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

ABSTRACT

We show that the Platinum gamma-ray burst (GRB) data compilation, probing the redshift range 0.553 ≤ z ≤ 5.0, obeys a cosmological-model-independent three-parameter Fundamental Plane (Dainotti) correlation and so is standardizable. While they probe the largely unexplored z ∼ 2.3–5 part of cosmological redshift space, the GRB cosmological parameter constraints are consistent with, but less precise than, those from a combination of baryon acoustic oscillation (BAO) and Hubble parameter [H(z)] data. In order to increase the precision of GRB-only cosmological constraints, we exclude common GRBs from the larger Amati-correlated A118 data set composed of 118 GRBs and jointly analyse the remaining 101 Amati-correlated GRBs with the 50 Platinum GRBs. This joint 151 GRB data set probes the largely unexplored z ∼ 2.3–8.2 region; the resulting GRB-only cosmological constraints are more restrictive, and consistent with, but less precise than, those from H(z)  + BAO data.

1. INTRODUCTION

Currently accelerated cosmological expansion and other cosmological observations are reasonably well accommodated in the spatially flat Λ cold dark matter (ΛCDM) model (Peebles 1984) with of the current cosmological energy budget being a time-independent cosmological constant (Λ), being non-relativistic CDM, and most of the remaining being non-relativistic baryonic matter (see e.g. Farooq et al. 2017; Scolnic et al. 2018; Planck Collaboration 2020; eBOSS Collaboration 2021). In this paper, we also study cosmological models with a little spatial curvature or dynamical dark energy, since the observations do not rule out such models, and since some sets of measurements seem mutually incompatible when analysed in the spatially flat ΛCDM model (see e.g. Di Valentino, Melchiorri & Silk 2021b; Perivolaropoulos & Skara 2021).

It is still unclear whether this incompatibility is evidence against the spatially flat ΛCDM model or is caused by unidentified systematic errors in one of the established cosmological probes or by evolution of the parameters themselves with the redshift (Dainotti et al. 2021b, 2022). Newer, alternate cosmological probes could help alleviate this issue. Recent examples of such probes include reverberation-mapped quasar (QSO) measurements that reach to redshift z ∼ 1.9 (Czerny et al. 2021; Khadka et al. 2021a,b; Yu et al. 2021; Zajaček et al. 2021), H ii starburst galaxy measurements that reach to z ∼ 2.4 (Mania & Ratra 2012; Chávez et al. 2014; González-Morán et al. 2019, 2021; Cao, Ryan & Ratra 2020, 2022a; Cao et al. 2021a; Johnson, Sangwan & Shankaranarayanan 2022; Mehrabi et al. 2022), QSO angular size measurements that reach to z ∼ 2.7 (Cao et al. 2017, 2020, 2021a; Ryan, Chen & Ratra 2019; Lian et al. 2021; Zheng et al. 2021), QSO flux measurements that reach to z ∼ 7.5 (Risaliti & Lusso 2015, 2019; Khadka & Ratra 2020a,b, 2021, 2022; Lusso et al. 2020; Yang, Banerjee & Ó Colgáin 2020; Li et al. 2021; Lian et al. 2021; Luongo et al. 2021; Rezaei, Solà Peracaula & Malekjani 2021; Zhao & Xia 2021),¹ and the main subject of this paper, gamma-ray burst (GRB) measurements that reach to z ∼ 8.2 (Amati et al. 2008, 2019; Cardone, Capozziello & Dainotti 2009; Cardone et al. 2010; Samushia & Ratra 2010; Dainotti et al. 2011, 2013a,b; Postnikov et al. 2014; Wang, Dai & Liang 2015; Wang et al. 2016, 2022; Fana Dirirsa et al. 2019; Khadka & Ratra 2020c; Hu, Wang & Dai 2021; Dai et al. 2021; Demianski et al. 2021; Khadka et al. 2021c; Luongo et al. 2021; Luongo & Muccino 2021; Cao et al. 2021a). Some of these probes might eventually allow for a reliable extension of the Hubble diagram to z ∼ 3–4, well beyond the reach of Type Ia supernovae. GRBs have been detected to z ∼ 9.4 (Cucchiara et al. 2011), and might be detectable to z = 20 (Lamb & Reichart 2000), so in principle GRBs could act as a cosmological probe to higher redshifts than 8.2.

Cao, Khadka & Ratra (2022b) recently used the A118 two-parameter Amati-correlated GRB data set (Khadka et al. 2021c) and the long GRB whose plateau phase is dominated by magnetic dipole radiation (MD-LGRB) and gravational wave emission (GW-LGRB), and short GRB whose plateau phase is dominated by magnetic dipole radiation (MD-SGRB) two-parameter Dainotti-correlated GRB data sets (Hu et al. 2021; Wang et al. 2022) to constrain cosmological parameters.² The circularity problem was circumvented by simultaneously constraining cosmological model parameters and GRB correlation parameters (see Dainotti et al. 2013b for a more extended discussion), and the cosmological model independence of the GRB correlation parameters shows that these GRBs are standardizable (Khadka & Ratra 2020c; Cao et al. 2022b). Not only does the simultaneous fitting method circumvent the circularity problem, it also allows for the derivation of unbiased GRB-only constraints (unlike the constraints derived from GRBs that have been calibrated by using other data, which are correlated with the calibrating data), which can be straightforwardly used to compare with other constraints derived from other data, such as H(z)  + BAO data, as we have done here. The A118, MD-LGRB, GW-LGRB, and MD-SGRB GRBs provide cosmological constraints that are mostly compatible with those determined using better-established cosmological probes (Cao et al. 2022b).

Here, we use the new Platinum compilation of 50 long GRBs, spanning 0.553 ≤ z ≤ 5.0, that obey the three-parameter Fundamental Plane (Dainotti) correlation between the peak prompt luminosity, the luminosity at the end of the plateau emission, and its rest-frame duration (Dainotti et al. 2016, 2017, 2020) to constrain cosmological model parameters and GRB correlation parameters. The Platinum sample is listed in Table A1 of Appendix A. For this data set, measured quantities for a GRB are redshift z, characteristic time-scale , which marks the end of the plateau emission, the measured gamma-ray energy flux F_X at and the prompt peak flux F_peak over a 1 s interval, and the X-ray spectral index of the plateau phase β′. This sample spans the redshift range 0.553 ≤ z ≤ 5.0. We find that the Platinum GRBs are standardizable through the Dainotti correlation and they also provide cosmological parameter constraints compatible with those from better-established cosmological probes, as well as with those derived from A118 GRB data. We also combine the Platinum data set with the 101 non-overlapping Amati-correlated GRBs (A101) from the A118 data set to perform a joint (Platinum + A101) analysis. We find that this joint GRB data set provides slightly more restrictive cosmological constraints (in which the twice as large A101 data set dominates the statistics and so plays a more dominant role) that are consistent with those from a combined analysis of baryon acoustic oscillation (BAO) and Hubble parameter [H(z)] data. However, the cosmological constraints from Platinum, A118, A101, and Platinum + A101 data are less restrictive (precise) than those from H(z)  + BAO data because there are more parameters to be constrained in the GRB cases but not yet enough precise-enough data points to determine more precise GRB constraints. A joint analysis of the H(z) + BAO + Platinum + A101 data results in slightly more restrictive cosmological constraints relative to those from just H(z)  + BAO data.

Our paper is organized as follows. We summarize the cosmological models we use in Section 2 and outline the data sets adopted in Section 3. We then describe our analysis methods in Section 4 and discuss results in Section 5. We summarize our conclusions in Section 6.

2. COSMOLOGICAL MODELS

We constrain cosmological model parameters and GRB correlation parameters in six spatially flat and non-flat dark energy cosmological models.³ The essential cosmological quantity for constraining purposes, for data we use, is the Hubble parameter, , with H₀ being the Hubble constant and the expansion rate as a function of redshift z and the cosmological parameters . Here, we consider one massive and two massless neutrino species, with the effective number of relativistic neutrino species N_eff = 3.046 and the total neutrino mass ∑m_ν = 0.06 eV. The non-relativistic neutrino physical energy density parameter is , where h is the reduced Hubble constant in units of 100 . With the baryonic (Ω_bh²) and CDM (Ω_ch²) physical energy density parameters as free cosmological parameters to be constrained, the derived non-relativistic matter density parameter is therefore Ω_m0 = (Ω_νh² + Ω_bh² + Ω_ch²)/h².

In the flat and non-flat ΛCDM models, the expansion rate function

(1)

where Ω_Λ = 1 − Ω_m0 − Ω_k0 is the cosmological constant dark energy density parameter and Ω_k0 is the curvature energy density parameter. In the non-flat ΛCDM model, the free parameters being constrained are H₀, Ω_bh², Ω_ch², and Ω_k0, whereas in the flat ΛCDM model, Ω_k0 = 0 is implied.

In the flat and non-flat XCDM parametrizations,

(2)

where w_X and Ω_X = 1 − Ω_m0 − Ω_k0 are the equation-of-state parameter and the dynamical dark energy density parameter of the X-fluid, respectively. In the non-flat XCDM parametrization, the free parameters being constrained are H₀, Ω_bh², Ω_ch², Ω_k0, and w_X, whereas in the flat XCDM parametrization, Ω_k0 = 0 is implied.

In the flat and non-flat ϕCDM models (Peebles & Ratra 1988; Ratra & Peebles 1988; Pavlov et al. 2013),⁴

(3)

where the scalar field, ϕ, dynamical dark energy density parameter

(4)

and can be numerically computed by solving the Friedmann equation (3) and the equation of motion of the scalar field

(5)

We assume an inverse power-law scalar field potential energy density

(6)

In these equations, an overdot and a prime denote a derivative with respect to time and ϕ, respectively, m_p is the Planck mass, α is a positive constant, and the constant κ is determined by the shooting method in the Cosmic Linear Anisotropy Solving System (class) code (Blas, Lesgourgues & Tram 2011). In the non-flat ϕCDM model, the free parameters being constrained are H₀, Ω_bh², Ω_ch², Ω_k0, and α, whereas in the flat ϕCDM model, Ω_k0 = 0 is implied.

3. DATA

In this paper, we analyse two different GRB data sets as well as combinations of them and the joint H(z)  + BAO data set. These data sets are summarized in Table 1 and described below.⁵

Table 1.

Summary of data sets used.

Data set	N (Number of points)	Redshift range
Platinum	50	0.553 ≤ z ≤ 5.0
A118	118	0.3399 ≤ z ≤ 8.2
A101a	101	0.3399 ≤ z ≤ 8.2
Plat.  + A101	151	0.3399 ≤ z ≤ 8.2
H(z)	31	0.070 ≤ z ≤ 1.965
BAO	11	0.38 ≤ z ≤ 2.334

Note.^aExcluding from A118 those GRBs in common with Platinum [060418, 080721, 081008, 090418(A), 091020, 091029, 110213(A), 110818(A), 111008(A), 120811C, 120922(A), 121128(A), 131030A, 131105A, 140206A, 150314A, and 150403A].

• Platinum sample. This includes 50 long GRBs, which exhibit a plateau phase with an angle <41°, that do not have a flare during the plateau, and have a plateau with a duration longer than 500 s. The first criterion is based on evidence that the plateau angles are Gaussianly distributed and those with angle >41° are outliers; the second criterion allows one to eliminate cases contaminated by the presence of flaring activity; and the third criterion allows one to eliminate cases where prompt emission may mask the plateau to the point that the definition of the plateau is uncertain (Willingale et al. 2007, 2010). As discussed below, the Platinum GRBs obey the 3D Dainotti relation. The Platinum sample is listed in Table A1 of Appendix A. For this data set, measured quantities for a GRB are redshift z, characteristic time-scale , which marks the end of the plateau emission, the measured gamma-ray energy flux F_X at and the prompt peak flux F_peak over a 1 s interval, and the X-ray spectral index of the plateau phase β′. This sample spans the redshift range 0.553 ≤ z ≤ 5.0.

• A118 sample. This sample includes 118 long GRBs, listed in table 7 of Khadka et al. (2021c), that obey the 2D Amati relation. For this data set, measured quantities for a GRB are z, rest-frame spectral peak energy E_p, and measured bolometric fluence S_bolo, computed in the standard rest-frame energy band 1–10⁴ keV. This sample spans the redshift range 0.3399 ≤ z ≤ 8.2.

• A101 sample. The A118 data and the Platinum data have 17 common GRBs that are listed in the footnote of Table 1. We exclude these common GRBs from the A118 data set to form the A101 data set for joint analyses with the Platinum data set. This sample spans the redshift range 0.3399 ≤ z ≤ 8.2.

• Platinum  + A101 sample. This combination GRB sample includes 151 GRBs. This sample spans the redshift range 0.3399 ≤ z ≤ 8.2.

• H(z) and BAO data. There are 31 H(z) and 11 BAO measurements that have a redshift range 0.07 ≤ z ≤ 1.965 and 0.0106 ≤ z ≤ 2.33, respectively. The H(z) data are in table 2 of Ryan, Doshi & Ratra (2018) and the BAO data are in table 1 of Cao, Ryan & Ratra (2021b). We compare cosmological constraints from H(z)  + BAO data with those obtained from the GRB data sets, and also jointly analyse GRB and H(z)  + BAO data.

4. DATA ANALYSIS METHODOLOGY

Luminosity distance, D_L, as a function of z and cosmological parameters is given by

(7)

where the comoving distance is

(8)

c is the speed of light, and is the Hubble parameter that is described in Section 2 for each cosmological model.

For Platinum GRBs, the X-ray source rest-frame luminosity L_X, time at the end of the plateau emission, and the peak prompt luminosity L_peak are correlated through the three-parameter Fundamental Plane relation (Dainotti et al. 2016, 2017, 2020, 2021a)

(9)

where

(10)

(11)

C_o is the intercept parameter, and a and b are the slope parameters, with all three to be determined from data. F_X and F_peak are the measured gamma-ray energy flux (erg cm⁻² s⁻¹) at and in the peak of the prompt emission over a 1 s interval, respectively, and β′ is the X-ray spectral index of the plateau phase.

We compute L_X and L_peak as functions of cosmological parameters at the redshift of each GRB by using equations (7), (10), and (11). We then compute the natural log of the likelihood function (D’Agostini 2005)

(12)

where

(13)

with

(14)

Here, is the Platinum GRB data intrinsic scatter parameter, which also contains the unknown systematic uncertainty, and N is the number of data points.

For GRBs that obey the Amati correlation, a detailed description of the procedure can be found in section 4 of Cao et al. (2022b). Here, we denote its intrinsic scatter parameter as as opposed to the Platinum one, .

Detailed descriptions of the H(z)  + BAO data analysis procedure can be found in section 4 of Cao et al. (2020) and Cao et al. (2021b).

The flat priors of the free parameters are listed in Table 2. The best-fitting values and posterior distributions of all free parameters are obtained through maximizing the likelihood functions using the Markov chain Monte Carlo (MCMC) code MontePython (Brinckmann & Lesgourgues 2019), with the physics coded in class. The convergence of the MCMC chains for each free parameter is guaranteed by the Gelman–Rubin criterion (R − 1 < 0.05). The python package getdist (Lewis 2019) is used to compute the posterior means and uncertainties and plot the marginalized likelihood distributions and contours.

Table 2.

Flat priors of the constrained parameters.

Parameter		Prior
	Cosmological parameters
		[None, None]
Ωbh2 b		[0, 1]
Ωch2 c		[0, 1]
Ωk0		[−2, 2]
α		[0, 10]
w X		[−5, 0.33]
	GRB correlation parameters
a		[−5, 5]
b		[−5, 5]
Co		[−50, 50]
σint		[0, 5]
β		[0, 5]
γ		[0, 300]

Notes.^a. In all four GRB-only analyses, H₀ is set to be 70 , while in other cases, the prior range is irrelevant (unbounded).

^b In all four GRB-only analyses, Ω_bh² is set to be 0.0245, i.e. Ω_b = 0.05.

^c In all four GRB-only analyses, the Ω_c range is adjusted to ensure Ω_m0 ∈ [0, 1].

We use the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) to compare the goodness of fit of models with different numbers of parameters. Their definitions can be found in Cao et al. (2022b). We also compare the goodness of fit using the deviance information criterion (DIC) (Kunz, Trotta & Parkinson 2006; Amati et al. 2019) defined as

(15)

where is the number of effectively constrained parameters with brackets representing the average over the posterior distribution. We compute ΔAIC, ΔBIC, and ΔDIC differences of the other five cosmological models with respect to the flat ΛCDM reference model. Positive (negative) values of ΔAIC, ΔBIC, or ΔDIC indicate that the model under study fits the data worse (better) than does the reference model. Relative to the model with minimum AIC(BIC/DIC), ΔAIC(BIC/DIC) ∈ (0, 2] is said to be weak evidence against the candidate model, ΔAIC(BIC/DIC) ∈ (2, 6] is positive evidence against the candidate model, while ΔAIC(BIC/DIC) ∈ (6, 10] is strong evidence against the candidate model, with ΔAIC(BIC/DIC) > 10 being very strong evidence against the candidate model.

5. RESULTS

We show the posterior 1D probability distributions and 2D confidence regions of cosmological model and GRB correlation parameters for the six cosmological models in Figs 1–7, in grey (Platinum), green (A118 and A101), orange (Platinum  + A101), red [H(z)  + BAO], and blue [H(z) + BAO  + Platinum, in short HzBP, and H(z) + BAO + Platinum  + A101, in short HzBPA101]. The unmarginalized best-fitting parameter values, as well as the values of maximum likelihood , AIC, BIC, DIC, ΔAIC, ΔBIC, and ΔDIC, for all models and data combinations, are listed in Table 3. We list the marginalized posterior mean parameter values and uncertainties (±1σ error bars and 1σ or 2σ limits), for all models and data combinations, in Table 4.

Figure 1.

1D likelihood distributions and 1σ, 2σ, and 3σ 2D likelihood confidence contours for flat ΛCDM from various combinations of data. The zero-acceleration black dashed lines in panels (a) and (b) divide the parameter space into regions associated with currently accelerating (left) and currently decelerating (right) cosmological expansion.

Figure 7.

1D likelihood distributions and 1σ, 2σ, and 3σ 2D likelihood confidence contours for some cosmological parameters of the six cosmological models from Platinum data (grey), A101 data (green), and H(z)  + BAO data (red), which more clearly show the cosmological parameter overlaps between Platinum/A101 data and H(z)  + BAO data.

Table 3.

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

Model	Data set		Ωch2	Ωm0	Ωk0	w X/αb			a	b	Co		γ	β		AIC	BIC	DIC	ΔAIC	ΔBIC	ΔDIC
	Platinum	–	0.4560	0.982	–	–	–	0.322	−0.716	0.743	12.19	–	–	–	32.99	42.99	52.55	42.80	0.00	0.00	0.00
	A118	–	0.4088	0.886	–	–	–	–	–	–	–	0.402	50.02	1.099	128.72	136.72	147.81	136.05	0.00	0.00	0.00
	A101	–	0.2547	0.571	–	–	–	–	–	–	–	0.409	50.04	1.135	112.81	120.81	131.27	120.16	0.00	0.00	0.00
Flat	Plat.  + A101	–	0.3505	0.767	–	–	–	0.324	−0.694	0.753	11.60	0.413	49.98	1.125	146.08	162.08	186.22	162.07	0.00	0.00	0.00
ΛCDM	H(z)  + BAO	0.0240	0.1177	0.298	–	–	69.11	–	–	–	–	–	–	–	23.66	29.66	34.87	30.23	0.00	0.00	0.00
	HzBPd	0.0243	0.1169	0.298	–	–	69.04	0.325	−0.715	0.757	11.53	–	–	–	57.33	73.33	88.99	72.58	0.00	0.00	0.00
	HzBPA101e	0.0236	0.1155	0.296	–	–	68.71	0.324	−0.707	0.760	11.35	0.412	50.16	1.153	170.63	190.63	223.26	191.94	0.00	0.00	0.00
	Platinum	–	0.1178	0.292	−0.992	–	–	0.292	−0.700	0.829	7.59	–	–	–	24.51	36.51	47.98	51.00	−6.48	−4.57	8.20
	A118	–	0.4625	0.995	1.094	–	–	–	–	–	–	0.401	49.90	1.117	127.96	137.96	151.82	136.72	1.24	4.01	0.67
	A101	–	0.4606	0.991	1.993	–	–	–	–	–	–	0.405	49.78	1.151	111.08	121.08	134.16	120.04	0.27	2.89	−0.12
Non-flat	Plat.  + A101	–	0.4300	0.929	1.857	–	–	0.326	−0.712	0.746	11.99	0.400	49.75	1.167	144.65	162.65	189.80	161.75	0.57	3.58	−0.32
ΛCDM	H(z)  + BAO	0.0247	0.1141	0.295	0.023	–	68.78	–	–	–	–	–	–	–	23.59	31.59	38.54	32.21	1.93	3.67	1.99
	HzBPd	0.0245	0.1149	0.294	0.010	–	68.99	0.328	−0.719	0.752	11.81	–	–	–	57.31	73.31	93.48	74.65	−0.02	4.49	2.07
	HzBPA101e	0.0237	0.1115	0.295	0.032	–	67.91	0.328	−0.726	0.750	11.91	0.410	50.13	1.164	170.70	192.70	228.59	193.50	2.07	5.33	1.56
	Platinum	–	0.0809	0.216	–	0.137	–	0.323	−0.724	0.735	12.59	–	–	–	32.81	44.81	56.28	42.83	1.82	3.73	0.04
	A118	–	−0.0198	0.011	–	−0.139	–	–	–	–	–	0.402	50.04	1.112	128.42	138.42	152.28	136.65	1.70	4.47	0.61
	A101	–	−0.0221	0.006	–	−0.251	–	–	–	–	–	0.406	50.01	1.148	111.99	121.99	135.06	121.23	1.18	3.79	1.08
Flat	Plat.  + A101	–	0.1570	0.372	–	−0.193	–	0.317	−0.712	0.753	11.65	0.405	50.02	1.115	145.74	163.74	190.90	162.40	1.66	4.68	0.33
XCDM	H(z)  + BAO	0.0309	0.0870	0.280	–	−0.694	65.11	–	–	–	–	–	–	–	19.65	27.65	34.60	28.11	−2.01	−0.27	−2.11
	HzBPd	0.0302	0.0899	0.284	–	−0.711	65.22	0.320	−0.709	0.758	11.46	–	–	–	53.26	69.26	89.44	70.25	−4.07	0.45	−2.33
	HzBPA101e	0.0301	0.0891	0.282	–	−0.709	69.13	0.325	−0.724	0.756	11.63	0.408	50.17	1.150	166.30	188.30	224.19	188.98	−2.33	0.93	−2.96
	Platinum	–	0.0918	0.239	−0.760	−1.123	–	0.286	−0.761	0.788	10.04	–	–	–	24.58	38.58	51.97	51.01	−4.41	−0.58	8.21
	A118	–	0.4644	0.999	0.998	−1.137	–	–	–	–	–	0.398	49.91	1.113	127.97	139.97	156.60	137.16	3.25	8.79	1.11
	A101	–	0.4444	0.958	1.849	−1.222	–	–	–	–	–	0.410	49.73	1.164	111.14	123.14	138.83	120.59	2.33	7.56	0.44
Non-flat	Plat.  + A101	–	0.4120	0.892	1.526	−1.208	–	0.335	−0.707	0.758	11.34	0.405	49.77	1.163	144.69	164.69	194.86	162.75	2.61	8.64	0.67
XCDM	H(z)  + BAO	0.0295	0.0962	0.294	−0.159	−0.648	65.62	–	–	–	–	–	–	–	18.31	28.31	37.00	28.96	−1.35	2.13	−1.26
	HzBPd	0.0301	0.0933	0.290	−0.164	−0.640	65.39	0.324	−0.709	0.746	12.04	–	–	–	51.90	69.90	92.60	70.74	−3.43	3.61	−1.85
	HzBPA101e	0.0299	0.0883	0.284	−0.112	−0.640	64.65	0.319	−0.714	0.767	11.01	0.414	50.19	1.137	165.30	189.30	228.45	189.85	−1.33	5.19	−2.09
	Platinum	–	0.4615	0.993	–	4.896	–	0.322	−0.723	0.744	12.14	–	–	–	32.99	44.99	56.46	42.37	2.00	3.91	−0.43
	A118	–	0.2119	0.484	–	9.617	–	–	–	–	–	0.400	50.04	1.105	128.56	138.56	152.41	135.74	1.84	4.60	−0.31
	A101	–	0.0764	0.207	–	9.922	–	–	–	–	–	0.409	49.99	1.152	112.26	122.26	135.33	119.96	1.45	4.06	−0.20
Flat	Plat.  + A101	–	0.0936	0.242	–	7.330	–	0.326	−0.723	0.755	11.62	0.410	50.04	1.135	145.75	163.75	190.90	161.62	1.67	4.68	−0.46
ϕCDM	H(z)  + BAO	0.0332	0.0789	0.265	–	1.455	65.24	–	–	–	–	–	–	–	19.49	27.49	34.44	26.96	−2.17	−0.43	−3.27
	HzBPd	0.0343	0.0746	0.259	–	1.633	64.98	0.331	−0.714	0.765	11.10	–	–	–	53.12	69.12	89.29	69.20	−4.21	0.30	−3.39
	HzBPA101e	0.0374	0.0663	0.246	–	1.968	65.20	0.327	−0.703	0.759	11.38	0.412	50.10	1.172	166.28	188.28	224.17	187.99	−2.35	0.91	−3.95
	Platinum	–	0.4366	0.942	−0.905	0.061	–	0.325	−0.735	0.725	13.16	–	–	–	32.53	46.53	59.92	43.06	3.54	7.37	0.26
	A118	–	0.3331	0.731	0.234	5.269	–	–	–	–	–	0.402	50.02	1.111	128.42	140.42	157.04	136.67	3.70	9.23	0.63
	A101	–	0.2367	0.534	0.460	8.680	–	–	–	–	–	0.406	49.99	1.151	111.89	123.89	139.58	120.48	3.08	8.31	0.33
Non-flat	Plat.  + A101	–	0.3063	0.677	0.317	9.746	–	0.319	−0.728	0.742	12.27	0.412	49.96	1.140	145.40	165.40	195.57	162.44	3.32	9.35	0.36
ϕCDM	H(z)  + BAO	0.0344	0.0786	0.263	−0.149	2.014	65.71	–	–	–	–	–	–	–	18.15	28.15	36.84	27.39	−1.51	1.97	−2.84
	HzBPd	0.0326	0.0851	0.271	−0.176	1.826	66.10	0.322	−0.727	0.752	11.78	–	–	–	51.79	69.79	92.49	69.44	−3.54	3.50	−3.14
	HzBPA101e	0.0353	0.0730	0.257	−0.135	2.199	65.09	0.321	−0.729	0.758	11.52	0.403	50.16	1.144	165.08	189.08	228.23	188.65	−1.55	4.97	−3.29

Notes.^a In the four GRB-only cases, Ω_bh² is set to 0.0245.

^b w _X corresponds to flat/non-flat XCDM and α corresponds to flat/non-flat ϕCDM.

^c . In the four GRB-only cases, H₀ is set to 70 .

^d H(z) + BAO  + Platinum.

^e H(z) + BAO + Platinum + A101.

Table 4.

1D marginalized posterior mean values and uncertainties (±1σ error bars or 2σ limits) of the parameters for all models from various combinations of data.

Model	Data set		Ωch2	Ωm0	Ωk0	w X/αb			a	b	Co		γ	β
	Platinum	–	–		–	–	–		−0.714 ± 0.104	0.756 ± 0.083	11.52 ± 4.45	–	–	–
	A118	–	–	>0.256	–	–	–	–	–	–	–		50.09 ± 0.25	1.109 ± 0.089
	A101	–	–		–	–	–	–	–	–	–		50.03 ± 0.28	1.140 ± 0.098
Flat	Plat.  + A101	–	–		–	–	–		−0.717 ± 0.102	0.751 ± 0.083	11.80 ± 4.45		50.03 ± 0.27	1.138 ± 0.095
ΛCDM	H(z)  + BAO	0.0243 ± 0.0029			–	–	69.27 ± 1.85	–	–	–	–	–	–	–
	HzBPe	0.0242 ± 0.0029			–	–	69.24 ± 1.81		−0.711 ± 0.104	0.762 ± 0.081	11.23 ± 4.35	–	–	–
	HzBPA101f				–	–	69.20 ± 1.79		−0.711 ± 0.104	0.762 ± 0.080	11.23 ± 4.32		50.13 ± 0.26	1.159 ± 0.094
	Platinum	–	–			–	–		−0.727 ± 0.109			–	–	–
	A118	–	–	>0.295		–	–	–	–	–	–		50.00 ± 0.26	1.123 ± 0.088
	A101	–	–	>0.212		–	–	–	–	–	–		49.93 ± 0.27	1.156 ± 0.096
Non-flat	Plat.  + A101	–	–	>0.235		–	–		−0.711 ± 0.100	0.759 ± 0.080			49.93 ± 0.27	1.154 ± 0.095
ΛCDM	H(z)  + BAO		0.1127 ± 0.0195	0.293 ± 0.025		–	68.76 ± 2.36	–	–	–	–	–	–	–
	HzBPe			0.294 ± 0.024		–	68.80 ± 2.32		−0.710 ± 0.104	0.763 ± 0.080	11.18 ± 4.30	–	–	–
	HzBPA101f			0.293 ± 0.023		–	68.62 ± 2.24		−0.710 ± 0.102	0.764 ± 0.079	11.15 ± 4.27		50.13 ± 0.26	1.161 ± 0.093
	Platinum	–	–		–	<−0.078	–		−0.717 ± 0.103	0.749 ± 0.083	11.90 ± 4.45	–	–	–
	A118	–	–		–		–	–	–	–	–			1.106 ± 0.090
	A101	–	–		–		–	–	–	–	–			1.134 ± 0.097
Flat	Plat.  + A101	–	–		–	<−0.089	–		−0.718 ± 0.105	0.749 ± 0.084	11.94 ± 4.54			1.133 ± 0.094
XCDM	H(z)  + BAO				–			–	–	–	–	–	–	–
	HzBPe				–				−0.711 ± 0.104	0.763 ± 0.081	11.24 ± 4.32	–	–	–
	HzBPA101f				–				−0.711 ± 0.102	0.762 ± 0.080	11.21 ± 4.38		50.14 ± 0.26	1.159 ± 0.093
	Platinum	–	–				–		−0.727 ± 0.109			–	–	–
	A118	–	–	>0.257			–	–	–	–	–		50.01 ± 0.27	1.120 ± 0.088
	A101	–	–	>0.193			–	–	–	–	–		49.91 ± 0.30	1.157 ± 0.098
Non-flat	Plat.  + A101	–	–	>0.207			–		−0.711 ± 0.103	0.759 ± 0.081			49.91 ± 0.29	1.154 ± 0.096
XCDM	H(z)  + BAO			0.294 ± 0.028				–	–	–	–	–	–	–
	HzBPe			0.294 ± 0.028					−0.713 ± 0.103	0.758 ± 0.082	11.46 ± 4.44	–	–	–
	HzBPA101f			0.290 ± 0.027	−0.106 ± 0.128				−0.713 ± 0.103	0.758 ± 0.081	11.44 ± 4.38		50.14 ± 0.26	1.151 ± 0.094
	Platinum	–	–		–	–	–		−0.715 ± 0.102	0.753 ± 0.081	11.68 ± 4.33	–	–	–
	A118	–	–		–	–	–	–	–	–	–		50.05 ± 0.25	1.110 ± 0.089
	A101	–	–		–	–	–	–	–	–	–		49.99 ± 0.27	1.140 ± 0.097
Flat	Plat.  + A101	–	–		–	–	–		−0.716 ± 0.103	0.751 ± 0.082	11.77 ± 4.39		49.99 ± 0.26	1.137 ± 0.095
ϕCDM	H(z)  + BAO			0.268 ± 0.024	–		65.12 ± 2.18	–	–	–	–	–	–	–
	HzBPe			0.267 ± 0.024	–				−0.711 ± 0.104	0.762 ± 0.081	11.24 ± 4.37	–	–	–
	HzBPA101f				–				−0.711 ± 0.103	0.762 ± 0.082			50.14 ± 0.26	1.158 ± 0.093
	Platinum	–	–			–	–		−0.717 ± 0.105	0.751 ± 0.084		–	–	–
	A118	–	–				–	–	–	–	–		50.05 ± 0.25	1.110 ± 0.090
	A101	–	–				–	–	–	–	–		49.99 ± 0.27	1.146 ± 0.098
Non-flat	Plat.  + A101	–	–			–	–		−0.715 ± 0.103	0.754 ± 0.083	11.64 ± 4.42		49.98 ± 0.26	1.142 ± 0.097
ϕCDM	H(z)  + BAO							–	–	–	–	–	–	–
	HzBPe						65.57 ± 2.27		−0.713 ± 0.104	0.760 ± 0.081	11.37 ± 4.35	–	–	–
	HzBPA101f						65.39 ± 2.24		−0.712 ± 0.103	0.760 ± 0.082	11.36 ± 4.40		50.14 ± 0.26	1.153 ± 0.094

Notes.^a In the four GRB-only cases, Ω_bh² is set to 0.0245.

^b w _X corresponds to flat/non-flat XCDM and α corresponds to flat/non-flat ϕCDM.

^c . In the four GRB-only cases, H₀ is set to 70 .

^d This is the 1σ limit. The 2σ limit is set by the prior and not shown here.

^e H(z) + BAO  + Platinum.

^f H(z) + BAO + Platinum  + A101.

Figure 2.

Same as Fig. 1 but for non-flat ΛCDM. The zero-acceleration black dashed lines divide the parameter space into regions associated with currently accelerating (below left) and currently decelerating (above right) cosmological expansion.

Figure 3.

1D likelihood distributions and 1σ, 2σ, and 3σ 2D likelihood confidence contours for flat XCDM from various combinations of data. The zero-acceleration black dashed lines divide the parameter space into regions associated with currently accelerating (either below left or below) and currently decelerating (either above right or above) cosmological expansion. The magenta dashed lines represent w_X = −1, i.e. flat ΛCDM.

Figure 4.

Same as Fig. 3 but for non-flat XCDM. The zero-acceleration black dashed lines are computed for the third cosmological parameter set to the H(z)  + BAO data best-fitting values listed in Table 3, and divide the parameter space into regions associated with currently accelerating (either below left or below) and currently decelerating (either above right or above) cosmological expansion. The crimson dash–dotted lines represent flat hypersurfaces, with closed spatial hypersurfaces either below or to the left. The magenta dashed lines represent w_X = −1, i.e. non-flat ΛCDM.

Figure 5.

1D likelihood distributions and 1σ, 2σ, and 3σ 2D likelihood confidence contours for flat ϕCDM from various combinations of data. The zero-acceleration black dashed lines divide the parameter space into regions associated with currently accelerating (below left) and currently decelerating (above right) cosmological expansion. The α = 0 axes correspond to flat ΛCDM.

Figure 6.

Same as Fig. 5 but for non-flat ϕCDM. The zero-acceleration black dashed lines are computed for the third cosmological parameter set to the H(z)  + BAO data best-fitting values listed in Table 3, and divide the parameter space into regions associated with currently accelerating (below left) and currently decelerating (above right) cosmological expansion. The crimson dash–dotted lines represent flat hypersurfaces, with closed spatial hypersurfaces either below or to the left. The α = 0 axes correspond to non-flat ΛCDM.

5.1. Constraints from Platinum, A118, A101, and Platinum + A101 data

As in Khadka et al. (2021c) and Cao et al. (2022b), in the four GRB-only cases here, we set H₀ = 70 and Ω_b = 0.05.

The constraints on the Platinum GRB correlation parameters in the six different cosmological models are mutually consistent, so the 3D Dainotti correlation Platinum data set is standardizable. The constraints on the intrinsic scatter parameter are almost identical, ranging from (flat ϕCDM) to (non-flat ΛCDM). The constraints on the slope a range from a low of −0.727 ± 0.109 (non-flat ΛCDM and non-flat XCDM) to a high of −0.714 ± 0.104 (flat ΛCDM), the constraints on the slope b range from a low of (non-flat XCDM) to a high of 0.756 ± 0.083 (flat ΛCDM), and the constraints on the intercept C_o range from a low of 11.52 ± 4.45 (flat ΛCDM) to a high of (non-flat XCDM), with central values of each pair being 0.09σ, 0.15σ, and 0.16σ away from each other, respectively. We note that a compilation of (the two-parameter) Dainotti correlation GRBs, the 31 MD-LGRBs of Wang et al. (2022), have a somewhat smaller intrinsic dispersion, σ_int ∼ 0.303–0.306, table 5 of Cao et al. (2022b), than the of the 50 Platinum GRBs.⁶ A possible reason for the smaller scatter of the two-parameter MD-LGRB sample is due to how the sample was chosen. In principle, one can retain fewer GRBs that lie exactly on the plane, or are much closer to the plane, thus reducing the scatter. However, further investigation is needed in order to draw a definite conclusion. Probably, as a consequence of the smaller σ_int, the MD-LGRB constraints on Ω_m0 and Ω_k0 are more restrictive than the constraints from Platinum data.

Compared with the analyses in Cao et al. (2022b) where we neglect massive neutrinos (setting Ω_νh² = 0), here we include a non-zero Ω_νh². The constraints on the A118 GRB correlation parameters here are almost identical to those listed in table 8 of Cao et al. (2022b) and are cosmological-model-independent implying that the A118 GRBs are standardizable. For the 118 GRBs’ A118 data, , larger than the of the 50 Platinum GRBs. The constraints on the A101 GRB correlation parameters are also cosmological-model-independent, so A101 GRBs are also standardizable. The constraints on the A101 intrinsic scatter parameter range from a low of (non-flat ΛCDM) to a high of (flat ΛCDM and flat XCDM), which are slightly (0.2σ at most) higher than those from A118 data; the constraints on the slope β range from a low of 1.140 ± 0.098 (flat ΛCDM) to a high of 1.157 ± 0.098 (non-flat XCDM), which are slightly (0.28σ at most) higher than those from A118 data; and the constraints on the intercept γ range from a low of 49.91 ± 0.30 (non-flat XCDM) to a high of (flat XCDM), which are slightly (0.25σ at most) lower than those from A118 data, with central values of each pair being 0.06σ, 0.12σ, and 0.43σ away from each other, respectively.

Below, we discuss cosmological parameter constraints in more detail. From Fig. 7, we see that the overlap of the constraints on the cosmological parameters indicates that Platinum data and A101 data cosmological constraints are mutually consistent and so these two data sets can be jointly analysed. The Platinum  + A101 data combination is also standardizable with cosmological-model-independent constraints on GRB correlation parameters that are consistent (well within 1σ) with those from both Platinum data and A101 data individually.

We find that in the flat ΛCDM model and the flat XCDM parametrization, all GRB data more favour currently accelerating cosmological expansion. In the non-flat ΛCDM model, all but the Platinum GRBs more favour currently decelerating cosmological expansion. In the non-flat XCDM parametrization, in the Ω_k0–Ω_m0 parameter subspace, all but the Platinum GRBs more favour currently decelerating cosmological expansion, while in the w_X–Ω_m0 (w_X–Ω_k0) parameter subspace, all (all but the A101) GRBs more favour currently accelerating cosmological expansion. In the flat and non-flat ϕCDM models, currently decelerating cosmological expansion is slightly more favoured.

Cosmological constraints from Platinum data are less restrictive than those from A118 and A101 data, which contain a little more than twice as many GRBs than the Platinum compilation. Comparing the Platinum  + A101 constraints with the Platinum constraints and with the A101 constraints, we see that A101 data, with double the number of GRBs compared to Platinum, play a more dominant role in the combination.

In the flat and non-flat ΛCDM models, the Platinum, A118, A101, and Platinum  + A101 2σ constraints on Ω_m0 are mutually consistent and also consistent with those of H(z)  + BAO, with the 2σ lower limits in the flat (non-flat) ΛCDM model being None (None), >0.256 (>0.295), >0.191 (>0.212), and >0.216 (>0.235), respectively. In the flat ΛCDM model, the A101 constraint on Ω_m0 is more consistent (than the Platinum and Platinum  + A101 constraints) with that from H(z) + BAO data, differing by only 1.18σ. In both models, Platinum  + A101 data favour a higher 1σ lower limit of Ω_m0 than do Platinum data or A101 data. In the non-flat ΛCDM model, the constraints on Ω_k0 from the four GRB data sets are mutually consistent within 1σ, with all but Platinum data slightly favouring open spatial hypersurfaces. Platinum and A118 data are consistent with flat hypersurfaces within 1σ, while A101 and Platinum  + A101 data are 1.44σ and 1.17σ, respectively, away from flat. The posterior mean value of Ω_k0 from A101 data is 1.36σ away from that from H(z)  + BAO data, while those from the other three GRB data sets are consistent with that from H(z)  + BAO data within 1σ.

In the flat and non-flat XCDM parametrizations, the Platinum, A118, A101, and Platinum  + A101 2σ constraints on Ω_m0 are mutually consistent and also consistent with those of H(z)  + BAO, with the 2σ lower limits in the flat (non-flat) XCDM parametrization being None (None), >0.181 (>0.257), >0.148 (>0.193), and >0.151 (>0.207), respectively. In flat XCDM, the A101 constraint on Ω_m0 is consistent with that from H(z) + BAO within 1σ. Platinum  + A101 data also favour a higher 1σ lower limit of Ω_m0 than do Platinum or A101 data in both XCDM parametrizations. The constraints on w_X are weak, thus affected by the w_X prior, and consistent with each other. They mildly favour phantom dark energy, but Λ is less than 1σ away, except for the flat XCDM Platinum and Platinum  + A101 cases (where Λ is still within 2σ), as is the case for the w_X constraints from H(z)  + BAO data.⁷ In non-flat XCDM, the constraints on Ω_k0 from the four GRB data sets are mutually consistent within 1σ, and they slightly favour open hypersurfaces. Platinum, A118, and Platinum  + A101 data are consistent with flat hypersurfaces within 1σ, while A101 data are 1.10σ away from flat. The posterior mean value of Ω_k0 from A101 data is 1.24σ away from that from H(z)  + BAO data, while those from the other three GRB data sets are consistent with that from H(z)  + BAO data within 1σ.

In the flat and non-flat ϕCDM models, the Platinum, A118, A101, and Platinum  + A101 constraints on Ω_m0 are mutually consistent within 1σ and the A101 and Platinum  + A101 (Platinum and A118) constraints are within 1σ (2σ) of those from H(z)  + BAO data. Only A118 and A101 data provide (very weak) constraints on α with Λ being more than 1σ away but within 2σ. In non-flat ϕCDM, the constraints on Ω_k0 from the four GRB data sets are mutually consistent and consistent with that from H(z)  + BAO data and flat hypersurfaces are within 1σ.

From the ΔAIC and ΔBIC values listed in Table 3, in the Platinum case, non-flat ΛCDM is the most favoured model, while the evidence against other models is positive, strong, and very strong (non-flat ϕCDM). In the A118, A101, and Platinum  + A101 cases, the flat ΛCDM model is the most favoured model and, except for non-flat XCDM and non-flat ϕCDM (with strong BIC evidence against them), the evidence against other models is either weak or positive. However, based on ΔDIC, in theses cases, flat ϕCDM is the most favoured model; in the A118, A101, and Platinum  + A101 cases, the evidence against other models is weak, whereas in the Platinum case, the evidence against other models is either weak or strong (non-flat ΛCDM and non-flat XCDM).

5.2. Constraints from H(z) + BAO data in combination with Platinum (HzBP) and Platinum  + A101 (HzBPA101) data

Since the H(z) + BAO, Platinum, and Platinum  + A101 cosmological constraints are mutually consistent, we jointly analyse combinations of these data to determine more restrictive constraints on cosmological and GRB correlation parameters. We find that the new constraints on GRB correlation parameters are more restrictive but change less than 1σ compared to those from the GRB-only analyses. The correlation parameter constraints remain cosmological-model-independent, indicating again that these GRBs are standardizable. In what follows, we discuss how the H(z)  + BAO data cosmological parameter constraints are altered when these GRB data are jointly analysed with H(z)  + BAO data.

Compared to the H(z)  + BAO constraints, the HzBP constraints on Ω_m0 are almost unchanged in all models, but there are small changes in other cosmological parameter constraints. The constraints on H₀ are slightly tightened, with posterior means being 0.01–0.03σ smaller or larger from model to model; the constraints on w_X are slightly shifted away from Λ, with posterior means being ∼0.04σ larger; the constraints on α are slightly shifted away from Λ, with posterior means being 0.02σ (0.05σ) larger in flat (non-flat) ϕCDM; and the constraints on Ω_k0 are slightly shifted away from flat, with posterior means being 0.06σ (0.10σ) smaller in non-flat XCDM (ϕCDM).

Compared to H(z)  + BAO, the HzBPA101 data provide mildly different constraints on Ω_m0 in all models, but the effects on other cosmological parameter constraints are more noticeable. The constraints on H₀ are slightly tightened, with posterior means being 0.03–0.10σ smaller from model to model; the constraints on w_X are slightly shifted away from Λ, with posterior means being 0.11σ (0.08σ) larger in flat (non-flat) XCDM; the constraints on α are slightly shifted away from Λ, with posterior means being 0.07σ (0.08σ) larger in flat (non-flat) ϕCDM; and the constraints on Ω_k0 are slightly shifted towards (away from) flat, with posterior means being 0.05σ (0.02σ) larger (smaller) in non-flat XCDM (ϕCDM), respectively.

From the ΔAIC, ΔBIC, and ΔDIC values listed in Table 3, in both the HzBP and the HzBPA101 cases, following the patterns of the H(z)  + BAO case, flat ϕCDM is the most favoured model, but the evidence against other models is only either weak or positive. As expected, the better-established H(z)  + BAO data play the dominant role in these combined analyses and therefore currently accelerating cosmological expansion is favoured.

6. CONCLUSION

We have analysed the 50 Platinum GRBs that obey the three-parameter Fundamental Plane (or Dainotti) relation, using six different cosmological models. By simultaneously constraining cosmological model and GRB correlation parameters, our approach circumvents the circularity problem. We find that the Platinum GRB correlation parameters are cosmological-model-independent, so the Platinum sample is standardizable through the three-parameter Dainotti correlation and can be used to constrain cosmological parameters. Since the cosmological constraints from Platinum data are consistent with those from H(z)  + BAO data, we have combined Platinum and H(z)  + BAO data to perform a joint (HzBP) analysis and find mild changes of the cosmological parameter constraints relative to those from H(z)  + BAO data, with the central values in the two cases agreeing to two significant figures.

We have also reanalysed the A118 GRBs that obey the Amati (E_p–E_iso) correlation (Khadka et al. 2021c; Cao et al. 2022b), because of different, improved modelling of neutrino physics here, and, by excluding the 17 overlapping (with Platinum) GRBs from the larger A118 data set to form the truncated A101 data set, we have also performed a joint analysis of Platinum and A101 data. The twice as large A101 data set plays a dominant role in the joint Platinum + A101 analysis, and the cosmological constraints from Platinum + A101 data are closer to those from A101 data. The results show that the Platinum  + A101 GRBs are also standardizable and their cosmological constraints are also more consistent (than the A118 ones) with those from H(z)  + BAO data. The joint analyses of H(z) + BAO and Platinum + A101 (HzBPA101) data show that Platinum  + A101 data do have more impact on the joint constraints than do Platinum data; however, H(z)  + BAO data still play the dominant role.

Current compilations of GRBs provide some improvements on the cosmological constraints determined using H(z)  + BAO data. Importantly, GRBs are the only currently reliable probes of the z ∼ 3–8 part of cosmological redshift space and so are well worth improving upon. With the upcoming SVOM mission (Cordier 2019) in 2023, and possibly THESEUS (Amati et al. 2021) in 2037 if approved in the new ESA call, there soon will be larger GRB data sets over a wider range of redshifts that will allow for improved GRB cosmological constraints.

APPENDIX A: PLATINUM GRB DATA

Table A1.

50 Platinum GRB samples.

GRB	z		log FX (erg cm−2 s−1)	β′	F peak (10−8 erg cm−2 s−1)
060418	1.49		−9.79296 ± 0.04904	1.98	49.9 ± 1.63
060605	3.8			1.835	4.73 ± 0.693
060708	1.92			2.485	6.89 ± 0.796
060714	2.71		−10.8586 ± 0.0352	1.87	9.13 ± 0.549
060814	0.84		−10.9108 ± 0.0353	1.97	60.6 ± 1.48
060906	3.685			2.06	12.2 ± 1.18
061121	1.314	3.78753 ± 0.01703		1.9485	196 ± 2.43
061222A	2.088		−9.95369 ± 0.02311	1.86	73.3 ± 1.51
070110	2.352			2.186	4.73 ± 0.648
070306	1.4959	4.86926 ± 0.02824	−11.2737 ± 0.0335	1.831	20.1 ± 0.899
070508	0.82	3.02525 ± 0.01189		1.764	224 ± 3.07
070521	0.553	3.55243 ± 0.04844		1.98	8.71 ± 1.2
070529	2.4996		−10.2468 ± 0.0552	1.76	11.1 ± 1.84
080310	2.4266		−11.39909559 ± 0.04247343	1.878	7.52 ± 0.893
080430	0.767			4.18	18.2 ± 0.808
080721	2.6			1.735	193 ± 10.3
081008	1.967	3.855508334 ± 0.056285684	−10.79325557 ± 0.06052989	1.84	10.5 ± 0.813
081221	2.26	2.931957486 ± 0.031176651	−9.354590372 ± 0.024985500	1.991	147 ± 2.46
090418A	1.608			1.98	15.8 ± 1.66
091018	0.971			1.91	59.1 ± 1.22
091020	1.71	2.952650108 ± 0.045271255	−9.712875280 ± 0.034480687	1.895	36.4 ± 1.71
091029	2.752		−11.34682089 ± 0.02563352	2.064	10.9 ± 0.786
100219A	4.7	4.721899099 ± 0.103773463		1.46	3.15 ± 0.721
110213A	1.46			2.1905	7.15 ± 1.88
110818A	3.36	3.846274153 ± 0.059648416	−11.28292779 ± 0.05514905	1.83	14.1 ± 1.5
111008A	5	3.989933926 ± 0.035676696		1.829	54.1 ± 3.91
120118B	2.943			2.01	13.8 ± 1.32
120404A	2.88			1.69	7.62 ± 0.109
120811C	2.67		−10.14322956 ± 0.05294686	1.65	26 ± 1.1
120922A	3.1			2.17	10.6 ± 0.757
121128A	2.2			1.9455	104 ± 2.1
131030A	1.29			1.693	265 ± 4.61
131105A	1.686			1.94	26.8 ± 0.204
140206A	2.7			1.672	169 ± 2.62
140419A	3.956	3.68383 ± 0.02442		1.678	41.8 ± 1.28
140506A	0.889	3.30686 ± 0.06882	−9.90438 ± 0.05516	1.9	86.7 ± 4.56
140509A	2.4			1.86	11.8 ± 1.77
140629A	2.3	2.85608 ± 0.06161		1.815	35 ± 1.87
150314A	1.758			1.73	381 ± 6.06
150403A	2.06			1.679	171 ± 3.81
150910A	1.36	3.85419 ± 0.02868		1.6645	8.28 ± 2.18
151027A	0.81	3.99907 ± 0.01731	−9.93519 ± 0.02221	1.9235	58.2 ± 2.88
160121A	1.96		−11.1726 ± 0.0694	2.02	7.79 ± 0.823
160227A	2.38	4.47751 ± 0.03885		1.679	4.88 ± 0.688
160327A	4.99	3.76407 ± 0.06033	−11.2238 ± 0.0673	1.78	12.5 ± 0.877
170202A	3.645			2.04	39.2 ± 1.79
170705A	2.01	3.63746 ± 0.06603		1.66	113 ± 2.41
180329B	1.998	4.02296 ± 0.04314		1.77	9.09 ± 2.02
190106A	1.86			2.9365	33.6 ± 1.3
190114A	3.37			1.86	4.12 ± 0.953

DATA AVAILABILITY

The data underlying this paper are listed in Table A1.