# Standardizing Dainotti-correlated gamma-ray bursts, and using them with
standardized Amati-correlated gamma-ray bursts to constrain cosmological model
parameters

Shulei Cao,${}^{\href https://orcid.org/0000-0003-2421-70711}$ Narayan
Khadka,${}^{\href https://orcid.org/0000-0001-5512-27161}$ Bharat
Ratra${}^{\href https://orcid.org/0000-0002-7307-07261}$
1Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan,
KS 66502, USA E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We show that each of the three Dainotti-correlated gamma-ray burst (GRB) data
sets recently compiled by Wang et al. and Hu et al., that together probe the
redshift range $0.35\leq z\leq 5.91$, obey cosmological-model-independent
Dainotti correlations and so are standardizable. We use these GRB data in
conjunction with the best currently-available Amati-correlated GRB data, that
probe $0.3399\leq z\leq 8.2$, to constrain cosmological model parameters. The
resulting cosmological constraints are weak, providing lower limits on the
non-relativistic matter density parameter, mildly favoring non-zero spatial
curvature, and largely consistent with currently accelerated cosmological
expansion as well as with constraints determined from better-established data.

###### keywords:

cosmological parameters – dark energy – cosmology: observations – gamma-ray
bursts

††pubyear: 2021††pagerange: Standardizing Dainotti-correlated gamma-ray
bursts, and using them with standardized Amati-correlated gamma-ray bursts to
constrain cosmological model parameters–Standardizing Dainotti-correlated
gamma-ray bursts, and using them with standardized Amati-correlated gamma-ray
bursts to constrain cosmological model parameters

##  1 Introduction

The observed currently accelerated cosmological expansion indicates that — if
general relativity provides an accurate description of gravitation on
cosmological scales — dark energy must contribute significantly to the current
cosmological energy budget. The simpler spatially-flat $\Lambda$CDM model
(Peebles, 1984) is consistent with this and other observations. Fits of this
model to most better-established cosmological data suggest that a time-
independent cosmological constant $(\Lambda)$ provides $\sim 70\%$ of the
current cosmological energy budget, non-relativistic cold dark matter (CDM)
provides $\sim 25\%$, and non-relativistic baryonic matter provides most of
the remaining $\sim 5\%$ (see, e.g. Farooq et al., 2017; Scolnic et al., 2018;
Planck Collaboration, 2020; eBOSS Collaboration, 2021). While the spatially-
flat $\Lambda$CDM model is consistent with most observations (see, e.g. Di
Valentino et al., 2021b; Perivolaropoulos & Skara, 2021), observational data
do not strongly rule out a little spatial curvature or dynamical dark energy.
In this paper, in addition to the spatially-flat $\Lambda$CDM model, we also
study spatially non-flat and dynamical dark energy models.

Observational astronomy now provides many measurements that can be used to
test cosmological models. Largely, these data are either at low or at high
redshift. So cosmological models are mostly tested at low and high redshifts,
remaining poorly tested in the intermediate redshift regime. The highest
redshift of the better-established low-redshift data, $\sim 2.3$, is reached
through baryon acoustic oscillation (BAO) observations; the high redshift
region, $z\sim 1100$, is probed by better-established cosmic microwave
background anisotropy data. Fits of these better-established data to
cosmological models provide mostly mutually consistent results. However, for a
better understanding of our Universe, it is necessary to also test
cosmological models in the intermediate redshift range of $2.3\lesssim
z\lesssim 1100$. Some progress has been achieved: methods that test
cosmological models in the intermediate redshift region include the use of H
ii starburst galaxy measurements which reach to $z\sim 2.4$ (Mania & Ratra,
2012; Chávez et al., 2014; González-Morán et al., 2019, 2021; Cao et al.,
2020, 2021a, 2021b; Johnson et al., 2021), quasar angular size measurements
which reach to $\sim 2.7$ (Cao et al., 2017; Ryan et al., 2019; Cao et al.,
2020, 2021b; Zheng et al., 2021; Lian et al., 2021), and quasar flux
measurements which reach to $\sim 7.5$ (Risaliti & Lusso, 2015, 2019; Khadka &
Ratra, 2020a, b, 2021a, 2021b; Yang et al., 2020; Lusso et al., 2020; Zhao &
Xia, 2021; Li et al., 2021; Lian et al., 2021; Rezaei et al., 2021; Luongo et
al., 2021).111Note that in the latest Lusso et al. (2020) quasar flux
compilation, their assumed UV–X-ray correlation is valid only to a much lower
redshift, $z\sim 1.5-1.7$, and so these quasars can be used to derive only
lower-$z$ cosmological constraints (Khadka & Ratra, 2021a, b).

Gamma-ray burst (GRB) measurements are another high redshift probe and reach
to $z\sim 8.2$ (Amati et al., 2008, 2019; Salvaterra et al., 2009; Tanvir et
al., 2009; Samushia & Ratra, 2010; Cardone et al., 2010; Dainotti et al.,
2013a; Wang et al., 2015, 2016; Dainotti & Del Vecchio, 2017; Fana Dirirsa et
al., 2019; Khadka & Ratra, 2020c; Demianski et al., 2021; Khadka et al.,
2021b; Luongo et al., 2021; Luongo & Muccino, 2021). While there are quite a
few Amati correlation long GRBs that have been used to constrain cosmological
parameters, currently only a smaller fraction of 118 such GRBs (hereafter
A118) that cover the redshift range $0.3399\leq z\leq 8.2$ (Khadka & Ratra,
2020c; Khadka et al., 2021a) are reliable enough to be used to constrain
cosmological parameters. To date, this is the lower-$z$ data set used to
constrain cosmological parameters that spans the widest range of redshifts.
These A118 data provide cosmological constraints which are consistent with
those obtained from the better-established cosmological probes but the GRB
constraints are significantly less restrictive. To obtain tighter cosmological
constraints using GRB data, we need to make use of more GRBs.

Recently Wang et al. (2021) and Hu et al. (2021) have compiled smaller GRB
data sets that together probe the redshift range $0.35\leq z\leq 5.91$. These
are GRBs whose plateau phase luminosity $L_{0}$ and spin-down characteristic
time $t_{b}$ are correlated through the Dainotti ($L_{0}-t_{b}$) correlation
(Dainotti et al., 2013b, 2017). This correlation between $L_{0}$ and $t_{b}$
allows one to use these GRBs for cosmological purposes. These GRBs can be
classified in two categories depending on whether the plateau phase is
dominated by magnetic dipole (MD) radiation or gravitational wave (GW)
emission (Wang et al., 2021; Hu et al., 2021). In this paper we use long and
short GRBs whose plateau phase is dominated by MD radiation (hereafter MD-
LGRBs and MD-SGRBs) and long GRBs whose plateau phase is dominated by GW
emission (hereafter GW-LGRBs). All three sets of GRBs obey the Dainotti
correlation but each set can have different correlation parameters. We use the
three individual GRB data sets, as well as some combinations of them, to
constrain cosmological model parameters and Dainotti correlation parameters
simultaneously.222The advantage of fitting cosmological and GRB correlation
parameters simultaneously is that the fitting process is free from the
circularity problem. More specifically, this procedure allows us to determine
whether the GRB correlation parameters depend on the assumed cosmological
model and so determine whether the GRBs are standardizable. We find that these
GRBs are standardizable, as was assumed in Wang et al. (2021) and Hu et al.
(2021). However, cosmological constraints obtained from these Dainotti
correlation GRB data sets are very weak.

When we combine the MD-LGRB or GW-LGRB data sets with the 115 non-overlapping
Amati correlation GRBs from the A118 data set, they slightly tighten the
constraints from the 115 Amati correlation GRBs, but not significantly so.
Each of the individual Amati or Dainotti correlation GRB data sets, as well as
combinations of these GRB data sets, mostly provide only lower limits on the
current value of the non-relativistic matter energy density parameter
$\Omega_{\rm m0}$ and the resulting cosmological parameter constraints are
mostly consistent with those obtained from better-established cosmological
data.

In this paper, we use a combination of Hubble parameter ($H(z)$) and BAO data,
$H(z)$ \+ BAO, results as a proxy for better-established data results, to
compare with our GRB data results. Qualitatively, results from the individual
GRB data sets, as well as those from combinations of GRB data sets, are
consistent with those from the $H(z)$ \+ BAO data which favor $\Omega_{\rm
m0}\sim 0.3$, but there are a few combinations of GRB data sets with
constraints on $\Omega_{\rm m0}$ being more than $2\sigma$ away from 0.3 in
the $\Lambda$CDM models.

This paper is structured as follows. In Sec. 2 we summarize the cosmological
models we use. In Sec. 3 we describe the data sets we analyze. In Sec. 4 we
summarize our analyses techniques. In Sec. 5 we present our results. We
conclude in Sec. 6.

##  2 Cosmological models

In this paper we derive cosmological parameter constraints in six different
general-relativity cosmological dark energy models. Three of them assume
spatially-flat geometry while the other three allow non-flat spatial
geometry.333For recent discussions of constraints on non-flat models see Chen
et al. (2016), Rana et al. (2017), Ooba et al. (2018a, c), Yu et al. (2018),
Park & Ratra (2019a, c), Wei (2018), DES Collaboration (2019), Li et al.
(2020), Handley (2019), Efstathiou & Gratton (2020), Di Valentino et al.
(2021a), Velasquez-Toribio & Fabris (2020), Vagnozzi et al. (2021a); Vagnozzi
et al. (2021b), KiDS Collaboration (2021), Arjona & Nesseris (2021), Dhawan et
al. (2021), and references therein. These models are used to predict the
luminosity distance for a GRB at a given redshift. For this purpose the
fundamental quantity is the expansion rate of the Universe, or the Hubble
parameter, $H(z)$, a function of cosmological parameters and redshift.

The Hubble parameter in all six models we use can be written as

$H(z)=H_{0}\sqrt{\Omega_{\rm m0}(1+z)^{3}+\Omega_{\rm k0}(1+z)^{2}+\Omega_{\rm
DE}(z)},$ (1)

where $H_{0}$ is the Hubble constant and $\Omega_{\rm k0}$ is the current
value of the spatial curvature energy density parameter. In analyses of the
$H(z)$ \+ BAO data set we express $\Omega_{\rm m0}$ in terms of the current
value of the baryonic matter energy density parameter $(\Omega_{b})$ and the
current value of the cold dark matter energy density parameter $(\Omega_{c})$
through the equation $\Omega_{\rm m0}$ = $\Omega_{b}$ \+ $\Omega_{c}$. In four
of the six models, the dark energy density parameter term is expressed as
$\Omega_{\rm DE}(z)$ = $\Omega_{\rm DE0}(1+z)^{1+w_{\rm DE}}$, where
$\Omega_{\rm DE0}$ is the current value of the dark energy density parameter
and $w_{\rm DE}$ is the dark energy equation of state parameter.

In the $\Lambda$CDM models $w_{\rm DE}=-1$ and $\Omega_{\rm DE}$ =
$\Omega_{\rm DE0}$ = $\Omega_{\Lambda}$ is the cosmological constant dark
energy density parameter and is time-independent. The current values of the
three energy density parameters are related by the energy budget equation,
$\Omega_{\rm m0}+\Omega_{\rm k0}+\Omega_{\Lambda}=1$. In the spatially-flat
$\Lambda$CDM model we choose to constrain $\Omega_{\rm m0}$ and $H_{0}$ while
in the spatially non-flat $\Lambda$CDM model we constrain $\Omega_{\rm m0}$,
$\Omega_{\rm k0}$, and $H_{0}$. For analyses which involve $H(z)$ \+ BAO data,
instead of $\Omega_{\rm m0}$ we constrain $\Omega_{b}h^{2}$ and
$\Omega_{c}h^{2}$; here $h$ is the Hubble constant in units of 100 km s-1
Mpc-1.

In the XCDM parametrizations the equation of state for the dynamical dark
energy X-fluid is $P_{\rm X}=w_{\rm X}\rho_{\rm X}$ where $P_{\rm X}$,
$\rho_{\rm X}$, and $w_{\rm X}$ are the pressure, energy density, and equation
of state parameter for the dynamical dark energy X-fluid, and $\Omega_{\rm
DE0}$ = $\Omega_{\rm X0}$ is the current value of the X-fluid dynamical dark
energy density parameter. In this case the current values of the three energy
density parameters are related by $\Omega_{\rm m0}+\Omega_{\rm k0}+\Omega_{\rm
X0}=1$. The X-fluid energy density decreases with time when $w_{\rm X}$
satisfies the conditions $-1<w_{\rm X}<0$. In the spatially-flat XCDM
parametrization we choose to constrain $\Omega_{\rm m0}$, $w_{\rm X}$, and
$H_{0}$ while in the non-flat XCDM parametrization we constrain $\Omega_{\rm
m0}$, $\Omega_{\rm k0}$, $w_{\rm X}$, and $H_{0}$. For analyses which involve
$H(z)$ \+ BAO data, instead of $\Omega_{\rm m0}$ we constrain
$\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$. When $w_{\rm X}=-1$ the XCDM
parametrizations reduce to the $\Lambda$CDM models.

In the $\phi$CDM models the dynamical dark energy is a scalar field $\phi$
(Peebles & Ratra, 1988; Ratra & Peebles, 1988; Pavlov et al.,
2013).444Discussions of observational constraints on the $\phi$CDM model can
be traced through Chen et al. (2017), Zhai et al. (2017), Solà Peracaula et
al. (2018, 2019), Ooba et al. (2018b, 2019), Park & Ratra (2018, 2019b, 2020),
Sangwan et al. (2018), Singh et al. (2019), Ureña-López & Roy (2020), Sinha &
Banerjee (2021), Cao et al. (2021c), Khadka et al. (2021a), Xu et al. (2021),
de Cruz Perez et al. (2021) and references therein. In this model $\Omega_{\rm
DE}$ = $\Omega_{\phi}(z,\alpha)$, the scalar field dynamical dark energy
density parameter, is determined by the scalar field potential energy density,
for which we assume an inverse power law form,

$V(\phi)=\frac{1}{2}\kappa m_{p}^{2}\phi^{-\alpha}.$ (2)

Here $m_{p}$ is the Planck mass, $\alpha$ is a positive parameter, and
$\kappa$ is a constant whose value is determined by using the shooting method
to ensure that the current energy budget equation $\Omega_{\rm m0}+\Omega_{\rm
k0}+\Omega_{\phi}(z=0,\alpha)=1$ is satisfied.

In the $\phi$CDM models the dynamics of a spatially homogeneous scalar field
$\phi$ is governed by two coupled non-linear ordinary differential equations.
The first is the dark energy scalar field equation of motion

$\ddot{\phi}+3\bigg{(}\frac{\dot{a}}{a}\bigg{)}\dot{\phi}-\frac{1}{2}\alpha\kappa
m_{p}^{2}\phi^{-\alpha-1}=0,$ (3)

and the second is the Friedmann equation

$\bigg{(}\frac{\dot{a}}{a}\bigg{)}^{2}=\frac{8\pi}{3m_{p}^{2}}(\rho_{\rm
m}+\rho_{\phi})-\frac{k}{a^{2}},$ (4)

where $a$ is the scale factor and an overdot denotes a time derivative. In eq.
(4), ${-k}/{a^{2}}$ is the spatial curvature term with $\Omega_{\rm k0}=0$,
$>0$, $<0$ corresponding to $k=0$, $-1$, $+1$, respectively, and $\rho_{\rm
m}$ and $\rho_{\phi}$ are the non-relativistic matter and scalar field energy
densities where

$\rho_{\phi}=\frac{m_{p}^{2}}{32\pi}\bigg{(}\dot{\phi}^{2}+\kappa
m_{p}^{2}\phi^{-\alpha}\bigg{)}.$ (5)

By solving eqs. (3) and (4) numerically we can compute $\rho_{\phi}$ and then
compute $\Omega_{\phi}(z,\alpha)$ by using the expression

$\Omega_{\phi}(z,\alpha)=\frac{8\pi\rho_{\phi}}{3m^{2}_{p}H^{2}_{0}}.$ (6)

In the spatially-flat $\phi$CDM model we choose to constrain $\Omega_{\rm
m0}$, $\alpha$, and $H_{0}$ while in the non-flat $\phi$CDM model we constrain
$\Omega_{\rm m0}$, $\Omega_{\rm k0}$, $\alpha$, and $H_{0}$. For analyses
which involve $H(z)$ \+ BAO data, instead of $\Omega_{\rm m0}$ we constrain
$\Omega_{b}h^{2}$ and $\Omega_{c}h^{2}$. When $\alpha=0$ the $\phi$CDM models
reduce to the $\Lambda$CDM models.

##  3 Data

In this paper, we analyze four different GRB data sets as well as some
combinations of these data sets. We also use a joint $H(z)$ \+ BAO data set.
These data sets are summarized in Table 1 and described in what follows.555In
this table and elsewhere, for compactness, we sometimes use ML, MS, and GL as
abbreviations for the MD-LGRB, MD-SGRB, and GW-LGRB data sets compiled by Wang
et al. (2021) and Hu et al. (2021).

  * MD-LGRB sample. This includes 31 long GRBs, with burst duration longer than 2 seconds, listed in Table 1 of Wang et al. (2021). For this data set, measured quantities for a GRB are redshift $z$, X-ray flux $F_{0}$, characteristic time scale $t_{b}$, and spectral index during the plateau phase $\beta^{\prime}$.666ML, MS, and GL data error bars on $F_{0}$ and $t_{b}$ are mostly asymmetric. We symmetrize these error bars using the method applied in Wang et al. (2021) and Hu et al. (2021), with the symmetrized error bar $\sigma=\sqrt{(\sigma_{u}^{2}+\sigma_{d}^{2})/2}$, where $\sigma_{u}$ and $\sigma_{d}$ are the asymmetric upper and lower error bars. This sample probes the redshift range $1.45\leq z\leq 5.91$.

  * MD-SGRB sample. This includes 5 short GRBs, with burst duration shorter than 2 seconds, listed in Table 1 of Hu et al. (2021). For this data set, measured quantities for a GRB are the same as those for the MD-LGRB sample. This data set probes the redshift range $0.35\leq z\leq 2.6$.

  * GW-LGRB sample. This includes 24 long GRBs listed in Table 1 of Hu et al. (2021). For this data set, measured quantities for a GRB are the same as those for the MD-LGRB sample. This sample probes the redshift range $0.55\leq z\leq 4.81$.

  * A118 sample. This sample include 118 long GRBs listed in Table 7 of Khadka et al. (2021a). For this data set, measured quantities for a GRB are $z$, rest-frame spectral peak energy $E_{\rm p}$, and measured bolometric fluence $S_{\rm bolo}$, computed in the standard rest-frame energy band $1-10^{4}$ keV. This sample probes the redshift range $0.3399\leq z\leq 8.2$.

The A118 data and the MD-LGRB data sets have 3 common GRBs, GRB060526,
GRB081008, and GRB090516. We exclude these common GRBs from the A118 data set
to form the A115 data set for joint analyses with the MD-LGRB data set. There
are also 3 common GRBs between the A118 data set and the GW-LGRB data set,
GRB060206, GRB091029, and GRB131105A. We exclude these common GRBs from the
A118 data set to form the A115′ data set for joint analyses with the GW-LGRB
data set.

  * H(z) and BAO data. In addition to the GRB data, we also use 31 $H(z)$ and 11 BAO measurements. These $H(z)$ and BAO measurements probe the redshift range $0.07\leq z\leq 1.965$ and $0.0106\leq z\leq 2.33$, respectively. The $H(z)$ data are in Table 2 of Ryan et al. (2018) and the BAO data are in Table 1 of Khadka & Ratra (2020c). We use cosmological constraints from the better-established $H(z)$ \+ BAO data to compare with those obtained from the GRB data sets.

##  4 Data Analysis Methodology

Data set |  $N$ (Number of points) | Redshift range
---|---|---
ML | 31 | $1.45\leq z\leq 5.91$
MS | 5 | $0.35\leq z\leq 2.6$
GL | 24 | $0.55\leq z\leq 4.81$
MS + GL | 29 | $0.35\leq z\leq 4.81$
A118 | 118 | $0.3399\leq z\leq 8.2$
A115a | 115 | $0.3399\leq z\leq 8.2$
A115′b | 115 | $0.3399\leq z\leq 8.2$
$H(z)$ | 31 | $0.070\leq z\leq 1.965$
BAO | 11 | $0.38\leq z\leq 2.334$

  * a

Excluding from A118 those GRBs in common with MD-LGRB (GRB060526, GRB081008,
and GRB090516).

  * b

Excluding from A118 those GRBs in common with GW-LGRB (GRB060206, GRB091029,
and GRB131105A).

Table 1: Summary of data sets used.

Parameter |  | Prior
---|---|---
| Cosmological Parameters |
$H_{0}$ a |  | [None, None]
$\Omega_{b}h^{2}$ b |  | [0, 1]
$\Omega_{c}h^{2}$ c |  | [0, 1]
$\Omega_{\rm k0}$ |  | [-2, 2]
$\alpha$ |  | [0, 10]
$w_{\rm X}$ |  | [-5, 0.33]
| GRB Nuisance Parametersd |
$k$ |  | [-10, 10]
$b$ e |  | [0, 10]
$\sigma_{\rm int}$ |  | [0, 5]
$\beta$ |  | [0, 5]
$\gamma$ |  | [0, 300]

  * a

$\rm{km\ s^{-1}\ Mpc^{-1}}$. In the GRB alone cases, $H_{0}$ is set to be 70
$\rm{km\ s^{-1}\ Mpc^{-1}}$, while in the $H(z)$ \+ BAO case, the prior range
is irrelevant (unbounded).

  * b

In the GRB alone cases, $\Omega_{b}h^{2}$ is set to be 0.0245, i.e.
$\Omega_{b}=0.05$.

  * c

In the GRB alone cases, $\Omega_{c}\in[-0.05,0.95]$ to ensure $\Omega_{\rm
m0}\in[0,1]$.

  * d

Note that $k$, $b$, and $\sigma_{\rm int}$ of MD-LGRBs are different from
those of MD-SGRBs/GW-LGRBs, but with the same prior ranges.

  * e

$b<0$ values are possible for MD-SGRBs (due to fewer data points) but, as
discussed below, requiring $b\geq 0$ does not have significant consequences.

Table 2: Flat priors of the constrained parameters.

For GRBs which obey the Dainotti correlation the luminosity of the plateau
phase is (Dainotti et al., 2008; Dainotti et al., 2010, 2011)

$L_{0}=\frac{4\pi D_{L}^{2}F_{0}}{(1+z)^{1-\beta^{\prime}}},$ (7)

where $F_{0}$ is the GRB X-ray flux, $\beta^{\prime}$ is the spectral index in
the plateau phase, and $D_{L}$ is the luminosity distance.

$D_{L}$, as a function of redshift $z$ and cosmological parameters p, is given
by

$\frac{H_{0}\sqrt{\left|\Omega_{\rm
k0}\right|}D_{L}(z,\textbf{\emph{p}})}{c(1+z)}=\begin{cases}{\rm
sinh}\left[g(z,\textbf{\emph{p}})\right]&\text{if}\ \Omega_{\rm k0}>0,\\\
\vspace{1mm}g(z,\textbf{\emph{p}})&\text{if}\ \Omega_{\rm k0}=0,\\\
\vspace{1mm}{\rm sin}\left[g(z,\textbf{\emph{p}})\right]&\text{if}\
\Omega_{\rm k0}<0,\end{cases}$ (8)

where

$g(z,\textbf{\emph{p}})=H_{0}\sqrt{\left|\Omega_{\rm
k0}\right|}\int^{z}_{0}\frac{dz^{\prime}}{H(z^{\prime},\textbf{\emph{p}})},$
(9)

$c$ is the speed of light, and $H(z,\textbf{\emph{p}})$ is the Hubble
parameter that is described in Sec. 2 for each cosmological model.

For these GRBs the luminosity of the plateau phase $L_{0}$ and the
characteristic time scale $t_{b}$ are correlated through the Dainotti or
luminosity-time relation

$y\equiv\log\left(\frac{L_{0}}{10^{47}\
\mathrm{erg/s}}\right)=k\log\frac{t_{b}}{10^{3}(1+z)\ \mathrm{s}}+b\equiv
kx+b,$ (10)

where $\log=\log_{10}$ and the slope $k$ and the intercept $b$ are free
parameters to be determined from the data.

We predict $L_{0}$ as a function of cosmological parameters p at the redshift
of each GRB by using eqs. (7), (8), and (10). We then compare predicted and
measured values of $L_{0}$ by using the natural log of the likelihood function
(D’Agostini, 2005)

$\ln\mathcal{L}_{\rm GRB}=-\frac{1}{2}\Bigg{[}\chi^{2}_{\rm
GRB}+\sum^{N}_{i=1}\ln\left(2\pi(\sigma_{\rm
int}^{2}+\sigma_{{y_{i}}}^{2}+k^{2}\sigma_{{x_{i}}}^{2})\right)\Bigg{]},$ (11)

where

$\chi^{2}_{\rm
GRB}=\sum^{N}_{i=1}\bigg{[}\frac{(y_{i}-kx_{i}-b)^{2}}{(\sigma_{\rm
int}^{2}+\sigma_{{y_{i}}}^{2}+k^{2}\sigma_{{x_{i}}}^{2})}\bigg{]}.$ (12)

Here $N$ is the number of data points (e.g., for MD-LGRB $N=31$), and
$\sigma_{\rm int}$ is the intrinsic scatter parameter (which also contains the
unknown systematic uncertainty).

For GRBs which obey the Amati correlation the rest frame isotropic radiated
energy $E_{\rm iso}$ is

$E_{\rm iso}=\frac{4\pi D_{L}^{2}}{1+z}S_{\rm bolo},$ (13)

where $S_{\rm bolo}$ is the bolometric fluence.

For these GRBs the rest frame peak photon energy $E_{\rm p}$ and $E_{\rm iso}$
are correlated through the Amati (or $E_{\rm p}-E_{\rm iso}$) relation (Amati
et al., 2008; Amati et al., 2009)

$\log E_{\rm iso}=\gamma+\beta\log E_{\rm p},$ (14)

where the intercept $\gamma$ and the slope $\beta$ are free parameters to be
determined from the data. Note that the peak energy $E_{\rm p}=(1+z)E_{\rm
p,obs}$ where $E_{\rm p,obs}$ is the observed peak energy.

We predict $E_{\rm iso}$ as a function of cosmological parameters p at the
redshift of each GRB by using eqs. (8), (13), and (14). We then compare
predicted and measured values of $E_{\rm iso}$ by using the natural log of the
likelihood function (D’Agostini, 2005)

$\ln\mathcal{L}_{\rm A118}=-\frac{1}{2}\Bigg{[}\chi^{2}_{\rm
A118}+\sum^{N}_{i=1}\ln\left(2\pi(\sigma_{\rm
int}^{2}+\sigma_{{y^{\prime}_{i}}}^{2}+\beta^{2}\sigma_{{x^{\prime}_{i}}}^{2})\right)\Bigg{]},$
(15)

where

$\chi^{2}_{\rm A118}=\sum^{N}_{i=1}\bigg{[}\frac{(y^{\prime}_{i}-\beta
x^{\prime}_{i}-\gamma)^{2}}{(\sigma_{\rm
int}^{2}+\sigma_{{y^{\prime}_{i}}}^{2}+\beta^{2}\sigma_{{x^{\prime}_{i}}}^{2})}\bigg{]}.$
(16)

Here $x^{\prime}=\log(E_{\rm p}/{\rm keV})$,
$\sigma_{x^{\prime}}=\sigma_{E_{\rm p}}/(E_{\rm p}\ln 10)$,
$y^{\prime}=\log(E_{\rm iso}/{\rm erg})$, and $\sigma_{\rm int}$ is the
intrinsic scatter parameter, which also contains the unknown systematic
uncertainty.

The $H(z)$ \+ BAO data analyses follow the method described in Sec. 4 of
Khadka & Ratra (2021a).

We maximize the likelihood function using the Markov chain Monte Carlo (MCMC)
method as implemented in the MontePython code (Brinckmann & Lesgourgues, 2019)
and determine the best-fitting and posterior mean values and the corresponding
uncertainties for all free parameters. We assure convergence of the MCMC
chains for each free parameter from the Gelman-Rubin criterion ($R-1<0.05$).
Flat priors used for the free parameters are given in Table 2.

The Akaike Information Criterion ($AIC$) and the Bayesian Information
Criterion ($BIC$) are used to compare the goodness of fit of models with
different numbers of parameters. These are

$AIC=-2\ln\mathcal{L}_{\rm max}+2n,$ (17)

and

$BIC=-2\ln\mathcal{L}_{\rm max}+n\ln N.$ (18)

In these equations, $\mathcal{L}_{\rm max}$ is the maximum value of the
relevant likelihood function and $n$ is the number of free parameters of the
model under consideration.

##  5 Results

###  5.1 Constraints from ML, MS, and GL data

Table 3: One-dimensional marginalized posterior means and 68.27% limits of the
Dainotti correlation parameters for the ML, GL, and MS data sets using the
flat $\Lambda$CDM model with $\Omega_{\rm m0}=0.3$ and $H_{0}=70$ $\rm{km\
s^{-1}\ Mpc^{-1}}$, and comparison with the results given in Wang et al.
(2021) and Hu et al. (2021).

Data set | Source | $k$ | $b$ | $\sigma_{\rm int}$
---|---|---|---|---
| a | $-1.02^{+0.09}_{-0.08}$ | $1.72^{+0.07}_{-0.07}$ | –
ML | b | $-1.026\pm 0.085$ | $1.726\pm 0.074$ | $0.303^{+0.032}_{-0.050}$
| c | $-1.026\pm 0.086$ | $1.726\pm 0.074$ | $0.303^{+0.032}_{-0.050}$
| d | $-1.77^{+0.20}_{-0.20}$ | $0.66^{+0.01}_{-0.01}$ | $0.42^{+0.08}_{-0.06}$
GL | b | $-1.753^{+0.187}_{-0.208}$ | $0.642^{+0.100}_{-0.071}$ | $0.428^{+0.053}_{-0.086}$
| c | $-1.769\pm 0.205$ | $0.656\pm 0.094$ | $0.431^{+0.054}_{-0.088}$
| d | $-1.38^{+0.17}_{-0.19}$ | $0.33^{+0.17}_{-0.16}$ | $0.35^{+0.20}_{-0.12}$
MS | b | $-1.381^{+0.209}_{-0.213}$ | $0.327^{+0.195}_{-0.189}$ | $0.420^{+0.086}_{-0.242}$
| c | $-1.397^{+0.247}_{-0.241}$ | $0.354^{+0.195}_{-0.242}$ | $0.525^{+0.044}_{-0.358}$

  * a

Results from Wang et al. (2021) with the prior ranges of the parameters being
$k\in(-1.3,-0.75)$, $b\in(1.4,2.0)$, and $\sigma_{\rm int}\in(0.1,0.6)$ for
ML.

  * b

Our results with the same prior ranges of the parameters as Wang et al. (2021)
or Hu et al. (2021).

  * c

Our results with wider prior ranges of the parameters $k\in[-10,10]$,
$b\in[0,10]$ ($b\in[-0.5,10]$), and $\sigma_{\rm int}\in[0,3]$ for ML and GL
(MS).

  * d

Results from Hu et al. (2021) with the prior ranges of the parameters being
$k\in(-2.2,-1)$, $b\in(0.1,0.8)$, and $\sigma_{\rm int}\in(0.01,0.8)$ for GL
and being $k\in(-2.1,-0.55)$, $b\in(-0.5,1.0)$, and $\sigma_{\rm
int}\in(0.01,1)$ for MS.

In Table 3 we list Dainotti correlation parameters computed using the ML, GL,
and MS data sets. These are computed in the flat $\Lambda$CDM model with
$\Omega_{\rm m0}=0.3$ and $H_{0}=70$ $\rm{km\ s^{-1}\ Mpc^{-1}}$, the same
model and parameter values used in Wang et al. (2021) and Hu et al. (2021).
The first line of parameter values in each of the three subpanels of Table 3
are taken from these papers.777Wang et al. (2021) do not list a value for
$\sigma_{\rm int}$ in the ML case. To compare to these results, we used emcee
(Foreman-Mackey et al., 2013) to compute the values listed in the second and
third lines of each subpanel. Comparing the first and second lines in each
subpanel, we find that they are consistent, except: i) for the GL case our $b$
uncertainties are larger than those of Hu et al. (2021); and, ii) for the MS
case we have larger $b$ and $k$ error bars and a larger central value of
$\sigma_{\rm int}$ than those of Hu et al. (2021), but they agree within
1$\sigma$. In the third line of each subpanel we list results obtained
assuming wider prior ranges of the parameters. We find that the ML results do
not change, the GL results are shifted closer to those of Hu et al. (2021),
except for the values of $\sigma_{\rm int}$, and the MS results are shifted
away from those of Hu et al. (2021) with larger error bars, especially for
$\sigma_{\rm int}$.

(a) MD-LGRB (b) MD-SGRB

(c) GW-LGRB and MD-SGRB (d) MD-SGRB + GW-LGRB

Figure 1: $L_{0}-t_{b}$ correlations for MD-LGRB, MD-SGRB, GW-LGRB, and MD-
SGRB + GW-LGRB data using the flat $\Lambda$CDM model. The MD-LGRB, MD-SGRB,
and GW-LGRB data with error bars are shown in black, dark red, and dark green,
respectively. The solid lines are the $L_{0}-t_{b}$ correlations with
posterior mean values for the slopes and intercepts listed in Table 5, for MD-
LGRB (blue), MD-SGRB (red), GW-LGRB (green), and MD-SGRB + GW-LGRB (purple)
data. The $1\sigma$ and $3\sigma$ confidence regions are the dark and light
colored shaded regions with the uncertainties propagated from those of $k$ and
$b$ (without considering $\sigma_{\rm int}$).

We now record and discuss results when these data sets are used to jointly
constrain the Dainotti parameters and the cosmological parameters of the six
spatially-flat and non-flat dark energy cosmological models. Figure 1 shows
the flat $\Lambda$CDM Dainotti correlations for the ML, MS, GL, and MS + GL
data sets. The unmarginalized best-fitting results and the one-dimensional
(1D) posterior mean values and uncertainties are reported in Tables 4 and 5,
respectively. The corresponding posterior 1D probability distributions and
two-dimensional (2D) confidence regions of these parameters are shown in Figs.
2–4, in blue (ML), gray (MS), green (GL), pink (ML + MS), violet (ML + GL),
orange (MS + GL), and red ($H(z)$ \+ BAO, as a baseline). Note that $H_{0}=70$
$\rm{km\ s^{-1}\ Mpc^{-1}}$ and $\Omega_{b}=0.05$ are applied in the GRB
cases.

Model | Data set | $\Omega_{b}h^{2}$ | $\Omega_{c}h^{2}$ | $\Omega_{\mathrm{m0}}$ | $\Omega_{\mathrm{k0}}$ | $w_{\mathrm{X}}$ | $\alpha$ |  $H_{0}$a | $\sigma_{\mathrm{int,\,\textsc{ml}}}$ | $b_{\mathrm{\textsc{ml}}}$ | $k_{\mathrm{\textsc{ml}}}$ | $\sigma_{\mathrm{int,\,\textsc{ms}}}$ | $b_{\mathrm{\textsc{ms}}}$ | $k_{\mathrm{\textsc{ms}}}$ | $\sigma_{\mathrm{int,\,\textsc{gl}}}$ | $b_{\mathrm{\textsc{gl}}}$ | $k_{\mathrm{\textsc{gl}}}$ | $\sigma_{\mathrm{int,\,\textsc{ms+gl}}}$ | $b_{\mathrm{\textsc{ms+gl}}}$ | $k_{\mathrm{\textsc{ms+gl}}}$ | $-2\ln\mathcal{L}_{\mathrm{max}}$ | $AIC$ | $BIC$ | $\Delta AIC$ | $\Delta BIC$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  $H(z)$ \+ BAO | 0.0239 | 0.1187 | 0.298 | – | – | – | 69.13 | – | – | – | – | – | – | – | – | – | – | – | – | 23.66 | 29.66 | 34.87 | 0.00 | 0.00
| ML | 0.0245 | 0.4645 | 0.998 | – | – | – | 70 | 0.275 | 1.383 | $-1.010$ | – | – | – | – | – | – | – | – | – | 8.68 | 16.68 | 22.41 | 0.00 | 0.00
| MS | 0.0245 | 0.4651 | 0.999 | – | – | – | 70 | – | – | – | 0.159 | 0.105 | $-1.310$ | – | – | – | – | – | – | $-3.49$ | 4.51 | 2.95 | 0.00 | 0.00
Flat $\Lambda$CDM | GL | 0.0245 | 0.4641 | 0.997 | – | – | – | 70 | – | – | – | – | – | – | 0.370 | 0.359 | $-1.675$ | – | – | – | 22.94 | 30.94 | 35.65 | 0.00 | 0.00
| MS + GL | 0.0245 | 0.4652 | 0.999 | – | – | – | 70 | – | – | – | – | – | – | – | – | – | 0.361 | 0.313 | $-1.545$ | 25.59 | 33.59 | 38.30 | 0.00 | 0.00
| ML + GL | 0.0245 | 0.4642 | 0.997 | – | – | – | 70 | 0.274 | 1.378 | $-1.008$ | – | – | – | 0.362 | 0.358 | $-1.708$ | – | – | – | 31.66 | 45.66 | 59.71 | 0.00 | 0.00
| ML + MS | 0.0245 | 0.463 | 0.995 | – | – | – | 70 | 0.274 | 1.388 | $-1.013$ | 0.148 | 0.080 | $-1.322$ | – | – | – | – | – | – | 5.34 | 19.34 | 30.42 | 0.00 | 0.00
|  $H(z)$ \+ BAO | 0.0247 | 0.1140 | 0.294 | 0.029 | – | – | 68.68 | – | – | – | – | – | – | – | – | – | – | – | – | 23.60 | 31.60 | 38.55 | 1.94 | 3.68
| ML | 0.0245 | 0.4410 | 0.950 | $-0.973$ | – | – | 70 | 0.268 | 1.316 | $-0.967$ | – | – | – | – | – | – | – | – | – | 7.48 | 17.48 | 24.65 | 0.80 | 2.24
| MS | 0.0245 | 0.2727 | 0.607 | $-1.730$ | – | – | 70 | – | – | – | 0.007 | 0.023 | $-0.919$ | – | – | – | – | – | – | $-15.81$ | $-5.81$ | $-7.76$ | $-10.32$ | $-10.71$
Non-flat $\Lambda$CDM | GL | 0.0245 | 0.4640 | 0.997 | $-1.703$ | – | – | 70 | – | – | – | – | – | – | 0.329 | 0.238 | $-1.377$ | – | – | – | 17.00 | 27.00 | 32.89 | $-3.94$ | $-2.76$
| MS + GL | 0.0245 | 0.4649 | 0.999 | $-1.738$ | – | – | 70 | – | – | – | – | – | – | – | – | – | 0.304 | 0.220 | $-1.303$ | 16.02 | 26.02 | 31.91 | $-7.57$ | $-6.39$
| ML + GL | 0.0245 | 0.4330 | 0.934 | $-1.288$ | – | – | 70 | 0.269 | 1.248 | $-0.953$ | – | – | – | 0.336 | 0.330 | $-1.529$ | – | – | – | 26.79 | 42.79 | 58.84 | $-2.87$ | $-0.87$
| ML + MS | 0.0245 | 0.4625 | 0.994 | $-1.130$ | – | – | 70 | 0.259 | 1.278 | $-0.966$ | 0.136 | 0.159 | $-1.249$ | – | – | – | – | – | – | 2.91 | 18.91 | 31.58 | $-0.43$ | 1.14
|  $H(z)$ \+ BAO | 0.0304 | 0.0891 | 0.281 | – | $-0.701$ | – | 65.18 | – | – | – | – | – | – | – | – | – | – | – | – | 19.65 | 27.65 | 34.60 | $-2.01$ | $-0.27$
| ML | 0.0245 | 0.0327 | 0.117 | – | 0.133 | – | 70 | 0.275 | 1.288 | $-0.997$ | – | – | – | – | – | – | – | – | – | 8.14 | 18.14 | 25.31 | 1.46 | 2.90
| MS | 0.0245 | 0.0939 | 0.242 | – | 0.141 | – | 70 | – | – | – | 0.160 | 0.054 | $-1.285$ | – | – | – | – | – | – | $-4.23$ | 5.77 | 3.81 | 1.26 | 0.86
Flat XCDM | GL | 0.0245 | 0.0035 | 0.057 | – | 0.139 | – | 70 | – | – | – | – | – | – | 0.364 | 0.259 | $-1.651$ | – | – | – | 21.97 | 31.97 | 37.86 | 1.03 | 2.21
| MS + GL | 0.0245 | 0.0058 | 0.062 | – | 0.141 | – | 70 | – | – | – | – | – | – | – | – | – | 0.346 | 0.248 | $-1.518$ | 23.92 | 33.92 | 39.81 | 0.33 | 1.51
| ML + GL | 0.0245 | 0.0735 | 0.200 | – | 0.143 | – | 70 | 0.273 | 1.300 | $-1.015$ | – | – | – | 0.359 | 0.273 | $-1.636$ | – | – | – | 30.20 | 46.20 | 62.26 | 0.54 | 6.55
| ML + MS | 0.0245 | $-0.0207$ | 0.008 | – | 0.137 | – | 70 | 0.278 | 1.269 | $-0.984$ | 0.175 | 0.031 | $-1.282$ | – | – | – | – | – | – | 3.99 | 19.99 | 32.65 | 0.65 | 2.23
|  $H(z)$ \+ BAO | 0.0290 | 0.0980 | 0.295 | $-0.152$ | $-0.655$ | – | 65.59 | – | – | – | – | – | – | – | – | – | – | – | – | 18.31 | 28.31 | 37.00 | $-1.35$ | 2.13
| ML | 0.0245 | 0.1525 | 0.361 | $-1.893$ | 0.036 | – | 70 | 0.269 | 0.949 | $-0.976$ | – | – | – | – | – | – | – | – | – | 7.39 | 19.39 | 27.99 | 2.71 | 5.58
| MS | 0.0245 | 0.3596 | 0.784 | $-1.915$ | $-1.108$ | – | 70 | – | – | – | 0.057 | 0.098 | $-0.995$ | – | – | – | – | – | – | $-13.61$ | $-1.61$ | $-3.95$ | $-6.12$ | $-6.90$
Non-flat XCDM | GL | 0.0245 | 0.0378 | 0.127 | $-0.174$ | $-4.518$ | – | 70 | – | – | – | – | – | – | 0.327 | 1.237 | $-1.299$ | – | – | – | 16.61 | 28.61 | 35.68 | $-2.33$ | 0.03
| MS + GL | 0.0245 | 0.4212 | 0.910 | $-1.218$ | $-2.308$ | – | 70 | – | – | – | – | – | – | – | – | – | 0.298 | 0.403 | $-1.248$ | 15.65 | 27.65 | 34.72 | $-5.94$ | $-3.58$
| ML + GL | 0.0245 | 0.3594 | 0.783 | $-0.789$ | $-4.432$ | – | 70 | 0.269 | 1.449 | $-0.927$ | – | – | – | 0.338 | 0.576 | $-1.429$ | – | – | – | 26.04 | 44.04 | 62.11 | $-1.62$ | 2.40
| ML + MS | 0.0245 | 0.1499 | 0.306 | $-1.993$ | $0.130$ | – | 70 | 0.281 | 0.881 | $-0.978$ | 0.095 | $-0.164$ | $-1.206$ | – | – | – | – | – | – | $-0.04$ | 17.96 | 32.21 | $-1.38$ | 1.79
|  $H(z)$ \+ BAO | 0.0333 | 0.0788 | 0.264 | – | – | 1.504 | 65.20 | – | – | – | – | – | – | – | – | – | – | – | – | 19.49 | 27.49 | 34.44 | $-2.17$ | $-0.43$
| ML | 0.0245 | 0.4651 | 0.999 | – | – | 5.225 | 70 | 0.275 | 1.383 | $-1.011$ | – | – | – | – | – | – | – | – | – | 8.68 | 18.68 | 25.85 | 2.00 | 3.44
| MS | 0.0245 | 0.4649 | 0.999 | – | – | 8.046 | 70 | – | – | – | 0.160 | 0.099 | $-1.306$ | – | – | – | – | – | – | $-3.49$ | 6.51 | 4.56 | 2.00 | 1.61
Flat $\phi$CDM | GL | 0.0245 | 0.4641 | 0.997 | – | – | 4.299 | 70 | – | – | – | – | – | – | 0.372 | 0.360 | $-1.674$ | – | – | – | 22.94 | 32.94 | 38.83 | 2.00 | 3.18
| MS + GL | 0.0245 | 0.4653 | 1.000 | – | – | 6.323 | 70 | – | – | – | – | – | – | – | – | – | 0.359 | 0.314 | $-1.545$ | 25.59 | 35.59 | 41.48 | 2.00 | 3.18
| ML + GL | 0.0245 | 0.4648 | 0.999 | – | – | 7.886 | 70 | 0.274 | 1.375 | $-1.013$ | – | – | – | 0.371 | 0.333 | $-1.647$ | – | – | – | 31.72 | 47.72 | 63.78 | 2.06 | 4.07
| ML + MS | 0.0245 | 0.4611 | 0.991 | – | – | 6.029 | 70 | 0.268 | 1.371 | $-0.997$ | 0.148 | 0.113 | $-1.317$ | – | – | – | – | – | – | 5.32 | 21.32 | 33.99 | 1.98 | 3.57
|  $H(z)$ \+ BAO | 0.0334 | 0.0816 | 0.266 | $-0.147$ | – | 1.915 | 65.70 | – | – | – | – | – | – | – | – | – | – | – | – | 18.15 | 28.15 | 36.84 | $-1.51$ | 1.97
| ML | 0.0245 | 0.4558 | 0.980 | $-0.980$ | – | 0.423 | 70 | 0.266 | 1.296 | $-0.973$ | – | – | – | – | – | – | – | – | – | 7.48 | 19.48 | 28.09 | 2.80 | 5.68
| MS | 0.0245 | 0.4482 | 0.965 | $-0.964$ | – | 8.262 | 70 | – | – | – | 0.132 | 0.003 | $-1.261$ | – | – | – | – | – | – | $-5.25$ | 6.75 | 4.40 | 2.24 | 1.45
Non-flat $\phi$CDM | GL | 0.0245 | 0.4644 | 0.998 | $-0.993$ | – | 0.173 | 70 | – | – | – | – | – | – | 0.340 | 0.337 | $-1.547$ | – | – | – | 20.15 | 32.15 | 39.22 | 1.21 | 3.57
| MS + GL | 0.0245 | 0.4601 | 0.989 | $-0.982$ | – | 0.011 | 70 | – | – | – | – | – | – | – | – | – | 0.315 | 0.310 | $-1.484$ | 21.46 | 33.46 | 40.53 | $-0.13$ | 2.23
| ML + GL | 0.0245 | 0.4465 | 0.961 | $-0.950$ | – | 0.232 | 70 | 0.255 | 1.290 | $-0.969$ | – | – | – | 0.348 | 0.356 | $-1.600$ | – | – | – | 28.04 | 46.04 | 64.11 | 0.38 | 4.40
| ML + MS | 0.0245 | 0.4628 | 0.995 | $-0.936$ | – | 8.517 | 70 | 0.281 | 1.202 | $-0.994$ | 0.152 | 0.027 | $-1.291$ | – | – | – | – | – | – | 2.71 | 20.71 | 34.96 | 1.36 | 4.53

  * a

$\rm{km\ s^{-1}\ Mpc^{-1}}$. In the GRB only cases, $H_{0}$ is set to be 70
$\rm{km\ s^{-1}\ Mpc^{-1}}$.

Table 4: Unmarginalized best-fitting parameter values for all models from
various combinations of data.

Model | Data set | $\Omega_{b}h^{2}$ | $\Omega_{c}h^{2}$ | $\Omega_{\mathrm{m0}}$ | $\Omega_{\mathrm{k0}}$ | $w_{\mathrm{X}}$ | $\alpha$ |  $H_{0}$a | $\sigma_{\mathrm{int,\,\textsc{ml}}}$ | $b_{\mathrm{\textsc{ml}}}$ | $k_{\mathrm{\textsc{ml}}}$ | $\sigma_{\mathrm{int,\,\textsc{ms}}}$ | $b_{\mathrm{\textsc{ms}}}$ | $k_{\mathrm{\textsc{ms}}}$ | $\sigma_{\mathrm{int,\,\textsc{gl}}}$ | $b_{\mathrm{\textsc{gl}}}$ | $k_{\mathrm{\textsc{gl}}}$ | $\sigma_{\mathrm{int,\,\textsc{ms+gl}}}$ | $b_{\mathrm{\textsc{ms+gl}}}$ | $k_{\mathrm{\textsc{ms+gl}}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  $H(z)$ \+ BAO | $0.0241\pm 0.0029$ | $0.1193^{+0.0082}_{-0.0090}$ | $0.299^{+0.017}_{-0.019}$ | – | – | – | $69.30\pm 1.84$ | – | – | – | – | – | – | – | – | – | – | – | –
| ML | – | – | $>0.188$ | – | – | – | – | $0.305^{+0.035}_{-0.053}$ | $1.552^{+0.108}_{-0.189}$ | $-1.017\pm 0.090$ | – | – | – | – | – | – | – | – | –
| MS | – | – | $0.520^{+0.379}_{-0.253}$ | – | – | – | – | – | – | – | $0.695^{+0.044}_{-0.550}$ | $0.437^{+0.073}_{-0.400}$ | $-1.450^{+0.362}_{-0.258}$ | – | – | – | – | – | –
Flat $\Lambda$CDM | GL | – | – | $>0.202$ | – | – | – | – | – | – | – | – | – | – | $0.429^{+0.059}_{-0.094}$ | $0.495^{+0.120}_{-0.173}$ | $-1.720\pm 0.219$ | – | – | –
| MS + GL | – | – | $>0.293$ | – | – | – | – | – | – | – | – | – | – | – | – | – | $0.412^{+0.052}_{-0.079}$ | $0.421^{+0.101}_{-0.141}$ | $-1.577\pm 0.155$
| ML + GL | – | – | $>0.294$ | – | – | – | – | $0.301^{+0.033}_{-0.051}$ | $1.507^{+0.095}_{-0.151}$ | $-1.015\pm 0.089$ | – | – | – | $0.424^{+0.056}_{-0.090}$ | $0.465^{+0.113}_{-0.149}$ | $-1.708\pm 0.210$ | – | – | –
| ML + MS | – | – | $>0.206$ | – | – | – | – | $0.302^{+0.034}_{-0.052}$ | $1.542^{+0.104}_{-0.184}$ | $-1.016\pm 0.091$ | $0.613^{+0.010}_{-0.479}$ | $0.379^{+0.049}_{-0.350}$ | $-1.426^{+0.312}_{-0.210}$ | – | – | – | – | – | –
|  $H(z)$ \+ BAO | $0.0253^{+0.0041}_{-0.0050}$ | $0.1135^{+0.0196}_{-0.0197}$ | $0.293\pm 0.025$ | $0.039^{+0.102}_{-0.115}$ | – | – | $68.75^{+2.37}_{-2.36}$ | – | – | – | – | – | – | – | – | – | – | – | –
| ML | – | – | $>0.241$ | $-0.131^{+0.450}_{-0.919}$ | – | – | – | $0.304^{+0.035}_{-0.053}$ | $1.478^{+0.123}_{-0.166}$ | $-1.000\pm 0.096$ | – | – | – | – | – | – | – | – | –
| MS | – | – | $0.564^{+0.426}_{-0.149}$ | $0.066^{+1.002}_{-1.199}$ | – | – | – | – | – | – | $0.670^{+0.024}_{-0.533}$ | $0.413^{+0.047}_{-0.393}$ | $-1.430^{+0.360}_{-0.239}$ | – | – | – | – | – | –
Non-flat $\Lambda$CDM | GL | – | – | $>0.290$ | $-0.762^{+0.271}_{-0.888}$ | – | – | – | – | – | – | – | – | – | $0.402^{+0.057}_{-0.090}$ | $0.407^{+0.136}_{-0.160}$ | $-1.536\pm 0.252$ | – | – | –
| MS + GL | – | – | $>0.391$ | $-1.165^{+0.225}_{-0.519}$ | – | – | – | – | – | – | – | – | – | – | – | – | $0.357^{+0.046}_{-0.070}$ | $0.337^{+0.110}_{-0.127}$ | $-1.382^{+0.164}_{-0.163}$
| ML + GL | – | – | $>0.338$ | $-0.737^{+0.299}_{-0.547}$ | – | – | – | $0.300^{+0.033}_{-0.051}$ | $1.386^{+0.138}_{-0.154}$ | $-0.966\pm 0.093$ | – | – | – | $0.397^{+0.053}_{-0.086}$ | $0.437^{+0.110}_{-0.142}$ | $-1.588^{+0.215}_{-0.214}$ | – | – | –
| ML + MS | – | – | $>0.270$ | $-0.300^{+0.380}_{-0.836}$ | – | – | – | $0.302^{+0.034}_{-0.052}$ | $1.457^{+0.128}_{-0.163}$ | $-0.993\pm 0.095$ | $0.518^{+0.009}_{-0.393}$ | $0.339^{+0.056}_{-0.301}$ | $-1.385^{+0.271}_{-0.186}$ | – | – | – | – | – | –
|  $H(z)$ \+ BAO | $0.0296^{+0.0046}_{-0.0052}$ | $0.0939^{+0.0194}_{-0.0171}$ | $0.284^{+0.023}_{-0.021}$ | – | $-0.754^{+0.155}_{-0.107}$ | – | $65.89^{+2.41}_{-2.71}$ | – | – | – | – | – | – | – | – | – | – | – | –
| ML | – | – | $>0.123$ | – | $-2.456^{+2.567}_{-2.180}$ | – | – | $0.306^{+0.036}_{-0.054}$ | $1.611^{+0.113}_{-0.277}$ | $-1.014\pm 0.092$ | – | – | – | – | – | – | – | – | –
| MS | – | – | $0.520^{+0.340}_{-0.276}$ | – | $-2.494^{+1.264}_{-2.050}$ | – | – | – | – | – | $0.704^{+0.035}_{-0.564}$ | $0.497^{+0.086}_{-0.458}$ | $-1.441^{+0.366}_{-0.259}$ | – | – | – | – | – | –
Flat XCDM | GL | – | – | $>0.141$ | – | $<-0.046$ | – | – | – | – | – | – | – | – | $0.428^{+0.058}_{-0.092}$ | $0.556^{+0.127}_{-0.256}$ | $-1.706\pm 0.215$ | – | – | –
| MS + GL | – | – | $>0.192$ | – | $<0.028$ | – | – | – | – | – | – | – | – | – | – | – | $0.409^{+0.052}_{-0.078}$ | $0.470^{+0.108}_{-0.206}$ | $-1.570\pm 0.155$
| ML + GL | – | – | $>0.164$ | – | $<0.022$ | – | – | $0.300^{+0.033}_{-0.051}$ | $1.566^{+0.099}_{-0.239}$ | $-1.012\pm 0.087$ | – | – | – | $0.424^{+0.057}_{-0.092}$ | $0.531^{+0.116}_{-0.240}$ | $-1.700\pm 0.213$ | – | – | –
| ML + MS | – | – | $>0.142$ | – | $<-0.072$ | – | – | $0.303^{+0.034}_{-0.052}$ | $1.597^{+0.108}_{-0.255}$ | $-1.013\pm 0.089$ | $0.562^{+0.010}_{-0.431}$ | $0.409^{+0.066}_{-0.377}$ | $-1.408^{+0.288}_{-0.196}$ | – | – | – | – | – | –
|  $H(z)$ \+ BAO | $0.0290^{+0.0052}_{-0.0055}$ | $0.0990^{+0.0214}_{-0.0215}$ | $0.293\pm 0.028$ | $-0.116\pm 0.134$ | $-0.700^{+0.138}_{-0.083}$ | – | $65.96^{+2.32}_{-2.55}$ | – | – | – | – | – | – | – | – | – | – | – | –
| ML | – | – | $>0.174$ | $-0.262^{+0.580}_{-0.724}$ | $-2.000^{+2.117}_{-1.264}$ | – | – | $0.305^{+0.036}_{-0.054}$ | $1.462^{+0.194}_{-0.196}$ | $-0.996\pm 0.097$ | – | – | – | – | – | – | – | – | –
| MS | – | – | $0.552^{+0.442}_{-0.152}$ | $0.134^{+0.793}_{-0.987}$ | $-2.234^{+2.159}_{-0.969}$ | – | – | – | – | – | $0.733^{+0.031}_{-0.596}$ | $0.464^{+0.052}_{-0.444}$ | $-1.433^{+0.394}_{-0.270}$ | – | – | – | – | – | –
Non-flat XCDM | GL | – | – | $>0.194$ | $-0.615^{+0.470}_{-0.685}$ | $-2.212^{+2.186}_{-0.962}$ | – | – | – | – | – | – | – | – | $0.403^{+0.058}_{-0.092}$ | $0.480^{+0.177}_{-0.223}$ | $-1.532^{+0.259}_{-0.260}$ | – | – | –
| MS + GL | – | – | $>0.268$ | $-0.920^{+0.460}_{-0.386}$ | $-2.323^{+2.085}_{-1.095}$ | – | – | – | – | – | – | – | – | – | – | – | $0.358^{+0.047}_{-0.072}$ | $0.452^{+0.185}_{-0.203}$ | $-1.363\pm 0.175$
| ML + GL | – | – | $>0.196$ | $-0.696^{+0.484}_{-0.408}$ | $-2.158^{+2.254}_{-1.424}$ | – | – | $0.298^{+0.033}_{-0.051}$ | $1.422^{+0.201}_{-0.221}$ | $-0.968\pm 0.092$ | – | – | – | $0.397^{+0.054}_{-0.087}$ | $0.480^{+0.205}_{-0.224}$ | $-1.561^{+0.220}_{-0.219}$ | – | – | –
| ML + MS | – | – | $>0.198$ | $-0.375^{+0.458}_{-0.611}$ | $-2.235^{+2.289}_{-2.007}$ | – | – | $0.301^{+0.034}_{-0.052}$ | $1.470^{+0.184}_{-0.183}$ | $-0.988^{+0.095}_{-0.094}$ | $0.525^{+0.007}_{-0.409}$ | $0.397^{+0.079}_{-0.364}$ | $-1.367^{+0.275}_{-0.193}$ | – | – | – | – | – | –
|  $H(z)$ \+ BAO | $0.0321^{+0.0056}_{-0.0039}$ | $0.0823^{+0.0186}_{-0.0183}$ | $0.268\pm 0.024$ | – | – | $1.467^{+0.637}_{-0.866}$ | $65.24^{+2.15}_{-2.35}$ | – | – | – | – | – | – | – | – | – | – | – | –
| ML | – | – | $>0.148$ | – | – | – | – | $0.304^{+0.035}_{-0.053}$ | $1.493^{+0.093}_{-0.143}$ | $-1.017\pm 0.089$ | – | – | – | – | – | – | – | – | –
| MS | – | – | $0.514^{+0.365}_{-0.275}$ | – | – | – | – | – | – | – | $0.597^{+0.029}_{-0.457}$ | $0.355^{+0.052}_{-0.331}$ | $-1.425^{+0.311}_{-0.221}$ | – | – | – | – | – | –
Flat $\phi$CDM | GL | – | – | $>0.148$ | – | – | – | – | – | – | – | – | – | – | $0.428^{+0.059}_{-0.094}$ | $0.444^{+0.112}_{-0.141}$ | $-1.710\pm 0.218$ | – | – | –
| MS + GL | – | – | $>0.222$ | – | – | – | – | – | – | – | – | – | – | – | – | – | $0.408^{+0.050}_{-0.076}$ | $0.384^{+0.092}_{-0.111}$ | $-1.573\pm 0.151$
| ML + GL | – | – | $>0.235$ | – | – | – | – | $0.301^{+0.033}_{-0.051}$ | $1.463^{+0.086}_{-0.119}$ | $-1.015\pm 0.088$ | – | – | – | $0.423^{+0.056}_{-0.091}$ | $0.423^{+0.104}_{-0.121}$ | $-1.701\pm 0.209$ | – | – | –
| ML + MS | – | – | $>0.166$ | – | – | – | – | $0.303^{+0.034}_{-0.053}$ | $1.485^{+0.092}_{-0.137}$ | $-1.017\pm 0.090$ | $0.517^{+0.001}_{-0.385}$ | $0.304^{+0.033}_{-0.283}$ | $-1.410^{+0.268}_{-0.173}$ | – | – | – | – | – | –
|  $H(z)$ \+ BAO | $0.0319^{+0.0062}_{-0.0037}$ | $0.0848^{+0.0181}_{-0.0220}$ | $0.271^{+0.025}_{-0.028}$ | $-0.074^{+0.104}_{-0.110}$ | – | $1.653^{+0.685}_{-0.856}$ | $65.46^{+2.31}_{-2.29}$ | – | – | – | – | – | – | – | – | – | – | – | –
| ML | – | – | $>0.207$ | $-0.163^{+0.355}_{-0.317}$ | – | – | – | $0.303^{+0.035}_{-0.053}$ | $1.448^{+0.120}_{-0.165}$ | $-1.011\pm 0.091$ | – | – | – | – | – | – | – | – | –
| MS | – | – | $0.505^{+0.310}_{-0.313}$ | $0.017^{+0.387}_{-0.375}$ | – | – | – | – | – | – | $0.589^{+0.023}_{-0.455}$ | $0.352^{+0.048}_{-0.331}$ | $-1.432^{+0.314}_{-0.207}$ | – | – | – | – | – | –
Non-flat $\phi$CDM | GL | – | – | $>0.207$ | $-0.193^{+0.364}_{-0.347}$ | – | – | – | – | – | – | – | – | – | $0.422^{+0.057}_{-0.092}$ | $0.408^{+0.121}_{-0.149}$ | $-1.693^{+0.215}_{-0.214}$ | – | – | –
| MS + GL | – | – | $>0.313$ | $-0.293^{+0.330}_{-0.359}$ | – | – | – | – | – | – | – | – | – | – | – | – | $0.397^{+0.050}_{-0.076}$ | $0.343^{+0.101}_{-0.120}$ | $-1.546\pm 0.150$
| ML + GL | – | – | $>0.340$ | $-0.327^{+0.321}_{-0.345}$ | – | – | – | $0.299^{+0.033}_{-0.051}$ | $1.391^{+0.107}_{-0.136}$ | $-1.004\pm 0.088$ | – | – | – | $0.414^{+0.055}_{-0.088}$ | $0.374^{+0.113}_{-0.128}$ | $-1.668^{+0.207}_{-0.208}$ | – | – | –
| ML + MS | – | – | $>0.238$ | $-0.198^{+0.351}_{-0.321}$ | – | – | – | $0.301^{+0.034}_{-0.052}$ | $1.435^{+0.114}_{-0.156}$ | $-1.010\pm 0.089$ | $0.507^{+0.003}_{-0.379}$ | $0.292^{+0.025}_{-0.281}$ | $-1.393^{+0.258}_{-0.173}$ | – | – | – | – | – | –

  * a

$\rm{km\ s^{-1}\ Mpc^{-1}}$. In the GRB only cases, $H_{0}$ is set to be 70
$\rm{km\ s^{-1}\ Mpc^{-1}}$.

Table 5: One-dimensional marginalized posterior mean values and uncertainties
($\pm 1\sigma$ error bars or $2\sigma$ limits) of the parameters for all
models from various combinations of data.

ML, MS, and GL GRB data have almost cosmological-model independent Dainotti
parameters. This means that it is not unreasonable to treat the ML, MS, and GL
GRBs as standardizable candles, as was assumed in Wang et al. (2021) and Hu et
al. (2021).

In the ML case (with subscript “ML” in the first line of Tables 4, 5, 7, and
8), the slope $k$ ranges from a high of $-0.996\pm 0.097$ (non-flat XCDM) to a
low of $-1.017\pm 0.090$ (flat $\Lambda$CDM), the intercept $b$ ranges from a
high of $1.611^{+0.113}_{-0.277}$ (flat XCDM) to a low of
$1.448^{+0.120}_{-0.165}$ (non-flat $\phi$CDM), and the intrinsic scatter
$\sigma_{\rm int}$ ranges from a high of $0.306^{+0.036}_{-0.054}$ (flat XCDM)
to a low of $0.303^{+0.035}_{-0.053}$ (non-flat $\phi$CDM), with central
values of each pair being $0.16\sigma$, $0.54\sigma$, and $0.05\sigma$ away
from each other, respectively.

In the MS case (with subscript “MS” in the first line of Tables 4 and 5) with
prior range of $b\in[0,10]$, the slope $k$ ranges from a high of
$-1.425^{+0.311}_{-0.221}$ (flat $\phi$CDM) to a low of
$-1.450^{+0.362}_{-0.258}$ (flat $\Lambda$CDM), the intercept $b$ ranges from
a high of $0.497^{+0.086}_{-0.458}$ (flat XCDM) to a low of
$0.352^{+0.048}_{-0.331}$ (non-flat $\phi$CDM), and the intrinsic scatter
$\sigma_{\rm int}$ ranges from a high of $0.733^{+0.031}_{-0.596}$ (non-flat
XCDM) to a low of $0.589^{+0.023}_{-0.455}$ (non-flat $\phi$CDM), with central
values of each pair being $0.06\sigma$, $0.54\sigma$, and $0.05\sigma$ away
from each other, respectively.888Note, however, that the lower error bars of
$b$ and $\sigma_{\rm int}$ are considerably larger than the upper ones due to
cut-off prior ranges of the former and skewed distributions of the latter.
Therefore here we also consider the MS case with wider prior range of
$b\in[-10,10]$, which are not listed in the tables due to their insignificant
differences. Because the lowest and highest values of $k$, $b$, and
$\sigma_{\rm int}$ from these two MS cases differ from each other at only
$0.22\sigma$, $0.56\sigma$, and $0.37\sigma$, respectively, and the
constraints of the cosmological parameters are also within $1\sigma$ range,
the prior range of $b\in[0,10]$ is an acceptable choice.

In the GL case (with subscript “GL” in the first line of Tables 4, 5, 7, and
8), the slope $k$ ranges from a high of $-1.532^{+0.259}_{-0.260}$ (non-flat
XCDM) to a low of $-1.720\pm 0.219$ (flat $\Lambda$CDM), the intercept $b$
ranges from a high of $0.556^{+0.127}_{-0.256}$ (flat XCDM) to a low of
$0.407^{+0.136}_{-0.160}$ (non-flat $\Lambda$CDM), and the intrinsic scatter
$\sigma_{\rm int}$ ranges from a high of $0.429^{+0.059}_{-0.094}$ (flat
$\Lambda$CDM) to a low of $0.402^{+0.057}_{-0.090}$ (non-flat $\phi$CDM), with
central values of each pair being $0.55\sigma$, $0.51\sigma$, and $0.25\sigma$
away from each other, respectively.

Figure 1, panel (c), shows that the GL and MS GRBs obey the same Dainotti
correlation in the flat $\Lambda$CDM model, within the uncertainties.999It is
unclear if this is more than just a coincidence, as the plateau phases in the
two cases are dominated by GW emission (GL) and MD radiation (MS),
respectively. Table 6 shows that the differences between the GL and MS
Dainotti parameters in all six cosmological models are within $1\sigma$. GL
and MS GRBs however follow a different Dainotti correlation than the ML GRBs.
Given the similarity of the GL and MS Dainotti correlation parameters, it is
not unreasonable to use just three (not six) correlation parameters in joint
analyses of MS and GL data (with subscript “MS+GL” in the first line of Tables
4 and 5). In this case, the slope $k$ ranges from a high of $-1.363\pm 0.175$
(non-flat XCDM) to a low of $-1.577\pm 0.155$ (flat $\Lambda$CDM), the
intercept $b$ ranges from a high of $0.470^{+0.108}_{-0.206}$ (flat XCDM) to a
low of $0.337^{+0.110}_{-0.127}$ (non-flat $\Lambda$CDM), and the intrinsic
scatter $\sigma_{\rm int}$ ranges from a high of $0.412^{+0.052}_{-0.079}$
(flat $\Lambda$CDM) to a low of $0.357^{+0.046}_{-0.070}$ (non-flat
$\Lambda$CDM), with central values of each pair being $0.92\sigma$,
$0.57\sigma$, and $0.60\sigma$ away from each other, respectively. In contrast
to the GL case, the MS + GL case tightens the constraints a little bit, with
smaller error bars, and prefers lower values of $b$ and $\sigma_{\rm int}$,
and higher values of $k$. When we jointly analyze ML + GL and ML + MS, the
constraints on the Dainotti parameters follow the same pattern as that of MS +
GL against GL.

(a) Flat $\Lambda$CDM (b) Non-flat $\Lambda$CDM

(c) Flat XCDM (d) Non-flat XCDM

(e) Flat $\phi$CDM (f) Non-flat $\phi$CDM

Figure 2: One-dimensional likelihoods and 1$\sigma$, 2$\sigma$, and 3$\sigma$
two-dimensional likelihood confidence contours from MD-LGRB (blue), MD-SGRB
(gray), MD-LGRB + MD-SGRB (pink), and $H(z)$ \+ BAO (red) data for all six
models. The zero-acceleration lines are shown as black dashed lines, which
divide the parameter space into regions associated with currently-accelerating
and currently-decelerating cosmological expansion. In the non-flat XCDM and
non-flat $\phi$CDM cases, the zero-acceleration lines are computed for the
third cosmological parameter set to the $H(z)$ \+ BAO data best-fitting values
listed in Table 4. The crimson dash-dot lines represent flat hypersurfaces,
with closed spatial hypersurfaces either below or to the left. The magenta
lines represent $w_{\rm X}=-1$, i.e. flat or non-flat $\Lambda$CDM models. The
$\alpha=0$ axes correspond to flat and non-flat $\Lambda$CDM models in panels
(e) and (f), respectively.

(a) Flat $\Lambda$CDM (b) Non-flat $\Lambda$CDM

(c) Flat XCDM (d) Non-flat XCDM

(e) Flat $\phi$CDM (f) Non-flat $\phi$CDM

Figure 3: One-dimensional likelihoods and 1$\sigma$, 2$\sigma$, and 3$\sigma$
two-dimensional likelihood confidence contours from MD-LGRB (blue), GW-LGRB
(green), MD-LGRB + GW-LGRB (violet), and $H(z)$ \+ BAO (red) data for all six
models. The zero-acceleration lines are shown as black dashed lines, which
divide the parameter space into regions associated with currently-accelerating
and currently-decelerating cosmological expansion. In the non-flat XCDM and
non-flat $\phi$CDM cases, the zero-acceleration lines are computed for the
third cosmological parameter set to the $H(z)$ \+ BAO data best-fitting values
listed in Table 4. The crimson dash-dot lines represent flat hypersurfaces,
with closed spatial hypersurfaces either below or to the left. The magenta
lines represent $w_{\rm X}=-1$, i.e. flat or non-flat $\Lambda$CDM models. The
$\alpha=0$ axes correspond to flat and non-flat $\Lambda$CDM models in panels
(e) and (f), respectively.

(a) Flat $\Lambda$CDM (b) Non-flat $\Lambda$CDM

(c) Flat XCDM (d) Non-flat XCDM

(e) Flat $\phi$CDM (f) Non-flat $\phi$CDM

Figure 4: One-dimensional likelihoods and 1$\sigma$, 2$\sigma$, and 3$\sigma$
two-dimensional likelihood confidence contours from MD-SGRB (gray), GW-LGRB
(green), MD-SGRB + GW-LGRB (orange), and $H(z)$ \+ BAO (red) data for all six
models, without subscripts on $\sigma_{\rm int}$, $k$, and $b$. The zero-
acceleration lines are shown as black dashed lines, which divide the parameter
space into regions associated with currently-accelerating and currently-
decelerating cosmological expansion. In the non-flat XCDM and non-flat
$\phi$CDM cases, the zero-acceleration lines are computed for the third
cosmological parameter set to the $H(z)$ \+ BAO data best-fitting values
listed in Table 4. The crimson dash-dot lines represent flat hypersurfaces,
with closed spatial hypersurfaces either below or to the left. The magenta
lines represent $w_{\rm X}=-1$, i.e. flat or non-flat $\Lambda$CDM models. The
$\alpha=0$ axes correspond to flat and non-flat $\Lambda$CDM models in panels
(e) and (f), respectively.

The constraints on the cosmological parameters are very loose for all of these
cases. In the flat $\Lambda$CDM model, the highest $2\sigma$ lower limit of
$\Omega_{\rm m0}$ among these cases is $\Omega_{\rm m0}>0.294$ of the ML + GL
data. In the non-flat $\Lambda$CDM model, the highest $2\sigma$ lower limit of
$\Omega_{\rm m0}$ is $\Omega_{\rm m0}>0.391$ of the MS + GL case, which is
inconsistent with that of the $H(z)$ \+ BAO case. The MS data favor open
hypersufaces while all other cases favor closed hypersufaces, with the favored
spatial geometries for GL, ML + GL, and MS + GL data being more than $1\sigma$
(or even $2\sigma$) away from flat geometry. In the flat XCDM parametrization,
the highest $2\sigma$ lower limit of $\Omega_{\rm m0}$ is $\Omega_{\rm
m0}>0.192$ for the MS + GL data, and the constraints on the X-fluid equation
of state parameter $w_{\rm X}$ are very loose, with the highest $1\sigma$
upper limit being $0.111$ for the ML case. In the non-flat XCDM
parametrization, the highest $2\sigma$ lower limit of $\Omega_{\rm m0}$ is
$\Omega_{\rm m0}>0.268$ for the MS + GL data, and the constraints on $w_{\rm
X}$ are very loose, with the highest $1\sigma$ upper limit being $0.238$ for
the MS + GL data. The favored spatial geometries for these cases follow the
same pattern as that for non-flat $\Lambda$CDM, but with larger upper limits
of $\Omega_{\rm k0}$ except for the MS data. In the flat $\phi$CDM model, the
highest $2\sigma$ lower limit of $\Omega_{\rm m0}$ is $\Omega_{\rm m0}>0.235$
for the ML + GL data. In the non-flat $\phi$CDM model, the highest $2\sigma$
lower limit of $\Omega_{\rm m0}$ is $\Omega_{\rm m0}>0.340$ for the ML + GL
case, which is inconsistent with that of the $H(z)$ \+ BAO data. Except for
the MS case, closed spatial hypersurfaces are favored, but only in the ML + GL
case is flat geometry slightly more than $1\sigma$ away. There are no
constraints on $\alpha$ from these GRB data.

In the $\Lambda$CDM and XCDM cases, all GRB data combinations more favor
currently accelerating cosmological expansion. They however more favor
currently decelerating cosmological expansion in the $\phi$CDM models, in the
$\Omega_{\rm m0}-\alpha$ and $\Omega_{\rm m0}-\Omega_{\rm k0}$ parameter
subspaces.

From the $AIC$ and $BIC$ values we compute $\Delta AIC$ and $\Delta BIC$
values with respect to the flat $\Lambda$CDM model. These are listed in the
last two columns of Table 4. In the ML case, flat $\Lambda$CDM is the most
favored model but there is only weak or positive evidence against any other
model. In the MS case non-flat $\Lambda$CDM model is the most favored model
and, except for non-flat XCDM (with positive evidence against it), the other
models are very strongly disfavored. In the GL case, non-flat $\Lambda$CDM is
again the most favored model, while the evidence against the others are mostly
positive, except for non-flat $\phi$CDM (with strong $BIC$ evidence against
it). In the MS + GL case, similar to the MS case, non-flat $\Lambda$CDM is the
most favored model and, except for non-flat XCDM (with weak $AIC$ and positive
$BIC$ evidence against it), the others are strongly disfavored. In the ML + GL
case, non-flat $\Lambda$CDM is the most favored model but, except for flat
XCDM (with strong $BIC$ evidence against it), the evidence against the other
models is either weak or positive. In the ML + MS case, the best candidates
are non-flat XCDM based on $AIC$ and flat $\Lambda$CDM based on $BIC$, while
the evidence against the other models is either weak or positive.

Table 6: MD-SGRB and GW-LGRB data $L_{0}-t_{b}$ correlation parameters (and $\sigma_{\rm int}$) differences. Model | $\Delta\sigma_{\rm int}$ | $\Delta k$ | $\Delta b$
---|---|---|---
Flat $\Lambda$CDM | $0.48\sigma$ | $0.31\sigma$ | $0.80\sigma$
Non-flat $\Lambda$CDM | $0.50\sigma$ | $0.31\sigma$ | $0.01\sigma$
Flat XCDM | $0.49\sigma$ | $0.79\sigma$ | $0.22\sigma$
Non-flat XCDM | $0.55\sigma$ | $0.26\sigma$ | $0.07\sigma$
Flat $\phi$CDM | $0.37\sigma$ | $0.92\sigma$ | $0.59\sigma$
Non-flat $\phi$CDM | $0.36\sigma$ | $0.87\sigma$ | $0.36\sigma$

###  5.2 Constraints from A118, A115 (and jointly with ML), and A115′ (and
jointly with GL) data

Model | Data set | $\Omega_{c}h^{2}$ | $\Omega_{\mathrm{m0}}$ | $\Omega_{\mathrm{k0}}$ | $w_{\mathrm{X}}$ | $\alpha$ | $\sigma_{\mathrm{int,\,\textsc{ml}}}$ | $b_{\mathrm{\textsc{ml}}}$ | $k_{\mathrm{\textsc{ml}}}$ | $\sigma_{\mathrm{int}}$ | $\gamma$ | $\beta$ | $\sigma_{\mathrm{int,\,\textsc{gl}}}$ | $b_{\mathrm{\textsc{gl}}}$ | $k_{\mathrm{\textsc{gl}}}$ | $-2\ln\mathcal{L}_{\mathrm{max}}$ | $AIC$ | $BIC$ | $\Delta AIC$ | $\Delta BIC$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| A118 | 0.4089 | 0.884 | – | – | – | – | – | – | 0.401 | 50.02 | 1.099 | – | – | – | 128.72 | 136.72 | 147.81 | 0.00 | 0.00
| ML | 0.4645 | 0.998 | – | – | – | 0.275 | 1.383 | $-1.010$ | – | – | – | – | – | – | 8.68 | 16.68 | 22.41 | 0.00 | 0.00
| A115 | 0.4172 | 0.901 | – | – | – | – | – | – | 0.405 | 50.01 | 1.099 | – | – | – | 127.97 | 135.97 | 146.95 | 0.00 | 0.00
Flat $\Lambda$CDM | ML + A115 | 0.4540 | 0.977 | – | – | – | 0.274 | 1.400 | $-1.019$ | 0.407 | 50.00 | 1.097 | – | – | – | 136.70 | 150.70 | 171.59 | 0.00 | 0.00
| GL | 0.4641 | 0.997 | – | – | – | – | – | – | – | – | – | 0.370 | 0.359 | $-1.675$ | 22.94 | 30.94 | 35.65 | 0.00 | 0.00
| A115′ | 0.4629 | 0.995 | – | – | – | – | – | – | 0.403 | 50.01 | 1.091 | – | – | – | 126.34 | 134.34 | 145.32 | 0.00 | 0.00
| GL + A115′ | 0.4652 | 0.999 | – | – | – | – | – | – | 0.402 | 49.96 | 1.110 | 0.370 | 0.363 | $-1.666$ | 149.34 | 163.34 | 183.88 | 0.00 | 0.00
| A118 | 0.4622 | 0.993 | 0.907 | – | – | – | – | – | 0.400 | 49.92 | 1.115 | – | – | – | 127.96 | 137.96 | 151.82 | 1.24 | 4.01
| ML | 0.4410 | 0.950 | $-0.973$ | – | – | 0.268 | 1.316 | $-0.967$ | – | – | – | – | – | – | 7.48 | 17.48 | 24.65 | 0.80 | 2.24
| A115 | 0.4631 | 0.995 | 1.014 | – | – | – | – | – | 0.403 | 49.90 | 1.118 | – | – | – | 127.18 | 137.18 | 150.90 | 1.21 | 3.95
Non-flat $\Lambda$CDM | ML + A115 | 0.4637 | 0.996 | 0.062 | – | – | 0.283 | 1.383 | $-1.006$ | 0.410 | 50.01 | 1.088 | – | – | – | 136.74 | 152.74 | 176.61 | 2.04 | 5.02
| GL | 0.4640 | 0.997 | $-1.703$ | – | – | – | – | – | – | – | – | 0.329 | 0.238 | $-1.377$ | 17.00 | 27.00 | 32.89 | $-3.94$ | $-2.76$
| A115′ | 0.4647 | 0.998 | 0.729 | – | – | – | – | – | 0.404 | 49.94 | 1.110 | – | – | – | 125.93 | 135.93 | 149.65 | 1.59 | 4.33
| GL + A115′ | 0.4544 | 0.977 | $-0.244$ | – | – | – | – | – | 0.402 | 50.00 | 1.100 | 0.362 | 0.364 | $-1.674$ | 149.29 | 165.29 | 188.77 | 1.95 | 4.89
| A118 | $-0.0115$ | 0.027 | – | $-0.098$ | – | – | – | – | 0.399 | 50.04 | 1.102 | – | – | – | 128.43 | 138.43 | 152.29 | 1.71 | 4.48
| ML | 0.0327 | 0.117 | – | 0.133 | – | 0.275 | 1.288 | $-0.997$ | – | – | – | – | – | – | 8.14 | 18.14 | 25.31 | 1.46 | 2.90
| A115 | $-0.0197$ | 0.010 | – | $-0.102$ | – | – | – | – | 0.407 | 50.01 | 1.115 | – | – | – | 127.66 | 137.66 | 151.38 | 1.69 | 4.43
Flat XCDM | ML + A115 | 0.4065 | 0.880 | – | $-4.386$ | – | 0.273 | 1.431 | $-1.008$ | 0.403 | 50.05 | 1.093 | – | – | – | 136.70 | 152.70 | 176.57 | 2.00 | 4.98
| GL | 0.0035 | 0.057 | – | 0.139 | – | – | – | – | – | – | – | 0.364 | 0.259 | $-1.651$ | 21.97 | 31.97 | 37.86 | 1.03 | 2.21
| A115′ | 0.0119 | 0.074 | – | $-0.082$ | – | – | – | – | 0.402 | 50.03 | 1.100 | – | – | – | 126.22 | 136.22 | 149.94 | 1.88 | 4.62
| GL + A115′ | 0.2817 | 0.625 | – | 0.122 | – | – | – | – | 0.401 | 49.98 | 1.084 | 0.376 | 0.330 | $-1.672$ | 149.21 | 165.21 | 188.69 | 1.87 | 4.81
| A118 | 0.4603 | 0.989 | 0.841 | $-1.048$ | – | – | – | – | 0.399 | 49.93 | 1.112 | – | – | – | 127.99 | 139.99 | 156.61 | 3.27 | 8.80
| ML | 0.1525 | 0.361 | $-1.893$ | 0.036 | – | 0.269 | 0.949 | $-0.976$ | – | – | – | – | – | – | 7.39 | 19.39 | 27.99 | 2.71 | 5.58
| A115 | 0.4602 | 0.989 | 0.955 | $-1.097$ | – | – | – | – | 0.404 | 49.91 | 1.115 | – | – | – | 127.18 | 139.18 | 155.65 | 3.21 | 8.70
Non-flat XCDM | ML + A115 | 0.3647 | 0.794 | 0.002 | $-4.000$ | – | 0.269 | 1.478 | $-1.015$ | 0.404 | 50.05 | 1.107 | – | – | – | 136.75 | 154.75 | 181.60 | 4.05 | 10.01
| GL | 0.0378 | 0.127 | $-0.174$ | $-4.518$ | – | – | – | – | – | – | – | 0.327 | 1.237 | $-1.299$ | 16.61 | 28.61 | 35.68 | $-2.33$ | 0.03
| A115′ | 0.4643 | 0.997 | 0.783 | $-0.956$ | – | – | – | – | 0.403 | 49.94 | 1.112 | – | – | – | 125.92 | 137.92 | 154.39 | 3.58 | 9.07
| GL + A115′ | 0.4349 | 0.938 | $-0.255$ | $-0.043$ | – | – | – | – | 0.410 | 50.03 | 1.063 | 0.358 | 0.296 | $-1.616$ | 149.16 | 167.16 | 193.57 | 3.82 | 9.69
| A118 | 0.2301 | 0.520 | – | – | 9.936 | – | – | – | 0.402 | 50.02 | 1.109 | – | – | – | 128.56 | 138.56 | 152.41 | 1.84 | 4.60
| ML | 0.4651 | 0.999 | – | – | 5.225 | 0.275 | 1.383 | $-1.011$ | – | – | – | – | – | – | 8.68 | 18.68 | 25.85 | 2.00 | 3.44
| A115 | 0.2065 | 0.471 | – | – | 9.932 | – | – | – | 0.405 | 50.04 | 1.106 | – | – | – | 127.79 | 137.79 | 151.52 | 1.82 | 4.57
Flat $\phi$CDM | ML + A115 | 0.4563 | 0.981 | – | – | 2.762 | 0.273 | 1.391 | $-1.021$ | 0.405 | 49.99 | 1.099 | – | – | – | 136.70 | 152.70 | 176.57 | 2.00 | 4.98
| GL | 0.4641 | 0.997 | – | – | 4.299 | – | – | – | – | – | – | 0.372 | 0.360 | $-1.674$ | 22.94 | 32.94 | 38.83 | 2.00 | 3.18
| A115′ | 0.3334 | 0.730 | – | – | 9.652 | – | – | – | 0.402 | 50.02 | 1.096 | – | – | – | 126.29 | 136.29 | 150.01 | 1.95 | 4.69
| GL + A115′ | 0.4484 | 0.965 | – | – | 8.745 | – | – | – | 0.403 | 50.05 | 1.077 | 0.372 | 0.348 | $-1.663$ | 149.36 | 165.36 | 188.83 | 2.02 | 4.95
| A118 | 0.3245 | 0.712 | 0.245 | – | 8.862 | – | – | – | 0.400 | 50.01 | 1.116 | – | – | – | 128.42 | 140.42 | 157.04 | 3.70 | 9.23
| ML | 0.4558 | 0.980 | $-0.980$ | – | 0.423 | 0.266 | 1.296 | $-0.973$ | – | – | – | – | – | – | 7.48 | 19.48 | 28.09 | 2.80 | 5.68
| A115 | 0.3217 | 0.706 | 0.290 | – | 3.150 | – | – | – | 0.406 | 50.05 | 1.108 | – | – | – | 127.64 | 139.64 | 156.11 | 3.67 | 9.16
Non-flat $\phi$CDM | ML + A115 | 0.3044 | 0.671 | $-0.076$ | – | 8.893 | 0.274 | 1.406 | $-1.001$ | 0.404 | 50.03 | 1.095 | – | – | – | 136.77 | 154.77 | 181.62 | 4.07 | 10.03
| GL | 0.4644 | 0.998 | $-0.993$ | – | 0.173 | – | – | – | – | – | – | 0.340 | 0.337 | $-1.547$ | 20.15 | 32.15 | 39.22 | 1.21 | 3.57
| A115′ | 0.3787 | 0.823 | 0.161 | – | 7.940 | – | – | – | 0.404 | 50.02 | 1.105 | – | – | – | 126.21 | 138.21 | 154.68 | 3.87 | 9.36
| GL + A115′ | 0.4420 | 0.952 | $-0.266$ | – | 8.343 | – | – | – | 0.400 | 50.00 | 1.077 | 0.372 | 0.310 | $-1.622$ | 149.18 | 167.18 | 193.59 | 3.84 | 9.71

  * a

In these GRB cases, $\Omega_{b}$ and $H_{0}$ are set to be 0.05 and 70
$\rm{km\ s^{-1}\ Mpc^{-1}}$, respectively.

Table 7: Unmarginalized best-fitting parameter values for all models from
various combinations of data.a

Model | Data set | $\Omega_{\mathrm{m0}}$ | $\Omega_{\mathrm{k0}}$ | $w_{\mathrm{X}}$ | $\alpha$ | $\sigma_{\mathrm{int,\,\textsc{ml}}}$ | $b_{\mathrm{\textsc{ml}}}$ | $k_{\mathrm{\textsc{ml}}}$ | $\sigma_{\mathrm{int}}$ | $\gamma$ | $\beta$ | $\sigma_{\mathrm{int,\,\textsc{gl}}}$ | $b_{\mathrm{\textsc{gl}}}$ | $k_{\mathrm{\textsc{gl}}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| A118 | $>0.247$ | – | – | – | – | – | – | $0.412^{+0.027}_{-0.033}$ | $50.09\pm 0.26$ | $1.110\pm 0.090$ | – | – | –
| ML | $>0.188$ | – | – | – | $0.305^{+0.035}_{-0.053}$ | $1.552^{+0.108}_{-0.189}$ | $-1.017\pm 0.090$ | – | – | – | – | – | –
| A115 | $0.630^{+0.352}_{-0.135}$ | – | – | – | – | – | – | $0.417^{+0.028}_{-0.035}$ | $50.09\pm 0.26$ | $1.112\pm 0.093$ | – | – | –
Flat $\Lambda$CDM | ML + A115 | $>0.298$ | – | – | – | $0.301^{+0.033}_{-0.051}$ | $1.515^{+0.101}_{-0.151}$ | $-1.015\pm 0.088$ | $0.416^{+0.027}_{-0.034}$ | $50.07\pm 0.25$ | $1.111\pm 0.089$ | – | – | –
| GL | $>0.202$ | – | – | – | – | – | – | – | – | – | $0.429^{+0.059}_{-0.094}$ | $0.495^{+0.120}_{-0.173}$ | $-1.720\pm 0.219$
| A115′ | $>0.264$ | – | – | – | – | – | – | $0.414^{+0.028}_{-0.034}$ | $50.10\pm 0.26$ | $1.107\pm 0.090$ | – | – | –
| GL + A115′ | $>0.339$ | – | – | – | – | – | – | $0.413^{+0.027}_{-0.034}$ | $50.08\pm 0.25$ | $1.104\pm 0.089$ | $0.423^{+0.056}_{-0.090}$ | $0.458^{+0.112}_{-0.139}$ | $-1.705\pm 0.210$
| A118 | $>0.287$ | $0.694^{+0.626}_{-0.848}$ | – | – | – | – | – | $0.412^{+0.027}_{-0.034}$ | $50.01\pm 0.26$ | $1.121\pm 0.090$ | – | – | –
| ML | $>0.241$ | $-0.131^{+0.450}_{-0.919}$ | – | – | $0.304^{+0.035}_{-0.053}$ | $1.478^{+0.123}_{-0.166}$ | $-1.000\pm 0.096$ | – | – | – | – | – | –
| A115 | $>0.275$ | $0.698^{+0.639}_{-0.857}$ | – | – | – | – | – | $0.417^{+0.028}_{-0.034}$ | $50.00\pm 0.27$ | $1.124\pm 0.092$ | – | – | –
Non-flat $\Lambda$CDM | ML + A115 | $>0.346$ | $0.352^{+0.427}_{-0.830}$ | – | – | $0.304^{+0.034}_{-0.052}$ | $1.486^{+0.096}_{-0.136}$ | $-1.019\pm 0.091$ | $0.416^{+0.028}_{-0.034}$ | $50.03\pm 0.26$ | $1.113\pm 0.091$ | – | – | –
| GL | $>0.290$ | $-0.762^{+0.271}_{-0.888}$ | – | – | – | – | – | – | – | – | $0.402^{+0.057}_{-0.090}$ | $0.407^{+0.136}_{-0.160}$ | $-1.536\pm 0.252$
| A115′ | $>0.299$ | $0.599^{+0.582}_{-0.887}$ | – | – | – | – | – | $0.414^{+0.028}_{-0.034}$ | $50.02\pm 0.26$ | $1.117\pm 0.090$ | – | – | –
| GL + A115′ | $>0.381$ | $0.214^{+0.428}_{-0.855}$ | – | – | – | – | – | $0.414^{+0.027}_{-0.034}$ | $50.05\pm 0.26$ | $1.103\pm 0.090$ | $0.423^{+0.056}_{-0.090}$ | $0.432^{+0.111}_{-0.132}$ | $-1.701\pm 0.212$
| A118 | $0.599^{+0.350}_{-0.175}$ | – | $-2.440^{+1.779}_{-1.715}$ | – | – | – | – | $0.412^{+0.028}_{-0.034}$ | $50.15^{+0.26}_{-0.30}$ | $1.106\pm 0.089$ | – | – | –
| ML | $>0.123$ | – | $-2.456^{+2.567}_{-2.180}$ | – | $0.306^{+0.036}_{-0.054}$ | $1.611^{+0.113}_{-0.277}$ | $-1.014\pm 0.092$ | – | – | – | – | – | –
| A115 | $0.589^{+0.357}_{-0.184}$ | – | $-2.411^{+1.797}_{-1.729}$ | – | – | – | – | $0.417^{+0.028}_{-0.035}$ | $50.14^{+0.27}_{-0.31}$ | $1.109\pm 0.092$ | – | – | –
Flat XCDM | ML + A115 | $>0.191$ | – | $<-0.041$ | – | $0.300^{+0.033}_{-0.051}$ | $1.562^{+0.103}_{-0.220}$ | $-1.012\pm 0.088$ | $0.417^{+0.028}_{-0.034}$ | $50.13^{+0.27}_{-0.30}$ | $1.108\pm 0.090$ | – | – | –
| GL | $>0.141$ | – | $<-0.046$ | – | – | – | – | – | – | – | $0.428^{+0.058}_{-0.092}$ | $0.556^{+0.127}_{-0.256}$ | $-1.706\pm 0.215$
| A115′ | $0.605^{+0.394}_{-0.126}$ | – | $-2.391^{+1.826}_{-1.758}$ | – | – | – | – | $0.414^{+0.028}_{-0.034}$ | $50.15^{+0.27}_{-0.30}$ | $1.103\pm 0.092$ | – | – | –
| GL + A115′ | $>0.205$ | – | $<-0.017$ | – | – | – | – | $0.413^{+0.027}_{-0.034}$ | $50.14^{+0.26}_{-0.30}$ | $1.101\pm 0.088$ | $0.421^{+0.055}_{-0.090}$ | $0.512^{+0.113}_{-0.210}$ | $-1.698\pm 0.208$
| A118 | $>0.246$ | $0.590^{+0.476}_{-0.796}$ | $-2.358^{+2.032}_{-1.154}$ | – | – | – | – | $0.412^{+0.027}_{-0.033}$ | $50.01\pm 0.28$ | $1.121\pm 0.091$ | – | – | –
| ML | $>0.174$ | $-0.262^{+0.580}_{-0.724}$ | $-2.000^{+2.117}_{-1.264}$ | – | $0.305^{+0.036}_{-0.054}$ | $1.462^{+0.194}_{-0.196}$ | $-0.996\pm 0.097$ | – | – | – | – | – | –
| A115 | $>0.240$ | $0.563^{+0.498}_{-0.796}$ | $-2.290^{+2.146}_{-1.032}$ | – | – | – | – | $0.418^{+0.028}_{-0.034}$ | $50.01\pm 0.28$ | $1.122\pm 0.093$ | – | – | –
Non-flat XCDM | ML + A115 | $>0.231$ | $0.202^{+0.376}_{-0.635}$ | $-2.155^{+2.224}_{-1.156}$ | – | $0.302^{+0.033}_{-0.051}$ | $1.489^{+0.122}_{-0.167}$ | $-1.016\pm 0.089$ | $0.417^{+0.027}_{-0.034}$ | $50.05\pm 0.28$ | $1.109\pm 0.090$ | – | – | –
| GL | $>0.194$ | $-0.615^{+0.470}_{-0.685}$ | $-2.212^{+2.186}_{-0.962}$ | – | – | – | – | – | – | – | $0.403^{+0.058}_{-0.092}$ | $0.480^{+0.177}_{-0.223}$ | $-1.532^{+0.259}_{-0.260}$
| A115′ | $>0.236$ | $0.451^{+0.469}_{-0.789}$ | $-2.210^{+2.208}_{-0.946}$ | – | – | – | – | $0.415^{+0.028}_{-0.034}$ | $50.03\pm 0.28$ | $1.114\pm 0.091$ | – | – | –
| GL + A115′ | $>0.226$ | $0.014^{+0.408}_{-0.604}$ | $-2.080^{+2.201}_{-1.138}$ | – | – | – | – | $0.415^{+0.027}_{-0.033}$ | $50.09^{+0.27}_{-0.30}$ | $1.096\pm 0.090$ | $0.416^{+0.055}_{-0.088}$ | $0.446^{+0.142}_{-0.183}$ | $-1.682\pm 0.208$
| A118 | $0.568^{+0.332}_{-0.230}$ | – | – | – | – | – | – | $0.411^{+0.027}_{-0.033}$ | $50.05\pm 0.25$ | $1.110\pm 0.089$ | – | – | –
| ML | $>0.148$ | – | – | – | $0.304^{+0.035}_{-0.053}$ | $1.493^{+0.093}_{-0.143}$ | $-1.017\pm 0.089$ | – | – | – | – | – | –
| A115 | $0.565^{+0.339}_{-0.228}$ | – | – | – | – | – | – | $0.416^{+0.028}_{-0.034}$ | $50.04\pm 0.25$ | $1.113\pm 0.091$ | – | – | –
Flat $\phi$CDM | ML + A115 | $>0.198$ | – | – | – | $0.302^{+0.033}_{-0.051}$ | $1.477^{+0.091}_{-0.126}$ | $-1.016\pm 0.088$ | $0.416^{+0.027}_{-0.034}$ | $50.03\pm 0.25$ | $1.110\pm 0.089$ | – | – | –
| GL | $>0.148$ | – | – | – | – | – | – | – | – | – | $0.428^{+0.059}_{-0.094}$ | $0.444^{+0.112}_{-0.141}$ | $-1.710\pm 0.218$
| A115′ | $0.586^{+0.391}_{-0.156}$ | – | – | – | – | – | – | $0.414^{+0.028}_{-0.034}$ | $50.06\pm 0.25$ | $1.106\pm 0.089$ | – | – | –
| GL + A115′ | $>0.231$ | – | – | – | – | – | – | $0.413^{+0.027}_{-0.033}$ | $50.05\pm 0.24$ | $1.102\pm 0.088$ | $0.423^{+0.055}_{-0.090}$ | $0.426^{+0.105}_{-0.119}$ | $-1.703\pm 0.210$
| A118 | $0.560^{+0.256}_{-0.247}$ | $-0.002^{+0.294}_{-0.286}$ | – | $5.203^{+3.808}_{-2.497}$ | – | – | – | $0.412^{+0.027}_{-0.033}$ | $50.05\pm 0.25$ | $1.111^{+0.089}_{-0.090}$ | – | – | –
| ML | $>0.207$ | $-0.163^{+0.355}_{-0.317}$ | – | – | $0.303^{+0.035}_{-0.053}$ | $1.448^{+0.120}_{-0.165}$ | $-1.011\pm 0.091$ | – | – | – | – | – | –
| A115 | $0.546^{+0.260}_{-0.253}$ | $0.011^{+0.299}_{-0.291}$ | – | – | – | – | – | $0.417^{+0.028}_{-0.034}$ | $50.04\pm 0.26$ | $1.115\pm 0.092$ | – | – | –
Non-flat $\phi$CDM | ML + A115 | $0.630^{+0.286}_{-0.181}$ | $-0.108^{+0.296}_{-0.263}$ | – | – | $0.301^{+0.033}_{-0.051}$ | $1.453^{+0.115}_{-0.139}$ | $-1.013\pm 0.089$ | $0.416^{+0.027}_{-0.034}$ | $50.03\pm 0.25$ | $1.106\pm 0.090$ | – | – | –
| GL | $>0.207$ | $-0.193^{+0.364}_{-0.347}$ | – | – | – | – | – | – | – | – | $0.422^{+0.057}_{-0.092}$ | $0.408^{+0.121}_{-0.149}$ | $-1.693^{+0.215}_{-0.214}$
| A115′ | $0.581^{+0.254}_{-0.247}$ | $-0.033^{+0.296}_{-0.290}$ | – | $5.215^{+3.853}_{-2.429}$ | – | – | – | $0.414^{+0.028}_{-0.034}$ | $50.05\pm 0.25$ | $1.106\pm 0.091$ | – | – | –
| GL + A115′ | $0.666^{+0.327}_{-0.106}$ | $-0.169^{+0.317}_{-0.270}$ | – | – | – | – | – | $0.413^{+0.027}_{-0.034}$ | $50.04\pm 0.25$ | $1.096\pm 0.089$ | $0.419^{+0.055}_{-0.089}$ | $0.401^{+0.115}_{-0.128}$ | $-1.686\pm 0.209$

  * a

In these GRB cases, $\Omega_{b}$ and $H_{0}$ are set to be 0.05 and 70
$\rm{km\ s^{-1}\ Mpc^{-1}}$, respectively.

Table 8: One-dimensional marginalized posterior mean values and uncertainties
($\pm 1\sigma$ error bars or $2\sigma$ limits) of the parameters for all
models from various combinations of data.a

The A118 data set was previously studied by Khadka et al. (2021b). Here we
analyze it along with the truncated A115 and A115′ data sets, which are also
used in joint analyses with the ML and GL data sets. The constraints from
these data sets on the GRB correlation parameters and on the cosmological
model parameters are presented in Tables 7 and 8. The corresponding posterior
1D probability distributions and 2D confidence regions of these parameters are
shown in Figs. 5 and 6, in gray (A118), red (A115 and A115′), green (ML and
GL), and purple (ML + A115 and GL + A115′). Note that these analyses assume
$H_{0}=70$ $\rm{km\ s^{-1}\ Mpc^{-1}}$ and $\Omega_{b}=0.05$.

The constraints from A115 data and from ML data, and from A115′ data and from
GL data, are not mutually inconsistent, so it is not unreasonable to examine
joint ML + A115 and GL + A115′ constraints. The ML data have the smallest
intrinsic dispersion, $\sim 0.30-0.31$, with A115, A115′, and GL having larger
intrinsic dispersion, $\sim 0.40-0.43$

The constraints on the Amati parameters are quite cosmological-model-
independent for these GRB data sets. In the A118 case, the slope $\beta$
ranges from a high of $1.121\pm 0.091$ (non-flat XCDM) to a low of $1.106\pm
0.089$ (flat XCDM), the intercept $\gamma$ ranges from a high of
$50.15^{+0.26}_{-0.30}$ (flat XCDM) to a low of $50.01\pm 0.26$ (non-flat
$\Lambda$CDM), and the intrinsic scatter $\sigma_{\rm int}$ ranges from a high
of $0.412^{+0.028}_{-0.034}$ (flat XCDM) to a low of $0.411^{+0.027}_{-0.033}$
(flat $\phi$CDM), with central values of each pair being $0.12\sigma$,
$0.35\sigma$, and $0.02\sigma$ away from each other, respectively.

In the A115 case, the slope $\beta$ ranges from a high of $1.124\pm 0.092$
(non-flat $\Lambda$CDM) to a low of $1.109\pm 0.092$ (flat XCDM), the
intercept $\gamma$ ranges from a high of $50.14^{+0.27}_{-0.31}$ (flat XCDM)
to a low of $50.00\pm 0.27$ (non-flat $\Lambda$CDM), and the intrinsic scatter
$\sigma_{\rm int}$ ranges from a high of $0.418^{+0.028}_{-0.034}$ (non-flat
XCDM) to a low of $0.416^{+0.028}_{-0.034}$ (flat $\phi$CDM), with central
values of each pair being $0.12\sigma$, $0.34\sigma$, and $0.05\sigma$ away
from each other, respectively.

In the A115′ case, the slope $\beta$ ranges from a high of $1.117\pm 0.090$
(non-flat $\Lambda$CDM) to a low of $1.103\pm 0.092$ (flat XCDM), the
intercept $\gamma$ ranges from a high of $50.15^{+0.27}_{-0.30}$ (flat XCDM)
to a low of $50.02\pm 0.26$ (non-flat $\Lambda$CDM), and the intrinsic scatter
$\sigma_{\rm int}$ ranges from a high of $0.415^{+0.028}_{-0.034}$ (non-flat
XCDM) to a low of $0.414^{+0.028}_{-0.034}$ (the others), with central values
of each pair being $0.11\sigma$, $0.33\sigma$, and $0.02\sigma$ away from each
other, respectively.

The lowest and highest values of $\beta$, $\gamma$, and $\sigma_{\rm int}$
from the A118, A115, and A115′ cases differ from each other at $0.16\sigma$,
$0.37\sigma$, and $0.16\sigma$, respectively. This implies that excluding
three GRBs from A118 does not significantly affect the constraints on the
Amati parameters.

In the joint analysis of ML and A115 (ML + A115) data, $\beta$ ranges from a
high of $1.113\pm 0.091$ (non-flat $\Lambda$CDM) to a low of $1.106\pm 0.090$
(non-flat $\phi$CDM), $\gamma$ ranges from a high of $50.13^{+0.27}_{-0.30}$
(flat XCDM) to a low of $50.03\pm 0.25$ (flat and non-flat $\phi$CDM), and
$\sigma_{\rm int}$ ranges from a high of $0.417^{+0.028}_{-0.034}$ (flat XCDM)
to a low of $0.416^{+0.027}_{-0.034}$ (flat $\Lambda$CDM, and flat and non-
flat $\phi$CDM), with central values of each pair being $0.05\sigma$,
$0.26\sigma$, and $0.02\sigma$ away from each other, respectively; also $k$
ranges from a high of $-1.012\pm 0.088$ (flat XCDM) to a low of $-1.019\pm
0.091$ (non-flat $\Lambda$CDM), $b$ ranges from a high of
$1.562^{+0.103}_{-0.220}$ (flat XCDM) to a low of $1.453^{+0.115}_{-0.139}$
(non-flat $\phi$CDM), and $\sigma_{\mathrm{int,\,\textsc{ml}}}$ ranges from a
high of $0.304^{+0.034}_{-0.052}$ (non-flat $\Lambda$CDM) to a low of
$0.300^{+0.033}_{-0.051}$ (flat XCDM), with central values of each pair being
$0.06\sigma$, $0.44\sigma$, and $0.06\sigma$ away from each other,
respectively. The lowest and highest values of $\beta$, $\gamma$, and
$\sigma_{\rm int}$ from the A115 and ML + A115 cases differ from each other at
$0.14\sigma$, $0.32\sigma$, and $0.05\sigma$, respectively; also those of $k$,
$b$, and $\sigma_{\mathrm{int,\,\textsc{ml}}}$ differ from each other at
$0.17\sigma$, $0.53\sigma$, and $0.09\sigma$, respectively.

In the joint analysis of GL and A115′ (GL + A115′) data, $\beta$ ranges from a
high of $1.104\pm 0.089$ (flat $\Lambda$CDM) to a low of $1.096\pm 0.089$
(non-flat $\phi$CDM), $\gamma$ ranges from a high of $50.14^{+0.26}_{-0.30}$
(flat XCDM) to a low of $50.04\pm 0.25$ (non-flat $\phi$CDM), and $\sigma_{\rm
int}$ ranges from a high of $0.415^{+0.027}_{-0.033}$ (non-flat XCDM) to a low
of $0.413^{+0.027}_{-0.034}$ (flat $\Lambda$CDM, flat XCDM, and non-flat
$\phi$CDM), with central values of each pair being $0.06\sigma$, $0.26\sigma$,
and $0.05\sigma$ away from each other, respectively; also $k$ ranges from a
high of $-1.682\pm 0.208$ (non-flat XCDM) to a low of $-1.705\pm 0.210$ (flat
$\Lambda$CDM), $b$ ranges from a high of $0.512^{+0.113}_{-0.210}$ (flat XCDM)
to a low of $0.401^{+0.115}_{-0.128}$ (non-flat $\phi$CDM), and
$\sigma_{\mathrm{int,\,\textsc{gl}}}$ ranges from a high of
$0.423^{+0.056}_{-0.090}$ (flat and non-flat $\Lambda$CDM) to a low of
$0.416^{+0.055}_{-0.088}$ (non-flat XCDM), with central values of each pair
being $0.08\sigma$, $0.46\sigma$, and $0.07\sigma$ away from each other,
respectively. The lowest and highest values of $\beta$, $\gamma$, and
$\sigma_{\rm int}$ from the A115′ and GL + A115′ cases differ from each other
at $0.17\sigma$, $0.30\sigma$, and $0.05\sigma$, respectively; also those of
$k$, $b$, and $\sigma_{\mathrm{int,\,\textsc{gl}}}$ differ from each other at
$0.52\sigma$, $0.47\sigma$, and $0.20\sigma$, respectively.

(a) Flat $\Lambda$CDM (b) Non-flat $\Lambda$CDM

(c) Flat XCDM (d) Non-flat XCDM

(e) Flat $\phi$CDM (f) Non-flat $\phi$CDM

Figure 5: One-dimensional likelihoods and 1$\sigma$, 2$\sigma$, and 3$\sigma$
two-dimensional likelihood confidence contours from MD-LGRB (green), A118
(gray), A115 (red), and MD-LGRB + A115 (purple) data for all six models. The
zero-acceleration lines are shown as black dashed lines, which divide the
parameter space into regions associated with currently-accelerating and
currently-decelerating cosmological expansion. In the non-flat XCDM and non-
flat $\phi$CDM cases, the zero-acceleration lines are computed for the third
cosmological parameter set to the $H(z)$ \+ BAO data best-fitting values
listed in Table 7. The crimson dash-dot lines represent flat hypersurfaces,
with closed spatial hypersurfaces either below or to the left. The magenta
lines represent $w_{\rm X}=-1$, i.e. flat or non-flat $\Lambda$CDM models. The
$\alpha=0$ axes correspond to flat and non-flat $\Lambda$CDM models in panels
(e) and (f), respectively.

(a) Flat $\Lambda$CDM (b) Non-flat $\Lambda$CDM

(c) Flat XCDM (d) Non-flat XCDM

(e) Flat $\phi$CDM (f) Non-flat $\phi$CDM

Figure 6: One-dimensional likelihoods and 1$\sigma$, 2$\sigma$, and 3$\sigma$
two-dimensional likelihood confidence contours from GW-LGRB (green), A118
(gray), A11$5^{\prime}$ (red), and GW-LGRB + A11$5^{\prime}$ (purple) data for
all six models. The zero-acceleration lines are shown as black dashed lines,
which divide the parameter space into regions associated with currently-
accelerating and currently-decelerating cosmological expansion. In the non-
flat XCDM and non-flat $\phi$CDM cases, the zero-acceleration lines are
computed for the third cosmological parameter set to the $H(z)$ \+ BAO data
best-fitting values listed in Table 7. The crimson dash-dot lines represent
flat hypersurfaces, with closed spatial hypersurfaces either below or to the
left. The magenta lines represent $w_{\rm X}=-1$, i.e. flat or non-flat
$\Lambda$CDM models. The $\alpha=0$ axes correspond to flat and non-flat
$\Lambda$CDM models in panels (e) and (f), respectively.

Judging from the constraints on the parameters of the Amati and Dainotti
correlations, those from the joint analyses do not deviate much from the
individual cases. We next focus on constraints on cosmological model
parameters.

Similar to GRB data from Sec. 5.1, these data more favor currently
accelerating cosmological expansion in the $\Lambda$CDM and XCDM cases, but
also more favor currently decelerating cosmological expansion in the $\phi$CDM
models, in the $\Omega_{\rm m0}-\alpha$ and $\Omega_{\rm m0}-\Omega_{\rm k0}$
parameter subspaces.

In the flat $\Lambda$CDM model, the A118, A115, and A115′ $2\sigma$
constraints on $\Omega_{\rm m0}$ are mutually consistent (A115 with
$\Omega_{\rm m0}>0.241$), but they favor higher values of $\Omega_{\rm m0}$
than do ML and GL while the joint ML + A115 and GL + A115′ cases favor even
higher values of $>0.298$ and $>0.339$, respectively. In fact, this is also
true for the ML + GL case ($\Omega_{\rm m0}>0.294$).

In the non-flat $\Lambda$CDM model, the A118, A115, and A115′ $2\sigma$
constraints on $\Omega_{\rm m0}$ are also mutually consistent, but they favor
slightly higher values of $\Omega_{\rm m0}$ than in the flat $\Lambda$CDM
case, with $\Omega_{\rm m0}>0.299$ in the A115′ non-flat $\Lambda$CDM case.
The joint ML + A115, GL + A115′, and ML + GL cases favor higher $\Omega_{\rm
m0}$ $2\sigma$ limits of $>0.346$, $>0.381$, and $>0.338$, respectively. The
A118, A115, ML + A115, A115′, and GL + A115′ data mildly favor open
hypersurfaces in the non-flat $\Lambda$CDM model, being less than $1\sigma$
away from flatness.

In the flat and non-flat XCDM parametrizations, the $2\sigma$ constraints on
$\Omega_{\rm m0}$ are mutually consistent in all cases, where in the flat XCDM
parametrization, the $2\sigma$ limits are $\Omega_{\rm m0}>0.181$ (A118),
$>0.170$ (A115), and $>0.185$ (A115′). The constraints on $w_{\rm X}$ are very
loose, and thus affected by the $w_{\rm X}$ prior, and consistent with each
other in all cases, and mildly favor phantom dark energy (but $\Lambda$ is
less than $1\sigma$ away). In the non-flat XCDM parametrization, the A118,
A115, ML + A115, A115′, and GL + A115′ data also mildly favor open
hypersurfaces, with flatness being less than $1\sigma$ away.

In the flat $\phi$CDM model, the A118, A115, and A115′ constraints on
$\Omega_{\rm m0}$ are mutually consistent, with $2\sigma$ limits of
$\Omega_{\rm m0}>0.149$ (A118), $>0.145$ (A115), and $>0.159$ (A115′), which
are consistent with the other cases. These GRB data do not provide constraints
on $\alpha$ in the flat $\phi$CDM model, while in the non-flat $\phi$CDM
model, A118 and A115′ provide constraints of $\alpha=5.203^{+3.808}_{-2.497}$
and $\alpha=5.215^{+3.853}_{-2.429}$, respectively. Note that $\alpha=0$ is
still within $2\sigma$ for both cases. Similar trends hold for non-flat
$\phi$CDM $\Omega_{\rm m0}$ constraints, but ML + A115 and GL + A115′ data
constraints posterior mean values are larger than for the individual data
sets. The $2\sigma$ limits are $\Omega_{\rm m0}>0.183$ (A118), $\Omega_{\rm
m0}=0.546^{+0.449}_{-0.384}$ (A115), $\Omega_{\rm m0}>0.198$ (A115′), $>0.251$
(ML + A115), and $>0.286$ (GL + A115′). Except for the A115 data, non-flat
$\phi$CDM constraints favor closed hypersurfaces (unlike non-flat $\Lambda$CDM
and non-flat XCDM), but with flatness well within $1\sigma$ for all cases.

The $\Delta AIC$ and $\Delta BIC$ values with respect to the flat $\Lambda$CDM
model are listed in the last two columns of Table 7. In all cases (except for
the GL case, which is discussed in Sec. 5.1 above), the flat $\Lambda$CDM
model is the most favored model but the evidence against the other models are
either weak or positive, except that, based on $BIC$, the evidence against
non-flat XCDM and non-flat $\phi$CDM are strong. For ML + A115 data the non-
flat $\phi$CDM model is very strongly disfavored with $\Delta BIC=10.03$.

In summary, while the joint analyses do slightly tighten the constraints, the
improvements relative to those from A118 data alone are not significant.

##  6 Conclusion

We have used six different cosmological models in analyses of the three (ML,
MS, and GL) Dainotti ($L_{0}-t_{b}$) correlation GRB data sets compiled by
Wang et al. (2021) and Hu et al. (2021). We find for each data sets, as well
as the MS + GL, ML + GL, and ML + MS combinations, that the GRB correlation
parameters are independent of cosmological model. Our results thus indicate
that these GRBs are standardizable through the Dainotti correlation and so can
be used to constrain cosmological parameters, justifying the assumption made
by Wang et al. (2021) and Hu et al. (2021). These results also mean that the
circularity problem does not affect cosmological parameter constraints derived
from these GRB data.

In contrast to Wang et al. (2021) and Hu et al. (2021) we do not use $H(z)$
data to calibrate these GRB data, instead we use these data to derive GRB only
cosmological constraints. We find that ML, MS, GL, MS + GL, ML + GL, and ML +
MS GRBs provide only weak restrictions on cosmological parameters.

We have also used the more-restrictive ML and GL Dainotti data sets in joint
analyses with the largest available reliable compilation of Amati ($E_{\rm
p}-E_{\rm iso}$) correlation A118 GRB data (Khadka et al., 2021b), but
excluding three overlapping GRBs from the A118 data in the joint analyses.
While the joint analyses do result in slightly tighter constraints, typically
with larger lower limits on $\Omega_{\rm m0}$ than those from the ML, GL, or
A118 data alone, the improvements relative to the A118 data constraints are
not significant.

Current GRB data provide quite weak constraints on cosmological parameters but
do favor currently accelerated cosmological expansion in the $\Lambda$CDM
models and the XCDM parametrizations. We hope that in the near future there
will be more and better-quality GRB measurements that will result in more
restrictive GRB cosmological constraints. GRBs probe a very wide range of
cosmological redshift space, a significant part of which is as yet unprobed,
so it is worth putting effort into further developing GRB cosmological
constraints.

## Acknowledgements

We thank F. Y. Wang and J. P. Hu for valuable discussions. This research was
supported in part by DOE grant DE-SC0011840. Part of the computation for this
project was performed on the Beocat Research Cluster at Kansas State
University.

## Data availability

MD-LGRB data are available in Wang et al. (2021) and MD-SGRB and GW-LGRB data
are available in Hu et al. (2021).

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