Bulk viscous cosmological models with a cosmological constant: Observational constraints

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Table C.1.

Summary of best-fit paramaters values at 68% CL for both ΛvCDM and ΛvCDM + Ω_K scenarios with constant bulk viscosity (m = 0), using different datasets.

Dataset	H₀	Ω_K0	Ω_b0h²	Ω_vc0h²	M	${\tilde{ζ}}_{0}$ $Mathematical equation: $ \tilde{\zeta}_0 $$	ζ₀ (×10⁶)	χ_min, i²	Δχ_min²
Dataset	(km s⁻¹ Mpc⁻¹)						(Pa s)
	ΛvCDM (m = 0)

PPS	73.1 ± 1.1	–	0.037 ± 0.010	0.135 ± 0.009	−19.265 ± 0.032	$0 . 017_{- 0.017}^{+ 0.032}$ $Mathematical equation: $ 0.017^{+0.032}_{-0.017} $$	$0 . 72_{- 0.72}^{+ 1.35}$ $Mathematical equation: $ 0.72^{+1.35}_{-0.72} $$	1451.96	0.07
PPS + DESI	73.3 ± 1.1	–	0.028 ± 0.002	$0 . 137_{- 0.006}^{+ 0.007}$ $Mathematical equation: $ 0.137^{+0.007}_{-0.006} $$	−19.262 ± 0.032	$0 . 011_{- 0.011}^{+ 0.021}$ $Mathematical equation: $ 0.011^{+0.021}_{-0.011} $$	$0 . 47_{- 0.47}^{+ 0.89}$ $Mathematical equation: $ 0.47^{+0.89}_{-0.47} $$	1464.63	1.05

Base 1	$67 . 94_{- 0.92}^{+ 0.97}$ $Mathematical equation: $ 67.94^{+0.97}_{-0.92} $$	–	0.030 ± 0.010	0.127 ± 0.010	$- 19 . 426_{- 0.027}^{+ 0.026}$ $Mathematical equation: $ -19.426^{+0.026}_{-0.027} $$	$0 . 092_{- 0.066}^{+ 0.075}$ $Mathematical equation: $ 0.092^{+0.075}_{-0.066} $$	$3 . 62_{- 2.60}^{+ 2.95}$ $Mathematical equation: $ 3.62^{+2.95}_{-2.60} $$	1420.81	6.55
Base 1 + CC	68.0 ± 0.9	–	$0 . 029_{- 0.009}^{+ 0.010}$ $Mathematical equation: $ 0.029^{+0.010}_{-0.009} $$	0.127 ± 0.010	$- 19 . 425_{- 0.026}^{+ 0.025}$ $Mathematical equation: $ -19.425^{+0.025}_{-0.026} $$	$0 . 088_{- 0.063}^{+ 0.075}$ $Mathematical equation: $ 0.088^{+0.075}_{-0.063} $$	$3 . 46_{- 2.48}^{+ 2.95}$ $Mathematical equation: $ 3.46^{+2.95}_{-2.48} $$	1435.77	6.76
Base 1 + CC + R22	$69 . 92_{- 0.74}^{+ 0.71}$ $Mathematical equation: $ 69.92^{+0.71}_{-0.74} $$	–	0.027 ± 0.010	0.125 ± 0.010	$- 19 . 373_{- 0.020}^{+ 0.019}$ $Mathematical equation: $ -19.373^{+0.019}_{-0.020} $$	$0 . 082_{- 0.061}^{+ 0.073}$ $Mathematical equation: $ 0.082^{+0.073}_{-0.061} $$	$3 . 32_{- 2.47}^{+ 2.95}$ $Mathematical equation: $ 3.32^{+2.95}_{-2.47} $$	1453.00	3.75

Base 2	$70 . 39_{- 0.73}^{+ 0.75}$ $Mathematical equation: $ 70.39^{+0.75}_{-0.73} $$	–	0.027 ± 0.010	0.125 ± 0.010	−19.357 ± 0.020	$0 . 088_{- 0.063}^{+ 0.072}$ $Mathematical equation: $ 0.088^{+0.072}_{-0.063} $$	$3 . 59_{- 2.57}^{+ 2.93}$ $Mathematical equation: $ 3.59^{+2.93}_{-2.57} $$	1494.40	4.12
Base 2 + CC	$70 . 30_{- 0.72}^{+ 0.73}$ $Mathematical equation: $ 70.30^{+0.73}_{-0.72} $$	–	0.027 ± 0.010	0.125 ± 0.010	$- 19 . 360_{- 0.019}^{+ 0.020}$ $Mathematical equation: $ -19.360^{+0.020}_{-0.019} $$	$0 . 085_{- 0.061}^{+ 0.070}$ $Mathematical equation: $ 0.085^{+0.070}_{-0.061} $$	$3 . 46_{- 2.48}^{+ 2.85}$ $Mathematical equation: $ 3.46^{+2.85}_{-2.48} $$	1510.30	3.68
Base 2 + CC + R22	$71 . 05_{- 0.60}^{+ 0.62}$ $Mathematical equation: $ 71.05^{+0.62}_{-0.60} $$	–	0.026 ± 0.010	0.124 ± 0.010	−19.340 ± 0.016	$0 . 081_{- 0.058}^{+ 0.071}$ $Mathematical equation: $ 0.081^{+0.071}_{-0.058} $$	$3 . 33_{- 2.38}^{+ 2.92}$ $Mathematical equation: $ 3.33^{+2.92}_{-2.38} $$	1516.14	2.62

Base 3	69.53 ± 0.56	–	0.021 ± 0.002	$0 . 124_{- 0.007}^{+ 0.009}$ $Mathematical equation: $ 0.124^{+0.009}_{-0.007} $$	$- 19 . 379_{- 0.017}^{+ 0.016}$ $Mathematical equation: $ -19.379^{+0.016}_{-0.017} $$	$0 . 030_{- 0.026}^{+ 0.038}$ $Mathematical equation: $ 0.030^{+0.038}_{-0.026} $$	$1 . 21_{- 1.05}^{+ 1.53}$ $Mathematical equation: $ 1.21^{+1.53}_{-1.05} $$	1511.96	1.18
Base 3 + CC	$69 . 52_{- 0.57}^{+ 0.54}$ $Mathematical equation: $ 69.52^{+0.54}_{-0.57} $$	–	0.021 ± 0.002	$0 . 124_{- 0.007}^{+ 0.009}$ $Mathematical equation: $ 0.124^{+0.009}_{-0.007} $$	−19.379 ± 0.016	$0 . 030_{- 0.026}^{+ 0.036}$ $Mathematical equation: $ 0.030^{+0.036}_{-0.026} $$	$1 . 21_{- 1.05}^{+ 1.45}$ $Mathematical equation: $ 1.21^{+1.45}_{-1.05} $$	1526.91	1.18
Base 3 + CC + R22	$70 . 17_{- 0.51}^{+ 0.49}$ $Mathematical equation: $ 70.17^{+0.49}_{-0.51} $$	–	0.022 ± 0.002	$0 . 122_{- 0.007}^{+ 0.009}$ $Mathematical equation: $ 0.122^{+0.009}_{-0.007} $$	$- 19 . 361_{- 0.015}^{+ 0.014}$ $Mathematical equation: $ -19.361^{+0.014}_{-0.015} $$	$0 . 026_{- 0.023}^{+ 0.035}$ $Mathematical equation: $ 0.026^{+0.035}_{-0.023} $$	$1 . 06_{- 0.93}^{+ 1.42}$ $Mathematical equation: $ 1.06^{+1.42}_{-0.93} $$	1537.12	0.05

	ΛvCDM + Ω_K (m = 0)

PPS	73.3 ± 1.2	$0 . 196_{- 0.090}^{+ 0.092}$ $Mathematical equation: $ 0.196^{+0.092}_{-0.090} $$	0.026 ± 0.011	0.124 ± 0.011	$- 19 . 247_{- 0.034}^{+ 0.033}$ $Mathematical equation: $ -19.247^{+0.033}_{-0.034} $$	$0 . 048_{- 0.047}^{+ 0.084}$ $Mathematical equation: $ 0.048^{+0.084}_{-0.047} $$	$2 . 04_{- 1.99}^{+ 3.56}$ $Mathematical equation: $ 2.04^{+3.56}_{-1.99} $$	1451.72	0.31
PPS + DESI	73.3 ± 1.1	$0 . 062_{- 0.035}^{+ 0.037}$ $Mathematical equation: $ 0.062^{+0.037}_{-0.035} $$	0.031 ± 0.003	0.129 ± 0.009	$- 19 . 255_{- 0.033}^{+ 0.032}$ $Mathematical equation: $ -19.255^{+0.032}_{-0.033} $$	$0 . 019_{- 0.019}^{+ 0.029}$ $Mathematical equation: $ 0.019^{+0.029}_{-0.019} $$	$0 . 81_{- 0.81}^{+ 1.23}$ $Mathematical equation: $ 0.81^{+1.23}_{-0.81} $$	1463.20	2.48

Base 1	$66 . 1_{- 1.1}^{+ 1.2}$ $Mathematical equation: $ 66.1^{+1.2}_{-1.1} $$	$0 . 181_{- 0.074}^{+ 0.073}$ $Mathematical equation: $ 0.181^{+0.073}_{-0.074} $$	0.023 ± 0.011	0.121 ± 0.011	$- 19 . 470_{- 0.030}^{+ 0.032}$ $Mathematical equation: $ -19.470^{+0.032}_{-0.030} $$	$0 . 195_{- 0.095}^{+ 0.094}$ $Mathematical equation: $ 0.195^{+0.094}_{-0.095} $$	$7 . 46_{- 3.64}^{+ 3.60}$ $Mathematical equation: $ 7.46^{+3.60}_{-3.64} $$	1417.36	10.00
Base 1 + CC	66.3 ± 1.1	$0 . 174_{- 0.070}^{+ 0.074}$ $Mathematical equation: $ 0.174^{+0.074}_{-0.070} $$	0.023 ± 0.011	0.121 ± 0.011	$- 19 . 466_{- 0.029}^{+ 0.030}$ $Mathematical equation: $ -19.466^{+0.030}_{-0.029} $$	$0 . 188_{- 0.094}^{+ 0.091}$ $Mathematical equation: $ 0.188^{+0.091}_{-0.094} $$	$7 . 20_{- 3.60}^{+ 3.49}$ $Mathematical equation: $ 7.20^{+3.49}_{-3.60} $$	1433.17	9.36
Base 1 + CC + R22	$69 . 56_{- 0.83}^{+ 0.81}$ $Mathematical equation: $ 69.56^{+0.81}_{-0.83} $$	$0 . 049_{- 0.058}^{+ 0.059}$ $Mathematical equation: $ 0.049^{+0.059}_{-0.058} $$	0.025 ± 0.011	0.123 ± 0.011	$- 19 . 381_{- 0.022}^{+ 0.021}$ $Mathematical equation: $ -19.381^{+0.021}_{-0.022} $$	$0 . 109_{- 0.074}^{+ 0.081}$ $Mathematical equation: $ 0.109^{+0.081}_{-0.074} $$	$4 . 39_{- 2.98}^{+ 3.26}$ $Mathematical equation: $ 4.39^{+3.26}_{-2.98} $$	1453.65	3.10

Base 2	$70 . 09_{- 0.90}^{+ 0.87}$ $Mathematical equation: $ 70.09^{+0.87}_{-0.90} $$	0.040 ± 0.061	0.025 ± 0.011	0.124 ± 0.011	$- 19 . 365_{- 0.022}^{+ 0.023}$ $Mathematical equation: $ -19.365^{+0.023}_{-0.022} $$	$0 . 111_{- 0.075}^{+ 0.082}$ $Mathematical equation: $ 0.111^{+0.082}_{-0.075} $$	$4 . 50_{- 3.04}^{+ 3.33}$ $Mathematical equation: $ 4.50^{+3.33}_{-3.04} $$	1494.81	3.71
Base 2 + CC	$70 . 00_{- 0.85}^{+ 0.84}$ $Mathematical equation: $ 70.00^{+0.84}_{-0.85} $$	0.025 ± 0.011	0.025 ± 0.011	0.123 ± 0.011	$- 19 . 366_{- 0.022}^{+ 0.021}$ $Mathematical equation: $ -19.366^{+0.021}_{-0.022} $$	$0 . 105_{- 0.072}^{+ 0.080}$ $Mathematical equation: $ 0.105^{+0.080}_{-0.072} $$	$4 . 25_{- 2.91}^{+ 3.24}$ $Mathematical equation: $ 4.25^{+3.24}_{-2.91} $$	1510.80	3.18
Base 2 + CC + R22	71.06 ± 0.70	$- 0 . 002_{- 0.053}^{+ 0.056}$ $Mathematical equation: $ -0.002^{+0.056}_{-0.053} $$	0.026 ± 0.011	0.125 ± 0.011	−19.341 ± 0.018	$0 . 078_{- 0.059}^{+ 0.077}$ $Mathematical equation: $ 0.078^{+0.077}_{-0.059} $$	$3 . 16_{- 2.43}^{+ 3.17}$ $Mathematical equation: $ 3.16^{+3.17}_{-2.43} $$	1515.16	3.60

Base 3	$69 . 42_{- 0.67}^{+ 0.68}$ $Mathematical equation: $ 69.42^{+0.68}_{-0.67} $$	0.009 ± 0.032	0.021 ± 0.002	0.124 ± 0.009	$- 19 . 381_{- 0.019}^{+ 0.018}$ $Mathematical equation: $ -19.381^{+0.018}_{-0.019} $$	$0 . 033_{- 0.028}^{+ 0.040}$ $Mathematical equation: $ 0.033^{+0.040}_{-0.028} $$	$1 . 33_{- 1.12}^{+ 1.61}$ $Mathematical equation: $ 1.33^{+1.61}_{-1.12} $$	1512.06	1.08
Base 3 + CC	$69 . 38_{- 0.65}^{+ 0.67}$ $Mathematical equation: $ 69.38^{+0.67}_{-0.65} $$	$0 . 009_{- 0.031}^{+ 0.032}$ $Mathematical equation: $ 0.009^{+0.032}_{-0.031} $$	0.021 ± 0.002	0.124 ± 0.009	−19.382 ± 0.018	$0 . 032_{- 0.028}^{+ 0.040}$ $Mathematical equation: $ 0.032^{+0.040}_{-0.028} $$	$1 . 29_{- 1.12}^{+ 1.61}$ $Mathematical equation: $ 1.29^{+1.61}_{-1.12} $$	1526.91	1.18
Base 3 + CC + R22	$70 . 28_{- 0.58}^{+ 0.60}$ $Mathematical equation: $ 70.28^{+0.60}_{-0.58} $$	$- 0 . 014_{- 0.029}^{+ 0.030}$ $Mathematical equation: $ -0.014^{+0.030}_{-0.029} $$	$0 . 0219_{- 0.0019}^{+ 0.0018}$ $Mathematical equation: $ 0.0219^{+0.0018}_{-0.0019} $$	$0 . 126_{- 0.008}^{+ 0.009}$ $Mathematical equation: $ 0.126^{+0.009}_{-0.008} $$	−19.358 ± 0.016	$0 . 024_{- 0.024}^{+ 0.034}$ $Mathematical equation: $ 0.024^{+0.034}_{-0.024} $$	$0 . 98_{- 0.98}^{+ 1.38}$ $Mathematical equation: $ 0.98^{+1.38}_{-0.98} $$	1536.15	1.02

Notes. All Δ are computed relative to the standard ΛCDM model for the same combination of data sets. Here, Δχ_min² ≡ χ_min, ΛCDM − χ_min, i, where the index i denotes each proposed bulk viscous model. Positive values of Δχ_min² indicate that the bulk viscous model provides a better fit to the data. The viscosity parameter ζ₀ is calculated using Eq. (15).

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