at the last scattering surface at the redshift $z\simeq
1100$. So we use the Planck distance priors from the Planck 2015 data sets
[9]. The proper angular diameter distance $D_{A}(z)$, CMB shift parameter
${\cal R}$, comoving sound horizon size $r_{s}(z)$ and acoustic scale
parameter $\ell_{A}$ are respectively given by [71, 72]

$D_{A}(z)=\frac{1}{H_{0}(1+z)}\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})}~{},$
(33) ${\cal R}=H_{0}\sqrt{\Omega_{m0}}~{}(1+z_{*})D_{A}(z_{*})~{},$ (34)
$r_{s}(z)=\frac{1}{H_{0}}\int_{0}^{x}\frac{dx^{\prime}}{x^{\prime}E(x^{\prime})\sqrt{3(1+\bar{R}_{b}x^{\prime})}}$
(35)

and

$\ell_{A}=(1+z_{*})~{}\frac{\pi D_{A}(z_{*})}{r_{s}(z_{*})}$ (36)

where $\bar{R}_{b}x=0.75\rho_{b}/\rho_{\gamma}$. The $\rho_{b}$ and
$\rho_{\gamma}$ are respectively the baryon energy density and photon energy
density at the present epoch. Also, the baryon density is
$\omega_{b}=\Omega_{b0}h^{2}$ [71, 73]. The $z_{*}$ is the redshift in the CMB
power spectrum at the last scattering surface, given by the fitting formula
[71, 73]

$z_{*}=1048\left[1+0.00124\left(\Omega_{b0}h^{2}\right)^{-0.738}\right]\left[1+\frac{0.0783\left(\Omega_{b0}h^{2}\right)^{-0.238}}{1+39.5\left(\Omega_{b0}h^{2}\right)^{-0.76}}~{}\left(\Omega_{m0}h^{2}\right)^{\frac{0.560}{1+21.1\left(\Omega_{b0}h^{2}\right)^{1.81}}}\right]$
(37)

The $\chi^{2}$ function for CMB measurement, is defined by [9, 71]

$\chi^{2}_{CMB}=\triangle
S_{i}\left[Cov_{CMB}^{-1}(S_{i},S_{j})\right]\triangle S_{j},~{}~{}\triangle
S_{i}=S_{i}^{theory}-S_{i}^{obs}$ (38)

where $S_{1}=\ell_{A}$, $S_{2}={\cal R}$, $S_{3}=\omega_{b}$ and $Cov^{-1}$ is
the inverse covariance matrix. The CMB data does not provide the full Planck
information, but it is an optimum way of studying several dark energy models.

###  5.3 BAO Data

Eisenstein et al. [4, 5] proposed the Baryon Acoustic Oscillation (BAO)
observational data analysis. BAO refers to the imprint left by relativistic
sound waves in the early universe, providing an observable to the late-time
large-scale structure. BAO measures the structures in the universe from very
large scales at different times (i.e., redshifts). For BAO measurement, the
effective distance measure $D_{V}(z)$ is defined by [71, 72]

$D_{V}(z)=\left[(1+z)^{2}D_{A}^{2}(z)~{}\frac{z}{H(z)}\right]^{1/3}$ (39)

The comoving sound horizon size $r_{s}(z_{d})$ is given in equation (35),
where $z_{d}$ is the redshift at the drag epoch given by [71, 4]

$z_{d}=\frac{1291\left(\Omega_{m0}h^{2}\right)^{0.251}}{1+0.659\left(\Omega_{m0}h^{2}\right)^{0.828}}~{}\left[1+0.313\left(\Omega_{m0}h^{2}\right)^{-0.419}\left(\Omega_{b0}h^{2}\right)^{0.238\left(\Omega_{m0}h^{2}\right)^{0.223}}\left[1+0.607\left(\Omega_{m0}h^{2}\right)^{0.674}\right]\right]$
(40)

For our data analysis, we use four data points: (i) for 6dF Galaxy Survey
[74], $r_{s}(z_{d})/D_{V}(0.106)=0.336\pm 0.015$; (ii) for SDSS-DR7 [75],
$D_{V}(0.15)=(664\pm 25Mpc)(r_{d}/r_{d,fid})$; (iii) for BOSS-DR11 [76],
$D_{V}(0.32)=(1264\pm 25Mpc)(r_{d}/r_{d,fid})$ and (iv) for BOSS-DR11 [76],
$D_{V}(0.57)=(2056\pm 20Mpc)(r_{d}/r_{d,fid})$. Th $\chi^{2}$ function for BAO
measurement, is defined by [71]

$\chi^{2}_{BAO}=\triangle
S_{i}\left[Cov_{BAO}^{-1}(S_{i},S_{j})\right]\triangle S_{j},~{}~{}\triangle
S_{i}=S_{i}^{theory}-S_{i}^{obs}$ (41)

###  5.4 SNIa Data

The most important and key class of cosmological probes is the standard
candles, in which most known standard candles are Supernovae Type Ia (SNIa).
For SNIa observation, the luminosity distance $d_{L}(z)$ and distance modulus
$\mu(z)$ are given by [3, 2, 77, 78]

$d_{L}(z)=(1+z)\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})}$ (42)

and

$\mu(z)=5\log_{10}\left[\frac{d_{L}(z)/H_{0}}{1~{}MPc}\right]+25$ (43)

From SDSS-II sample, SNLS sample and few high redshift samples from HST
(Hubble Space Telescope) with 740 data points [79] with redshift $z$ lies in
the region $0.01\leq z\leq 1.30$, the $\chi^{2}$ function for JLA (joint
light-curve analysis) compilation can be written as [80, 81, 82, 83]

$\chi^{2}_{SN}=A_{SN}-\frac{B_{SN}^{2}}{C_{SN}}$ (44)

where

$A_{SN}=\sum_{i=1}^{740}\frac{[\mu_{th}(z_{i})-\mu_{obs}(z_{i})]^{2}}{\sigma^{2}(z_{i})}~{},~{}B_{SN}=\sum_{i=1}^{740}\frac{[\mu_{th}(z_{i})-\mu_{obs}(z_{i})]}{\sigma^{2}(z_{i})}~{},~{}C_{SN}=\sum_{i=1}^{740}\frac{1}{\sigma^{2}(z_{i})}$
(45)

The total $\chi^{2}$ function for OHD+CMB+BAO+SNIa joint data analysis can be
written as

$\displaystyle\chi^{2}=\chi^{2}_{OHD}+\chi^{2}_{CMB}+\chi^{2}_{BAO}+\chi^{2}_{SN}$
(46)

Using the $\chi^{2}$ minimum test, we can determine the best fit values of the
parameters of MCJG and MCAG in the FRW universe. Since the MCJG and MCAG
models have a different number of parameters, so it is very difficult to a
fair comparison between these DE models by directly comparing their $\chi^{2}$
values. Obviously, a DE model involves more parameters that have a lower value
of $\chi^{2}$. So to discuss the fair model comparison, we need to study the
information criteria in the following subsection.

###  5.5 Information Criteria

Since the MCJG and MCAG have several parameters, so to discuss the analysis,
we need to know the information criteria (IC). The Akaike Information Criteria
(AIC) [84], Bayesian Information Criteria (BIC) [85] and Deviance Information
Criterion (DIC) [86, 87] are more popular among ICs. Since AIC was derived by
an approximate minimization of Kullback-Leibler information, so AIC is an
asymptotically unbiased estimator of Kullback-Leibler information. In the
standard assumption of Gaussian errors, the corresponding Gaussian estimator
for the AIC can be written as [88, 89, 87, 90] $\text{AIC}=-2\ln({\cal
L}_{max})+2\kappa+\frac{2\kappa(\kappa+1)}{N-\kappa-1}$ where ${\cal L}_{max}$
is the maximum likelihood function, $\kappa$ is the number of model
parameters, and $N$ is the number of data points used in the data fit. Since
$N\gg 1$, so the above expression reduces to the original AIC version:
$\text{AIC}=-2\ln({\cal L}_{max})+2\kappa$. Also, the Bayesian estimator for
BIC can be defined as [88, 89, 87] $\text{BIC}=-2\ln({\cal L}_{max})+\kappa\ln
N$. We see that the BIC is similar to AIC with a different second term. For
the given set of models, the deviations of the IC values are
$\triangle\text{AIC}=\text{AIC}_{model}-\text{AIC}_{min}=\triangle\chi^{2}_{min}+2\triangle\kappa$
and
$\triangle\text{BIC}=\text{BIC}_{model}-\text{BIC}_{min}=\triangle\chi^{2}_{min}+\triangle\kappa\ln
N$. For data analysis, the extensively favorable range of
$\triangle\text{AIC}$ is $(0,2)$. The less support range of
$\triangle\text{AIC}$ is $(4,7)$, but when $\triangle\text{AIC}>10$, the model
provides no support. For a model, $\triangle\text{BIC}$ of 2 provides positive
evidence against the model with higher BIC while $\triangle\text{BIC}$ of 6
supports strong evidence. On the other hand, the concepts of both Bayesian
statistics and information theory, the DIC can be defined as [86, 87]
$\text{DIC}=2\overline{D(\theta)}-D(\bar{\theta})$ where
$D(\theta)=-2\ln({\cal L}(\theta))+C$. Here $C$ is a standard constant
depending only on the data which will vanish from any derived quantity, $D$ is
the deviance of the likelihood function, and bar denotes the mean value of the
posterior distribution. For deviation of the DIC value
$\triangle\text{DIC}=\text{DIC}_{model}-\text{DIC}_{min}$, the lower value
$(<2)$ provides strong support of the model, instead of the higher value of
$\triangle\text{DIC}$ (less support). It should be noted that the AIC and BIC
count all the involved parameters of the model, but DIC counts only the number
of parameters of the model that contribute to the fit in an actual way.

Fig.1 Fig.2

Fig.3 Fig.4
Figs.1 - 4: Contour plots of $h$ vs $\Omega_{m0}$, $\alpha$ vs $A_{s}$, $A$ vs
$A_{s}$ and $k$ vs $A_{s}$ for MCJG by OHD+CMB+BAO+SNIa joint data analysis
for different confidence levels $1\sigma$ (blue), $2\sigma$ (red) and
$3\sigma$ (black).

Fig.5 Fig.6

Fig.7 Fig.8

Fig.9

Figs.5 - 9: Contour plots of $h$ vs $\Omega_{m0}$, $\alpha$ vs $B_{s}$, $A$ vs
$B_{s}$, $e$ vs $B_{s}$ and $c$ vs $B_{s}$ for MCAG by OHD+CMB+BAO+SNIa joint
data analysis for different confidence levels $1\sigma$ (green), $2\sigma$
(purple) and $3\sigma$ (orange).

###  5.6 Results

In the MCJG and MCAG models, the best fit values of the parameters can be
obtained by $\chi^{2}$ minimum test using the maximum likelihood estimation.
The parameter of the radiation density is given by the formula
$\Omega_{r0}=\Omega_{m0}/(1+z_{eq})$ where $z_{eq}=2.5\times
10^{4}\Omega_{m0}h^{2}(T_{cmb}/2.7K)^{-4}$. Now we choose
$\Omega_{rad0}=8.14\times 10^{-5}$. Using OHD+CMB+BAO+SNIa joint data
analysis, the best fit parameters for MCJG and MCAG modes are obtained by (i)
$\Omega_{m0}=0.328$, $h=0.657$, $\alpha=0.066$, $A=0.297$, $A_{s}=0.586$,
$k=0.521$, $\chi^{2}_{min}=657.381$ and (ii) $\Omega_{m0}=0.317$, $h=0.631$,
$\alpha=0.063$, $A=0.276$, $B_{s}=0.527$, $e=0.172$, $c=0.392$,
$\chi^{2}_{min}=632.197$ respectively. The best fit results with errors of the
model parameters are also given in Tables 1 and 2. In the figures 1 - 4, we
have drawn the contour plots of $h$ vs $\Omega_{m0}$, $\alpha$ vs $A_{s}$, $A$
vs $A_{s}$ and $k$ vs $A_{s}$ for MCJG by OHD+CMB+BAO+SNIa joint data analysis
for $1\sigma$, $2\sigma$ and $3\sigma$ confidence levels. Also in the figures
5 - 9, we have plotted the contours of $h$ vs $\Omega_{m0}$, $\alpha$ vs
$B_{s}$, $A$ vs $B_{s}$, $e$ vs $B_{s}$ and $c$ vs $B_{s}$ for MCAG model.
Since $\Lambda$CDM model is the reference model, so the values of
$\triangle$AIC, $\triangle$BIC and $\triangle$DIC for MCJG and MCAG models can
be measured relative to the $\Lambda$CDM model. For $\Lambda$CDM model, we
know that $\triangle$AIC=$\triangle$BIC=$\triangle$DIC=0 [71]. But for MCJG
and MCAG models, we found (i) $\triangle$AIC=1.293, $\triangle$BIC=5.824,
$\triangle$DIC=1.096 and (ii) $\triangle$AIC=1.131, $\triangle$BIC=5.795,
$\triangle$DIC=1.072 which lie on the favorable ranges. So we can conclude
that the MCJG and MCAG models are observationally viable models.

Model | $\Omega_{m0}$ | $h$ | $\alpha$ | $A$ | $A_{s}$ | $k$ | $\chi^{2}_{min}$
---|---|---|---|---|---|---|---
MCJG | $0.328^{+0.049}_{-0.047}$ | $0.657^{+0.028}_{-0.027}$ | $0.066^{+0.0025}_{-0.0026}$ | $0.297^{+0.011}_{-0.012}$ | $0.586^{+0.09}_{-0.08}$ | $0.521^{+0.019}_{-0.021}$ | $657.381$

Table 1: Best fit results of the MCJG model parameters by OHD+CMB+BAO+SNIa
joint data analysis.

Model | $\Omega_{m0}$ | $h$ | $\alpha$ | $A$ | $B_{s}$ | $e$ | $c$ | $\chi^{2}_{min}$
---|---|---|---|---|---|---|---|---
MCAG | $0.317^{+0.043}_{-0.049}$ | $0.631^{+0.024}_{-0.023}$ | $0.063^{+0.0025}_{-0.0023}$ | $0.276^{+0.012}_{-0.013}$ | $0.527^{+0.075}_{-0.077}$ | $0.172^{+0.007}_{-0.008}$ | $0.392^{+0.018}_{-0.017}$ | $632.197$

Table 2: Best fit results of the MCAG model parameters by OHD+CMB+BAO+SNIa
joint data analysis.

##  6 Evolutions of Cosmological and Cosmographical Parameters

In this section, we’ll study the nature of the cosmological and cosmographical
parameters due to taking the best fit values of the parameters of MCJG and
MCAG models. The propagations of MCJG and MCAG for the evolution of the
universe can be determined by the equation of state parameter $W$. From figure
10, we see that $W$ decreases from positive level $W>0$ to $W\sim 0$, i.e.,
radiation to dust models can be generated for both MCJG and MCAG. Then $W$
decreases and keeps negative sign ($-1<W<0$), i.e., quintessence models can be
generated. Here $W<-1/3$ shows the dark energy phase of the universe. Finally,
$W$ reaches to $-1$ as $z\sim-1$, i.e., the universe goes to the $\Lambda$CDM
stages due to the contributions of MCJG and MCAG. At the early stage of the
universe, the value of $W$ for MCAG is higher than MCJG. Obviously, at $z=0$,
the value of $W$ lies within $-1$ and $-0.8$. So at present, our universe is
accelerating, which may be caused by both MCJG and MCAG. Since $W\nless-1$, so
phantom crossing cannot occur in our considered MCJG and MCAG models.

In order to deceleration or acceleration phase of the universe, we need to
study the deceleration parameter, which is given as
$q=-\frac{\ddot{a}}{aH^{2}}$. For the deceleration phase of the universe,
$q>0$ and for the acceleration phase of the universe, $q<0$. The deceleration
parameter has been drawn in figure 11 for both MCJG and MCAG. As time passes,
we see that $q$ decreases from positive level to negative level, i.e.,
deceleration to acceleration transition occurs to the universe. There are
another parameters named as cosmographical parameters for understanding the
past and future evolution of the universe. The cosmography is the study of
scale factor by expanding it by Taylor series with respect to the cosmic time.
The cosmographical parameter like jerk ($J$), snap ($S$), and lerk ($L$)
parameters are [91, 92, 93, 94]

$\displaystyle J=\frac{\dddot{a}}{aH^{3}}=(1+z)\frac{dq}{dz}+q(1+2q),$ (47)
$\displaystyle S=\frac{a^{(4)}}{aH^{4}}=-(1+z)\frac{dJ}{dz}-J(2+3q),$ (48)
$\displaystyle L=\frac{a^{(5)}}{aH^{5}}=-(1+z)\frac{dS}{dz}-S(3+4q)$ (49)

So the cosmographical parameters are higher-order derivatives of deceleration
parameter $q$. We have drawn the jerk ($J$), snap ($S$) and lerk ($L$)
parameters in figures 12, 13 and 14 respectively for both MCJG and MCAG
models. The parameters $J$ and $L$ both decrease sharply from some positive
values to near-zero upto $z\sim 1$ and then increase sharply upto some
positive values. The sharpness of decrease and increase curves for MCAG is
higher than MCJG. On the other hand, the parameter $S$ increases from negative
level to positive level, and the transition occurs near $z=1$. The sharpness
of decrease and increase curves for MCAG is higher than MCJG.

Another important parameters are statefinder parameters which are
distinguishing different dark energy models. The statefinder parameters
$\\{r,s\\}$ are defined as [95, 96]

$r=\frac{\dddot{a}}{aH^{3}}~{},~{}s=\frac{r-1}{3(q-0.5)}$ (50)

For $\Lambda$CDM model, $\\{r,s\\}=\\{0,1\\}$ and for SCDM model,
$\\{r,s\\}=\\{1,1\\}$. The $\\{r,s\\}$ trajectory is drawn in figure 15. We
see that when $r$ increases, $s$ always decreases. Two branches of the
trajectory found in the diagram. Two branches intersect at $r=1,~{}s=0$. For
$s>0$, the value of $r$ is $<1$. In this region, the lower value of $r$ is
$\sim 0.4$ and the corresponding upper value of $s$ is $\sim 0.2$. But for
$s<0$, the value of $r$ is $>1$ for both MCJG and MCAG models.

The $Om$ diagnostic of dark energy is the first derivative of the scale factor
through the Hubble parameter, was introduced to differentiate $\Lambda$CDM
from other DE models, which is defined as [97]

$Om(x)=\frac{E^{2}(x)-1}{x^{3}-1}$ (51)

where $x=z+1$. For $\Lambda CDM$ model, $Om=\Omega_{0m}=$ constant. The $Om$
diagnostic against redshift $z$ is plotted in figure 16. We see that in the
region $z>0$, the $Om$ parameter decreases from some positive values to nearly
zero around $z\sim 0$. But for $z<0$, the $Om$ parameter obeys the negative
signature for both MCJG and MCAG.

Finally, we can examine whether the MCJG and MCAG are stable or not for the
best fit values of the parameters. So to study the stability of MCJG and MCAG
against small perturbation, we can write the squared speed of sound in the
form $v_{s}^{2}=\frac{dp}{d\rho}$. If $0<v_{s}^{2}<1$, we can say that the
model is classically stable while $v_{s}^{2}<0$ or $v_{s}^{2}>1$ represents
classically unstable. The square speed of sound $v_{s}^{2}$ is drawn in figure
17. We observe that $v_{s}^{2}$ decreases smoothly but lies within 0.2 and
0.82. So we can conclude that MCJG and MCAG are classically stable due to
taking the best fit values of the model’s parameters.

Fig.10 Fig.11

Fig.12 Fig.13

Fig.14 Fig.15

Fig.16 Fig.17

Figs.10 - 17: Plots of $W$, $q$, $J$, $S$, $L$ vs $z$, $s$ vs $r$, $Om$ and
$v_{s}^{2}$ vs $z$ for MCJG (red line) and MCAG (blue line).

##  7 Discussions and Conclusions

We have considered the flat FRW model of the universe and reviewed the
modified Chaplygin gas as the fluid source. Associated with the scalar field
model, we have determined the Hubble parameter as a generating function in
terms of the scalar field. Instead of hyperbolic function, we have taken
Jacobi elliptic function and Abel function in the generating function and
obtained modified Chaplygin-Jacobi gas (MCJG) and modified Chaplygin-Abel gas
(MCAG) equation of states, respectively. Next, we have assumed that the
universe filled in dark matter, radiation, and dark energy. The sources of
dark energy candidates are assumed as MCJG and MCAG. We have constrained the
model parameters by recent observational data analysis. Using $\chi^{2}$
minimum test (maximum likelihood estimation), we have determined the best fit
values of the model parameters by OHD+CMB+BAO+SNIa joint data analysis. In the
figures 1 - 4, we have drawn the contour plots of $h$ vs $\Omega_{m0}$,
$\alpha$ vs $A_{s}$, $A$ vs $A_{s}$ and $k$ vs $A_{s}$ for MCJG model in
$1\sigma$, $2\sigma$ and $3\sigma$ confidence levels. Also, in figures 5 - 9,
we have plotted the contours of $h$ vs $\Omega_{m0}$, $\alpha$ vs $B_{s}$, $A$
vs $B_{s}$, $e$ vs $B_{s}$ and $c$ vs $B_{s}$ for MCAG model. To examine the
viability of the MCJG and MCAG models, we have determined the values of the
deviations of information criteria like $\triangle$AIC, $\triangle$BIC and
$\triangle$DIC. Since $\Lambda$CDM model is the reference model, so the values
of $\triangle$AIC, $\triangle$BIC, and $\triangle$DIC for MCJG and MCAG models
can be measured relative to the $\Lambda$CDM model. Our computational values
of $\triangle$AIC, $\triangle$BIC, and $\triangle$DIC lie on the favorable
ranges. So our constructed MCJG and MCAG are observationally favorable models.

The evolutions of cosmological and cosmographical parameters (like equation of
state, deceleration, jerk, snap, lerk, statefinder, $Om$ diagnostic) have been
studied using best-fit values of MCJG and MCAG parameters. From figure 10, we
have seen that the EoS parameter $W$ decreases from positive level $W>0$ to
$W=-1$, i.e., radiation to $\Lambda$CDM models can be generated for both MCJG
and MCAG. But at the early stage of the universe, the value of $W$ for MCAG is
higher than MCJG. At present, our universe is accelerating, which may be
caused by MCJG and MCAG. Since $W\nless-1$, so phantom divide cannot occur in
our considered MCJG and MCAG models. Also, for both MCJG and MCAG models, from
figure 11, we have seen that the deceleration parameter $q$ decreases from
positive level to negative level, i.e., deceleration to acceleration phase
transition occurs to the universe. We have drawn the jerk ($J$), snap ($S$),
and lerk ($L$) parameters in figures 12, 13, and 14, respectively, for both
MCJG and MCAG models. The parameters $J$ and $L$ decrease sharply from some
positive values to near-zero upto $z\sim 1$ and then increase sharply upto
some positive values. The sharpness of decrease and increase curves for MCAG
is higher than MCJG. On the other hand, $S$ increases from negative level to
positive level, and the transition occurs near $z=1$. The sharpness of
decrease and increase of the trajectories for MCAG is higher than MCJG. From
figure 15, we have seen that when $r$ increases, $s$ always decreases. Two
branches of the trajectory found in the $(r,s)$ diagram. Two branches
intersect at $r=1,~{}s=0$. For $s>0$, the value of $r$ is $<1$. In this
region, the lower value of $r$ is $\sim 0.4$ and the corresponding upper value
of $s$ is $\sim 0.2$. But for $s<0$, the value of $r$ is $>1$ for both MCJG
and MCAG models. On the other hand, from figure 16, we have seen that in the
region $z>0$, the $Om$ parameter decreases from some positive values to nearly
zero around $z\sim 0$, while for $z<0$, $Om$ parameter obeys negative
signature for both MCJG and MCAG. To check the classical stability of the
models, we have examined the values of square speed of sound $v_{s}^{2}$ in
the interval $(0,1)$ for expansion of the universe. From figure 17, we have
observed that $v_{s}^{2}$ decreases smoothly but lies within 0.2 and 0.82. So
we can conclude that MCJG and MCAG are both classically stable.

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