Bulk viscous cosmological models with a cosmological constant: Observational constraints

Open Access

Table B.1.

Summary of parameter constraints and model comparison between ΛCDM and curved ΛCDM at 68% CL.

Datasets	ΛCDM ( ${\tilde{ζ}}_{0} = 0$ $Mathematical equation: $ \tilde{\zeta}_0 = 0 $$ )

	H₀	Ω_K0	M	χ_min²	AIC_ΛCDM	ΔAIC	BIC_ΛCDM	ΔBIC	DIC_ΛCDM	ΔDIC	lnℰ₀	lnℬ_0i
	(km s⁻¹ Mpc⁻¹)
PPS	73.2 ± 1.1	–	$- 19 . 262_{- 0.031}^{+ 0.032}$ $Mathematical equation: $ -19.262^{+0.032}_{-0.031} $$	1452.03	1460.03	0	1481.68	0	1458.81	0	−4.49	0
PPS + DESI	$73 . 4_{- 1.0}^{+ 1.1}$ $Mathematical equation: $ 73.4^{+1.1}_{-1.0} $$	–	$- 19 . 257_{- 0.032}^{+ 0.030}$ $Mathematical equation: $ -19.257^{+0.030}_{-0.032} $$	1465.68	1473.68	0	1495.36	0	1473.60	0	−5.26	0

Base 1	$67 . 97_{- 0.91}^{+ 0.94}$ $Mathematical equation: $ 67.97^{+0.94}_{-0.91} $$	–	−19.426 ± 0.026	1427.36	1435.36	0	1456.92	0	1433.04	0	−4.62	0
Base 1 + CC	$68 . 05_{- 0.90}^{+ 0.89}$ $Mathematical equation: $ 68.05^{+0.89}_{-0.90} $$	–	−19.424 ± 0.025	1442.53	1450.53	0	1472.17	0	1448.60	0	−4.72	0
Base 1 + CC + R22	$69 . 94_{- 0.72}^{+ 0.69}$ $Mathematical equation: $ 69.94^{+0.69}_{-0.72} $$	–	$- 19 . 373_{- 0.020}^{+ 0.019}$ $Mathematical equation: $ -19.373^{+0.019}_{-0.020} $$	1456.75	1464.75	0	1486.39	0	1462.85	0	−4.99	0

Base 2	$70 . 41_{- 0.71}^{+ 0.72}$ $Mathematical equation: $ 70.41^{+0.72}_{-0.71} $$	–	$- 19 . 358_{- 0.019}^{+ 0.020}$ $Mathematical equation: $ -19.358^{+0.020}_{-0.019} $$	1498.52	1506.52	0	1528.24	0	1504.38	0	−4.94	0
Base 2 + CC	$70 . 32_{- 0.70}^{+ 0.71}$ $Mathematical equation: $ 70.32^{+0.71}_{-0.70} $$	–	−19.360 ± 0.019	1513.98	1521.98	0	1543.78	0	1519.83	0	−5.00	0
Base 2 + CC + R22*	$71 . 08_{- 0.61}^{+ 0.60}$ $Mathematical equation: $ 71.08^{+0.60}_{-0.61} $$	–	−19.340 ± 0.016	1518.76	1526.76	0	1548.56	0	1524.46	0	−5.14	0

Base 3	$69 . 58_{- 0.54}^{+ 0.56}$ $Mathematical equation: $ 69.58^{+0.56}_{-0.54} $$	–	−19.377 ± 0.016	1513.14	1521.14	0	1542.89	0	1521.01	0	−6.23	0
Base 3 + CC	$69 . 58_{- 0.55}^{+ 0.54}$ $Mathematical equation: $ 69.58^{+0.54}_{-0.55} $$	–	−19.378 ± 0.016	1528.09	1536.09	0	1557.92	0	1535.94	0	−6.31	0
Base 3 + CC + R22	$70 . 20_{- 0.48}^{+ 0.49}$ $Mathematical equation: $ 70.20^{+0.49}_{-0.48} $$	–	−19.360 ± 0.014	1537.17	1545.17	0	1567.00	0	1545.04	0	−6.34	0

Datasets	ΛCDM + Ω_K ( ${\tilde{ζ}}_{0} = 0$ $Mathematical equation: $ \tilde{\zeta}_0 = 0 $$ )

	H₀	Ω_K0	M	χ_min²	AIC_i	ΔAIC	BIC_i	ΔBIC	DIC_i	ΔDIC	lnℰ_i	lnℬ_0i
	(km s⁻¹ Mpc⁻¹)

PPS	73.4 ± 1.1	$0 . 147_{- 0.078}^{+ 0.077}$ $Mathematical equation: $ 0.147^{+0.077}_{-0.078} $$	$- 19 . 245_{- 0.033}^{+ 0.034}$ $Mathematical equation: $ -19.245^{+0.034}_{-0.033} $$	1451.73	1461.73	−1.7	1488.79	−7.11	1458.45	0.36	−7.32	2.83
PPS + DESI	73.5 ± 1.1	$0 . 052_{- 0.034}^{+ 0.033}$ $Mathematical equation: $ 0.052^{+0.033}_{-0.034} $$	$- 19 . 251_{- 0.033}^{+ 0.031}$ $Mathematical equation: $ -19.251^{+0.031}_{-0.033} $$	1463.99	1473.99	−0.31	1501.09	−5.73	1473.00	0.60	−9.87	4.61

Base 1	66.7 ± 1.1	$0 . 120_{- 0.065}^{+ 0.068}$ $Mathematical equation: $ 0.120^{+0.068}_{-0.065} $$	$- 19 . 457_{- 0.031}^{+ 0.030}$ $Mathematical equation: $ -19.457^{+0.030}_{-0.031} $$	1418.54	1428.54	6.82	1455.49	1.43	1427.47	5.57	−7.69	3.07
Base 1 + CC	66.8 ± 1.1	$0 . 118_{- 0.066}^{+ 0.067}$ $Mathematical equation: $ 0.118^{+0.067}_{-0.066} $$	$- 19 . 454_{- 0.029}^{+ 0.030}$ $Mathematical equation: $ -19.454^{+0.030}_{-0.029} $$	1434.05	1444.05	6.48	1471.10	1.07	1442.77	5.83	−7.72	3.00
Base 1 + CC + R22	$69 . 78_{- 0.79}^{+ 0.84}$ $Mathematical equation: $ 69.78^{+0.84}_{-0.79} $$	$0 . 017_{- 0.055}^{+ 0.053}$ $Mathematical equation: $ 0.017^{+0.053}_{-0.055} $$	$- 19 . 376_{- 0.021}^{+ 0.022}$ $Mathematical equation: $ -19.376^{+0.022}_{-0.021} $$	1456.07	1466.07	−1.32	1493.12	−6.73	1463.49	−0.64	−7.92	2.93

Base 2	$70 . 37_{- 0.86}^{+ 0.83}$ $Mathematical equation: $ 70.37^{+0.83}_{-0.86} $$	0.004 ± 0.055	−19.359 ± 0.022	1498.16	1508.16	−1.64	1535.32	−7.08	1505.59	−1.21	−7.83	2.89
Base 2 + CC	$70 . 28_{- 0.83}^{+ 0.82}$ $Mathematical equation: $ 70.28^{+0.82}_{-0.83} $$	$0 . 006_{- 0.055}^{+ 0.054}$ $Mathematical equation: $ 0.006^{+0.054}_{-0.055} $$	−19.361 ± 0.021	1513.65	1523.65	−1.68	1550.90	−7.12	1521.02	−1.19	−7.91	2.91
Base 2 + CC + R22	$71 . 22_{- 0.67}^{+ 0.68}$ $Mathematical equation: $ 71.22^{+0.68}_{-0.67} $$	$- 0 . 027_{- 0.050}^{+ 0.051}$ $Mathematical equation: $ -0.027^{+0.051}_{-0.050} $$	$- 19 . 338_{- 0.017}^{+ 0.018}$ $Mathematical equation: $ -19.338^{+0.018}_{-0.017} $$	1518.73	1528.73	−1.98	1555.99	−7.43	1525.97	−1.51	−8.01	2.87

Base 3	$69 . 60_{- 0.64}^{+ 0.67}$ $Mathematical equation: $ 69.60^{+0.67}_{-0.64} $$	−0.002 ± 0.030	−19.377 ± 0.018	1513.19	1523.19	−2.05	1550.38	−7.49	1522.53	−1.52	−11.04	4.81
Base 3 + CC	69.59 ± 0.64	$- 0 . 001_{- 0.030}^{+ 0.029}$ $Mathematical equation: $ -0.001^{+0.029}_{-0.030} $$	$- 19 . 378_{- 0.017}^{+ 0.018}$ $Mathematical equation: $ -19.378^{+0.018}_{-0.017} $$	1528.12	1538.12	−2.03	1565.41	−7.49	1537.46	−1.52	−11.14	4.83
Base 3 + CC + R22	$70 . 39_{- 0.54}^{+ 0.57}$ $Mathematical equation: $ 70.39^{+0.57}_{-0.54} $$	−0.021 ± 0.028	$- 19 . 357_{- 0.015}^{+ 0.016}$ $Mathematical equation: $ -19.357^{+0.016}_{-0.015} $$	1536.70	1546.70	−1.53	1573.99	−6.99	1545.99	−0.95	−11.21	4.87

Notes. We shown information criteria ΔIC = IC_ΛCDM − IC_{ΛCDM + Ω_K}, where ΔIC > 2 indicates preference for the curved model; and Bayesian evidence via lnℬ_0i = lnℰ_ΛCDM − lnℰ_{ΛCDM + Ω_K}, where lnℬ_0i > 1 favors ΛCDM.

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