Standardizing the gamma-ray burst as a standard candle and applying it to cosmological probes: Constraints on the two-component dark energy model

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Volume 689 (September 2024)

A&A, 689 (2024) A165

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Table 3.

Constraints on the cosmological parameters of different models for different GRB samples and correlations.

Flat ωCDM model	Ω_m	ω	H₀	BIC	AIC
SNe Ia+X-ray (L_b, z − T_b, z)	$0 . 353_{- 0.034}^{+ 0.030}$ $0.353_{-0.034}^{+0.030}$	$- 1 . 211_{- 0.137}^{+ 0.126}$ $-1.211_{-0.137}^{+0.126}$	$70 . 32_{- 0.333}^{+ 0.340}$ $70.32_{-0.333}^{+0.340}$	1120.00	1105.05
SNe Ia+optical (L_b, z − T_b, z)	$0 . 356_{- 0.036}^{+ 0.031}$ $0.356_{-0.036}^{+0.031}$	$- 1 . 222_{- 0.145}^{+ 0.133}$ $-1.222_{-0.145}^{+0.133}$	$70 . 34_{- 0.338}^{+ 0.349}$ $70.34_{-0.338}^{+0.349}$	1125.33	1110.31
SNe Ia+X-ray+optical (L_b, z − T_b, z)	$0 . 359_{- 0.033}^{+ 0.029}$ $0.359_{-0.033}^{+0.029}$	$- 1 . 235_{- 0.136}^{+ 0.127}$ $-1.235_{-0.136}^{+0.127}$	$70 . 36_{- 0.337}^{+ 0.343}$ $70.36_{-0.337}^{+0.343}$	1159.20	1144.11
SNe Ia+X-ray (L_b, z − T_b, z − E_p, i)	$0 . 350_{- 0.038}^{+ 0.033}$ $0.350_{-0.038}^{+0.033}$	$- 1 . 197_{- 0.145}^{+ 0.135}$ $-1.197_{-0.145}^{+0.135}$	$70 . 30_{- 0.339}^{+ 0.348}$ $70.30_{-0.339}^{+0.348}$	1087.70	1072.75
SNe Ia+optical (L_b, z − T_b, z − E_p, i)	$0 . 352_{- 0.038}^{+ 0.032}$ $0.352_{-0.038}^{+0.032}$	$- 1 . 205_{- 0.144}^{+ 0.136}$ $-1.205_{-0.144}^{+0.136}$	$70 . 31_{- 0.338}^{+ 0.345}$ $70.31_{-0.338}^{+0.345}$	1108.36	1093.36
SNe Ia+X-ray+optical (L_b, z − T_b, z − E_p, i)	$0 . 352_{- 0.036}^{+ 0.032}$ $0.352_{-0.036}^{+0.032}$	$- 1 . 206_{- 0.143}^{+ 0.133}$ $-1.206_{-0.143}^{+0.133}$	$70 . 31_{- 0.339}^{+ 0.345}$ $70.31_{-0.339}^{+0.345}$	1109.97	1094.88

Non-flat ωCDM model	Ω_m	Ω_Λ	ω	H₀	BIC	AIC

SNe Ia+X-ray (L_b, z − T_b, z)	$0 . 343_{- 0.039}^{+ 0.037}$ $0.343_{-0.039}^{+0.037}$	$0 . 718_{- 0.189}^{+ 0.181}$ $0.718_{-0.189}^{+0.181}$	$- 1 . 109_{- 0.371}^{+ 0.201}$ $-1.109_{-0.371}^{+0.201}$	$70 . 29_{- 0.335}^{+ 0.361}$ $70.29_{-0.335}^{+0.361}$	1127.53	1107.60
SNe Ia+optical (L_b, z − T_b, z)	$0 . 343_{- 0.051}^{+ 0.044}$ $0.343_{-0.051}^{+0.044}$	$0 . 673_{- 0.223}^{+ 0.221}$ $0.673_{-0.223}^{+0.221}$	$- 1 . 182_{- 0.534}^{+ 0.248}$ $-1.182_{-0.534}^{+0.248}$	$70 . 35_{- 0.342}^{+ 0.376}$ $70.35_{-0.342}^{+0.376}$	1133.71	1113.71
SNe Ia+X-ray + optical (L_b, z − T_b, z)	$0 . 351_{- 0.039}^{+ 0.036}$ $0.351_{-0.039}^{+0.036}$	$0 . 756_{- 0.188}^{+ 0.162}$ $0.756_{-0.188}^{+0.162}$	$- 1 . 072_{- 0.326}^{+ 0.171}$ $-1.072_{-0.326}^{+0.171}$	$70 . 30_{- 0.326}^{+ 0.345}$ $70.30_{-0.326}^{+0.345}$	1165.99	1145.88
SNe Ia+X-ray (L_b, z − T_b, z − E_p, i)	$0 . 321_{- 0.060}^{+ 0.049}$ $0.321_{-0.060}^{+0.049}$	$0 . 571_{- 0.201}^{+ 0.262}$ $0.571_{-0.201}^{+0.262}$	$- 1 . 337_{- 0.733}^{+ 0.365}$ $-1.337_{-0.733}^{+0.365}$	$70 . 37_{- 0.366}^{+ 0.408}$ $70.37_{-0.366}^{+0.408}$	1096.22	1076.29
SNe Ia+optical (L_b, z − T_b, z − E_p, i)	$0 . 330_{- 0.057}^{+ 0.047}$ $0.330_{-0.057}^{+0.047}$	$0 . 611_{- 0.214}^{+ 0.245}$ $0.611_{-0.214}^{+0.245}$	$- 1 . 267_{- 0.650}^{+ 0.310}$ $-1.267_{-0.650}^{+0.310}$	$70 . 36_{- 0.356}^{+ 0.391}$ $70.36_{-0.356}^{+0.391}$	1116.78	1096.77
SNe Ia+X-ray+optical (L_b, z − T_b, z − E_p, i)	$0 . 331_{- 0.054}^{+ 0.047}$ $0.331_{-0.054}^{+0.047}$	$0 . 615_{- 0.210}^{+ 0.244}$ $0.615_{-0.210}^{+0.244}$	$- 1 . 265_{- 0.621}^{+ 0.311}$ $-1.265_{-0.621}^{+0.311}$	$70 . 35_{- 0.353}^{+ 0.388}$ $70.35_{-0.353}^{+0.388}$	1118.90	1098.79

Flat X₁X₂CDM model	Ω_m0	Ω_x1	ω₁	ω₂	H₀	BIC	AIC

SNe Ia+X-ray (L_b, z − T_b, z)	$0 . 338_{- 0.078}^{+ 0.049}$ $0.338_{-0.078}^{+0.049}$	$0 . 535_{- 0.400}^{+ 0.214}$ $0.535_{-0.400}^{+0.214}$	$- 1 . 205_{- 0.535}^{+ 0.412}$ $-1.205_{-0.535}^{+0.412}$	$- 1 . 240_{- 1.470}^{+ 0.728}$ $-1.240_{-1.470}^{+0.728}$	$70 . 37_{- 0.366}^{+ 0.378}$ $70.37_{-0.366}^{+0.378}$	1137.91	1112.99
SNe Ia+optical (L_b, z − T_b, z)	$0 . 336_{- 0.100}^{+ 0.053}$ $0.336_{-0.100}^{+0.053}$	$0 . 515_{- 0.387}^{+ 0.225}$ $0.515_{-0.387}^{+0.225}$	$- 1 . 233_{- 0.682}^{+ 0.490}$ $-1.233_{-0.682}^{+0.490}$	$- 1 . 236_{- 1.461}^{+ 0.684}$ $-1.236_{-1.461}^{+0.684}$	$70 . 39_{- 0.366}^{+ 0.390}$ $70.39_{-0.366}^{+0.390}$	1149.66	1124.65
SNe Ia+X-ray + optical (L_b, z − T_b, z)	$0 . 348_{- 0.076}^{+ 0.050}$ $0.348_{-0.076}^{+0.050}$	$0 . 553_{- 0.423}^{+ 0.218}$ $0.553_{-0.423}^{+0.218}$	$- 1 . 237_{- 0.524}^{+ 0.368}$ $-1.237_{-0.524}^{+0.368}$	$- 1 . 211_{- 1.514}^{+ 0.802}$ $-1.211_{-1.514}^{+0.802}$	$70 . 40_{- 0.369}^{+ 0.379}$ $70.40_{-0.369}^{+0.379}$	1175.04	1149.90
SNe Ia+X-ray (L_b, z − T_b, z − E_p, i)	$0 . 324_{- 0.108}^{+ 0.058}$ $0.324_{-0.108}^{+0.058}$	$0 . 515_{- 0.380}^{+ 0.217}$ $0.515_{-0.380}^{+0.217}$	$- 1 . 195_{- 0.703}^{+ 0.503}$ $-1.195_{-0.703}^{+0.503}$	$- 1 . 205_{- 1.531}^{+ 0.715}$ $-1.205_{-1.531}^{+0.715}$	$70 . 37_{- 0.381}^{+ 0.399}$ $70.37_{-0.381}^{+0.399}$	1113.08	1088.16
SNe Ia+optical (L_b, z − T_b, z − E_p, i)	$0 . 328_{- 0.104}^{+ 0.056}$ $0.328_{-0.104}^{+0.056}$	$0 . 516_{- 0.387}^{+ 0.218}$ $0.516_{-0.387}^{+0.218}$	$- 1 . 207_{- 0.666}^{+ 0.493}$ $-1.207_{-0.666}^{+0.493}$	$- 1 . 213_{- 1.549}^{+ 0.713}$ $-1.213_{-1.549}^{+0.713}$	$70 . 39_{- 0.377}^{+ 0.391}$ $70.39_{-0.377}^{+0.391}$	1132.63	1107.62
SNe Ia+X-ray+optical (L_b, z − T_b, z − E_p, i)	$0 . 326_{- 0.108}^{+ 0.057}$ $0.326_{-0.108}^{+0.057}$	$0 . 533_{- 0.399}^{+ 0.205}$ $0.533_{-0.399}^{+0.205}$	$- 1 . 192_{- 0.647}^{+ 0.517}$ $-1.192_{-0.647}^{+0.517}$	$- 1 . 251_{- 1.622}^{+ 0.735}$ $-1.251_{-1.622}^{+0.735}$	$70 . 37_{- 0.369}^{+ 0.387}$ $70.37_{-0.369}^{+0.387}$	1135.70	1110.56

non-flat X₁X₂CDM model	Ω_m0	Ω_x1	Ω_x2	ω₁	ω₂	H₀	BIC	AIC

SNe Ia+X-ray (L_b, z − T_b, z)	$0 . 294_{- 0.114}^{+ 0.059}$ $0.294_{-0.114}^{+0.059}$	$0 . 396_{- 0.288}^{+ 0.338}$ $0.396_{-0.288}^{+0.338}$	$0 . 403_{- 0.301}^{+ 0.343}$ $0.403_{-0.301}^{+0.343}$	$- 1 . 030_{- 0.907}^{+ 0.497}$ $-1.030_{-0.907}^{+0.497}$	$- 1 . 002_{- 0.802}^{+ 0.493}$ $-1.002_{-0.802}^{+0.493}$	$70 . 35_{- 0.359}^{+ 0.397}$ $70.35_{-0.359}^{+0.397}$	1157.23	1127.33
SNe Ia+optical (L_b, z − T_b, z)	$0 . 288_{- 0.114}^{+ 0.068}$ $0.288_{-0.114}^{+0.068}$	$0 . 386_{- 0.274}^{+ 0.351}$ $0.386_{-0.274}^{+0.351}$	$0 . 412_{- 0.292}^{+ 0.362}$ $0.412_{-0.292}^{+0.362}$	$- 1 . 015_{- 1.010}^{+ 0.503}$ $-1.015_{-1.010}^{+0.503}$	$- 0 . 983_{- 0.911}^{+ 0.461}$ $-0.983_{-0.911}^{+0.461}$	$70 . 37_{- 0.354}^{+ 0.404}$ $70.37_{-0.354}^{+0.404}$	1155.65	1125.64
SNe Ia+X-ray+optical (L_b, z − T_b, z)	$0 . 295_{- 0.121}^{+ 0.063}$ $0.295_{-0.121}^{+0.063}$	$0 . 436_{- 0.323}^{+ 0.339}$ $0.436_{-0.323}^{+0.339}$	$0 . 414_{- 0.310}^{+ 0.341}$ $0.414_{-0.310}^{+0.341}$	$- 0 . 959_{- 0.758}^{+ 0.473}$ $-0.959_{-0.758}^{+0.473}$	$- 0 . 994_{- 0.782}^{+ 0.483}$ $-0.994_{-0.782}^{+0.483}$	$70 . 33_{- 0.339}^{+ 0.386}$ $70.33_{-0.339}^{+0.386}$	1198.98	1168.81
SNe Ia+X-ray (L_b, z − T_b, z − E_p, i)	$0 . 278_{- 0.113}^{+ 0.068}$ $0.278_{-0.113}^{+0.068}$	$0 . 335_{- 0.255}^{+ 0.348}$ $0.335_{-0.255}^{+0.348}$	$0 . 360_{- 0.255}^{+ 0.348}$ $0.360_{-0.255}^{+0.348}$	$- 1 . 135_{- 1.155}^{+ 0.578}$ $-1.135_{-1.155}^{+0.578}$	$- 1 . 053_{- 1.084}^{+ 0.572}$ $-1.053_{-1.084}^{+0.572}$	$70 . 38_{- 0.362}^{+ 0.418}$ $70.38_{-0.362}^{+0.418}$	1119.34	1089.45
SNe Ia+optical (L_b, z − T_b, z − E_p, i)	$0 . 279_{- 0.115}^{+ 0.067}$ $0.279_{-0.115}^{+0.067}$	$0 . 325_{- 0.31}^{+ 0.337}$ $0.325_{-0.31}^{+0.337}$	$0 . 365_{- 0.254}^{+ 0.354}$ $0.365_{-0.254}^{+0.354}$	$- 1 . 133_{- 1.181}^{+ 0.615}$ $-1.133_{-1.181}^{+0.615}$	$- 1 . 049_{- 1.073}^{+ 0.552}$ $-1.049_{-1.073}^{+0.552}$	$70 . 39_{- 0.367}^{+ 0.413}$ $70.39_{-0.367}^{+0.413}$	1137.60	1107.59
SNe Ia+X-ray+optical (L_b, z − T_b, z − E_p, i)	$0 . 284_{- 0.111}^{+ 0.066}$ $0.284_{-0.111}^{+0.066}$	$0 . 337_{- 0.242}^{+ 0.358}$ $0.337_{-0.242}^{+0.358}$	$0 . 376_{- 0.266}^{+ 0.340}$ $0.376_{-0.266}^{+0.340}$	$- 1 . 113_{- 1.178}^{+ 0.572}$ $-1.113_{-1.178}^{+0.572}$	$- 1 . 047_{- 0.992}^{+ 0.510}$ $-1.047_{-0.992}^{+0.510}$	$70 . 38_{- 0.360}^{+ 0.409}$ $70.38_{-0.360}^{+0.409}$	1141.46	1111.29

Notes. The AIC and BIC values of different cases are also shown.

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