$\tilde{D}(G_{1},G_{2})$, by constraining the variable $P$ to be not only
orthogonal, but also $P=U_{2}MU_{1}^{T}$ where M is the solution (i.e.
“matching”, in the terminology of that section) to a Linear Assignment problem
with costs given by a function of the eigenvalues of $L(G_{1})$ and
$L(G_{2})$. In this appendix we derive a similar lower bound on the distance.

As in Theorem 4.1, let $G_{1}$ and $G_{2}$ be undirected graphs with
Laplacians $L_{1}=L(G_{1})$ and $L_{2}=L(G_{2})$, and let $\alpha>0$ be
constant. By Equation (56), we have

$\displaystyle\tilde{D}^{2}(G_{1},G_{2})$
$\displaystyle=\inf_{\alpha>0}\inf_{P^{T}P=I}\left(\sum\limits_{i=1}^{n_{2}}\sum\limits_{j=1}^{n_{1}}p_{ij}^{2}{\left(\frac{1}{\alpha}{\lambda^{(1)}_{j}}-\alpha{\lambda^{(2)}_{i}}\right)}^{2}\right).$
(101)

Previously, we derived an upper bound on $\tilde{D}$ by constraining $P$ to be
not only orthogonal, but related to a constrained matching problem between the
two lists of eigenvalues:

$\begin{array}[]{ll@{}ll}\tilde{D}^{2}(G_{1},G_{2})\leq\inf_{\alpha>0}\inf_{M}&{\left|\left|\frac{1}{\alpha}M\Lambda_{1}-\alpha\Lambda_{2}M\right|\right|}^{2}_{F}&\\\
\text{subject to}&\displaystyle\sum\limits_{i=1}^{n_{2}}m_{ij}\leq
1,&j=1\ldots n_{1}\\\ &\displaystyle\sum\limits_{j=1}^{n_{1}}m_{ij}\leq
1,&i=1\ldots n_{2}\\\ &m_{ij}\geq 0&i=1\ldots n_{2},j=1\ldots n_{1},\\\
\end{array}$ (102)

where $\Lambda_{1}$ and $\Lambda_{2}$ are diagonal matrices of the eigenvalues
of $L_{1}$ and $L_{2}$ respectively. In that proof, we used the explicit map
$\tilde{P}=U_{2}^{T}PU_{1}$ as a change of basis; we then converted the
constraints on $P$ into equivalent constraints on $\tilde{P}$, and imposed
additional constraints so that the resulting optimization (a linear assignment
problem) was an upper bound. We show in this section that a more loosely
constrained assignment problem is a lower bound on $\tilde{D}^{2}$. We do this
by computing the same mapping $\tilde{P}=U_{2}^{T}PU_{1}$ and then dropping
some of the constraints on $\tilde{P}$ (which is equivalent to dropping
constraints on $P$, yielding a lower bound). The constraint $P^{T}P=I$ is the
conjunction of $n_{1}^{2}$ constraints on the column vectors of $P$: if
$\mathbf{p}_{i}$ is the $i$th column of $P$, then $P^{T}P=I$ is equivalent to:

$\displaystyle\mathbf{p}_{i}\cdot\mathbf{p}_{i}=1$ $\displaystyle\forall
i=1\ldots n_{1}$ (103) $\displaystyle\mathbf{p}_{i}\cdot\mathbf{p}_{i}=0$
$\displaystyle\forall i=1\ldots n_{1},j=1\ldots i-1,i+1\ldots n_{1},$ (104)

If we discard the constraints in Equation (104), we are left with the
constraint that every column of $p$ must have unit norm.

Construct the “spectral lower bound matching” matrix $P^{(\text{SLB})}$ as
follows:

$P^{(\text{SLB})}_{i,j}=\begin{cases}1&\text{if i =
}\arg\min_{k}{\left(\frac{1}{\alpha}\lambda^{(1)}_{j}-\alpha\lambda^{(k)}_{k}\right)}^{2}\\\
0&\text{otherwise.}\end{cases}$ (105)

For any $\alpha$, this matrix is the solution to a matching problem (less
constrained than the original optimization over all $P$) where each
$\lambda^{(1)}_{j}$ is assigned to the closest $\lambda^{(2)}_{i}$, allowing
collisions. It clearly satisfies the constraints in Equation (103), but may
violate those in Equation (104). Thus, we have

$\displaystyle\tilde{D}^{2}(G_{1},G_{2})$
$\displaystyle=\inf_{\alpha>0}\inf_{P^{T}P=I}\left(\sum\limits_{i=1}^{n_{2}}\sum\limits_{j=1}^{n_{1}}p_{ij}^{2}{\left(\frac{1}{\alpha}{\lambda^{(1)}_{j}}-\alpha{\lambda^{(2)}_{i}}\right)}^{2}\right).$
(106)
$\displaystyle\geq\tilde{D}^{2}\left(G_{1},G_{2}\left|P^{(\text{SLB})}\right.\right)$

Various algorithms exist to rapidly find the member of a set of points which
is closest to some reference point (for example, KD-Trees
bentley1975multidimensional ). For any $\alpha$, the spectral lower bound can
be calculated by an outer loop over alpha and an inner loop which applies one
of these methods. We do not consider joint optimization of the lower bound
over $P$ and $\alpha$ in this work.


# Gamma ray burst constraints on cosmological models from the improved Amati
correlation

Yang Liu  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  Joint
Center for FAST Sciences Guizhou Normal University Node, Guiyang, Guizhou
550025, China  Xiaoyao Xie  Key Laboratory of Information and Computing
Science Guizhou Province, Guizhou Normal University, Guiyang, Guizhou 550025,
China  Joint Center for FAST Sciences Guizhou Normal University Node, Guiyang,
Guizhou 550025, 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
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

An improved Amati correlation was constructed in (ApJ 931 (2022) 50) by us
recently. In this paper, we further study constraints on the $\Lambda$CDM and
$w$CDM models from the gamma ray bursts (GRBs) standardized with the standard
and improved Amati correlations, respectively. By using the Pantheon type Ia
supernova sample to calibrate the latest A220 GRB data set, the GRB Hubble
diagram is obtained model-independently. We find that at the high redshift
region ($z>1.4$) the GRB distance modulus from the improved Amati correlation
is larger apparently than that from the standard Amati one. The GRB data from
the standard Amati correlation only give a lower bound limit on the present
matter density parameter $\Omega_{\mathrm{m0}}$, while the GRBs from the
improved Amati correlation constrain the $\Omega_{\mathrm{m0}}$ with the
$68\%$ confidence level to be $0.308^{+0.066}_{-0.230}$ and
$0.307^{+0.057}_{-0.290}$ in the $\Lambda$CDM and $w$CDM models, respectively,
which are consistent very well with those given by other current popular
observational data including BAO, CMB and so on. Once the $H(z)$ data are
added in our analysis, the constraint on the Hubble constant $H_{0}$ can be
achieved. We find that two different correlations provide slightly different
$H_{0}$ results but the marginalized mean values seem to be close to that from
the Planck 2018 CMB observations.

##  1 Introduction

Current observations, such as the type Ia supernovae (SN Ia) (Riess et al.,
1998; Perlmutter et al., 1999), the cosmic microwave background radiation
(CMB) (Spergel et al., 2003, 2007), and the baryon acoustic oscillation(BAO)
(Eisenstein et al., 2005), indicate that the cosmic expansion is accelerating.
To explain the origin of this peculiar phenomenon, commonly, a perfect fluid
with negative pressure in the Universe dubbed dark energy is introduced. The
nature of dark energy can be characterized by its equation of state (EoS) $w$.
The simplest candidate of dark energy is the cosmological constant $\Lambda$,
whose EoS is $w=-1$. The cosmological constant dark energy plus the cold dark
matter make up the $\Lambda$CDM cosmological model, and the $w$CDM model is
obtained if the EoS of dark energy is generalized from $-1$ to an arbitrary
constant $w$. The simple $\Lambda$CDM model fits with observations (eBOSS
Collaboration, 2021) very well on one hand; however, on the other hand, it is
plagued by the Hubble constant ($H_{0}$) tension (Perivolaropoulos & Skara,
2021; Valentino et al., 2021). Based on the $\Lambda$CDM model, the high-
redshift CMB data (Planck Collaboration, 2020) give a tight constraint on
$H_{0}$ ($67.4\pm 0.5~{}\mathrm{km~{}s^{-1}Mpc^{-1}}$), which deviates from
$H_{0}=73.2\pm 1.3~{}\mathrm{km~{}s^{-1}Mpc^{-1}}$ given by the low-redshift
SN Ia data (Riess et al., 2018a, b, 2021) more than 4$\sigma$. Although many
other observations such as the Hubble parameter $H(z)$ measurements, the BAO,
and the strong gravitational lenses have been used to discuss this $H_{0}$
tension, its origin is not yet determined (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; Cao & Ratra,
2022b). Since the farthest redshift of the popular SN Ia, $H(z)$ and BAO data
is about $2$, while the CMB data lie near the redshift of 1100, the
cosmological data during the region of middle redshift ($2\lesssim z\lesssim
1100$) may play an important role in understanding the origin of the $H_{0}$
tension.

Gamma ray bursts (GRBs) are extremely energetic and thus detectable at a
redshift up to $z\sim 9.4$ (Cucchiara et al., 2011). This implies that the
GRBs have potential to serve as a new cosmological probe to the cosmic
evolution in the middle redshift region. To standardize the GRBs and then use
them to constrain the cosmological models, many empirical correlations between
parameters of the light curves and/or spectra with the GRB luminosity or
energy have been proposed (Norris et al., 2000; Fenimore & Ramirez-Ruiz, 2000;
Amati et al., 2002; Yonetoku et al., 2004; Ghirlanda et al., 2004a; Liang &
Zhang, 2005; Firmani et al., 2006; Dainotti et al., 2008, 2016; Demianski et
al., 2017a; Wang et al., 2017, 2022; Hu et al., 2021; Luongo & Muccino, 2021;
Muccino et al., 2021; Cao et al., 2022c; Dainotti et al., 2022). However, to
calibrate GRB correlations, a cosmological model is usually applied and the
resulting correlations are later used to probe the cosmic evolution. Thus the
GRB cosmology suffers the so called circularity problem (Ghirlanda et al.,
2006; Wang et al., 2015). To avoid this problem, two different methods have
been proposed. The first one is the simultaneous fitting or global fitting
method (Ghirlanda et al., 2004b; Li et al., 2008), in which the coefficients
of the correlations and the parameters of a cosmological model are constrained
simultaneously from the GRB observations. The second one is the low-redshift
calibration (Liang et al., 2008, 2010; Kodama et al., 2008; Wei & Zhang.,
2009), which bases on the idea of distance ladder, as the GRB correlations are
calibrated by using other low redshift observations such as the $H(z)$ data
(Amati et al., 2019; Montiel et al., 2021) or the SN Ia data (Demianski et
al., 2017a, b, 2021). Up to now, the GRB data have been used widely to explore
the components of our Universe, the nature of dark energy, and the Hubble
constant tension (Demianski et al., 2017a, b, 2021; Liu & Wei, 2015; Wang et
al., 2016; Lin et al., 2016; Amati et al., 2019; Khadka & Ratra, 2020; Cao et
al., 2021b, 2022a).

Among the GRB empirical correlations, the Amati correlation is a very popular
one, which connects the spectral peak energy in the GRB cosmological rest-
frame and the isotropic equivalent radiated energy ($E_{p}-E_{\mathrm{iso}}$)
(Amati et al., 2002; Amati, 2006a, b; Amati et al., 2008, 2009; Amati &
Della., 2013). Recently, we proposed an improved Amati correlation (Liu et
al., 2022) by using the Gaussian copula which is a powerful statistical tool
capable of describing the dependence structures between multivariate random
variables, and has been applied to various fields by the astronomical
community (Benabed et al., 2009; Koen, 2009; Jiang et al., 2009; Scherrer et
al., 2010; Takeuchi, 2010; Yuan et al., 2018; Qin et al., 2020; Takeuchi &
Kono, 2020). In (Liu et al., 2022), by choosing the spatially flat
$\Lambda$CDM model with $\Omega_{\mathrm{m0}}=0.30$ and
$H_{0}=70~{}\mathrm{km~{}s^{-1}Mpc^{-1}}$ as the fiducial model, we utilize
the low-redshift ($z<1.4$) GRB data to calibrate the standard and improved
Amati correlations, and then extrapolate the results to the high-redshift GRB
data to achieve the GRB Hubble diagram, where $\Omega_{\mathrm{m0}}$ is the
present dimensionless matter density parameter. Using these calibrated GRBs to
constrain the flat $\Lambda$CDM model, we found that the improved Amati
correlation can give results well consistent with the fiducial model, while
the standard one can not. Thus, in (Liu et al., 2022), the reliability of the
improved Amati correlation was ascertained with a fiducial model, but its
cosmological application was not carried out. In this work, we will fill this
gap. In order to obtain the Hubble diagram of the latest A220 GRB samples
(Khadka et al., 2021) model-independently, we use the Pantheon SN Ia data
(Scolnic et al., 2018) to calibrate the standard and improved Amati
correlations, and then use these calibrated GRB data to constrain the
$\Lambda$CDM and $w$CDM models. Besides the GRB data, the $H(z)$ data set is
also added in our analysis to obtain a tight constraint on model parameters.

The rest of the paper is organized as follows: Section 2 introduces the
improved Amati correlation briefly, and standardizes the GRB samples by using
the method of the low-redshift calibration. Section 3 studies the constraints
on the $\Lambda$CDM and $w$CDM models from the GRB data and the
$\mbox{GRB}+H(z)$ data. Our conclusions are summarized in Section 4.

##  2 GRB Hubble diagram from low-redshift calibration

###  2.1 Improved and standard Amati correlations

The standard Amati correlation is proposed by Amati et al. (2002), which
describes a correlation between the spectral peak energy $E_{p}$ and the
isotropic equivalent radiated energy $E_{\mathrm{iso}}$ and has the form

$y_{\mathrm{Amati}}=a+bx$ (1)

with

$\displaystyle
y\equiv\log\frac{E_{\mathrm{iso}}}{1\mathrm{erg}},~{}x\equiv\log\frac{E_{p}}{300\mathrm{keV}}.$
(2)

Here “$\log$” denotes the logarithm to the base of 10, and $a$ and $b$ are
free coefficients. The spectral peak energy $E_{p}$ and the isotropic
equivalent radiated energy $E_{\mathrm{iso}}$ in Eq. (2) can be obtained
through

$\displaystyle E_{p}$ $\displaystyle=$ $\displaystyle
E_{p}^{\mathrm{obs}}(1+z),$ (3) $\displaystyle E_{\mathrm{iso}}$
$\displaystyle=$ $\displaystyle 4\pi d^{2}_{L}(z)S_{\mathrm{bolo}}(1+z)^{-1},$
(4)

when the luminosity distance $d_{L}(z)$ is known. Here $E_{P}^{\mathrm{obs}}$
is the observed peak energy of the GRB spectrum, and $S_{\mathrm{bolo}}$ is
the bolometric fluence, which is an observable.

The improved Amati correlation is derived from the copula function (Liu et
al., 2022). To construct the correlation between $x$ and $y$ from the Gaussian
copula, $x$ and $y$ are assumed to obey the Gaussian distributions, which are
represented as $f(x)$ and $g(y)$, respectively. In addition, the GRB is
assumed to obey a special redshift distribution $w(z)=ze^{-z}$ (Wang et al.,
2017). Then, we use $F(x)$, $G(y)$ and $W(z)$ to indicate the cumulative
distribution functions of $f(x)$, $g(y)$ and $w(z)$, respectively. According
to the Sklar’s theorem (Nelson, 2006), a joint distribution function $\bar{H}$
can be constructed by using the Gaussian copula $C$:

$\displaystyle\bar{H}(x,y,z;\bm{\theta})=C\left(F(x),G(y),W(z);\bm{\theta}\right).$
(5)

Here $\bm{\theta}$ denotes the parameters of the copula function. From the
density function of the joint distribution, we obtain the improved Amati
correlation, which takes the form (Liu et al., 2022)

$\displaystyle y_{\mathrm{copula}}$ $\displaystyle=$ $\displaystyle
a+b\,x+c\,\mathrm{erfc}^{-1}[2W(z)].$ (6)

Here $\mathrm{erfc}^{-1}$ is the inverse of complementary error function and
$W(z)=1-e^{-z}(1+z)$. Clearly, the improved Amati correlation has an extra
redshift-dependent term compared with the standard one given in Eq. (1). When
the coefficient $c$ is fixed to be zero, the improved Amati correlation
reduces to the standard one.

###  2.2  low-redshift calibration and GRB Hubble diagram

Eq. (4) indicates that to obtain the coefficients in the correlations from
observations the luminosity distance needs to be given. In (Liu et al., 2022),
this distance is given from the $\Lambda$CDM model and thus the results are
model-dependent. To achieve a model-independent GRB Hubble diagram, here we
utilize the method of the low-redshift calibration, and derive the luminosity
distance in Eq. (4) from the Pantheon SN Ia samples by using the idea that at
the same redshift the GRB has the same luminosity distance as the SN Ia.

In our analysis, we consider the A220 data set, which consists of A118 and
A102 data sets and contains 220 long GRBs spanning the redshift from 0.03 to
8.2 (Khadka et al., 2021; Wang et al., 2016; Fana Dirirsa et al., 2019;
Demianski et al., 2017a; Amati et al., 2019). This GRB data set is divided
into the low-redshift part ($z<1.4$), which has 79 data points, and the high-
redshift one. To determine the values of coefficients in Eqs. (1) and (6), we
use the linear interpolation to estimate the luminosity distance of 79 low-
redshift GRB data points by following the method given in Appendix E1 in
Betoule et al. (2014) to bin the 1048 Pantheon SN Ia data points into 35 bins
in the $\log z$ space (see Appendix A for details). The binned results and 79
low-redshift ($z<1.4$) GRBs calibrated by SN Ia are shown in Fig. 1. We must
point out here that although the redshift of Pantheon SN Ia spans to $z=2.26$,
only the luminosity distances at $z\leq 1.4$ are used to calibrate the GRB.
This is because there are only 6 SN Ia data points at $z>1.4$, which is too
few to estimate the luminosity distance accurately.

Figure 1:  Apparent magnitudes of Pantheon SN Ia (gray points) and 36
$\log$-spaced control points (red points). The green points are low-redshift
($z<1.4$) GRBs calibrated by SN Ia. The red dashed line denotes $z=1.4$.

With the improved Amati correlation as an example, the values of coefficients
($a$, $b$, $c$) in Eq. (6) can be obtained from the 79 low-redshift GRBs by
maximizing the D’Agostinis likelihood (D’Agostini, 2005):

$\displaystyle\mathcal{L}(\sigma_{\mathrm{int}},a,b,c)\propto\prod_{i}\frac{1}{\sqrt{\sigma_{\mathrm{int}}^{2}+\sigma_{y,i}^{2}+b^{2}\sigma_{x,i}^{2}}}\times\exp\left[-\frac{[y_{i}-y_{\mathrm{copula}}(x_{i},z_{i};a,b,c)]^{2}}{2\left(\sigma_{\mathrm{int}}^{2}+\sigma_{y,i}^{2}+b^{2}\sigma_{x,i}^{2}\right)}\right],$
(7)

where $\sigma_{x}$ and $\sigma_{y}$ are the uncertainties of $x$ and $y$,
respectively, and $\sigma_{\mathrm{int}}$ is the intrinsic uncertainty of
GRBs. From the error propagation equation, we find that $\sigma_{y}$ and
$\sigma_{x}$ can be derived from Eqs (3) and (4) as follows

$\displaystyle\sigma_{y}=\frac{1}{\ln
10}\frac{\sigma_{E_{\mathrm{iso}}}}{E_{\mathrm{iso}}},~{}\sigma_{x}=\frac{1}{\ln
10}\frac{\sigma_{E_{p}}}{E_{p}}$ (8)

with

$\displaystyle\sigma_{E_{\mathrm{iso}}}=4\pi
d_{L}^{2}\sigma_{S_{\mathrm{bolo}}}(1+z)^{-1}.$ (9)

Here $\sigma_{E_{p}}$ and $\sigma_{S_{\mathrm{bolo}}}$ are available in the
observations of GRBs. In our analysis, the CosmoMC code is used111The CosmoMC
code is available at https://cosmologist.info/cosmomc.. The obtained values of
$\sigma_{\mathrm{int}}$ and coefficients in Eqs. (1) and (6) are shown in Tab.
1. From it, one can see that the differences of $\sigma_{\mathrm{int}}$ and
$a$ between the standard and improved Amati correlations are negligible, while
the difference of $b$ is very significant. Furthermore, we find that the
redshift-dependent correlation is favored since the value of $c$ deviates from
zero at the $1\sigma$ confidence level (CL).

Table 1: Constraints on the standard and improved Amati correlations from low-
redshift calibration

Correlations | $\sigma_{\mathrm{int}}$ | $a$ | $b$ | $c$ | $-2\ln\mathcal{L}$
---|---|---|---|---|---
Standard Amati | 0.511(0.047) | 52.722(0.063) | 1.295(0.135) | - | 121.819
Improved Amati | 0.509(0.045) | 52.869(0.144) | 1.194(0.149) | -0.232(0.206) | 120.475

Note. — The unmarginalized best-fitted values with standard deviations of
standard and improved Amati correlations from 79 low-redshift GRBs.

Extrapolating the results from the low-redshift GRB data to the high-redshift
one, one can derive the GRB luminosity distance from observations and obtain
the corresponding distance modulus, which relates to the $d_{L}$ through

$\displaystyle\mu(z)=m-M=25+5\log\frac{d_{L}(z)}{\mathrm{Mpc}},$ (10)

where $m$ and $M$ denote the apparent and absolute magnitudes, respectively.
The uncertainty of the distance modulus is propagated from the uncertainties
of $S_{\mathrm{bolo}}$, $E_{\mathrm{iso}}$ and $E_{p}$:

$\displaystyle\sigma_{\mu}^{2}=\left(\frac{5}{2}\sigma_{\log\frac{E_{\mathrm{iso}}}{1\mathrm{erg}}}\right)^{2}+\left(\frac{5}{2\ln
10}\frac{\sigma_{S_{\mathrm{bolo}}}}{S_{\mathrm{bolo}}}\right)^{2},$ (11)

where

$\displaystyle\sigma_{\log\frac{E_{\mathrm{iso}}}{1\mathrm{erg}}}^{2}$
$\displaystyle=$ $\displaystyle\sigma_{\mathrm{int}}^{2}+\left(\frac{b}{\ln
10}\frac{\sigma_{E_{p}}}{E_{p}}\right)^{2}+\sum_{i=1}^{4}\left(\frac{\partial
y_{\mathrm{copula}}(x;\bm{\theta}_{c})}{\partial\theta_{i}}\right)^{2}C_{ii}$
(12) $\displaystyle+$ $\displaystyle
2\sum_{i=1}^{4}\sum_{j=i+1}^{4}\left(\frac{\partial
y_{\mathrm{copula}}(x;\bm{\theta}_{c})}{\partial\theta_{i}}\frac{\partial
y_{\mathrm{copula}}(x;\bm{\theta}_{c})}{\partial\theta_{j}}\right)C_{ij}.$

Here $\bm{\theta}_{c}=\left\\{\sigma_{\mathrm{int}},a,b,c\right\\}$, and
$C_{ij}$ is the covariance matrix of these fitted coefficients. We construct
the Hubble diagram of the GRBs and show it in Fig. (2a). The differences
between $\mu_{\mathrm{copula}}$ and $\mu_{\mathrm{Amati}}$ are shown in Fig.
(2b), which indicates that in the high-redshift region, the GRB distance
modulus from the improved Amati correlation is larger apparently than that
from the standard one.

Figure 2:  The left panel is the Hubble diagram of 220 long GRBs calibrated
from the standard and improved Amati correlations ($y_{\mathrm{Amati}}$ and
$y_{\mathrm{copula}}$), respectively. The red dashed line denotes $z=1.4$. The
right panel shows the differences between $\mu_{\mathrm{copula}}$ and
$\mu_{\mathrm{Amati}}$
($\Delta\mu=\mu_{\mathrm{copula}}-\mu_{\mathrm{Amati}}$).

##  3 Constraints on cosmological models

The distance modulus of the A220 GRB data set can be used to constrain
cosmological models by minimizing the $\chi^{2}$:

$\displaystyle\chi^{2}_{\mathrm{GRB}}=\sum_{i=1}^{N}\left[\frac{\mu_{\mathrm{GRB}}(z_{i})-\mu_{\mathrm{th}}(z_{i};\bm{p},\mu_{0})}{\sigma_{\mu_{i}}}\right]^{2},$
(13)

where $N=220$ is the number of the GRB data, $\mu_{\mathrm{GRB}}(z_{i})$ is
the distance modulus of the GRB data at redshift $z_{i}$ and
$\mu_{\mathrm{th}}$ denotes the theoretical value of the distance modulus,
which is given by a cosmological model with $\bm{p}$ representing the model
parameters. In Eq. (13), $\mu_{0}=25-5\log H_{0}$, which is a nuisance
parameter and is marginalized here by using the analytical method given in
(Nesseris & Perivolaropoulos, 2004). Then, the $\chi^{2}_{\mathrm{GRB}}$ is
modified to be

$\displaystyle\tilde{\chi}^{2}_{\mathrm{GRB}}=A-\frac{B^{2}}{C},$ (14)

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\left[\frac{\mu_{\mathrm{GRB}}(z_{i})-\mu_{\mathrm{th}}(z_{i};\bm{p},\mu_{0}=0)}{\sigma_{\mu_{i}}}\right]^{2},$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\frac{\mu_{\mathrm{GRB}}(z_{i})-\mu_{\mathrm{th}}(z_{i};\bm{p},\mu_{0}=0)}{\left(\sigma_{\mu_{i}}\right)^{2}},$
$\displaystyle C$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\left(\frac{1}{\sigma_{\mu_{i}}}\right)^{2}.$ (15)

Except the GRB data, we also consider 31 Hubble parameter measurements (Stern
et al., 2010; Moresco et al., 2012, 2016; Zhang et al., 2014; Moresco, 2015;
Ratsimbazafy et al., 2017; Ryan et al., 2018) spanning the redshift from 0.07
to 1.965, which are determined by the cosmic chronometric technique (Loeb,
1998; Jimenez & Loeb, 2002). For the $H(z)$ data set, the minimization of
$\chi^{2}$ method is also applicable:

$\displaystyle\chi^{2}_{H(z)}=\sum_{i=1}^{N}\left[\frac{H_{\mathrm{obs}}(z_{i})-H_{\mathrm{th}}(z_{i};\bm{p},H_{0})}{\sigma_{H_{i}}^{\mathrm{obs}}}\right]^{2}.$
(16)

Here $H_{\mathrm{obs}}(z_{i})$ is the Hubble parameter measurement at redshift
$z_{i}$ and $H_{\mathrm{th}}$ is the theoretical value of the Hubble
parameter. The constraints on cosmological models from the $\mbox{GRB}+H(z)$
data can be obtained by minimizing

$\displaystyle\chi^{2}_{\mathrm{total}}=\tilde{\chi}^{2}_{\mathrm{GRB}}+\chi^{2}_{H(z)}.$
(17)

We consider two different cosmological models: $\Lambda$CDM and $w$CDM. The
Hubble parameter $H(z)$ of the $w$CDM as a function of redshift $z$ has the
form

$\displaystyle\frac{H(z;\bm{p})^{2}}{H_{0}^{2}}=E^{2}(z;\bm{p})=\Omega_{\mathrm{m0}}(1+z)^{3}+(1-\Omega_{\mathrm{m0}})(1+z)^{3(1+w)},$
(18)

which reduces to that of the $\Lambda$CDM when $w=-1$. Thus, we have
$\bm{p}\equiv\\{\Omega_{\mathrm{m0}}\\}$ for the $\Lambda$CDM and
$\bm{p}\equiv\\{w,\Omega_{\mathrm{m0}}\\}$ for the $w$CDM. From Eq. (18), one
can achieve the luminosity distance $d_{L}(z,\bm{p})$

$\displaystyle
d_{L}(z;\bm{p})=\frac{(1+z)}{H_{0}}\int_{0}^{z}\frac{\mathrm{d}\tilde{z}}{E(\tilde{z};\bm{p})}.$
(19)

The probability density distributions of the model parameters of two
cosmological models are shown in Fig. (3), and the marginalized mean values
with 68% CL of these parameters are summarized in Tab. 2. For a comparison, we
consider both the standard and improved Amati correlations in standardizing
the GRB. For the standard Amati correlation, it is easy to see that the GRB
data only give a lower bound limit on the $\Omega_{\mathrm{m0}}$, which is
similar to the results obtained in (Khadka et al., 2021) with the simultaneous
fitting method. While, the GRB data from the improved Amati correlation can
constrain $\Omega_{\mathrm{m0}}$ more effectively, and the mean values of
$\Omega_{\mathrm{m0}}$ are $0.308^{+0.066}_{-0.230}$ and
$0.307^{+0.057}_{-0.290}$ in the $\Lambda$CDM and $w$CDM models, respectively.
These results are consistent very well with those given by other current
popular observation data including BAO, CMB and so on (Aubourg et al., 2015;
Scolnic et al., 2018; Planck Collaboration, 2020). Furthermore, we find that
for the $w$CDM model the GRB data from the standard Amati correlation only
give an upper bound limit on $w$ at the $68\%$ CL, but those from the improved
Amati correlation can give a tighter constraint.

When the $H(z)$ data are added in our analysis, the $\mbox{GRB}+H(z)$ give
tighter constraints $\Omega_{\mathrm{m0}}$ and $w$ than those from the GRB
only. We find that the GRB data from improved Amati correlation always favors
a smaller $\Omega_{\mathrm{m0}}$ than that from the standard Amati
correlation. The $H(z)+$GRB from the improved Amati correlation can give a
slightly tighter constraint on $w$ than the $H(z)+$GRB from the standard Amati
correlation. We also investigate the limit on $H_{0}$ from the $H(z)+$GRB. The
results are shown in Fig. (4) and summarized in Tab. 2. It is easy to see that
two different correlations provide slightly different constraints on the
Hubble constant, but the marginalized mean values seem to be close to the one
($67.4\pm 0.5~{}\mathrm{km~{}s^{-1}Mpc^{-1}}$) from the Planck 2018 CMB
observations (Planck Collaboration, 2020).

##  4 Conclusions

An improved Amati correlation was constructed from the Gaussian copula
function recently by us (Liu et al., 2022). In this paper, we compare the
constraints on cosmological models from GRB with the standard and improved
Amati correlations. To obtain model-independently the GRB Hubble diagram, we
use the Pantheon SN Ia data to calibrate the GRB data, and find that a
redshift evolutionary correlation is favored slightly since the constant $c$,
which is the coefficient of the redshift dependent term in the improved Amati
correlation, deviates from the zero at the $1\sigma$ CL. The distance modulus
of the GRB from the improved Amati correlation is apparently larger than that
from the standard Amati correlation at the high-redshift region ($z>1.4$).
Thus, when the effect of the redshift evolution is neglected, the distance
modulus of the GRBs will be underestimated at the high redshift region.

Using the GRBs to constrain the $\Lambda$CDM and $w$CDM models, we find that
the data based on the standard Amati correlation only give a lower bound limit
on $\Omega_{\mathrm{m0}}$ and an upper bound one on $w$, while those from the
improved Amati correlation can constrain $\Omega_{\mathrm{m0}}$ and $w$ more
tightly. The mean values of $\Omega_{\mathrm{m0}}$ with the $68\%$ CL are
$0.308^{+0.066}_{-0.230}$ and $0.307^{+0.057}_{-0.290}$ in the $\Lambda$CDM
and $w$CDM models, respectively, which are consistent very well with those
given by other current popular observational data including BAO, CMB and so on
(Aubourg et al., 2015; Scolnic et al., 2018; Planck Collaboration, 2020). When
the $H(z)$ data are added together to constrain $\Lambda$CDM and $w$CDM,
tighter limits on $\Omega_{\mathrm{m0}}$ and $w$ are obtained. Furthermore, a
constraint on $H_{0}$ is achieved. We find that two different correlations
provide marginally different $H_{0}$ results but the marginalized mean values
seem to be close to that from the Planck 2018 CMB observations (Planck
Collaboration, 2020).

Figure 3: Constraints on the $\Lambda$CDM and the $w$CDM model from the GRB
data and the $\mbox{GRB}+H(z)$ data.

Figure 4: Constraints on $H_{0}$ from the $\mbox{GRB}+H(z)$ data. The green
and red lines represent the results from the standard and improved Amati
correlations, respectively.  Table 2: Constraints on the $\Lambda$CDM and
$w$CDM from the GRB data and the $\mbox{GRB}+H(z)$ data.

Correlations | Models | Data Sets | $H_{0}$ | $\Omega_{\mathrm{m0}}$ | $w$
---|---|---|---|---|---
|  $\Lambda$CDM | A220 | - | $>0.651$ | -
$y_{\mathrm{Amati}}$ |  | A220+$H(z)$ | $66.195(3.105)^{+3.453}_{-3.139}$ | $0.370(0.067)^{+0.050}_{-0.079}$ | -
|  $w$CDM | A220 | - | $>0.573$ | $<-0.214$
|  | A220+$H(z)$ | $70.176(7.701)^{+5.161}_{-9.516}$ | $0.333(0.085)^{+0.076}_{-0.068}$ | $-1.400(0.655)^{+0.864}_{-0.496}$
|  $\Lambda$CDM | A220 | - | $0.308(0.192)^{+0.066}_{-0.230}$ | -
$y_{\mathrm{copula}}$ |  | A220+$H(z)$ | $68.543(2.954)^{+2.861}_{-2.933}$ | $0.314(0.056)^{+0.046}_{-0.063}$ | -
|  $w$CDM | A220 | - | $0.307(0.219)^{+0.057}_{-0.290}$ | $-1.244(0.781)^{+1.016}_{-0.463}$
|  | A220+$H(z)$ | $70.857(7.644)^{+4.310}_{-9.623}$ | $0.270(0.088)^{+0.090}_{-0.052}$ | $-1.203(0.625)^{+0.836}_{-0.327}$

Note. — The marginalized mean values, the standard deviations, and the $68\%$
CL.

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 Binned distance modulus of Pantheon SN Ia

Following the steps given in (Betoule et al., 2014), we use a piecewise linear
function to approximate the apparent magnitudes $m$ in the Pantheon SN Ia
data, which is defined on each segment $z_{i}<z<z_{i+1}$ as

$\displaystyle\bar{m}(z)=(1-\alpha)~{}m_{i}+\alpha~{}m_{i+1}$ (A1)

with $\alpha=\log(z/z_{i})/\log(z_{i+1}/z_{i})$, where $m_{i}$ is the apparent
magnitude at $z_{i}$. For the 1048 Pantheon SN Ia data points, we segment 35
bins with 36 $\log$-spaced control points in the redshift region $0.01<z<2.3$.
To determine the value of $m_{i}$ at each control point $z_{i}$, the
minimizing $\chi^{2}$ method is used:

$\displaystyle\chi^{2}=[\bm{\hat{m}}(z)-\bar{m}(z)]^{\dagger}C^{-1}[\bm{\hat{m}}(z)-\bar{m}(z)].$
(A2)

Here $C$ is the covariance matrix with $1048\times 1048$ elements, and
$\bm{\hat{m}}$ is a one-dimensional array with 1048 observations in the
Pantheon SN Ia sample. The results of 36 $\log$-spaced control points are
shown in Tab. 3. Since only the apparent magnitudes are obtained, we will set
the absolute magnitude to be $M=-19.36$ (Gómez-Valent, 2022) in our analysis
to obtain the luminosity distance of GRBs. If a different value of the
absolute magnitude is chosen, it will change the value of coefficient $a$, but
dose not affect the constraints on cosmological models.

Table 3: Binned apparent magnitudes of the Pantheon SN Ia data.

$z_{i}$ | $m_{i}$ | $\sigma_{m_{i}}$ | $z_{i}$ | $m_{i}$ | $\sigma_{m_{i}}$ | $z_{i}$ | $m_{i}$ | $\sigma_{m_{i}}$
---|---|---|---|---|---|---|---|---
0.010 | 13.912 | 0.143 | 0.065 | 17.966 | 0.049 | 0.416 | 22.469 | 0.028
0.012 | 14.132 | 0.131 | 0.075 | 18.293 | 0.051 | 0.486 | 22.836 | 0.030
0.014 | 14.604 | 0.088 | 0.088 | 18.696 | 0.040 | 0.568 | 23.272 | 0.030
0.016 | 14.751 | 0.059 | 0.103 | 19.094 | 0.032 | 0.664 | 23.632 | 0.041
0.019 | 15.208 | 0.076 | 0.120 | 19.386 | 0.026 | 0.775 | 24.081 | 0.036
0.022 | 15.483 | 0.053 | 0.140 | 19.825 | 0.025 | 0.905 | 24.503 | 0.041
0.025 | 15.839 | 0.041 | 0.164 | 20.079 | 0.025 | 1.058 | 24.893 | 0.073
0.030 | 16.205 | 0.041 | 0.192 | 20.530 | 0.022 | 1.235 | 25.432 | 0.117
0.035 | 16.550 | 0.036 | 0.224 | 20.881 | 0.023 | 1.443 | 25.570 | 0.150
0.040 | 16.863 | 0.047 | 0.261 | 21.231 | 0.020 | 1.686 | 26.246 | 0.224
0.047 | 17.234 | 0.050 | 0.305 | 21.664 | 0.021 | 1.969 | 26.130 | 0.295
0.055 | 17.536 | 0.044 | 0.356 | 22.042 | 0.022 | 2.300 | 26.971 | 0.297

Note. — The best-fitted value of binned apparent magnitude with standard
deviation at each control point $z_{i}$.

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