2021

[3]Geetanjali Sethi

1]Department of Physics, Indian Institute of Science Education and Research,
Mohali, India

2]Department of Physics and Astronomy, University of Potsdam, Potsdam, 10587,
Germany

3]Associate Professor,Department of Physics, St. Stephen’s College, University
of Delhi, Street, Delhi, 110007, Delhi, India

# Variable Chaplygin Gas: Constraints from Supernovae, GRB and Gravitational
Wave Merger Events

Ashley Chraya<EMAIL_ADDRESS>Yuvraj Muralichandran
<EMAIL_ADDRESS><EMAIL_ADDRESS>[  [  [

###### Abstract

We investigate the cosmological constraints on the Variable Chaplygin gas
model from the latest observational data: SCP Union 2.1 compilation dataset of
Type Ia supernovae (SNe Ia), Pantheon sample of SNe Ia, Platinum Sample of
Gamma Ray Bursts (GRB) and GWTC-3 of gravitational wave merger events.
Variable Chaplygin gas is a model of interacting dark matter and dark energy,
which interpolates from a dust-dominated era to a quintessence-dominated era.
The Variable Chaplygin gas model is shown to be compatible with Type Ia
Supernovae and gravitational merger data. We have obtained tighter constraints
on cosmological parameters $B_{s}$ and $n$, using the Pantheon sample. By
using the Markov chain Monte Carlo (MCMC) method on the Pantheon sample, we
obtain $B_{s}$=0.108 $\pm$ 0.034, n=1.157 $\pm$ 0.513 and $H_{0}$=70.020 $\pm$
0.407, for GRBs, we obtain $B_{s}$=0.20 $\pm$ 0.11, n=1.45 $\pm$ 1.40 and
$H_{0}$=70.41 $\pm$ 0.67

###### keywords:

Gravitational Wave Merger Events – Dark Energy – Maximum Likelihood Estimation

and on GWTC-3, we obtain $B_{s}$=0.130 $\pm$ 0.076, n=0.897 $\pm$ 1.182 and
$H_{0}$=69.838 $\pm$ 3.007. The combined constraints from the above data sets
are $B_{s}$=0.11 $\pm$ 0.03, n=1.14 $\pm$ 0.36 and $H_{0}$=70.34 $\pm$ 0.61

##  1 Introduction

Most of the time, the observations have the power to push the well-established
field to its most stringent tests. This is especially true in modern
cosmology, and one such observation was of the supernovae type-Ia (SN-Ia) by
two independent groups in 1998 (Perlmutter_1999, ; riess1998observational, ),
which established that the expansion of the universe is accelerating and two-
thirds of total energy density is of dark energy component with negative
pressure. This was followed by massive experimental efforts to find other
independent observational means to verify the expansion of the universe
theory. Through a series of independent findings such as the Baryon Acoustic
Oscillations (BAO), Cosmic Microwave Background Radiation (CMBR)
(Spergel_2003, ; miller1999measurement, ), large-scale structure (LSS)
(bahcall1999cosmic, ), and Gamma-ray Bursts (GRBs), the accelerated expansion
of the universe and the existence of dark energy gained traction in cosmology.

Regardless of the limitations of cosmological models based on the cosmological
principle, numerous cosmological models have been extremely successful in
explaining the observed universe. The simplest of such models, the Friedmann-
Lemaître-Robertson-Walker (FLRW) model, assumes cosmological principle,
Einstein’s general relativity and the Universe is composed of Baryonic matter
and radiation. Assuming the universe is homogeneous and isotropic, FLRW metric
is,

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

where k= curvature of the space which can take values $k=0,\pm 1$ and a(t) is
the scale factor. Even though this model was extremely successful there are
many aspects of our observed universe which cannot be captured by such a
simple model. For instance, it is not possible to explain the accelerated
expansion if assumptions of FLRW model i.e. cosmological principle, Einstein’s
general relativity and only Baryonic matter and radiation is taken into
consideration. Subsequently, other models were explored which can be broadly
classified into two classes. One class of models involve reforming the
geometry part of the Einstein’s equations. This includes generalisation of the
gravitational action such as f(R)-gravity and higher dimension spacetime. The
other class of models alter the matter component of the Universe in the
Einstein’s equations. If it is assumed that the universe is composed only of
Baryonic matter and radiation as it was assumed in FLRW model, due to the
gravitational pull of these components expansion rate of the universe would
have slowed down, therefore, the need for an additional entity to explain the
accelerated expansion of the Universe was realised. In this approach, exotic
matter with equation of state such that they don’t follow strong energy
conditions is added to the mass distribution of the Universe. This conclusion
is easily followed by second Friedmann equations. One possible approach to
constructing a viable model for dark energy is to associate it with a slowly
evolving and spatially homogenous scalar field $\phi$, called “quintessence”
(PhysRevD.37.3406, ; PhysRevLett.82.896, ; guo2007parametrizations, ) or two
coupled fields (bento2002two, ). Over a large class of potentials, the
Quintessence model gives the energy density convergent to its present value
for a wide range of initial conditions in the past and exhibit tracker
behaviour (sahni2000case, ; padmanabhan2003cosmological, ). Some of the models
based on adding exotic matter are quintessence, k-essence, tachyons,
barotropic fluid etc. Quintessence, k-essence are scalar field models, whereas
Dark Energy models include barotropic fluids whose pressure is a function of
energy density, $P=f(\rho)$. The relationship between pressure and energy
density determines the dynamics of the fluid.

Observational results of SN Ia, as well as the anisotropy of the CMBR power
spectrum and clustering estimates, show that our universe is also composed of
dark matter, in addition to dark energy, Baryonic matter and radiation. Among
the various models, the one that is most favourable is the standard model of
cosmology or $\Lambda$-Cold Dark Matter ($\Lambda$CDM) model, where $\Lambda$
represents the cosmological constant accounting for the vacuum energy or the
energy density of space. However, this model faces some serious problematic
issues such as the ﬁne tuning (unconventional small value) (RevModPhys.75.559,
; RevModPhys.61.1, ) and cosmic coincidence problems (why the dark matter and
the dark energy are of the same order today although the universe is in a
speedy expansion phase?). Similarly the ”Quintessence” models also suffer from
fine tuning problem.

The cosmological constant model has been transformed into a dynamical form in
several ways to remove these problematic issues. We can describe an
interaction between dark matter and dark energy in order to execute the
dynamical form of the cosmological constant dark energy
(barrow2006cosmologies, ; gonzalez2006dynamics, ; boehmer2008dynamics, ;
jamil2009constraining, ). Hence, as an alternative to both the cosmological
constant and quintessence, the accelerating expansion of the universe may be
explained by introducing a cosmic fluid component with an exotic equation of
state known as Chaplygin gas (CG) (kamenshchik2001alternative, ) which is an
example of barotropic fluid. Moreover, dark matter and dark energy has no
laboratory evidence for its existence directly which forces us to find a model
in which these two dark components are different manifestations of a single
cosmic fluid. CG model unifies the CDM and the $\Lambda$ models’ features into
a single component with an exotic equation of state. This versatility of the
models built on the Chaplygin gas becomes an attractive feature as these
models can explain both dark energy and dark matter with a single component,
which makes it unified dark matter/energy (UDME) model. The CG model has been
further extended also to the generalized (bilic2002unification, ;
saadat2014vol, ), modiﬁed (debnath2004role, ), Variable(chimento1996exact, )
and the extended (kahya2015higher, ) forms. The CG model arises from the
string Nambu–Goto action in the light-cone coordinate (jackiw2000particle, ;
pedram2008quantum, ). For the generalized Chaplygin gas (GCG) model, the
action can be written as a generalized Born–Infeld form (bento2002generalized,
). These models have been found to be consistent with the SNe Ia data
(gong2004constraints, ), CMB peak locations (carturan2003cosmological, ) and
other observational tests like gravitational lensing, cosmic age of old high
redshift objects etc. (dev2004constraints, ), as also with some combination of
some of them (PhysRevD.69.083503, ). This model has been shown to fit within
the standard structure formation scenarios (bilic2008transient, ;
bento2002generalized, ; fabris2002density, ). Generalized Chaplygin gas model
is shown to be consistent with the constraints provided by 21 cm absorption
line from EDGES detection along with CMB probe (Yang_2019, ). Therefore, the
Generalized Chaplygin gas model seems to be a good alternative to explain the
accelerated expansion of the universe. As the original Chaplygin gas model
produced oscillations or exponential blowup of matter power spectrum that are
inconsistent with observations, various modified models have been considered.
In this paper we consider the variable Chaplygin gas model that was proposed
VCGPrime  and constrained using SNeIa “gold” data Sethi_2006 . There are also
hybrid models like Viscous Generalized Chaplygin Gas (VGCG), that were
proposed to tackle the late accelerated expansion of the Universe. This kind
of hybrid models was originally proposed by Zhai:2005mu  and are able to avoid
causality problems that arise when only dissipative fluids are considered.

The parameters of variable Chaplygin gas model (VCG) has been constrained
using statistical analysis of SNeIa (Supernova Type Ia) Union 2.1 dataset from
Supernova Cosmology Project, Pantheon+ Dataset from SH0ES collaboration and
with the 03 run of GWTC. Formal introduction to the variable Chaplygin gas
model is provided briefly in section 2 and with description of the Pantheon
and GWTC-3 datasets used for constraining the parameters, in section 3 and the
description of GRBs in section 5. The statistical formalism to constrain
parameters is given in section 4. The analysis on the performance of VCG model
with dataset and the constraints obtained on the parameters of the model is
discussed in section 6, followed by conclusion of the study in section 7

##  2 VariableChaplygin Gas Model

The equation of state for the Chaplygin gas is $P=$-$A/\rho$ where A is a
positive constant. A generalized Chaplygin gas model is characterised by an
equation of state

$P_{ch}=-\bigg{(}\frac{A}{\rho_{ch}^{\alpha}}\bigg{)}$ (2)

where $\alpha$ is a constant such that $0<\alpha\leq 1$. The Chaplygin gas
model corresponds to $\alpha=1$. Taking time component of the energy-momentum
conservation equation $T_{,\mu}^{\mu\nu}=0$, we obtain the continuity
equation,

$\frac{\partial\rho}{\partial t}+3H(p+\rho)=0$ (3)

where H is the Hubble parameter, $H=\frac{\dot{a}}{a}$, $\rho$ = total density
of Universe. Using equation (3), the energy density of the Chaplygin gas
evolves as PhysRevD.66.043507

$\rho{}_{ch}=\bigg{(}A+\frac{B}{a^{3(1+\alpha)}}\bigg{)}^{\frac{1}{1+\alpha}}$
(4)

where $a$ is the scale factor in the current epoch of the universe, and $B$ is
the constant of integration. In the earlier epochs i.e. $a\ll 1$, equation (4)
translates to $\rho\propto a^{-3}$, therefore, the Chaplygin gas behaves like
Cold Dark Matter (CDM) and for late epochs where the scale factor of the
Universe is $a\gg 1$ equation (4) translates to $p=-\rho=constant$, therefore,
the behavior of the Chaplygin gas is like the cosmological constant at alter
times thus leading to an accelerated expansion of the Universe. Thus,
Chaplygin gas equation of state leads to a component which behaves as non
relativistic matter at early times and as cosmological constant equal to $8\pi
GA^{1/(1+\alpha)}$ at later stage.

The models based on Chaplygin gas haven’t faltered the interest it gained
among researches as they have shown promise in the past. However, the
Chaplygin gas model produces oscillations or an exponential blowup of matter
power spectrums that are inconsistent with observations (sandvik2004end, ).
This instability may be avoided by taking into account the combined effect of
shear and rotation, which slows down the collapse with respect to the simple
spherical collapse model (del2013shear, ). Subsequently a modification of
Chaplygin gas as proposed is VariableChaplygin gas (VCG) (VCGPrime, ). The VCG
is a modification of the Chaplygin gas model where the equation of the state
of the Chaplygin Gas is free flowing across the epochs.

The VCG emerges from the dynamics of a generalized d-brane in a (d + 1, 1)
spacetime and can be described by a complex scalar field $\phi$ whose action
can be written as a generalized Born-Infeld action (bento2002generalized, ).
Considering a Born-Infeld Lagrangian (sen2002tachyon, )

$\mathcal{L}_{\mathrm{BI}}=V(\phi)\sqrt{1+g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi}$
(5)

where $V(\phi)$ is the scalar potential. In a spatially flat FLRW universe,
the energy density and pressure are given by
$\rho=V(\phi)\left(1-\dot{\phi}^{2}\right)^{-1/2}$ and
$P=-V(\phi)\left(1-\dot{\phi}^{2}\right)^{1/2}$, respectively. Thus, the
corresponding equation of state of CG is given by

$P=-\frac{V^{2}(\phi)}{\rho}$ (6)

One can rewrite the self-interaction potential as a function of the cosmic
scale factor: $V^{2}(\phi)=A(a)$. Thus, VCG is characterised by the equation
of state:

$P{}_{ch}=-\frac{A(a)}{\rho_{ch}}$ (7)

where $A(a)=A_{0}a^{-n}$ is a positive function of the cosmological scale
factor $a$, $A_{0}$ and $n$ are constants. Using the energy conservation
equation, equation (3) in a flat Friedmann-Robertson-Walker universe and
equation (7), the VariableChaplygin gas density evolves as:

$\rho{}_{ch}=\sqrt{\frac{6}{6-n}\frac{A{}_{0}}{a{}^{n}}+\frac{B}{a{}^{6}}}$
(8)

where B is a constant of integration. Using Einstein field equations,
$G_{\alpha\beta}=8\pi GT_{\alpha\beta}$, and FLRW metric, we obtain Friedmann
equations

$\displaystyle H^{2}$ $\displaystyle=\frac{8\pi G}{3}\rho-\frac{k}{a^{2}}$ (9)
$\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle=-\frac{4\pi G}{3}(\rho+3p)$

where $\rho$ is baryonic matter density. If dark energy component is taken
into consideration, first Friedmann equation gives the expansion rate of the
Universe in terms of matter and radiation density, $\rho$, curvature, $k$, and
the cosmological constant, $\Lambda$, as

$H^{2}\equiv\bigg{(}\frac{\dot{a}}{a}\bigg{)}^{2}=\frac{8\pi
G}{3}\rho-\frac{k}{a^{2}}+\frac{\Lambda}{3}$ (10)

Assuming, critical density $\rho_{c}=\frac{3{H^{2}}}{8{\pi}G}$ and density
parameter $\Omega=\frac{\rho}{\rho_{c}}$, first Friedmann equation reads as

$\Omega_{b}(a)+\Omega_{k}(a)=1$ (11)

This relation extends directly to other models with several components. Here,
taking baryonic matter and radiation into account, Friedmann equation reads
as,

$H^{2}=H_{0}^{2}(\Omega_{m,0}a^{-3}+\Omega_{\gamma,0}a^{-4}+\Omega_{k,0}a^{-2})$
(12)

where $\Omega_{m,0}+\Omega_{\gamma,0}+\Omega_{k,0}=1$. Every other component
can be added when its behaviour with scale factor is known.

After reducing Eq. (10) to the case of the spatially flat Universe,

$H{}^{2}=\frac{8\pi G}{3}\rho$ (13)

where $H\equiv\dot{a}/a$ is the Hubble parameter. Therefore the acceleration
condition $\ddot{a}>0$ is equivalent to

$\left(\frac{12-3n}{6-n}\right)a^{6-n}>\frac{B}{A_{0}}$ (14)

In order to incorporate accelerated expansion of the Universe, the necessary
condition is $n<4$.

For $n=0$, the original Chaplygin gas behaviour is restored. The gas initially
behaves as dust-like matter ($\rho_{ch}\propto a^{-3}$) and later as a
cosmological constant ($p=-\rho=constant$). However, in the present case of
VCG the universe tends to be a quintessence-dominated ($n>0$)
(hannestad2002probing, ; guo2005parametrization, ) or phantom-dominated one
($n<0$) (caldwell2003phantom, ) with constant equation of state parameter
$w=$-$1+n/6$. The first term on the right hand side of Eq. (8) is initially
negligible so that the expression Eq. (8) can approximately be written as
$\rho\sim a^{-3}$ , which corresponds to a universe dominated by dust-like
matter.

The present value of energy density of the Variable Chaplygin gas

$\rho_{cho}=\sqrt{\frac{6}{6-n}A_{0}+B}$ (15)

where $a_{0}=1$. Defining a parameter, $B_{s}$,

$B_{s}=\frac{B}{6A_{0}/(6-n)+B}$ (16)

the energy density becomes

$\rho_{ch}(a)=\rho_{ch0}\left[\frac{B_{s}}{a^{6}}+\frac{1-B_{s}}{a^{n}}\right]^{1/2}$
(17)

###  2.1 Model

The Friedmann equation, using equation (17) and $a=\frac{1}{1+z}$ for a
Variable Chaplygin gas, becomes

$\begin{split}H^{2}=&\frac{8\pi
G}{3}\bigg{\\{}\rho_{r0}(1+z)^{4}+\rho_{b0}(1+z)^{3}+\\\
&\rho_{ch0}\Big{[}B_{s}(1+z)^{6}+(1-B_{s})(1+z)^{n}\Big{]}^{1/2}\bigg{\\}}\end{split}$
(18)

where $\rho_{r0}$ and $\rho_{b0}$ are the present values of energy densities
of radiation and baryons, respectively. Using111We have used the fact that for
a flat Universe, $\Omega_{b0}+\Omega_{r0}+\Omega_{ch0}=1$, i.e the total
matter density sums up to unity.

$\frac{\rho_{r0}}{\rho_{ch0}}=\frac{\Omega_{r0}}{\Omega_{ch0}}=\frac{\Omega_{r0}}{1-\Omega_{r0}-\Omega_{b0}}$
(19)

and

$\frac{\rho_{b0}}{\rho_{ch0}}=\frac{\Omega_{b0}}{\Omega_{ch0}}=\frac{\Omega_{b0}}{1-\Omega_{r0}-\Omega_{b0}},$
(20)

Equation (18) becomes,

$H^{2}=\Omega_{ch0}H_{0}^{2}a^{-4}X^{2}(a),$ (21)

where

$\begin{split}X^{2}(a)=&\frac{\Omega_{r0}}{1-\Omega_{r0}-\Omega_{b0}}+\\\
&\frac{\Omega_{b0}a}{1-\Omega_{r0}-\Omega_{b0}}+a^{4}\bigg{(}\frac{B_{s}}{a^{6}}+\frac{1-B_{s}}{a^{n}}\bigg{)}^{1/2}\end{split}$
(22)

##  3 Datasets

SNe Ia are crucial to understand expansion of the universe. We examine the
parameters of the VariableChaplygin gas model using a statistical analysis of
the most recent SNe Ia data from the Pantheon Sample (scolnic2018complete, ),
the Supernova Cosmology Project (SCP) Union 2.1 compilation (suzuki2012hubble,
) and GWTC-3 gravitational waves dataset. Pantheon sample is composed of 1048
SNe Ia data whose redshifts span from z = 0.01 up to z = 2.26. SCP Union 2.1
compilation is composed of 580 SNe Ia data whose redshifts span from z=0.623
up to z=1.415 providing redshift, distance moduli and associated errors in
distance moduli. The gravitational merger events are obtained from the GWOSC
(Gravitational Wave Open Science Center) which has the events obtained from
detectors at LIGO Hanford, LIGO Livingston and LIGO Virgo. The events are
collected across the three runs: O1 (from 12 September 2015 to 19 January
2016), O2 (from 30 November 2016 to 25 August 2017) and the O3 runs, O3a (from
1 April 2019 to 30 September 2019) and O3b (from 1 November 2019 to March
2020). The data set consist of 90 confirmed events in the GWTC-3 (LIGODATA, )
2021arXiv211103606T . The catalog contains events whose sources are black hole
binary mergers up to a redshift of 0.90. The merger events from the GWOSC
dataset (from O1 run to the recent O3 run) are used to constrain the
parameters of the Variable Chaplygin gas model. The redshift data from these
events were taken to predict luminosity distance and distance modulus using
Variable Chaplygin gas model and compare with the luminosity distance obtained
from the merger events using Bayesian inference.

##  4 Statistical Methodologies

In this work, we analyse luminosity distance cosmological observable.
Luminosity distance show an explicit dependence on the cosmological model
under consideration. Hence with this observable we can compare the model with
the experimentally observed value of such observable.

We have used the SNe-Ia and gravitational waves to constrain the parameters of
the Variable Chaplygin gas model. Using the Friedmann equation, in a flat
Universe, the luminosity distance is expressed as

$d_{L}(z,\mathbf{p})=c(1+z)\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime},\mathbf{p})}$
(23)

where, $\\{\mathbf{p}\\}$ denotes the set of all parameters describing the
cosmological model and $H(z,\mathbf{p})$ denotes the Hubble parameter as
defined in the model chosen. In our case we have taken into consideration the
contributions from radiation and baryons in addition to the Chaplygin gas.
Luminosity distance can also be expressed using (21) and $a=\frac{1}{1+z}$ as

$d_{L}=\frac{c(1+z)}{H_{0}}\int_{0}^{z}\frac{dz}{\Omega_{ch0}^{1/2}X(z)}$ (24)

The theoretical predicted value of distance modulus is obtained by

$\mu_{p}^{(l)}=5\log\left[\frac{d_{L}\left(z_{l}\right)}{\mathrm{Mpc}}\right]+25$
(25)

We can determine a best fit to the set of parameters by using a $\chi^{2}$
goodness fit test,

$\chi^{2}(\mathbf{p})=\sum_{l}\frac{\left[\mu_{p}^{(l)}\left(z_{l}\mid\mathbf{p}\right)-\mu_{0}^{(l)}\left(z_{l}\right)\right]^{2}}{\sigma_{l}^{2}}$
(26)

where $\mu_{p}^{(l)}(z_{l}\mid\mathbf{p}$) is the theoretical predicted value
of distance modulus at redshift $z_{l}$, given the set of parameters p, and
the sum is over all the observed data. Here $\sigma_{l}$ is the dispersion of
the distance modulus due to intrinsic and observational uncertainties. In the
case of SNe Ia, the errors $\sigma_{l}$ are Gaussian and hence, probability
density function for the parameters ($B_{s}$, $n$ and $H_{0}$) is

$p(\mathbf{s})\propto\exp\left(-\frac{\chi^{2}}{2}\right)$ (27)

with the parameter limits $0\leq B_{s}\leq 1$ and $n\leq 4$. The probability
density function of a parameter $\mathbf{p}_{i}$ is obtained by integrating
over all possible values of other parameters. To reduce the computation time,
we can analytically integrate over Hubble constant and hence with modified
$\chi^{2}$ statistics the fit is obtained by minimizing:

$\chi^{2}=\sum_{i}\bigg{[}\frac{\mu_{th}^{i}-\mu_{obs}^{i}}{\sigma_{i}}\bigg{]}^{2}-\frac{C_{1}}{C_{2}}\bigg{(}C_{1}+\frac{2}{5}\ln
10\bigg{)}-2\ln\mathit{h},$ (28)

here,

$C_{1}\equiv\sum_{i}\frac{\mu_{th}^{i}-\mu_{obs}^{i}}{\sigma_{i}^{2}},$ (29)

and

$C_{2}\equiv\sum_{i}\frac{1}{\sigma_{i}^{2}}.$ (30)

where $\sigma_{i}$ is the error associated with observed distance modulus
$\mu_{obs}$. The modified distance modulus in the above relation is

$\mu_{th}=5\log\frac{H_{0}d_{L}}{ch}+42.38$ (31)

$H_{0}$ is the value of Hubble parameter at present, $c$ is the speed of
light. The probability distribution function of the estimated parameters
(excluding h) is now obtained by using the modified $\chi^{2}$. It is
straightforward to check that the derivative of equation (28) with respect to
h is zero; hence our results are independent of the choice of h. We take
$h=0.70$. By minimizing the modified $\chi^{2}$ statistics we have found best
fit for $B_{s}$ and $n$. We have obtained $\chi^{2}$ contour plot with
confidence intervals for the Pantheon sample, see Fig. 1. As we are fitting
for two parameters, we use $\chi^{2}_{0.68}=\chi^{2}_{min}+2.30$,
$\chi^{2}_{0.95}=\chi^{2}_{min}+4.61$, $\chi^{2}_{0.99}=\chi^{2}_{min}+9.21$
for the value corresponding to 1 $\sigma$, 2 $\sigma$, 3 $\sigma$
respectively. We have also obtained $\chi^{2}$ contour plots with confidence
intervals for the Pantheon sample, SCP Union 2.1 compilation and GWTC-3
together, see Fig. 2. We are able to constrain the parameters of the model
more tightly with the Pantheon sample than SCP Union 2.1 compilation.
Therefore, for further analysis in this paper we will use the Pantheon sample
only. For gravitational wave dataset, the errors in the observables are non-
Gaussian, but we applied the above analysis for the gravitational wave dataset
taking average of the upper and lower error bars. As the errors on the
gravitational waves observables are too large we are not able to constrain the
parameters of the model with it. Though, we are able to show that parameter
values using $\chi^{2}$ by the Pantheon sample and GWTC-3 are consistent.

Figure 1: $\chi^{2}$ contour plot obtained for the Pantheon sample. Gradients
represents levels of $\sigma$ confidence interval range obtained from the
$\chi^{2}$ goodness test with two parameters $B_{s}$ and $n$ varying and fixed
Hubble Parameter. Results are shown in table 2.

The theoretical luminosity distance and distance modulus from the equation
(24) and equation (31) is compared to the observed values of distance
luminosity and distance modulus from the Pantheon, SCP Union 2.1 and GWTC-3
dataset respectively, see Fig. 3. We have shown that the model fits well and
the viability of the model with all three samples. This shows Variable
Chaplygin gas model (interacting dark energy and dark matter model) is capable
of explaining the observed expansion of the universe.

Figure 2: $\chi^{2}$ contour plots obtained for the Pantheon sample, SCP Union
2.1 compilation and GWTC-3 dataset with an overlap of confidence intervals for
all three observational samples. Results are shown in table 2.

Figure 3:  Viability of Variable Chaplygin gas model with Pantheon, SCP Union
2.1 and GWTC-3 datasets. Variation of theoretical prediction of distance
modulus (DM) (left) and luminosity distance (DL or LD) (right) from Variable
Chaplygin gas model with redshift where the parameters ($B_{s}$, $n$) are
determined by $\chi^{2}$ goodness test. This shows model fits well with the
Pantheon sample as well as with GWTC-3 dataset.

Equation 24, can also be written as

$d_{L}=\frac{X1}{H_{0}}$ (32)

where

$X1=c(1+z)\int_{0}^{z}\frac{dz}{\Omega_{ch0}^{1/2}X(z)}$ (33)

From the above equation, we inferred the value of the Hubble constant for all
three observational samples, see Fig. 4. For the case of GWTC-3 dataset,
errorbars are associated with both the variables, redshift and luminosity
distance, hence we use orthogonal distance regression method for linear
regression, by taking the average of the upper and lower error bar. The best
fit is obtained between the observed and the theoretical value by minimizing

$\sum_{i=1}^{n}\left[\frac{\left(y_{i}-\alpha-\beta
X_{i}\right)}{\eta}+\left(x_{i}-X_{i}\right)\right]$ (34)

where $y_{i}$ is the observed value of distance modulus or the luminosity
distance. $\alpha$-$\beta X_{i}$ is the value of distance modulus or
luminosity distance obtained by VCG Model with applied constraints. $X_{i}$ is
the absolute value of the other variable in study, redshift associated with
the merger event and $x_{i}$ is the redshift value including the error range.
$\eta$ is the ratio between the variance of the errorbars associated with the
variables under study. By accounting for the variance in the error bars
associated with the response variable, luminosity distance or the distance
modulus of the event and the predictor variable, the redshift of the merger
event, regression method is more sensitive towards the precision of both the
predictor and the variables and minimizes the discrepancies brought in by the
outliers in the dataset.

However the Orthogonal Regression can be only used when the errors are
normally distributed. In the case of the GWOSC dataset, the observed
luminosity distance and therefore the distance modulus of the gravitational
merger event is accompanied with non-Gaussian errors, therefore orthogonal
regression can not provide accurate constraints on parameter $B_{s}$ and $n$.
The non-Gaussian nature of the errorbars associated with the merger event
makes Maximum Likelihood Estimation an ideal method to constraint the
parameters under study.

Figure 4: Hubble Parameter Estimation from the Pantheon sample, SCP Union 2.1
compilation and GWTC-3 dataset. Results are shown in the last column of table
2

###  4.1 Maximum Likelihood Estimation (MLE)

The equation (28) which is for determining the $\chi^{2}$ goodness fit value
has a dependency on the $H_{0}$, which is fixed. This results in substituting
the Hubble parameter value from previous results to the equation. Therefore
$\chi^{2}$ goodness value will be affected by any potential errors associated
with the Hubble Parameter value from the previous results. This can not be
avoided in the modified $\chi^{2}$ statistics analysis as explained in the
previous section.
Maximum Likelihood Estimation (MLE) method can be utilized to infer the
parameters $B_{s}$, $n$ and $H_{0}$ together, unlike modified $\chi^{2}$
statistics equation (28), where we obtained parameters $B_{s}$ and $n$, and
was independent of $H_{0}$. This is extremely useful as even the $h_{0}$
parameter can also be determined using MLE along with $B_{s}$ and $n$,
therefore the result becomes self-contained. The assumption that has to be
taken is that the dataset are obtained from a same distribution family of the
given parameters of the dataset and each individual data point on the dataset
is not dependent on other data points to hold it’s current value. This is a
classic i.i.d. (Independent and Identically Distribution) assumption. The
maximum likelihood estimation determines the combination of parameters by
maximizing:

$\bigg{(}B_{s},n,h_{0}\bigg{)}_{MLE}=\\\
\underset{(B_{s},n,h_{0})}{\arg\max}\\\
\mbox{\Large$\Pi$}f(a_{i}\|(B_{s},n,h_{0}))$ (35)

here $\Pi f(a_{i}\|(B_{s},n,h_{0})_{MLE}$ is a summation of probability
function of every individual data point in the observed dataset. In general,
the probability function for the total dataset under a given combination of
parameters, $f(x_{1},x_{2},....x_{n}\|(B_{s},n,h_{0}))$ is required in
equation (35). Since, the dataset follows i.i.d. distribution, the probability
of combined data is equivalent to the summation of probability of each data
point in the given dataset.

Figure 5: Cosmological parameters contours for the Pantheon sample using
equation of distance modulus and Gaussian priors. Results are shown in table
4.

###  4.2 Maximum Likelihood Estimation using cobaya

Cobaya (COde for BAYesian Analysis) ((Torrado_2021, ; 2019ascl.soft10019T, ))
is a coding framework built for statistical modelling to find arbitary
posteriors from the given sets of parameters especially for Cosmology. cobaya
determines Maximum Likelihood Estimation for the dataset from the given
parameters using high advanced Monte Carlo samplers like MCMC from CosmoMC,
nested samplers from Polychord or from a wide range of samplers incorporated
in cobaya. cobaya removes the dependency on the Hubble parameter value
obtained from previous results and also significantly reduces the computing
time to find the combination of the parameters $B_{s}$, $n$ and $H_{0}$ for
which the probability for the given data distribution under MLE is maximum.
The $\chi^{2}$ goodness fit algorithm takes much longer duration to find the
combination of the parameters for which the $\chi^{2}$ goodness fit value is
minimum in spite the fact that the parameters which are kept arbitrary are
$B_{s}$ and $n$.

Figure 6: Cosmological parameters contours for the GWTC-3 sample using
equation of distance modulus with Gaussian priors and taking average of
assymmetrical error bars of GW results (Gaussian likelihood). Results are
shown in table 1 Figure 7: Cosmological parameters contours for the GWTC-3
sample using equation of distance modulus with Gaussian priors but taking
modified likelihood because the dataset of GW has assymetrical error bars.
Results are shown in table 1

Likelihood | $B_{s}$ | n | $H_{0}$
---|---|---|---
Non-Gaussian | $0.130\pm 0.079$ | $1.025\pm 1.120$ | $70.599\pm 3.227$
Gaussian | $0.130\pm 0.076$ | $0.897\pm 1.182$ | $69.838\pm 3.007$

Table 1: Constraints imposed on the parameters ($B_{s}$, $n$ & $H_{0}$) of VCG
Model from gravitational wave merger events using Gaussian and non-Gaussian
likelihood functions.

From figures 6 and 7, and table 1, we can conclude that cosmological
parameters values from Gaussian and non-Gaussian likelihoods fall within 1
sigma, hence, we would be using Gaussian likelihoods in further analysis for
GWTC-3 dataset. We used Gaussian priors to calculate cosmological results for
the Pantheon sample and GWTC-3 dataset as shown in figures 6 and 8. For
Gaussian priors, we considered mean value as the expectation value from the
$\chi^{2}$ contour plots, see figure 1 and 2, and we doubled the standard
deviation value $\sigma$ value which is then considered the new standard
deviation for the MCMC. To find these best fitting parameters we use the
D’Agostini d2005fits  Bayesian method, which takes into account the error bars
on all the axes. We are not considering SCP Union 2.1 compilation here as the
Pantheon sample has tighter constraints, as discussed before using $chi^{2}$
statistics. We have used modified likelihood function for the case of
asymmetric errors in the GWTC-3 dataset. We are again able to show that the
cosmological parameter values using the Pantheon sample and GWTC-3 dataset are
consistent, and we are able to constrain the parameters using both datasets
independently.

Figure 8: Cosmological parameters contours for the Pantheon sample and GWTC-3
together using equation of distance modulus, Gaussian priors and Gaussian
likelihood. Results are shown in table 4 Figure 9: Cosmological parameters
contours for the GRBs Platinum sample using equation of distance modulus and
Gaussian priors. Results are shown in table 4.

##  5 Gamma-ray Bursts 3D fundamental plane correlation as a cosmological tool

Gamma-ray bursts (GRBs) are incredibly powerful phenomena. They are the
brightest objects after the Big Bang, and some of the farthest astrophysical
objects ever detected (Paczynski1986, ; Planck2018, ; Kumar2015, ). These
features make GRBs promising cosmological tools, similar to supernovae type Ia
(SNe Ia). GRBs are extremely luminous, which allows them to be observed at
very large distances, corresponding to high redshifts. Indeed, GRBs have been
observed up to redshifts of 8.2 and 9.4 (tanvir2009, ; cucchiara2011, ), while
SNe Ia have only been observed up to redshift 2.26 (Rodney2015, ). However,
using GRBs as cosmological tools requires a full understanding of their
physical mechanisms. Both their energy emission mechanisms and progenitors are
still being studied by the scientific community.

There are two main scenarios for the birth of GRBs. The first scenario is the
explosion of a very massive star at the end of its lifetime (Narayan1992, ;
Woosley1993, ; MacFadyen1999, ; Nagataki2007, ; Nagataki2009, ). This is
followed by a core-collapse SNe (Stanek2003, ; MacFadyen2001, ). The second
scenario is the coalescence of two compact objects, like black holes (BHs) or
neutron stars (NSs) (Lattimer1976, ; Eichler1989, ; Li1998, ; Rowlinson2014, ;
Rea2015, ). The most probable frameworks for the central engine that powers
the GRB consider the following astrophysical objects: BHs, NSs, or fast
spinning newly born highly magnetized NSs magnetars (Usov1992, ; Liang2018, ;
Ai2018, ). GRBs can be divided into two main categories: short and long. Short
GRBs are associated with the merging of two compact objects, while long GRBs
are associated with the core collapse of a very massive star. GRB light curves
(LCs) can be divided into two main phases: the prompt emission and the
afterglow. The prompt emission is a rapid burst of gamma rays, while the
afterglow is a longer-lasting emission that can be detected in X-ray, optical,
and radio wavelengths. The plateau phase of GRBs is a flat part of the LC that
follows the prompt emission. It was discovered by the Neil Gehrels Swift
Observatory (Swift) and typically ranges from $10^{2}$ to $10^{5}$ seconds in
duration. The plateau phase is thought to be caused by the spin-down of a
newly born magnetar or the external shock model.

In the past decades, many efforts have been made to find possible correlations
between physical features of GRBs. One of these correlations is the so-called
Dainotti relation, which links the time at the end of the plateau emission
measured in the rest frame, $T^{*}_{X}$, with the corresponding X-ray
luminosity of the LC, $L_{X}$ (Dainotti2008, ). This correlation is
theoretically supported by the magnetar model. Its extension in three
dimensions has been discovered by adding the prompt peak luminosity,
$L_{\text{peak}}$, and is known as the fundamental plane correlation or the 3D
Dainotti relation (Dainotti2016, ; Dainotti2017c, ). It has also shown that
GRBs can be used as cosmological probes. The fundamental plane relation has
the following form:

$\log L_{X}=c+a\cdot\log T^{*}_{X}+b\cdot(\log L_{peak}),$ (36)

where $a$ and $b$ are the best-fit parameters given by the D’Agostini
procedure Dago05  linked to $\log T^{*}_{X}$ and $\log L_{peak}$,
respectively, while $c$ is the normalization. In this study, we follow the
same approach as in (dainotti2023gamma, ) and use the Platinum Sample of GRBs
as a cosmological probe. The Platinum Sample of GRBs is a compilation of the
highest-quality GRBs with well-measured redshifts. The GRBs in this sample are
well-suited for cosmological studies because they are bright and can be seen
to high redshifts. We further combine the GRB Platinum sample with the
Supernovae Ia Pantheon sample and GWTC-3 Catalog using a Bayesian framework,
and compute the cosmological parameters. This facilitates the examination of
whether the inclusion of GRBs in the analysis would corroborate the findings
observed when studying individual samples independently, as well as the extent
to which it could augment the accuracy of the cosmological parameters.

##  6 Analysis and Result

We used $\chi^{2}$ minimization to obtain the parameters $B_{s}$ and n of VCG
model using the Pantheon sample, SCP Union 2.1 compilation and through MLE for
GWTC-3 dataset. Subsequently, we used the obtained parameters to infer the
Hubble constant. Values of these parameters can be found in table 2. For
comparison and completeness, we also tested our $\chi^{2}$ program with the
Pantheon sample and obtained for Standard Model of Cosmology,
$\Omega_{m}=0.298\pm 0.0087$ and $H_{0}=71.985\pm 0.663$.

Figure 10: Cosmological parameters contours for the combined samples (SNIe
Pantheon+ GRBs Platinum + GWTC-3) using equation of distance modulus and
Gaussian priors. Results are shown in table 4.

Dataset | $B_{s}$ | n | $H_{0}$
---|---|---|---
Pantheon | $0.11^{+0.024,+0.043}_{-0.024,-0.037}$ | $1.260^{+0.205,+0.410}_{-0.308,-0.513}$ | $70.47^{+0.13}_{-0.13}$
SCP Union 2.1 | $0.096^{+0.049,+0.073}_{-0.043,-0.062}$ | $1.301^{+0.485,+0.623}_{-0.692,-0.969}$ | $69.41^{+0.39}_{-0.39}$
GWTC-3 | $0.213^{+0.355,+0.413}_{-0.203,-0.203}$ | $-1.744^{+4.170,4.667}_{-98.265,-98.265}$ | $68.13^{+0.19}_{-0.19}$

Table 2: Constraints imposed on the parameters ($B_{s}$, $n$ & $H_{0}$) of VCG
Model from Supernovae Events and gravitational wave merger events using
modified $\chi^{2}$ statistics governed by equation 28.

From figure 3, we can conclude that VCG model is a viable model which is
capable of explaining the observed expansion of the universe using the
Pantheon sample, SCP Union 2.1 compilation and GWTC-3 dataset. As the GWTC-3
dataset has asymmetrical error bars, we can’t use $\chi^{2}$ minimization
method to obtain the VCG model parameters for GWTC-3 dataset. Hence, we used
cobaya for MCMC computation using modified likelihood function and the
Gaussian priors. We used results obtained from $\chi^{2}$ in table 2 using
Pantheon sample as the priors and we let all the parameters vary
simultaneously. We performed the analysis using SNIe Pantheon sample, GWTC-3
catalog, GRBs Platinum sample and combining all three. We find the best fit
results of the constants in the fundamental plane relation are: $a=-0.88\pm
0.06$, $b=0.49\pm 0.01$, and $\sigma_{int}=1.06\pm 0.06$, where $\sigma_{int}$
is the intrinsic scatter of the correlation. The results are shown in table 4.
To complete our analysis in table 3, we present comparison of our results with
the VGCG and the GCG models also.

Dataset |  $\chi^{2}_{min}$/$dof$ | $h$ | $\Omega_{b0}h^{2}$ | $B_{s}$ | $\alpha$ | $\xi_{0}$
---|---|---|---|---|---|---
VGCG
SnIa | 1036.4/1043 | $0.70^{+0.18}_{-0.20}$ | $0.02242^{+0.00014}_{-0.00014}$ | $0.50^{+0.18}_{-0.21}$ | $0.93^{+0.69}_{-0.61}$ | $0.15^{+0.09}_{-0.09}$
GCG
SnIa | 1036.3/1044 | $0.71^{+0.20}_{-0.18}$ | $0.02242^{+0.00014}_{-0.00014}$ | $0.96^{+0.05}_{-0.04}$ | $1.13^{+0.44}_{-0.28}$ | -

Table 3: Results of constraining the cosmological parameters with Pantheon
dataset in other dissipative fluids models of cosmology like VGCG and GCG HER-
AL

Dataset | $B_{s}$ | n | $H_{0}$
---|---|---|---
Pantheon | $0.11\pm 0.03$ | $1.16\pm 0.51$ | $70.02\pm 0.41$
GWTC-3 | $0.13\pm 0.08$ | $0.90\pm 1.18$ | $69.84\pm 3.01$
GRBs | $0.20\pm 0.11$ | $1.45\pm 1.40$ | $70.41\pm 0.67$
Combined | $0.11\pm 0.03$ | $1.14\pm 0.36$ | $70.34\pm 0.61$

Table 4: Constraints imposed on the parameters ($B_{s}$, $n$ & $H_{0}$) of VCG
Model from Supernovae events, GRB events, GW merger events and combined of all
three datasets using distance modulus equation, Gaussian priors and Gaussian
likelihood.

##  7 Conclusions

The Variable Chaplygin gas model is indeed able to explain the evolution of
the universe in account for the gravitational waves from merger events. As the
VCG Model speculate, the Chaplygin gas is expected to behave like a non-
relativistic entity and later evolves to account for the accelerated expansion
observed in the current epoch of the Universe. The best fit obtained from the
VCG model with the GWTC-3 Dataset lies within confidence levels obtained in
analysis conducted by Sethi_2006  and VCGPrime  and also is compatible with
the constraints obtained from GRBs. The GWTC-3 dataset has also brought in
further constraints on the parameters $B_{s}$ and $n$. The constraints
obtained using GWTC-3 are expected to become better as the dataset becomes
bigger with more detections. These results are indeed limited to the fact that
locating sources of Gravitational waves is still subject to improvement in the
detection techniques.

##  8 Acknowledgement

We would acknowledge Isophote for providing us the opportunity to work in this
project. Dr. Geetanjali Sethi is also thankful to Principal, St. Stephen’s
College for allowing her to be a part of this project. Ashley Chraya is
thankful to Department of Physics, Indian Institute of Science Education and
Research, Mohali for enabling him to be a part of this project. We also thank
Dr. Akshay Rana, St. Stephen’s College for his inputs and suggestions.

## Declarations

### Funding and/or Conflicts of interests / Competing interests

The authors declare no conflict of interest.

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