Table 2.
Posterior mean and standard deviation for the sound horizon rd, H0rd and q0 inferred from late-time observations including H0LiCOW lensing observations, Pantheon SN sample, and BAO measurements (BOSS).
| Flat (Ωk = 0) |
||||
|---|---|---|---|---|
| Parameter | Model 1 | Model 2 | Model 3 | Model 4 (fΛCDM) |
| rd (Mpc) | 132.7 ± 4.2 | 132.9 ± 4.4 | 134.2 ± 4.4 | 136.9 ± 3.7 |
| H0rd (km s−1) | 10107 ± 147 | 10065 ± 150 | 10052 ± 152 | 10038 ± 136 |
| q0 | −0.7 ± 0.07 | −0.5 ± 0.2 | −0.4 ± 0.3 | −0.55 ± 0.03 |
| lnℒm.a.p. | −86.3 | −86.1 | −86.7 | −87.7 |
| BIC score | 193 | 196 | 198 | 192 |
| lnτ (PlanckΛCDM) | 6.6 (3.2σ) | 5.7 (2.9σ) | 5.0 (2.7σ) | 5.7 (2.9σ) |
| lnτ (PlanckΛCDM+Neff) | 6.3 (3.1σ) | 5.6 (2.9σ) | 4.9 (2.7σ) | 5.0 (2.7σ) |
| lnτ (Planck early DE) | 5.1 (2.8σ) | 4.4 (2.5σ) | 3.7 (2.3σ) | 3.7 (2.2σ) |
| Free Ωk |
||||
| Parameter | Model 1 | Model 2 | Model 3 | Model 4 (ΛCDM) |
| rd (Mpc) | 129.2 ± 5.7 | 130.6 ± 5.9 | 131.2 ± 6.1 | 137.2 ± 4.8 |
| H0rd (km s−1) | 10045 ± 155 | 10033 ± 157 | 10017 ± 160 | 10041 ± 156 |
| Ωk | 0.18 ± 0.2 | 0.13 ± 0.2 | 0.15 ± 0.2 | −0.01 ± 0.2 |
| q0 | −0.6 ± 0.1 | −0.4 ± 0.2 | −0.4 ± 0.3 | −0.56 ± 0.07 |
| lnℒm.a.p. | −86.1 | −85.9 | −86.4 | −87.7 |
| BIC score | 196 | 200 | 201 | 196 |
| lnτ (PlanckΛCDM) | 5.6 (2.9σ) | 4.6 (2.6σ) | 4.0 (2.4σ) | 4.2 (2.4σ) |
| lnτ (PlanckΛCDM+Neff) | 5.7 (2.9σ) | 4.7 (2.6σ) | 4.2 (2.4σ) | 3.9 (2.3σ) |
| lnτ (Planck early DE) | 4.5 (2.6σ) | 3.6 (2.2σ) | 3.1 (2.0σ) | 2.6 (1.8σ) |
Notes. The fit quality is summarised in terms of log-likelihood at the maximum posterior probability, lnℒm.a.p., and the Bayesian information criterion BIC = ln(N)k − 2ln(ℒm.a.p.), where N is the number of data points and k is the number of free parameters. The odds indicator τ quantifies the tension between rd and H0 measured from late-time observations and the Planck data (for the standard flat ΛCDM model and its two extensions with a free effective number of neutrinos or early dark energy).
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