Hubble tension and small-scale inhomogeneities on light propagation

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Volume 711 (July 2026)

A&A, 711 (2026) A199

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Table A.1.

Results from our statistical analysis for the ZKDR and mZKDR approximations and (FLRW) ΛCDM with Ω_k = 0 and α constant.

Model	Data	α	Ω_m	H₀	M_abs
ZKDR	H0LiCOW	$0 . 723_{- 0.060 (0.491)}^{+ 0.277 (0.277)}$ $Mathematical equation: $ 0.723_{-0.060(0.491)}^{+0.277(0.277)} $$	$0 . 387_{- 0.271 (0.337)}^{+ 0.121 (0.423)}$ $Mathematical equation: $ 0.387_{-0.271(0.337)}^{+0.121(0.423)} $$	$71 . 153_{- 1.873 (5.534)}^{+ 3.286 (4.942)}$ $Mathematical equation: $ 71.153_{-1.873(5.534)}^{+3.286(4.942)} $$	—
	PPS	$0 . 750_{- 0.072 (0.311)}^{+ 0.250 (0.250)}$ $Mathematical equation: $ 0.750_{-0.072(0.311)}^{+0.250(0.250)} $$	$0 . 360_{- 0.028 (0.050)}^{+ 0.026 (0.054)}$ $Mathematical equation: $ 0.360_{-0.028(0.050)}^{+0.026(0.054)} $$	$73 . 480_{- 0.984 (1.940)}^{+ 0.986 (1.987)}$ $Mathematical equation: $ 73.480_{-0.984(1.940)}^{+0.986(1.987)} $$	$- 19 . 242_{- 0.028 (0.058)}^{+ 0.028 (0.055)}$ $Mathematical equation: $ -19.242_{-0.028(0.058)}^{+0.028(0.055)} $$
	SH0ES+HF	$0 . 526_{- 0.175 (0.472)}^{+ 0.455 (0.464)}$ $Mathematical equation: $ 0.526_{-0.175(0.472)}^{+0.455(0.464)} $$	$0 . 410_{- 0.199 (0.322)}^{+ 0.136 (0.310)}$ $Mathematical equation: $ 0.410_{-0.199(0.322)}^{+0.136(0.310)} $$	$73 . 225_{- 1.307 (2.154)}^{+ 0.946 (2.396)}$ $Mathematical equation: $ 73.225_{-1.307(2.154)}^{+0.946(2.396)} $$	$- 19 . 247_{- 0.029 (0.059)}^{+ 0.030 (0.059)}$ $Mathematical equation: $ -19.247_{-0.029(0.059)}^{+0.030(0.059)} $$
	H0LiCOW+ PPS	$0 . 845_{- 0.040 (0.218)}^{+ 0.153 (0.155)}$ $Mathematical equation: $ 0.845_{-0.040(0.218)}^{+0.153(0.155)} $$	$0 . 349_{- 0.023 (0.042)}^{+ 0.020 (0.045)}$ $Mathematical equation: $ 0.349_{-0.023(0.042)}^{+0.020(0.045)} $$	$73 . 528_{- 0.863 (1.818)}^{+ 0.897 (1.653)}$ $Mathematical equation: $ 73.528_{-0.863(1.818)}^{+0.897(1.653)} $$	$- 19 . 242_{- 0.027 (0.050)}^{+ 0.025 (0.051)}$ $Mathematical equation: $ -19.242_{-0.027(0.050)}^{+0.025(0.051)} $$
		α	Ω_m	H₀	M_abs
mZKDR	H0LiCOW	$0 . 523_{- 0.317 (0.434)}^{+ 0.312 (0.477)}$ $Mathematical equation: $ 0.523_{-0.317(0.434)}^{+0.312(0.477)} $$	$0 . 525_{- 0.234 (0.394)}^{+ 0.230 (0.413)}$ $Mathematical equation: $ 0.525_{-0.234(0.394)}^{+0.230(0.413)} $$	$72 . 717_{- 1.698 (4.297)}^{+ 2.485 (3.971)}$ $Mathematical equation: $ 72.717_{-1.698(4.297)}^{+2.485(3.971)} $$	—
	PPS	$0 . 630_{- 0.223 (0.360)}^{+ 0.258 (0.370)}$ $Mathematical equation: $ 0.630_{-0.223(0.360)}^{+0.258(0.370)} $$	$0 . 457_{- 0.122 (0.145)}^{+ 0.050 (0.196)}$ $Mathematical equation: $ 0.457_{-0.122(0.145)}^{+0.050(0.196)} $$	$73 . 710_{- 1.118 (2.115)}^{+ 0.932 (1.919)}$ $Mathematical equation: $ 73.710_{-1.118(2.115)}^{+0.932(1.919)} $$	$- 19 . 239_{- 0.029 (0.057)}^{+ 0.031 (0.057)}$ $Mathematical equation: $ -19.239_{-0.029(0.057)}^{+0.031(0.057)} $$
	SH0ES+HF	$0 . 535_{- 0.255 (0.421)}^{+ 0.358 (0.464)}$ $Mathematical equation: $ 0.535_{-0.255(0.421)}^{+0.358(0.464)} $$	$0 . 509_{- 0.198 (0.374)}^{+ 0.228 (0.377)}$ $Mathematical equation: $ 0.509_{-0.198(0.374)}^{+0.228(0.377)} $$	$73 . 483_{- 1.144 (2.223)}^{+ 1.089 (2.163)}$ $Mathematical equation: $ 73.483_{-1.144(2.223)}^{+1.089(2.163)} $$	$- 19 . 245_{- 0.028 (0.058)}^{+ 0.030 (0.055)}$ $Mathematical equation: $ -19.245_{-0.028(0.058)}^{+0.030(0.055)} $$
	H0LiCOW+ PPS	$0 . 635_{- 0.118 (0.430)}^{+ 0.364 (0.365)}$ $Mathematical equation: $ 0.635_{-0.118(0.430)}^{+0.364(0.365)} $$	$0 . 463_{- 0.139 (0.158)}^{+ 0.035 (0.245)}$ $Mathematical equation: $ 0.463_{-0.139(0.158)}^{+0.035(0.245)} $$	$73 . 401_{- 0.880 (1.792)}^{+ 0.896 (1.754)}$ $Mathematical equation: $ 73.401_{-0.880(1.792)}^{+0.896(1.754)} $$	$- 19 . 249_{- 0.026 (0.054)}^{+ 0.026 (0.050)}$ $Mathematical equation: $ -19.249_{-0.026(0.054)}^{+0.026(0.050)} $$
			Ω_m	H₀	M_abs
ΛCDM	H0LiCOW	—	$0 . 403_{- 0.273 (0.353)}^{+ 0.108 (0.414)}$ $Mathematical equation: $ 0.403_{-0.273(0.353)}^{+0.108(0.414)} $$	$72 . 516_{- 1.823 (4.471)}^{+ 2.504 (4.127)}$ $Mathematical equation: $ 72.516_{-1.823(4.471)}^{+2.504(4.127)} $$	—
	PPS	—	$0 . 332_{- 0.018 (0.034)}^{+ 0.017 (0.036)}$ $Mathematical equation: $ 0.332_{-0.018(0.034)}^{+0.017(0.036)} $$	$73 . 594_{- 0.922 (2.036)}^{+ 1.101 (2.049)}$ $Mathematical equation: $ 73.594_{-0.922(2.036)}^{+1.101(2.049)} $$	$- 19 . 243_{- 0.029 (0.059)}^{+ 0.029 (0.060)}$ $Mathematical equation: $ -19.243_{-0.029(0.059)}^{+0.029(0.060)} $$
	SH0ES+HF	—	$0 . 394_{- 0.155 (0.287)}^{+ 0.143 (0.295)}$ $Mathematical equation: $ 0.394_{-0.155(0.287)}^{+0.143(0.295)} $$	$73 . 195_{- 1.103 (2.160)}^{+ 1.108 (2.195)}$ $Mathematical equation: $ 73.195_{-1.103(2.160)}^{+1.108(2.195)} $$	$- 19 . 248_{- 0.029 (0.055)}^{+ 0.029 (0.059)}$ $Mathematical equation: $ -19.248_{-0.029(0.055)}^{+0.029(0.059)} $$
	H0LiCOW+ PPS	—	$0 . 336_{- 0.018 (0.033)}^{+ 0.018 (0.036)}$ $Mathematical equation: $ 0.336_{-0.018(0.033)}^{+0.018(0.036)} $$	$73 . 685_{- 0.916 (1.809)}^{+ 0.891 (1.625)}$ $Mathematical equation: $ 73.685_{-0.916(1.809)}^{+0.891(1.625)} $$	$- 19 . 239_{- 0.023 (0.050)}^{+ 0.028 (0.050)}$ $Mathematical equation: $ -19.239_{-0.023(0.050)}^{+0.028(0.050)} $$

Notes. Data used in these analysis are from H0LiCOW gravitational lenses, luminosity distances reported by Pantheon⁺+SH0ES collaboration (PPS), and the ones employed by SH0ES. For each parameter, we present the mean value and the 68% (95%) confidence levels, or the upper limits obtained.

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