Determining H0 from a distance sum rule that combines gamma-ray bursts with observational Hubble data and strong gravitational lensing

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Volume 701 (September 2025)

A&A, 701 (2025) A220

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Table 1.

Comparison of the best-fit among the MCMC 1, MCMC 2, and MCMC 3 analyses and separate constraints of SGL and OHD.

	α₀ ≡ h₀	α₁ (or Ω_m)	α₂	a	b	σ
MCMC 1	$0 . 764_{- 0.048 (0.071)}^{+ 0.034 (0.062)}$ $0.764_{-0.048\,(0.071)}^{+0.034\,(0.062)}$	$0 . 868_{- 0.166 (0.271)}^{+ 0.169 (0.259)}$ $0.868_{-0.166\,(0.271)}^{+0.169\,(0.259)}$	$2 . 170_{- 0.283 (0.448)}^{+ 0.295 (0.471)}$ $2.170_{-0.283\,(0.448)}^{+0.295\,(0.471)}$	−	−	−
MCMC 2	$0 . 750_{- 0.040 (0.071)}^{+ 0.044 (0.071)}$ $0.750_{-0.040\,(0.071)}^{+0.044\,(0.071)}$	$0 . 902_{- 0.172 (0.274)}^{+ 0.156 (0.264)}$ $0.902_{-0.172\,(0.274)}^{+0.156\,(0.264)}$	$2 . 170_{- 0.289 (0.456)}^{+ 0.273 (0.451)}$ $2.170_{-0.289\,(0.456)}^{+0.273\,(0.451)}$	$1 . 650_{- 0.158 (0.276)}^{+ 0.119 (0.193)}$ $1.650_{-0.158\,(0.276)}^{+0.119\,(0.193)}$	$0 . 855_{- 0.089 (0.139)}^{+ 0.108 (0.191)}$ $0.855_{-0.089\,(0.139)}^{+0.108\,(0.191)}$	$0 . 331_{- 0.048 (0.072)}^{+ 0.066 (0.114)}$ $0.331_{-0.048\,(0.072)}^{+0.066\,(0.114)}$
MCMC 3	$0 . 752_{- 0.041 (0.066)}^{+ 0.040 (0.066)}$ $0.752_{-0.041\,(0.066)}^{+0.040\,(0.066)}$	$0 . 888_{- 0.161 (0.258)}^{+ 0.162 (0.265)}$ $0.888_{-0.161\,(0.258)}^{+0.162\,(0.265)}$	$2 . 300_{- 0.283 (0.439)}^{+ 0.266 (0.443)}$ $2.300_{-0.283\,(0.439)}^{+0.266\,(0.443)}$	$49 . 600_{- 0.345 (0.556)}^{+ 0.370 (0.598)}$ $49.600_{-0.345\,(0.556)}^{+0.370\,(0.598)}$	$0 . 835_{- 0.129 (0.221)}^{+ 0.145 (0.223)}$ $0.835_{-0.129\,(0.221)}^{+0.145\,(0.223)}$	$0 . 164_{- 0.023 (0.035)}^{+ 0.022 (0.036)}$ $0.164_{-0.023\,(0.035)}^{+0.022\,(0.036)}$

SGL	0.736 ± 0.017	−	−	−	−	−
OHD 1	$0 . 672_{- 0.051 (0.104)}^{+ 0.053 (0.106)}$ $0.672_{-0.051\,(0.104)}^{+0.053\,(0.106)}$	$1 . 025_{- 0.158 (0.316)}^{+ 0.151 (0.307)}$ $1.025_{-0.158\,(0.316)}^{+0.151\,(0.307)}$	$2 . 108_{- 0.224 (0.451)}^{+ 0.228 (0.453)}$ $2.108_{-0.224\,(0.451)}^{+0.228\,(0.453)}$	−	−	−
OHD 2	$0 . 674_{- 0.033 (0.068)}^{+ 0.032 (0.064)}$ $0.674_{-0.033\,(0.068)}^{+0.032\,(0.064)}$	$0 . 330_{- 0.062 (0.114)}^{+ 0.074 (0.165)}$ $0.330_{-0.062\,(0.114)}^{+0.074\,(0.165)}$	−	−	−	−

Notes. The best-fit results are given with 1σ (2σ) error bars. The first fit (MCMC 1) employs small redshift data only, from OHD and SGL catalogs. The second (MCMC 2) and third (MCMC 3), involve GRB data from two GRB correlations, namely, the Amati and Combo correlations, respectively. The SGL result on h₀ refers to the case with Ω_k = 0 obtained in Qi et al. (2021), whereas the results of OHD 1 and OHD 2 refer to the Bézier interpolation and the fit with the flat ΛCDM model of the OHD catalog, respectively. For OHD 2, the third column displays the constraint on the matter density parameter Ω_m.

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