# The improved Amati correlations from Gaussian copula

Yang Liu  Department of Physics and Synergistic Innovation Center for Quantum
Effects and Applications, Hunan Normal University, Changsha, Hunan 410081,
China  Fuyong Chen  Department of Physics and Synergistic Innovation Center
for Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan
410081, China  Nan Liang  Key Laboratory of Information and Computing Science
Guizhou Province, Guizhou Normal University, Guiyang, Guizhou 550025, China
Yunnan Observatory, Chinese Academy of Sciences, Kunming, Yunnan 650011, China
Zunli Yuan  Department of Physics and Synergistic Innovation Center for
Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan
410081, China  Hongwei Yu  Department of Physics and Synergistic Innovation
Center for Quantum Effects and Applications, Hunan Normal University,
Changsha, Hunan 410081, China  Puxun Wu  Department of Physics and Synergistic
Innovation Center for Quantum Effects and Applications, Hunan Normal
University, Changsha, Hunan 410081, China

###### Abstract

In this paper, we obtain two improved Amati correlations of the Gamma-Ray
burst (GRB) data via a powerful statistical tool called copula. After
calibrating, with the low-redshift GRB data, the improved Amati correlations
based on a fiducial $\Lambda$CDM model with $\Omega_{\mathrm{m0}}=0.3$ and
$H_{0}=70~{}\mathrm{km~{}s^{-1}Mpc^{-1}}$, and extrapolating the results to
the high-redshift GRB data, we obtain the Hubble diagram of GRB data points.
Applying these GRB data to constrain the $\Lambda$CDM model, we find that the
improved Amati correlation from copula can give a result well consistent with
$\Omega_{\mathrm{m0}}=0.3$, while the standard Amati and extended Amati
correlations do not. This results suggest that when the improved Amati
correlation from copula is used in the low-redshift calibration method, the
GRB data can be regarded as a viable cosmological explorer. However, the
Bayesian information criterion indicates that the standard Amati correlation
remains to be favored mildly since it has the least model parameters.
Furthermore, once the simultaneous fitting method rather than the low-redshift
calibration one is used, there is no apparent evidence that the improved Amati
correlation is better than the standard one. Thus, more works need to be done
in the future in order to compare different Amati correlations.

##  1 Introduction

The accelerating cosmic expansion was first discovered by two independent
teams in 1998 (Riess et al., 1998; Perlmutter et al., 1999) from observation
of Type Ia supernova (SNIa). This discovery was further confirmed by many
other observations, including the cosmic microwave background radiation (CMB)
(Spergel et al., 2003, 2007) and the baryon acoustic oscillation(BAO)
(Eisenstein et al., 2005). To explain this observational phenomenon, a
mysterious dark energy has been proposed to exist in our universe. The nature
of dark energy can be characterized by its equation of state (EoS) parameter.
The simplest candidate of dark energy is the cosmological constant $\Lambda$,
whose EoS parameter equals to $-1$, and the cosmological constant plus cold
dark matter ($\Lambda$CDM) model is consistent with observational data very
well. Based on the $\Lambda$CDM model, a tight constraint on the Hubble
constant $H_{0}$ can be given with very high-redshift CMB data (Aghanim et
al., 2019), which has a more than $4\sigma$ deviation from the value of
$H_{0}$ obtained directly from the very low-redshift SNIa data (Riess et al.,
2021, 2018a, 2018b). Some other low-redshift observational data such as the
Hubble parameter measurements $H(z)$, BAO, and the strong gravitational
lenses, have been used to explore the Hubble constant, and given a lower
$H_{0}$ than the one from the SNIa with large error bars which are reasonably
consistent with the value of Planck 2018 (Wu et al., 2017; Chen et al., 2017;
Abbott et al., 2018; Birrer et al., 2020; Cao et al., 2021a; Lin & Ishak,
2021; Khetan et al., 2021; Efstathiou, 2020; Freedman, 2021). The $H_{0}$
tension seems to suggest that the assumed $\Lambda$CDM model used to determine
the Hubble constant may be inconsistent with our present universe or there are
potentially unknown systematic errors in observational data. To precisely nail
down the property of dark energy and the possible origin of $H_{0}$ tension,
intermediate-redshift observational data are necessary, since the SNIa and BAO
only contain the redshift ranges of $z\lesssim 2$ and the CMB is at $z\sim
1100$.

Gamma-Ray bursts (GRBs) are the most violent and intriguing explosions in the
universe, whose power is dominant in the (sub-)$\mathrm{MeV}$ gamma-ray range
(Klebesadel et al., 1973). The highest isotropic energy of GRBs can be up to
$10^{54}~{}\mathrm{erg}$ and the detected redshift of GRBs reaches to $z\sim
9.4$. Thus, GRBs have the potential to be a farther tracker than SNIa to
explore the cosmic evolution. Many empirical luminosity correlations which are
relationships between parameters of the light curves and/or spectra with the
GRB luminosity or energy have been proposed to standardize GRB samples, such
as the time lag-isotropic peak luminosity correlation ($\tau_{lag}-L_{iso}$)
(Norris et al., 2000), the time variability-isotropic peak luminosity
correlation ($V-L_{iso}$) (Fenimore & Ramirez-Ruiz, 2000), the Amati
correlation (see the Appendix A for the details) which connects the spectral
peak energy in GRB cosmological rest-frame and the the isotropic equivalent
radiated energy ($E_{p}-E_{iso}$) (Amati et al., 2002), the correlation
between the spectral peak energy and the isotropic peak luminosity
($E_{p}-L_{iso}$) (Yonetoku et al., 2004)), the correlation between the
spectral peak energy and the collimated-corrected energy ($E_{p}-E_{\gamma}$)
(Ghirlanda et al., 2004a), and so on. Moreover, the issue of whether these
luminosity correlations are redshift dependent is investigated in (Basilakos &
Perivolaropoulos, 2008; Demianski et al., 2017; Wang et al., 2011; Li, 2007;
Lin et al., 2015, 2016; Khadka et al., 2021), and . For an example, Demianski
et al. (2017) introduced a power-law function to describe the evolutionary
function of the Amati correlation with redshift, and Wang et al. (2017)
selected two redshift dependent formulas to parameterize the coefficients in
the Amati correlation, which is called the extended Amati correlation in this
paper (the details can be found in the Appendix B). Since the luminosity
$L_{iso}$, the isotropic energy $E_{iso}$ and the collimated-corrected energy
$E_{\gamma}$ in GRBs are cosmology-dependent, the Hubble diagram can be
obtained from standardized GRB samples.

To use the GRBs as a cosmological probe, we need to calibrate GRB
correlations. Let us point out here that the cosmology-dependent calibration
method suffers the so-called circularity problem (Ghirlanda et al., 2006; Wang
et al., 2015). To avoid this problem, a low-redshift method (Liang et al.,
2008; Kodama et al., 2008; Wei & Zhang., 2009; Liang et al., 2010) to
calibrate the GRB correlations in a cosmology-independent way was proposed,
which uses other distance probes such as SNIa to calibrate the GRBs at low-
redshifts and then extrapolates the results to high-redshifts to constrain the
cosmological parameters. For this method, since the intrinsic dispersion of
SNIa data is very much smaller than the GRB intrinsic dispersion, the SNIa
data will dominate over the GRB data in a joint analysis of them on the
cosmological constraints and so the resulting cosmological constraints are
effectively from SNIa data.On the other hand, the simultaneous fitting or
global fitting (Ghirlanda et al., 2004b; Li et al., 2008) limits the
coefficients of the luminosity correlations and the parameters of cosmological
models simultaneously from the observational GRB data. In (Khadka & Ratra,
2020; Khadka et al., 2021), by simultaneously fitting cosmological and GRB
Amati correlation parameters and using a number of different cosmological
models, it has been found that the Amati correlation parameter values are
independent of the cosmological model, which seems to mean that there is no
circularity problem and that these GRB data sets are standardizable within the
error bars. Up to now, the GRB data has been used widely to investigate
different cosmological models (see recent Refs: Amati et al., 2019; Demianski
et al., 2021; Cao et al., 2021b, 2022; Hu et al., 2021; Wang et al., 2021;
Luongo & Muccino, 2021).

In this work we aim to improve the Amati correlation which has been widely
used in GRB cosmology. We introduce a powerful statistical tool called copula,
which is a specialized tool developed in modern statistics to describe the
complicated dependence structures between multivariate random variables.
Copula has been widely used in various areas such as mathematical finance and
hydrology in the past few decades. In recent years, it has gradually been
recognized by the astronomical community as a very useful tool to analyze
data. For example, Yuan et al. (2018) successfully determined the luminosity
function of the radio cores in active galactic nuclei via copula, which is
difficult to do if the traditional method is used, Koen (2009) studied, using
copula, the correlation between the GRB peak energy and the associated
supernova peak brightness, Benabed et al. (2009) proposed a new approximation
for the low multipole likelihood of the CMB temperature, and Jiang et al.
(2009) constructed a period-mass function for extrasolar planets. In addition,
the copula likelihood function was constructed for the convergence power
spectrum from the weak lensing surveys by Sato et al. (2010, 2011). Modeling
bivariate astronomical data with copula instead of the conventional gaussian
mixture method was proposed by Koen & Bere (2017). The copula function is also
useful in the study of galaxy luminosity functions and the large scale
structure fields of matter density (Takeuchi, 2010; Takeuchi et al., 2013;
Takeuchi & Kono, 2020; Scherrer et al., 2010; Qin et al., 2020). Based on the
3-dimensional Gaussian copula, we propose, in this work, two improved Amati
correlations and compare them with the standard Amati correlation and the
extended Amati correlation by using the latest GRB samples (Khadka et al.,
2021).

The frame of this paper is arranged as follows: Section 2 studies the issue of
how to construct a 3-dimensional probability density function (PDF) through a
Gaussian copula. Two improved Amati correlations are obtained in Section 3. In
Section 4, a comparison between the improved Amati correlations and the
(extended) Amati correlation is made by using both low-redshift calibration
and simultaneous fitting methods. The conclusions are given in Section 5.

##  2 The copula

Briefly speaking, copulas are functions that join or “couple” multivariate
distribution functions to their one-dimensional marginal distribution
functions (Nelson, 2006). Let $x$, $y$ and $z$ be three random variables, we
use $F(x)$, $G(y)$ and $W(z)$ to express their marginal cumulative
distribution functions (CDFs) respectively, and $H$ as the joint distribution
function of the three variables. According to Sklar’s theorem, if $F$, $G$ and
$W$ are continuous, there exists a unique copula $C$ such that

$H(x,y,z;\bm{\theta})=C\left(F(x),G(y),W(z);\bm{\theta}\right),$ (1)

where $\bm{\theta}$ denotes the parameters of the copula function $C$. Let
$u\equiv F(x),~{}v\equiv G(y)$, and $q\equiv W(z)$, the joint probability
density function (PDF) $h(x,y,z;\bm{\theta})$ can be obtained by

$\displaystyle h(x,y,z;\bm{\theta})$ $\displaystyle=$
$\displaystyle\frac{\partial^{3}H(x,y,z;\bm{\theta})}{\partial x\partial
y\partial z}$ (2) $\displaystyle=$
$\displaystyle\frac{\partial^{3}C(u,v,q;\bm{\theta})}{\partial u\partial
v\partial q}\frac{\partial u}{\partial x}\frac{\partial v}{\partial
y}\frac{\partial q}{\partial z}$ $\displaystyle=$ $\displaystyle
c[u,v,q;\bm{\theta}]f(x)g(y)w(z),$

where $f(x)$, $g(y)$ and $w(z)$ are the marginal PDFs of $x$, $y$ and $z$
respectively, and $c(u,v,q;\bm{\theta})$ is the density function of $C$.

One obvious advantage of Eq. (1) is that by using copulas one can model the
dependence structure and the marginal distributions separately. All the
information on the dependence between the three variables is carried by the
copula (see, e.g., Yuan et al., 2018). The next issue is to find a 3-dimension
optimal copula function and estimate its parameters to describe the observed
data in GRBs. The bivariate copulas are abundant and thus the 3-dimensional
copula functions are abundant too since most of the 2-dimensional copula can
be extended easily to the 3-dimensional case. Here we use the 3-dimensional
Gaussian copula function with linear correlation coefficient
$\bm{\theta}=\\{\rho_{1},\rho_{2},\rho_{3}\\}$ to model our data, i.e.,

$\displaystyle
C(u,v,q;\bm{\theta})=\Psi_{3}\left[\Psi_{1}^{-1}(u),\Psi_{1}^{-1}(v),\Psi_{1}^{-1}(q);\bm{\theta}\right],$
(3)

where $\Psi_{3}$ and $\Psi_{1}$ are the standard 3-dimensional Gaussian CDF
and 1-dimension Gaussian CDF respectively, and $\Psi_{1}^{-1}$ denotes the
inverse of $\Psi$. The density function of the Gaussian copula can be obtained
from

$\displaystyle c(u,v,q;\bm{\theta})$ $\displaystyle=$
$\displaystyle\frac{\partial^{3}\Psi_{3}\left[\Psi_{1}^{-1}(u),\Psi_{1}^{-1}(v),\Psi_{1}^{-1}(q);\bm{\theta}\right]}{\partial
u\partial v\partial q}$ (4) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\mathrm{det}\mathbf{\Sigma}}}\exp\left\\{-\frac{1}{2}\left[\mathbf{\Psi^{-1}}^{T}\left(\mathbf{\Sigma}^{-1}-\mathbf{I}\right)\mathbf{\Psi^{-1}}\right]\right\\},$

where
$\mathbf{\Psi^{-1}}\equiv\left[\Psi^{-1}(u),\Psi^{-1}(v),\Psi^{-1}(q)\right]^{T}$
, $\mathbf{I}$ stands for the identity matrix, and $\mathbf{\Sigma}^{-1}$
denotes the inverse of the covariance matrix $\mathbf{\Sigma}$, which reads

$\displaystyle\mathbf{\Sigma}\equiv\left(\begin{array}[]{ccccc}1{}&\rho_{1}{}&\rho_{2}\\\
\rho_{1}{}&1{}&\rho_{3}\\\ \rho_{2}{}&\rho_{3}{}&1\end{array}\right).$ (8)

The conditional PDF of $y$ denotes the probability of variable $y$ when $x$
and $z$ are fixed, which can be expressed as:

$\displaystyle g_{y}(y|x,z;\bm{\theta})$ $\displaystyle\equiv$
$\displaystyle\frac{h(x,y,z;\bm{\theta})}{h_{xz}(x,z;\rho_{2})}=\frac{c[u,v,q;\bm{\theta}]f(x)g(y)w(z)}{c[u,q;\rho_{2}]f(x)w(z)}=\frac{c[u,v,q;\bm{\theta}]}{c[u,q;\rho_{2}]}g(y),$
(9)

where $h_{xz}(x,z;\rho_{2})$ is constructed from a 2-dimensional Gaussian
copula with the correlation coefficient being $\rho_{2}$. If variable $y$
obeys a Gaussian distribution with the standard deviation being $\sigma_{y}$,
then $g_{y}$ can be expressed as

$\displaystyle g_{y}(y|x,z;\bm{\theta})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi\sigma_{y|x,z}^{2}}}\exp\left[-\frac{1}{2}S(x,y,z;\bm{\theta})\right],$
(10)

where

$\displaystyle\sigma_{y|x,z}^{2}$ $\displaystyle\equiv$
$\displaystyle\frac{\sigma_{y}^{2}\left(1-\rho_{1}^{2}-\rho_{2}^{2}-\rho_{3}^{2}+2\rho_{1}\rho_{2}\rho_{3}\right)}{1-\rho_{2}^{2}}.$
(11)

In Eq. (10), $S(x,y,z;\bm{\theta})$ is unknown if the marginal distributions
of variables $x$ and $z$ are undetermined. Due to that the variable $y$ obeys
the Gaussian distribution, the highest probability of $y$ corresponds to
$S(x,y,z;\bm{\theta})=0$.

##  3  The improved Amati correlation

Now, we use the copula function introduced in the above section to investigate
the correlation between $E_{p}$ and $E_{iso}$ of GRBs. After using $x$, $y$
and $z$ to denote $\log\frac{E_{p}}{300\mathrm{keV}}$,
$\log\frac{E_{iso}}{1\mathrm{erg}}$ and redshift $z$ of GRBs, respectively,
the marginal distribution of $x$, $y$ and $z$ need to be given in order to
determine $S(x,y,z;\bm{\theta})$. We assume that two Gaussian distributions
for $x=\log\frac{E_{p}}{300\mathrm{keV}}$ and
$y=\log\frac{E_{iso}}{1\mathrm{erg}}$ are

$\displaystyle
f(x;\bar{a}_{x},\sigma_{x})=\frac{1}{\sqrt{2\pi}\sigma_{x}}e^{-\frac{(\bar{a}_{x}-x)^{2}}{2\sigma_{x}^{2}}}$
,
$\displaystyle~{}g(y;\bar{a}_{y},\sigma_{y})=\frac{1}{\sqrt{2\pi}\sigma_{y}}e^{-\frac{(\bar{a}_{y}-y)^{2}}{2\sigma_{y}^{2}}}.$
(12)

Here $\bar{a}$ and $\sigma$ represent the mean value and the standard
deviation of the Gaussian distribution respectively. And the CDFs of $x$, $y$,
respectively, have the forms

$\displaystyle
F(x;\bar{a}_{x},\sigma_{x})=\int_{-\infty}^{x}f(\tilde{x};\bar{a}_{x},\sigma_{x})d\tilde{x},~{}G(y;\bar{a}_{y},\sigma_{y})=\int_{-\infty}^{y}g(\tilde{y};\bar{a}_{y},\sigma_{y})d\tilde{y}.$
(13)

We consider two different redshift distributions of GRB data:

  * •

the first distribution is a special form with the PDF being $w(z)=ze^{-z}$
(Wang et al., 2017). Thus, the corresponding CDF is:

$\displaystyle W(z)=1-e^{-z}(1+z).$ (14)

  * •

the second CDF of GRB’s redshift distribution is the empirical distribution
(Dekking et al., 2005):

$\displaystyle W_{N}(z)=\left\\{\begin{array}[]{ll}0&\mbox{ if }z<z_{1}\\\
i/n&\mbox{ if }z_{i}\leq z<z_{i+1}\\\ 1&\mbox{ if }z_{n}\leq z\
,\end{array}\right.$ (18)

where $\\{z_{1},z_{2},\dots,z_{n}\\}$ is an ordered list of redshifts of GRB
data, which satisfies $z_{1}\leq z_{2}\leq\dots\leq z_{n}$, and $n$ denotes
the number of data.

Since the first distribution of redshift $z$ of GRBs given in Eq. (14) is an
assumed special form, we need to evaluate whether the redshift of observed GRB
data obeys this form by using the Kolmogorov-Smirnov test (K-S test). The K-S
test bases on the distance measure $D$ which is defined to be

$\displaystyle D=\max|W_{N}(z)-W(z)|,$ (19)

where $W_{N}(z)$ and $W(z)$ are the empirical and assumed CDFs of redshift
distribution of GRBs respectively. Apparently, $D$ represents the maximum
deviation between two CDFs. In our K-S test, we set the significance level
$\alpha$ to be $\alpha=0.05$, where $\alpha\equiv 1-P(\lambda_{\alpha})$ and
$P(\lambda)$ is the the Kolmogorov distribution

$\displaystyle
P(\lambda)=\sum_{i=-\infty}^{+\infty}(-1)^{i}\exp\left(-2i^{2}\lambda^{2}\right)$
(20)

obeyed by the Kolmogorov stochasticity parameter $\lambda$. Here
$\lambda\equiv\sqrt{N}D$ and $N$ is the number of data. Then, a critical value
$D_{\alpha}=\frac{\lambda_{\alpha}}{\sqrt{N}}\simeq 0.09$ can be obtained,
which means that if the data satisfies the assumed distribution, the
probability that $D<D_{\alpha}$ is about $95\%$. Conversely, the probability
that $D>D_{\alpha}$ is only $5\%$. If $D>D_{\alpha}$, the assumed distribution
will be rejected since it is a rare event. Fig. 1 shows the CDFs of the
empirical distribution $W_{N}(z)$ from the latest 220 GRB data points (Khadka
et al., 2021) and the assumed distribution $W(z)$ used in our discussion. We
find that $D\simeq 0.06$ at $z\simeq 2$, which is less than $D_{\alpha}$. Thus
the assumed redshift distribution of GRB data passes the K-S test and is
acceptable.

Figure 1: The CDFs of the empirical and assumed redshift distributions of GRB
data. The maximum deviation between two CDFs is about 0.06 at $z\simeq 2$, and
is less than the critical value $D_{\alpha}\simeq 0.09$, which means that the
assumed distribution in our work passes the K-S test.

Substituting the CDFs given in Eqs. (13,14,18) into Eqs. (4-10),
$S(x,y,z;\bm{\theta})$ can be derived. However, the concrete expression is
very complicated, so we do not show it here. From the equation of
$S(x,y,z;\bm{\theta})=0$, we can obtain:

$\displaystyle y_{\mathrm{copula_{1}}}$ $\displaystyle=$ $\displaystyle
a+b\,x+c\,\mathrm{erfc}^{-1}[2W(z)],$ $\displaystyle y_{\mathrm{copula_{2}}}$
$\displaystyle=$ $\displaystyle a+b\,x+c\,\mathrm{erfc}^{-1}[2W_{N}(z)].$ (21)

Here, the subscripts $\mathrm{copula_{1}}$ and $\mathrm{copula_{2}}$ denote
the correlation relations from the copula method with the assumed redshift
distribution (Eq. (14)) and the empirical redshift distribution (Eq. (18)),
respectively, $\mathrm{erfc}$ is the complementary error function, and
coefficients $a$, $b$ and $c$ are defined as

$\displaystyle a$ $\displaystyle\equiv$
$\displaystyle\bar{a}_{y}-\frac{\left(\rho_{2}\rho_{3}-\rho_{1}\right)\bar{a}_{x}\sigma_{y}}{\left(\rho_{2}^{2}-1\right)\sigma_{x}},$
$\displaystyle b$ $\displaystyle\equiv$
$\displaystyle\frac{\left(\rho_{2}\rho_{3}-\rho_{1}\right)\sigma_{y}}{\left(\rho_{2}^{2}-1\right)\sigma_{x}},$
$\displaystyle c$ $\displaystyle\equiv$
$\displaystyle\frac{\sqrt{2}\sigma_{y}\left(\rho_{3}-\rho_{1}\rho_{2}\right)}{\rho_{2}^{2}-1}.$
(22)

Eq. (3) are different from the standard Amati correlation (Eq. (A1)) by a
redshift-dependent term, and this redshift-dependent correction term is also
different from that of the extended Amati correlation (Eq. (B2)). We name
these luminosity correlations from the Gaussian copula as the improved Amati
correlations, and they are the main results of our paper.

##  4 Hubble diagrams and GRB cosmology

###  4.1 The low-redshift calibration

To test the viability of the improved Amati correlations given in Eq. (3), we
use two GRB data samples: one is the latest GRB sample (A220) (Khadka et al.,
2021), which contains 220 long GRBs in the redshift range of $z\in[0.03,8.2]$,
and the other is the higher-quality A118 data set (Khadka & Ratra, 2020; Wang
et al., 2016; Dirirsa et al., 2019) contained in A220 with redshift range of
$z\in[0.34,8.2]$ since it has a tighter intrinsic scatter. These GRBs are
divided into the low-redshift part (79 and 20 GRBs at $z\in\left[0,1.4\right]$
in A220 and A118, respectively) and the high-redshift one (141 and 98 GRBs at
$z\in\left[1.4,8.2\right]$ in A220 and A118, respectively). We will use the
low-redshift GRB data of two data sets to determine the coefficients in
$y_{\mathrm{copula}}$, and then extrapolate these results to the high redshift
data to obtain their luminosity distances, which will then be used to
constrain the cosmological model. Since the isotropic equivalent radiated
energy $E_{iso}$ (see Eq. (A4)) is dependent on the luminosity distance, a
fiducial cosmological model needs to be chosen. Here, the spatially flat
$\Lambda$CDM with $\Omega_{\mathrm{m0}}=0.30$ and
$H_{0}=70~{}\mathrm{km~{}s^{-1}Mpc^{-1}}$ is chosen as the fiducial model,
where $\Omega_{\mathrm{m0}}$ is the dimensionless present matter density
parameter. Then, the allowed regions of $a$, $b$, $c$ and $\sigma_{int}$ can
be obtained by maximizing the D’Agostinis likelihood function

$\displaystyle\mathcal{L}(\sigma_{int},a,b,c)\propto\prod_{i=1}^{N}\frac{1}{\sqrt{\sigma_{int}^{2}+\sigma_{y,i}^{2}+b^{2}\sigma_{x,i}^{2}}}\times\exp\left[-\frac{[y_{i}-y_{\mathrm{copula}}(x_{i},z_{i};a,b,c)]^{2}}{2\left(\sigma_{int}^{2}+\sigma_{y,i}^{2}+b^{2}\sigma_{x,i}^{2}\right)}\right].$
(23)

Here $N=79$ or $20$, $\sigma_{int}$ is the intrinsic scatter of GRBs, and
$y_{i}$ denotes the observed $\log\frac{E_{iso}}{\mathrm{1erg}}$ of the low-
redshift GRBs. For a comparison, the Amati and extended Amati correlations,
which are denoted by $y_{\mathrm{Amati}}$ and $y_{\mathrm{exAmati}}$
respectively, are also investigated.  In our analysis, the CosmoMC code is
used111The CosmoMC code is available at https://cosmologist.info/cosmomc.. The
results are summarized in Tab. 1 and Tab. 2.

From two tables, one can see that in the case of the $\mathrm{c}opula_{1}$
correlation relation the value of intrinsic scatter $\sigma_{int}$ is always
slightly smaller than the one from the Amati relation, which means that the
quality of the correlation relation is improved slightly when the
$\mathrm{c}opula_{1}$ correlation relation is used. A comparison of Tab. 1 and
Tab. 2 reveals that the value of intrinsic scatter $\sigma_{int}$ from the
A118 GRB data set is smaller than the one from the A220 GRB data set. Thus,
the 118 data set is a higher quality one compared to the A220 data set, which
agrees with the results obtained in (Khadka et al., 2021). Furthermore, Tab. 1
shows that the coefficient $c$ in $y_{\mathrm{copula_{1}}}$(or
$y_{\mathrm{copula_{2}}}$) and coefficients $\alpha$ and $\beta$ in
$y_{\mathrm{exAmati}}$ are at most $1\sigma$ away from $0$, which indicates
that there is not strong support for nonzero values of these parameters from
the A220 data set. These results are consistent with what were obtained in
(Khadka et al., 2021) where the relation parameters $a$ and $b$ were found to
be independent of redshift within the error bars. This character may also be
found from the Pearson linear correlation coefficients $\rho_{1}$, $\rho_{2}$
and $\rho_{3}$, which denote the degree of linear correlation between
variables $x$, $y$ and $z$. We derive $\rho_{1}=0.771$, $\rho_{2}=0.493$,
$\rho_{3}=0.426$ from the 79 low-redshift GRBs in the A220 data set and
$\rho_{1}=0.781$, $\rho_{2}=0.381$, $\rho_{3}=0.377$ from the 20 low-redshift
GRBs in the A118 data set. Since $\rho_{2}$ and $\rho_{3}$ have values about
$0.4$ and they are smaller clearly than $\rho_{1}$, the linear correlations
between $\log\frac{E_{p}}{300\mathrm{keV}}$ and $z$, and
$\log\frac{E_{p}}{300\mathrm{keV}}$ and $z$ are weak, and they are weaker than
the linear correlation between $\log\frac{E_{p}}{300\mathrm{keV}}$ and
$\log\frac{E_{iso}}{1\mathrm{erg}}$.

To compare the correlation relations from copula function and the (extended)
Amati relation, we compute the values of the Akaike information criterion
(AIC)(Akaike, 1974, 1981) and the Bayesian information criterion
(BIC)(Schwarz, 1978), which, respectively, are defined as:

$\displaystyle\mathrm{AIC}$ $\displaystyle=$ $\displaystyle
2p-2\ln(\mathcal{L}),$ $\displaystyle\mathrm{BIC}$ $\displaystyle=$
$\displaystyle p\ln N-2\ln(\mathcal{L}),$ (24)

where $\mathcal{L}$ is the likelihood function, $p$ is the number of free
parameters in a model and $N$ is the number of data. Here we also compute the
$\Delta$AIC($\Delta$BIC), which denotes the difference of AIC(BIC) with
respect to the reference model (the standard Amati model here).
$0<\Delta\mathrm{AIC}(\Delta\mathrm{BIC})\leq 2$ indicates difficulty in
preferring a given model, $2<\Delta\mathrm{AIC}(\Delta\mathrm{BIC})\leq 6$
means mild evidence against the given model, and
$\Delta\mathrm{AIC}(\Delta\mathrm{BIC})>6$ suggests strong evidence against
the model. The obtained values of $\Delta$AIC and $\Delta$BIC are summarized
in Tabs. 1 and 2. We find that except for the case of the extended Amati
correlation whose $\Delta\mathrm{AIC}$ given by the low-redshift GRBs in the
A118 data set is larger than $2$, the AIC cannot determine the model preferred
by the data, while the BIC indicates that the standard Amati correlation
remains to be favored mildly since it has the least model parameters.

Table 1:

| Amati |  | extended Amati |  | $\mathrm{copula_{1}}$ |  | $\mathrm{copula_{2}}$
---|---|---|---|---|---|---|---
| Best-fit($\sigma$) | 0.68 CL |  | Best-fit($\sigma$) | 0.68 CL |  | Best-fit($\sigma$) | 0.68 CL |  | Best-fit($\sigma$) | 0.68 CL
$\sigma_{int}$ | $0.512(0.045)$ | ${}^{+0.054}_{-0.034}$ |  | $0.503(0.046)$ | ${}^{+0.063}_{-0.025}$ |  | $0.510(0.045)$ | ${}^{+0.054}_{-0.033}$ |  | $0.509(0.045)$ | ${}^{+0.060}_{-0.029}$
$a$ | $52.710(0.061)$ | ${}^{+0.060}_{-0.061}$ |  | $52.587(0.333)$ | ${}^{+0.324}_{-0.334}$ |  | $52.847(0.144)$ | ${}^{+0.145}_{-0.137}$ |  | $52.812(0.149)$ | ${}^{+0.141}_{-0.156}$
$b$ | $1.290(0.126)$ | ${}^{+0.126}_{-0.123}$ |  | $1.521(0.367)$ | ${}^{+0.422}_{-0.301}$ |  | $1.209(0.150)$ | ${}^{+0.145}_{-0.148}$ |  | $1.231(0.145)$ | ${}^{+0.156}_{-0.128}$
$c$ | $-$ | ${-}$ |  | $-$ | ${-}$ |  | $-0.217(0.207)$ | ${}^{+0.208}_{-0.198}$ |  | $-0.142(0.192)$ | ${}^{+0.200}_{-0.180}$
$\alpha$ | $-$ | ${-}$ |  | $0.319(0.729)$ | ${}^{+0.734}_{-0.702}$ |  | $-$ | ${-}$ |  | $-$ | ${-}$
$\beta$ | $-$ | ${-}$ |  | $-0.680(0.779)$ | ${}^{+0.624}_{-0.915}$ |  | $-$ | ${-}$ |  | $-$ | ${-}$
$-2\ln\mathcal{L}$ | 121.514 |  | 119.492 |  | 120.281 |  | 121.012
$\Delta$AIC | - |  | 1.978 |  | 0.767 |  | 1.498
$\Delta$BIC | - |  | 6.717 |  | 3.136 |  | 3.867

Note. — The best-fitted values, standard deviations, and the 68% confidence
level (CL) of coefficients of $y_{\mathrm{Amati}}$, $y_{\mathrm{exAmati}}$,
$y_{\mathrm{copula_{1}}}$ and $y_{\mathrm{copula_{2}}}$ from the 79 low-
redshift ($z<1.4$) long GRBs in A220 data set. Here
$\Delta\mathrm{AIC}(\Delta\mathrm{BIC})$ denotes the difference of AIC(BIC)
with standard Amati model.

Table 2:

| Amati |  | extended Amati |  | $\mathrm{copula_{1}}$ |  | $\mathrm{copula_{2}}$
---|---|---|---|---|---|---|---
| Best-fit($\sigma$) | 0.68 CL |  | Best-fit($\sigma$) | 0.68 CL |  | Best-fit($\sigma$) | 0.68 CL |  | Best-fit($\sigma$) | 0.68 CL
$\sigma_{int}$ | $0.429(0.089)$ | ${}^{+0.117}_{-0.044}$ |  | $0.438(0.099)$ | ${}^{+0.135}_{-0.041}$ |  | $0.418(0.098)$ | ${}^{+0.149}_{-0.025}$ |  | $0.430(0.097)$ | ${}^{+0.137}_{-0.035}$
$a$ | $52.860(0.111)$ | ${}^{+0.105}_{-0.110}$ |  | $52.730(0.609)$ | ${}^{+0.534}_{-0.6496}$ |  | $52.9586(0.282)$ | ${}^{+0.281}_{-0.257}$ |  | $52.935(0.282)$ | ${}^{+0.270}_{-0.266}$
$b$ | $0.990(0.205)$ | ${}^{+0.210}_{-0.187}$ |  | $1.525(1.158)$ | ${}^{+1.230}_{-1.168}$ |  | $0.967(0.231)$ | ${}^{+0.213}_{-0.229}$ |  | $0.976(0.235)$ | ${}^{+0.218}_{-0.230}$
$c$ | $-$ | ${-}$ |  | $-$ | ${-}$ |  | $-0.170(0.425)$ | ${}^{+0.394}_{-0.423}$ |  | $-0.103(0.368)$ | ${}^{+0.339}_{-0.360}$
$\alpha$ | $-$ | ${-}$ |  | $0.329(1.327)$ | ${}^{+1.405}_{-1.170}$ |  | $-$ | ${-}$ |  | $-$ | ${-}$
$\beta$ | $-$ | ${-}$ |  | $-1.136(2.270)$ | ${}^{+2.362}_{-2.476}$ |  | $-$ | ${-}$ |  | $-$ | ${-}$
$-2\ln\mathcal{L}$ | 23.374 |  | 22.938 |  | 23.119 |  | 23.240
$\Delta$AIC | - |  | 3.564 |  | 1.745 |  | 1.866
$\Delta$BIC | - |  | 5.555 |  | 2.741 |  | 2.862

Note. — The best-fitted values, standard deviations, and the 68% confidence
level (CL) of coefficients of $y_{\mathrm{Amati}}$, $y_{\mathrm{exAmati}}$,
$y_{\mathrm{copula_{1}}}$ and $y_{\mathrm{copula_{2}}}$ from the 20 low-
redshift ($z<1.4$) long GRBs in A118 data set.

Extrapolating directly the values of the coefficients in Tab. 1 and Tab. 2
from the low-redshift GRB data to the high-redshift samples we can construct
the Hubble diagram of GRBs. The distance modulus of GRBs and their errors are
obtained from Eqs. (A5) and (A10). As an example, in Fig. 2, we show the
Hubble diagrams of 220 and 118 long GRBs obtained from three different
correlations ($y_{\mathrm{Amati}}$, $y_{\mathrm{exAmati}}$ and
$y_{\mathrm{copula_{1}}}$), respectively. Here we do not plot the Hubble
diagram based on the $y_{\mathrm{copula_{2}}}$ correlation since it is very
similar to the one of $y_{\mathrm{copula_{1}}}$. The values of distance
modulus obtained from the copula method are less at low-redshift ($z\lesssim
1$) regions and larger at high-redshift ($z\gtrsim 1$) regions than the ones
from the standard Amati correlation because a correction exists in the right
hand side of Eq. (3).

Figure 2: The Hubble diagrams of 220 and 118 long GRBs calibrated from three
different correlations ($y_{\mathrm{Amati}}$, $y_{\mathrm{exAmati}}$ and
$y_{\mathrm{copula_{1}}}$). The red dashed line denotes $z=1.4$.

To test whether the GRB can be regarded as the viable cosmological indicator,
we can constrain the $\Lambda$CDM model from the distance modulus of GRBs
obtained in the above subsection, and check whether $\Omega_{\mathrm{m0}}=0.3$
is allowed after setting $H_{0}=70~{}\mathrm{km~{}s^{-1}Mpc^{-1}}$, which is
used in calibrating the improved Amati correlations from the low redshift
GRBs. The method of minimizing $\chi^{2}$ is used to constrain
$\Omega_{\mathrm{m0}}$. We consider two different samples for each data set:
the 141 high-redshift GRBs and the total 220 GRBs for A220, and the 98 high-
redshift GRBs and the total 118 GRBs for A118. We must emphasize here that
since $W_{N}(8.2)=1$ when the maximum redshift GRB data ($z=8.2$) is
considered, $\mathrm{erfc}^{-1}[2W_{N}(8.2)]=-\infty$ in $y_{copula_{2}}$ and
thus the data point with $z=8.2$ will be ignored when the $y_{copula_{2}}$ is
used to constrain $\Omega_{\mathrm{m0}}$.

The probability density plots of $\Omega_{\mathrm{m0}}$ for two data sets are
shown in Fig. 3 and Fig. 4, and the best-fitted values with the standard
deviation and $68\%$ confidence level (CL) are summarized in Tab. 3 and Tab.
4, respectively. It is easy to see that for all data sets the results from the
standard Amati correlation and the extended Amati correlation deviate
apparently from $\Omega_{\mathrm{m0}}=0.3$. The results from improved Amati
correlation based on an empirical distribution of redshift
($y_{\mathrm{copula_{2}}}$) are better than the ones from the (extended) Amati
correlation although they are $1\sigma$ away from $\Omega_{\mathrm{m0}}=0.3$,
while those from $y_{\mathrm{copula_{1}}}$ are always consistent with
$\Omega_{\mathrm{m0}}=0.3$ at the $1\sigma$ confidence level. Apparently, the
results from $y_{\mathrm{copula_{2}}}$ are not as good as those from
$y_{\mathrm{copula_{1}}}$. This is attributed to that the number of GRBs is
still inadequate to construct the empirical distribution precisely. Comparing
the constraint on $\Omega_{\mathrm{m0}}$ from the high-redshift and full-
redshift GRB data shown in Tabs. 3 and 4, we find that the values of
$\Omega_{\mathrm{m0}}$ from the full-redshift data are closer to $0.3$ than
those from the high-redshift data. This is because the low-redshift data are
calibrated with $\Omega_{\mathrm{m0}}=0.3$. When changing the GRB data from
the high-redshift region to the full-redshift one, the A220 data give the
maximum variation of $\Omega_{\mathrm{m0}}$ for the case of the extended Amati
correlation, which is about $12\%$. However, this variation is very small at
$y_{\mathrm{copula_{1}}}$ and $y_{\mathrm{copula_{2}}}$ cases. From Tabs. 3
and 4 and Figs. 3 and 4, one can also see that the values of
$\Omega_{\mathrm{m0}}$ from the total A220 data set are much closer to $0.3$
than those from the full-redshift A118 data set. This is attributed to that
the ratio of the number of the calibrated to the uncalibrated GRBs in the A220
data set, which is about 0.56 (79/141), is apparently larger than that in the
A118 data set, which is only 0.20 (20/98). In addition, It is easy to find
that our constraints on $\Omega_{\mathrm{m0}}$ from the A118 and A220 data
samples differ very significantly from what were obtained in (Khadka et al.,
2021) where the A118 and A220 data limit $\Omega_{\mathrm{m0}}>0.230$ and
$>0.455$, respectively, at the $95\%$ CL in the $\Lambda$CDM model. This
difference originates from that we utilize the low-redshift calibration method
while the simultaneous fitting was used in (Khadka et al., 2021).

Figure 3: The probability density plots of $\Omega_{\mathrm{m0}}$ in the
$\Lambda$CDM model with $H_{0}=70$ $\mathrm{km}~{}\mathrm{s^{-1}Mpc^{-1}}$.
The left and right panels show the results from the 141 high-redshift GRBs and
full-redshift 220 GRBs, respectively. The gray dashed line denotes the
$\Omega_{\mathrm{m0}}=0.3$ which is the value of fiducial model.

Figure 4: The probability density plots of $\Omega_{\mathrm{m0}}$ in the
$\Lambda$CDM model with $H_{0}=70$ $\mathrm{km}~{}\mathrm{s^{-1}Mpc^{-1}}$.
The left and right panels show the results from the 98 high-redshift GRBs and
full-redshift 118 GRBs, respectively. The gray dashed line denotes the
$\Omega_{\mathrm{m0}}=0.3$ which is the value of fiducial model.  Table 3:

| high-redshift |  | full-redshift
---|---|---|---
| $\Omega_{\mathrm{m0}}(\sigma)$ | 68%CL | $\chi^{2}$ |  | $\Omega_{\mathrm{m0}}(\sigma)$ | 68%CL | $\chi^{2}$
Amati | $0.649(0.106)$ | ${}^{+0.114}_{-0.096}$ | 90.535 |  | $0.589(0.086)$ | ${}^{+0.091}_{-0.077}$ | 168.851
extend Amati | $0.574(0.103)$ | ${}^{+0.107}_{-0.093}$ | 86.194 |  | $0.507(0.083)$ | ${}^{+0.083}_{-0.078}$ | 164.098
$\mathrm{copula_{1}}$ | $0.295(0.063)$ | ${}^{+0.067}_{-0.057}$ | 86.050 |  | $0.296(0.057)$ | ${}^{+0.058}_{-0.053}$ | 160.779
$\mathrm{copula_{2}}$ | $0.385(0.077)$ | ${}^{+0.080}_{-0.068}$ | 84.264 |  | $0.368(0.067)$ | ${}^{+0.071}_{-0.060}$ | 159.941

Note. — The best-fitted value of $\Omega_{\mathrm{m0}}$ with the standard
deviation $\sigma$ and the $68\%$ CL. The results are obtained from A220 data
set.

Table 4:

| high-redshift |  | full-redshift
---|---|---|---
| $\Omega_{\mathrm{m0}}(\sigma)$ | 68%CL | $\chi^{2}$ |  | $\Omega_{\mathrm{m0}}(\sigma)$ | 68%CL | $\chi^{2}$
Amati | $0.638(0.109)$ | ${}^{+0.119}_{-0.100}$ | 69.710 |  | $0.607(0.105)$ | ${}^{+0.116}_{-0.094}$ | 88.551
extend Amati | $0.579(0.133)$ | ${}^{+0.138}_{-0.123}$ | 50.431 |  | $0.540(0.120)$ | ${}^{+0.123}_{-0.108}$ | 66.905
$\mathrm{copula_{1}}$ | $0.390(0.101)$ | ${}^{+0.105}_{-0.086}$ | 46.979 |  | $0.381(0.094)$ | ${}^{+0.099}_{-0.081}$ | 64.609
$\mathrm{copula_{2}}$ | $0.466(0.117)$ | ${}^{+0.130}_{-0.097}$ | 46.563 |  | $0.445(0.103)$ | ${}^{+0.110}_{-0.090}$ | 63.794

Note. — The best-fitted value of $\Omega_{\mathrm{m0}}$ with the standard
deviation $\sigma$ and the $68\%$ CL. The results are obtained from A118 data
set.

###  4.2 The simultaneous fitting

To show clearly the difference of the results from the GRB low-redshift
calibration and the simultaneous fitting, now we follow the steps as in
(Khadka et al., 2021; Cao et al., 2022) to use the simultaneous fitting method
to constrain the cosmological parameter and the coefficients of four
correlation relations. After setting
$H_{0}=70~{}\mathrm{km~{}s^{-1}Mpc^{-1}}$, the constraints on
$\Omega_{\mathrm{m0}}$ and the coefficients of four correlation relations can
be obtained from the A118 and A220 data samples by using the D’Agostinis
likelihood function (Eq. 23).

Fig. 5 shows one dimensional probability density plots and contour plots of
$\Omega_{\mathrm{m0}}$ and the coefficients of four correlation relations, and
the marginalized mean values with the standard deviation and the $68\%$ CL are
summarized in Tab. 5. We find that the results from the standard Amati
correlation are slightly different from what were obtained in (Cao et al.,
2022). This is because the peak energy of the GRB 081121 data point, which is
released in (Dirirsa et al., 2019) and used in (Cao et al., 2022), is
different from the one given in Tab. 1 of (Amati et al., 2009), and it
actually corresponds to the distance modulus rather than the peak energy in
Tab. 4 of Wang et al. (2016)). Thus, there is an error in the peak energy of
the GRB 081121 used in (Cao et al., 2022). If this error were not corrected,
we would obtain the same result as Cao et al. (2022). In our analysis we have
corrected this error. Comparing Tab. 5 and Tabs. 1 and 2, we find that the
values of $c$ from the simultaneous fitting are smaller and closer to zero
than the ones from the low-redshift calibration. This seems to indicate that
the evolutionary character with redshift of Amati correlation becomes weaker
when more high redshift data are used. Fig. 5 and Tab. 5 show that when the
simultaneous fitting method is used, the results from the improved Amati
correlations are similar to those from the standard Amati correlation since
the GRB data favor a large value of $\Omega_{\mathrm{m0}}$ and can only give a
low bound limit on $\Omega_{\mathrm{m0}}$ although the low bound limits from
the improved Amati correlations are smaller clearly than the one from the
standard Amati correlation. If the extended Amati correlation is considered,
there is almost no constraint on $\Omega_{\mathrm{m0}}$ from GRB, which should
be attributed to that this relation has more coefficients. Thus, once the
simultaneous fitting method is used, we can not find that the improve Amati
correlations obtained in this paper are apparently better than the standard
and extended Amati correlations. These results are different significantly
from those obtained in the subsection above with the low-redshift calibration
method.

Figure 5:  The results of simultaneously fitting the flat $\Lambda$CDM model
and four correlations via the A118 (red) and A220 (blue) data sets. The upper
left, upper right, lower left and lower right panels denotes the Amati,
extended Amati, $\mathrm{copula_{1}}$ and $\mathrm{copula_{2}}$ correlations,
respectively.  Table 5:

Data set |  | Amati |  | extended Amati |  | $\mathrm{copula_{1}}$ |  | $\mathrm{copula_{2}}$
---|---|---|---|---|---|---|---|---
|  | Mean($\sigma$) | 0.68 CL |  | Mean($\sigma$) | 0.68 CL |  | Mean($\sigma$) | 0.68 CL |  | Mean($\sigma$) | 0.68 CL
| $\Omega_{\mathrm{m0}}$ | $>0.713$ | ${-}$ |  | $-$ | ${-}$ |  | $>0.554$ | ${-}$ |  | $>0.550$ | ${-}$
| $\sigma_{int}$ | $0.459(0.024)$ | ${}^{+0.022}_{-0.025}$ |  | $0.459(0.0.024)$ | ${}^{+0.021}_{-0.026}$ |  | $0.460(0.024)$ | ${}^{+0.022}_{-0.026}$ |  | $0.461(0.024)$ | ${}^{+0.022}_{-0.026}$
| $a$ | $52.630(0.068)$ | ${}^{+0.045}_{-0.079}$ |  | $52.462(0.158)$ | ${}^{+0.160}_{-0.140}$ |  | $52.685(0.118)$ | ${}^{+0.052}_{-0.130}$ |  | $52.692(0.127)$ | ${}^{+0.048}_{-0.140}$
A220 | $b$ | $1.286(0.072)$ | ${}^{+0.072}_{-0.072}$ |  | $1.434(0.228)$ | ${}^{+0.230}_{-0.230}$ |  | $1.258(0.078)$ | ${}^{+0.078}_{-0.078}$ |  | $1.262(0.079)$ | ${}^{+0.078}_{-0.078}$
| $c$ | $-$ | ${-}$ |  | $-$ | ${-}$ |  | $-0.066(0.074)$ | ${}^{+0.077}_{-0.057}$ |  | $-0.064(0.075)$ | ${}^{+0.076}_{-0.052}$
| $\alpha$ | $-$ | ${-}$ |  | $0.544(0.447)$ | ${}^{+0.240}_{-0.510}$ |  | $-$ | ${-}$ |  | $-$ | ${-}$
| $\beta$ | $-$ | ${-}$ |  | $-0.337(0.370)$ | ${}^{+0.370}_{-0.370}$ |  | $-$ | ${-}$ |  | $-$ | ${-}$
| $\Omega_{\mathrm{m0}}$ | $0.610(0.230)$ | ${}^{+0.340}_{-0.170}$ |  | $<0.680$ | ${-}$ |  | $>0.470$ | ${-}$ |  | $>0.459$ | ${-}$
| $\sigma_{int}$ | $0.391(0.028)$ | ${}^{+0.024}_{-0.031}$ |  | $0.394(0.028)$ | ${}^{+0.024}_{-0.031}$ |  | $0.392(0.028)$ | ${}^{+0.025}_{-0.031}$ |  | $0.394(0.029)$ | ${}^{+0.025}_{-0.031}$
| $a$ | $52.826(0.115)$ | ${}^{+0.082}_{-0.140}$ |  | $52.786(0.228)$ | ${}^{+0.240}_{-0.210}$ |  | $52.846(0.147)$ | ${}^{+0.067}_{-0.170}$ |  | $52.845(0.142)$ | ${}^{+0.075}_{-0.170}$
A118 | $b$ | $1.171(0.085)$ | ${}^{+0.085}_{-0.085}$ |  | $1.234(0.422)$ | ${}^{+0.418}_{-0.420}$ |  | $1.183(0.086)$ | ${}^{+0.086}_{-0.086}$ |  | $1.179(0.089)$ | ${}^{+0.089}_{-0.089}$
| $c$ | $-$ | ${-}$ |  | $-$ | ${-}$ |  | $0.017(0.097)$ | ${}^{+0.100}_{-0.068}$ |  | $0.010(0.084)$ | ${}^{+0.089}_{-0.069}$
| $\alpha$ | $-$ | ${-}$ |  | $0.198(0.520)$ | ${}^{+0.333}_{-0.587}$ |  | $-$ | ${-}$ |  | $-$ | ${-}$
| $\beta$ | $-$ | ${-}$ |  | $-0.097(0.629)$ | ${}^{+0.626}_{-0.625}$ |  | $-$ | ${-}$ |  | $-$ | ${-}$

Note. — The marginalized mean values, standard deviations, and the 68% CL of
the flat $\Lambda$CDM model parameter and coefficients of
$y_{\mathrm{Amati}}$, $y_{\mathrm{exAmati}}$, $y_{\mathrm{copula_{1}}}$ and
$y_{\mathrm{copula_{2}}}$ from the A220 and A118 data set using the
simultaneous fitting method.

##  5 conclusions

In this paper, we use the three dimensional Gaussian copula method to
investigate the luminosity correlation of GRB data. By assuming that the
logarithms of the special peek energy and the isotropic energy of GRBs satisfy
the Gaussian distributions and two different redshift distributions of GRB
data (one is the special form given in Eq. (14) and the other is empirical
distribution), we obtain two improved Amati correlations of GRB data
($y_{\mathrm{copula_{1}}}$ and $y_{\mathrm{copula_{2}}}$), which are
distinctively different from the standard Amati correlation and the extended
Amati correlation. After calibrating, with the low-redshift GRB data points
from A220 and A118 data sets respectively, these improved Amati correlations
based on a fiducial $\Lambda$CDM model with $\Omega_{\mathrm{m0}}=0.3$ and
$H_{0}=70~{}\mathrm{km~{}s^{-1}Mpc^{-1}}$, and extrapolating the results to
the high-redshift GRB data, we obtain the Hubble diagrams of 220 and 118 GRB
data points. Applying these GRB data to constrain the $\Lambda$CDM model, we
find that the results from the improved Amati correlations are apparently
better than those from the standard Amati and extended Amati correlations
although the BIC favors mildly the standard Amati correlation. The improved
Amati correlation based on the special redshift distribution of GRB data gives
the best result, which is always consistent with $\Omega_{\mathrm{m0}}=0.3$ at
the $1\sigma$ confidence level and is highly consistent with
$\Omega_{\mathrm{m0}}=0.3$ when A220 data set is used. These results seem to
indicate that when the improved Amati correlation with the special redshift
distribution ($y_{\mathrm{copula_{1}}}$) is used in the low-redshift
calibration, the GRB data can be regarded as a viable cosmological explorer.
However, the BIC indicates that the standard Amati correlation remains to be
favored mildly since it has the least model parameters. Furthermore, once the
simultaneous fitting method rather than the low-redshift calibration one is
used, we find that the constraints on $\Omega_{\mathrm{m0}}$ are weak and only
the low bound limit on $\Omega_{\mathrm{m0}}$ can be obtained. Although this
low bound limit from the improved Amati correlation is smaller than the one
from the standard Amati correlation, there is no apparent evidence that the
former is better than the latter. This result is different apparently from the
one from the low-redshift calibration method. Therefore, more works need to be
done in the future in order to compare different Amati correlations.

We appreciate very much the insightful comments and helpful suggestions by
anonymous referees. This work was supported in part by the NSFC under Grants
No. 12075084, No. 11690034, No. 11805063, No. 11775077, and 12073069, by the
Science and Technology Innovation Plan of Hunan province under Grant No.
2017XK2019, and by the Guizhou Provincial Science and Technology Foundation
(QKHJC-ZK[2021] Key 020).

##  Appendix A The Amati correlation

In 2002, Amati et al. found that in GRB observational data there is a positive
correlation between the spectral peak energy $E_{p}$ and the isotropic
equivalent radiated energy $E_{iso}$ (Amati, 2006a, b; Amati et al., 2008,
2009), and this correlation has the form

$y_{\mathrm{Amati}}=a+bx,$ (A1)

where

$\displaystyle
y\equiv\log\frac{E_{iso}}{1\mathrm{erg}},~{}x\equiv\log\frac{E_{p}}{300\mathrm{keV}},$
(A2)

intercept $a$ and slope $b$ are free coefficients, “$\log$” denotes the
logarithm to base 10, and

$\displaystyle E_{p}$ $\displaystyle=$ $\displaystyle E_{p}^{obs}(1+z),$ (A3)
$\displaystyle E_{iso}$ $\displaystyle=$ $\displaystyle 4\pi
d^{2}_{L}(z)S_{bolo}(1+z)^{-1}.$ (A4)

Here $z$ is the redshift, $E_{P}^{obs}$ is the observed peak energy of GRB
spectrum, $d_{L}(z)$ is the luminosity distance, and $S_{bolo}$ is the
bolometric fluence.

If the coefficients $a$ and $b$ are determined, the luminosity distance of GRB
data point can be obtained from Eqs. (A1, A3, A4). Then we can obtain the
distance modulus of GRB data, which is defined to be

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle
5\log\frac{d_{L}(z)}{\mathrm{Mpc}}+25.$ (A5)

If assuming a fiducial cosmological model, the values of coefficients $a$ and
$b$ in the Amati correlation can be obtained from the observational data by
using the following common fitting strategy (D’Agostini, 2005) :

$\displaystyle\mathcal{L}(\sigma_{int},a,b)\propto\prod_{i}\frac{1}{\sqrt{\sigma_{int}^{2}+\sigma_{yi}^{2}+b^{2}\sigma_{xi}^{2}}}\times\exp\left[-\frac{(y_{i}-a-bx_{i})^{2}}{2(\sigma_{int}^{2}+\sigma_{yi}^{2}+b^{2}\sigma_{xi}^{2})}\right],$
(A6)

where $\sigma_{x}$ and $\sigma_{y}$ are the uncertainties of $x$ and $y$,
respectively, and $\sigma_{int}$ is the intrinsic uncertainty of GRB. From the
well-known error propagation equation, one find that $\sigma_{y}$ and
$\sigma_{x}$ can be derived from Eqs. (A3,A4) and have the expressions:

$\displaystyle\sigma_{y}=\frac{1}{\ln
10}\frac{\sigma_{E_{iso}}}{E_{iso}},~{}\sigma_{x}=\frac{1}{\ln
10}\frac{\sigma_{E_{p}}}{E_{p}}$ (A7)

with

$\displaystyle\sigma_{E_{iso}}=4\pi d_{L}^{2}\sigma_{S_{bolo}}(1+z)^{-1}.$
(A8)

Here $\sigma_{E_{p}}$ and $\sigma_{S_{bolo}}$ are available in observations of
GRBs. Therefore, maximizing the likelihood function $\mathcal{L}$ (Eq. A6),
the allowed values of $a$, $b$, and $\sigma_{int}$ can be obtained. Then the
covariance matrix $C_{ij}$ of these fitted parameters can be approximately
evaluated from:

$(C^{-1})_{ij}(\bm{\theta}_{A})=\frac{\partial^{2}[-\ln\mathcal{L}(\bm{\theta}_{A})]}{\partial\theta_{i}\partial\theta_{j}}\arrowvert_{\bm{\theta}_{A}=\bm{\theta}_{b}},$
(A9)

where $\bm{\theta}_{A}=\left\\{\sigma_{int},a,b\right\\}$, and
$\bm{\theta}_{b}$ denote the best-fitted value of $a$, $b$ and $\sigma_{int}$.

Using the best fitted values of $a$ and $b$, we can get the luminosity
distance of GRBs and the corresponding distance modulus from Eq. (A5). By
using the error propagation equation, the uncertainty of distance modulus can
be derived from the following equation

$\displaystyle\sigma_{\mu}^{2}=\left(\frac{5}{2}\sigma_{\log\frac{E_{iso}}{1\mathrm{erg}}}\right)^{2}+\left(\frac{5}{2\ln
10}\frac{\sigma_{S_{bolo}}}{S_{bolo}}\right)^{2}.$ (A10)

Here

$\displaystyle\sigma_{\log\frac{E_{iso}}{1\mathrm{erg}}}^{2}$ $\displaystyle=$
$\displaystyle\sigma_{a}^{2}+\left(\sigma_{b}\log\frac{E_{p}}{300\mathrm{keV}}\right)^{2}+2\sum_{i=1}^{3}\sum_{j=i+1}^{3}\left(\frac{\partial
y_{\mathrm{Amati}}(x;\bm{\theta}_{A})}{\partial\theta_{i}}\frac{\partial
y_{\mathrm{Amati}}(x;\bm{\theta}_{A})}{\partial\theta_{j}}\right)C_{ij}$ (A11)
$\displaystyle+$ $\displaystyle\left(\frac{b}{\ln
10}\frac{\sigma_{E_{p}}}{E_{p}}\right)^{2}+\sigma_{int}^{2}.$

According to the distance modulus of GRBs, the cosmological model can be
constrained by minimizing $\chi^{2}$

$\chi^{2}=\sum_{i=1}^{N}\left[\frac{\mu_{obs}(z_{i})-\mu_{th}(z_{i};p)}{\sigma^{obs}_{\mu_{i}}}\right]^{2},$
(A12)

where $\mu_{obs}$ is the distance modulus of GRB and $\mu_{th}(z_{i};p)$ is
the theoretic value of distance modulus in cosmological model with $p$
representing the model parameters.

##  Appendix B The extended Amati correlation

The extended Amati correlation was proposed in (Wang et al., 2017) where the
authors used two formulas to parameterize the coefficients $a$ and $b$ in the
standard Amati correlation:

$\displaystyle a\rightarrow A=a+\alpha\frac{z}{1+z},~{}b\rightarrow
B=b+\beta\frac{z}{1+z},$ (B1)

where $\alpha$ and $\beta$ are two constants. Substituting these parameterized
formulas into the Eq. (A1), the extended Amati correlation can be obtained

$y_{\mathrm{exAmati}}=\left(a+\alpha\frac{z}{1+z}\right)+\left(b+\beta\frac{z}{1+z}\right)x.$
(B2)

Recently, Khadka et al. (2021) used the A220 GRB data set to limit $\alpha$
and $\beta$, and found that the Amati correlation is independent of redshift
within the error bars. The coefficient
$\bm{\theta}_{ex}=\\{\sigma_{int},a,b,\alpha,\beta\\}$ is also estimated from
D’Agostinis likelihood function. The uncertainty of
$\log\frac{E_{iso}}{\mathrm{1erg}}$ in Eq. (A10) can be obtained from

$\displaystyle\sigma_{\log\frac{E_{iso}}{1\mathrm{erg}}}^{2}$ $\displaystyle=$
$\displaystyle\sigma_{a}^{2}+\left(\sigma_{b}\log\frac{E_{p}}{300\mathrm{keV}}\right)^{2}+\left(\sigma_{\alpha}\frac{z}{1+z}\right)^{2}+\left(\sigma_{\beta}\frac{z}{1+z}\log\frac{E_{p}}{300\mathrm{keV}}\right)^{2}$
(B3) $\displaystyle+$ $\displaystyle
2\sum_{i=1}^{5}\sum_{j=i+1}^{5}\left(\frac{\partial
y_{\mathrm{exAmati}}(x;\bm{\theta}_{ex})}{\partial\theta_{i}}\frac{\partial
y_{\mathrm{exAmati}}(x;\bm{\theta}_{ex})}{\partial\theta_{j}}\right)C_{ij}$
$\displaystyle+$
$\displaystyle\left[\left(b+\beta\frac{z}{1+z}\right)\frac{1}{\ln
10}\frac{\sigma_{E_{p}}}{E_{p}}\right]^{2}+\sigma_{int}^{2}.$

## References

  * Abbott et al. (2018) Abbott, T. M. C., et al. 2018, MNRAS, 480, 3879
  * Aghanim et al. (2019) Aghanim, N., et al.  2020, A&A, 641, A1
  * Akaike (1974) Akaike, H.  1974, IEEE Trans. Autom. Control., 19, 716
  * Akaike (1981) Akaike, H. 1981, J. Econom., 16,3
  * Amati et al. (2002) Amati, L., et al. 2002, A&A, 390, 81
  * Amati (2006a) Amati, L. 2006a, Nuovo Cimento B, 121, 1081
  * Amati (2006b) Amati, L. 2006b, MNRAS, 372, 233
  * Amati et al. (2008) Amati, L., et al. 2008, MNRAS, 391, 577
  * Amati et al. (2019) Amati, L., D’Agostino, R., Luongo, O., Muccino, M., & Tantalo, M. 2019, MNRAS, 486, L46
  * Amati et al. (2009) Amati, L., Frontera, F., & Guidorzi, C. 2009, A&A, 508, 173
  * Basilakos & Perivolaropoulos (2008) Basilakos, S., & Perivolaropoulos, L., 2008, MNRAS, 391, 411
  * Benabed et al. (2009) Benabed, K., Cardoso, J.-F., Prunet, S., & Hivon, E. 2009, MNRAS, 400, 219
  * Birrer et al. (2020) Birrer, S., et al. 2020, A&A, 643, A165
  * Cao et al. (2022) Cao, S., Khadka, N., & Ratra, B. 2022, MNRAS, 510, 2928
  * Cao et al. (2021a) Cao, S., Ryan, J., & Ratra, B. 2021a, MNRAS, 504, 300
  * Cao et al. (2021b) Cao, S., Ryan, J., Khadka, N., & Ratra, B. 2021b, MNRAS, 501, 1520
  * Chen et al. (2017) Chen, Y., Kumar, S., & Ratra, B. 2017, ApJ, 835, 86
  * D’Agostini (2005) D’Agostini, G. 2005, arXiv:physics/0511182
  * Dekking et al. (2005) Dekking, F. M., et al., _A Modern Introduction to Probability and Statistics_. London, U.K.: Springer-Verlag, 2005
  * Demianski et al. (2017) Demianski, M., Piedipalumbo, E., Sawant, D., & Amati, L. 2017, A&A, 598, A112
  * Demianski et al. (2021) Demianski, M., Piedipalumbo, E., Sawant, D., & Amati, L. 2021, MNRAS, 506, 903
  * Dirirsa et al. (2019) Dirirsa, F. F, et al. 2019, ApJ, 887, 13
  * Efstathiou (2020) Efstathiou, G. 2020, arXiv:2007.10716v2
  * Eisenstein et al. (2005) Eisenstein, D. J., et al. 2005, ApJ, 633, 560.
  * Fenimore & Ramirez-Ruiz (2000) Fenimore, E. E., & Ramirez-Ruiz, E. 2000, arXiv:astro-ph/0004176
  * Freedman (2021) Freedman, W. L. 2021, ApJ, 919, 16
  * Ghirlanda et al. (2004a) Ghirlanda, G., Ghisellini, G., & Lazzati, D. 2004a, ApJ, 616, 331
  * Ghirlanda et al. (2004b) Ghirlanda, G., Ghisellini, G., Lazzati, D., & Firmani, C. 2004b, ApJ, 613, L13
  * Ghirlanda et al. (2006) Ghirlanda, G., Ghisellini, G.,& Firmani, C. 2006, New,J. Phys., 8, 123
  * Hu et al. (2021) Hu, J. P., Wang, F. Y., & Dai, Z. G. 2021, MNRAS, 507, 730
  * Jiang et al. (2009) Jiang, I.-G., Yeh, L.-C., Chang, Y.-C., & Hung, W.-L. 2009, AJ, 137, 329
  * Khadka et al. (2021) Khadka, N., Luongo, O., Muccino, M., & Ratra, B. 2021, J. Cosmology Astropart. Phys, 09, 042
  * Khadka & Ratra (2020) Khadka, N., & Ratra, B. 2020, MNRAS, 499, 391
  * Khetan et al. (2021) Khetan, N., et al. 2021, A&A, 647, A72
  * Klebesadel et al. (1973) Klebesadel, R. W., Strong, I. B., & Olson, R. A. 1973, ApJ, 182, L85
  * Kodama et al. (2008) Kodama, Y., et al. 2008, MNRAS, 391, L1
  * Koen (2009) Koen, C. 2009, MNRAS, 393, 1370
  * Koen & Bere (2017) Koen, C., & Bere,A. 2017, MNRAS, 471, 2771
  * Li (2007) Li, L.-X. 2007, MNRAS, 379, L55
  * Li et al. (2008) Li, H. et al. 2008, ApJ, 680, 92
  * Liang et al. (2008) Liang, N., Xiao, W. K., Liu, Y., & Zhang, S. N. 2008, ApJ, 685, 354
  * Liang et al. (2010) Liang, N., Wu, P. & Zhang, S. N. 2010, Phys. Rev. D, 81, 083518
  * Lin et al. (2015) Lin, H.-N., Li, X., Wang, S., & Chang, Z. 2015, MNRAS, 453, 128
  * Lin et al. (2016) Lin, H.-N., Li, X., Chang, Z. 2016, MNRAS, 455, 2131
  * Lin & Ishak (2021) Lin, W., & Ishak, M. 2021, J. Cosmology Astropart. Phys, 05, 009
  * Luongo & Muccino (2021) Luongo, O., & Muccino, M. 2021, arXiv:2110.14408
  * Nelson (2006) Nelson, R. B. 2006, An Introduction to Copulas (2nd ed.; New York: Springer)
  * Norris et al. (2000) Norris, J. P., Marani, G. F., & Bonnell, J. T. 2000, ApJ, 534, 248
  * Perlmutter et al. (1999) Perlmutter, S., et al.  1999, ApJ, 517, 565
  * Qin et al. (2020) Qin, J., Yu, Y., & Zhang, P. 2020, ApJ, 897, 105
  * Riess et al. (1998) Riess, A. G., et al.  1998, AJ, 116, 1009
  * Riess et al. (2018a) Riess, A. G., et al.  2018a, ApJ, 853, 126
  * Riess et al. (2018b) Riess, A. G., et al.  2018b, ApJ, 861, 126
  * Riess et al. (2021) Riess, A. G., et al. 2021, ApJ, 908, L6
  * Sato et al. (2010) Sato, M., Ichiki, K., & Takeuchi, T. T. 2010, Phys. Rev. Lett., 105, 251301
  * Sato et al. (2011) Sato, M., Ichiki, K., & Takeuchi, T. T. 2011, Phys. Rev. D, 83, 023501
  * Scherrer et al. (2010) Scherrer, R. J., Berlind, A. A., Mao, Q., & McBride, C. K. 2010, ApJ, 708, L9
  * Schwarz (1978) Schwarz, G. 1978, Annals of Stats, 6, 461
  * Spergel et al. (2003) Spergel, D. N., et al.  2003, ApJS, 148, 175
  * Spergel et al. (2007) Spergel, D. N., et al.  2007, ApJS, 170, 377
  * Takeuchi (2010) Takeuchi, T. T. 2010, MNRAS, 406, 1830
  * Takeuchi & Kono (2020) Takeuchi, T. T., & Kono, K. T. 2020, MNRAS, 498, 4365
  * Takeuchi et al. (2013) Takeuchi, T. T., Sakurai, A., Yuan, F.-T., Buat, V., & Burgarella, D. 2013, Earth Planet Sp, 65, 281
  * Wang et al. (2015) Wang, F. Y., Dai, Z. G., & Liang, E. W. 2015, New Astronomy Reviews, 67, 1
  * Wang et al. (2021) Wang, F. Y., Hu, J. P., Zhang, G. Q., & Dai, Z. G. 2022, ApJ, 924, 97
  * Wang et al. (2011) Wang, F.-Y., Qi, S., & Dai, Z.-G. 2011, MNRAS, 415, 3423
  * Wang et al. (2017) Wang, G.-J., Yu, H., Li, Z.-X., Xia, J.-Q., & Zhu Z.-H. 2017, ApJ, 836, 103
  * Wang et al. (2016) Wang, J. S., Wang, F. Y., Cheng, K. S. & Dai, Z. G. 2016, A&A, 585, A68
  * Wei & Zhang. (2009) Wei, H., Zhang, S. N. 2009, Eur. Phys. J. C, 63, 139
  * Wu et al. (2017) Wu, P.-X, Li, Z.-X, & Yu, H.-W. 2017, Front. Phys. 12, 129801
  * Yonetoku et al. (2004) Yonetoku, D., et al. 2004, ApJ, 609, 935
  * Yuan et al. (2018) Yuan, Z., Wang, J., Worrall, D. M., Zhang, B.-B., & Mao, J. 2018, ApJS, 239, 33