Bulk viscous cosmological models with a cosmological constant: Observational constraints

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Table C.2.

Summary of best-fit paramaters values at 68% CL for both ΛvCDM and ΛvCDM + Ω_K scenarios with free m, using different datasets.

Dataset	H₀	Ω_K0	Ω_b0h²	Ω_vc0h²	M	m	${\tilde{ζ}}_{0}$ $Mathematical equation: $ \tilde{\zeta}_0 $$	ζ₀ (×10⁶)	χ_min²	Δχ_min²
(km s⁻¹ Mpc⁻¹)							(Pa s)
	ΛvCDM (m free)

PPS	73.1 ± 1.1	–	$0 . 036_{- 0.009}^{+ 0.010}$ $Mathematical equation: $ 0.036^{+0.010}_{-0.009} $$	$0 . 135_{- 0.010}^{+ 0.009}$ $Mathematical equation: $ 0.135^{+0.009}_{-0.010} $$	$- 19 . 264_{- 0.034}^{+ 0.032}$ $Mathematical equation: $ -19.264^{+0.032}_{-0.034} $$	$- 1 . 11_{- 0.82}^{+ 1.39}$ $Mathematical equation: $ -1.11^{+1.39}_{-0.82} $$	$0 . 020_{- 0.019}^{+ 0.040}$ $Mathematical equation: $ 0.020^{+0.040}_{-0.019} $$	$0 . 85_{- 0.80}^{+ 1.69}$ $Mathematical equation: $ 0.85^{+1.69}_{-0.80} $$	1451.78	0.25
PPS + DESI	73.4 ± 1.1	–	0.028 ± 0.002	0.136 ± 0.006	−19.261 ± 0.031	$- 1 . 25_{- 0.63}^{+ 1.38}$ $Mathematical equation: $ -1.25^{+1.38}_{-0.63} $$	$0 . 012_{- 0.011}^{+ 0.026}$ $Mathematical equation: $ 0.012^{+0.026}_{-0.011} $$	$0 . 51_{- 0.47}^{+ 1.10}$ $Mathematical equation: $ 0.51^{+1.10}_{-0.47} $$	1463.19	2.49

Base 1	$67 . 3_{- 1.0}^{+ 1.1}$ $Mathematical equation: $ 67.3^{+1.1}_{-1.0} $$	–	0.022 ± 0.011	$0 . 127_{- 0.010}^{+ 0.010}$ $Mathematical equation: $ 0.127^{+0.010}_{-0.010} $$	−19.439 ± 0.028	$0 . 89_{- 0.60}^{+ 0.61}$ $Mathematical equation: $ 0.89^{+0.61}_{-0.60} $$	$0 . 032_{- 0.030}^{+ 0.050}$ $Mathematical equation: $ 0.032^{+0.050}_{-0.030} $$	$1 . 25_{- 1.17}^{+ 1.95}$ $Mathematical equation: $ 1.25^{+1.95}_{-1.17} $$	1416.83	10.53
Base 1 + CC	67.4 ± 1.0	–	0.030 ± 0.010	0.127 ± 0.010	$- 19 . 437_{- 0.028}^{+ 0.027}$ $Mathematical equation: $ -19.437^{+0.027}_{-0.028} $$	$0 . 91_{- 0.59}^{+ 0.63}$ $Mathematical equation: $ 0.91^{+0.63}_{-0.59} $$	$0 . 030_{- 0.029}^{+ 0.048}$ $Mathematical equation: $ 0.030^{+0.048}_{-0.029} $$	$1 . 17_{- 1.13}^{+ 1.87}$ $Mathematical equation: $ 1.17^{+1.87}_{-1.13} $$	1431.90	10.63
Base 1 + CC + R22	$69 . 76_{- 0.79}^{+ 0.78}$ $Mathematical equation: $ 69.76^{+0.78}_{-0.79} $$	–	0.025 ± 0.010	0.123 ± 0.010	−19.376 ± 0.021	$0 . 51_{- 1.03}^{+ 0.86}$ $Mathematical equation: $ 0.51^{+0.86}_{-1.03} $$	$0 . 040_{- 0.039}^{+ 0.064}$ $Mathematical equation: $ 0.040^{+0.064}_{-0.039} $$	$1 . 61_{- 1.57}^{+ 2.58}$ $Mathematical equation: $ 1.61^{+2.58}_{-1.57} $$	1451.68	5.07

Base 2	$70 . 19_{- 0.80}^{+ 0.82}$ $Mathematical equation: $ 70.19^{+0.82}_{-0.80} $$	–	$0 . 025_{- 0.010}^{+ 0.011}$ $Mathematical equation: $ 0.025^{+0.011}_{-0.010} $$	0.124 ± 0.010	−19.361 ± 0.021	$0 . 50_{- 0.93}^{+ 0.83}$ $Mathematical equation: $ 0.50^{+0.83}_{-0.93} $$	$0 . 043_{- 0.042}^{+ 0.064}$ $Mathematical equation: $ 0.043^{+0.064}_{-0.042} $$	$1 . 75_{- 1.71}^{+ 2.60}$ $Mathematical equation: $ 1.75^{+2.60}_{-1.71} $$	1493.43	5.09
Base 2 + CC	$70 . 13_{- 0.80}^{+ 0.82}$ $Mathematical equation: $ 70.13^{+0.82}_{-0.80} $$	–	0.024 ± 0.010	0.122 ± 0.010	−19.363 ± 0.021	$0 . 51_{- 0.98}^{+ 0.85}$ $Mathematical equation: $ 0.51^{+0.85}_{-0.98} $$	$0 . 042_{- 0.041}^{+ 0.061}$ $Mathematical equation: $ 0.042^{+0.061}_{-0.041} $$	$1 . 70_{- 1.66}^{+ 2.48}$ $Mathematical equation: $ 1.70^{+2.48}_{-1.66} $$	1509.09	4.89
Base 2 + CC + R22	$71 . 00_{- 0.66}^{+ 0.67}$ $Mathematical equation: $ 71.00^{+0.67}_{-0.66} $$	–	0.024 ± 0.010	$0 . 122_{- 0.009}^{+ 0.010}$ $Mathematical equation: $ 0.122^{+0.010}_{-0.009} $$	−19.342 ± 0.017	$0 . 23_{- 1.18}^{+ 0.83}$ $Mathematical equation: $ 0.23^{+0.83}_{-1.18} $$	$0 . 048_{- 0.047}^{+ 0.070}$ $Mathematical equation: $ 0.048^{+0.070}_{-0.047} $$	$1 . 97_{- 1.93}^{+ 2.88}$ $Mathematical equation: $ 1.97^{+2.88}_{-1.93} $$	1515.76	3.00

Base 3	$69 . 61_{- 0.62}^{+ 0.63}$ $Mathematical equation: $ 69.61^{+0.63}_{-0.62} $$	–	0.022 ± 0.002	$0 . 122_{- 0.006}^{+ 0.008}$ $Mathematical equation: $ 0.122^{+0.008}_{-0.006} $$	$- 19 . 378_{- 0.017}^{+ 0.018}$ $Mathematical equation: $ -19.378^{+0.018}_{-0.017} $$	$- 0 . 87_{- 0.84}^{+ 1.15}$ $Mathematical equation: $ -0.87^{+1.15}_{-0.84} $$	$0 . 029_{- 0.028}^{+ 0.050}$ $Mathematical equation: $ 0.029^{+0.050}_{-0.028} $$	$1 . 17_{- 1.12}^{+ 2.01}$ $Mathematical equation: $ 1.17^{+2.01}_{-1.12} $$	1511.89	1.25
Base 3 + CC	$69 . 59_{- 0.60}^{+ 0.61}$ $Mathematical equation: $ 69.59^{+0.61}_{-0.60} $$	–	$0 . 022_{- 0.002}^{+ 0.001}$ $Mathematical equation: $ 0.022^{+0.001}_{-0.002} $$	$0 . 122_{- 0.006}^{+ 0.008}$ $Mathematical equation: $ 0.122^{+0.008}_{-0.006} $$	−19.378 ± 0.017	$- 0 . 92_{- 0.80}^{+ 1.19}$ $Mathematical equation: $ -0.92^{+1.19}_{-0.80} $$	$0 . 030_{- 0.029}^{+ 0.050}$ $Mathematical equation: $ 0.030^{+0.050}_{-0.029} $$	$1 . 21_{- 1.17}^{+ 2.01}$ $Mathematical equation: $ 1.21^{+2.01}_{-1.17} $$	1527.00	1.09
Base 3 + CC + R22	$70 . 28_{- 0.52}^{+ 0.54}$ $Mathematical equation: $ 70.28^{+0.54}_{-0.52} $$	–	0.023 ± 0.002	$0 . 121_{- 0.006}^{+ 0.008}$ $Mathematical equation: $ 0.121^{+0.008}_{-0.006} $$	−19.359 ± 0.015	$- 1 . 31_{- 0.62}^{+ 1.03}$ $Mathematical equation: $ -1.31^{+1.03}_{-0.62} $$	$0 . 038_{- 0.037}^{+ 0.063}$ $Mathematical equation: $ 0.038^{+0.063}_{-0.037} $$	$1 . 55_{- 1.50}^{+ 2.56}$ $Mathematical equation: $ 1.55^{+2.56}_{-1.50} $$	1537.18	−0.01

	ΛvCDM + Ω_K (m free)

PPS	73.3 ± 1.2	$0 . 205_{- 0.094}^{+ 0.103}$ $Mathematical equation: $ 0.205^{+0.103}_{-0.094} $$	0.026 ± 0.011	0.124 ± 0.011	$- 19 . 247_{- 0.034}^{+ 0.033}$ $Mathematical equation: $ -19.247^{+0.033}_{-0.034} $$	$- 1 . 27_{- 0.72}^{+ 1.10}$ $Mathematical equation: $ -1.27^{+1.10}_{-0.72} $$	$0 . 073_{- 0.071}^{+ 0.137}$ $Mathematical equation: $ 0.073^{+0.137}_{-0.071} $$	$3 . 10_{- 3.01}^{+ 5.81}$ $Mathematical equation: $ 3.10^{+5.81}_{-3.01} $$	1451.76	0.27
PPS + DESI	73.4 ± 1.1	$0 . 062_{- 0.037}^{+ 0.040}$ $Mathematical equation: $ 0.062^{+0.040}_{-0.037} $$	0.031 ± 0.004	0.129 ± 0.011	$- 19 . 255_{- 0.039}^{+ 0.040}$ $Mathematical equation: $ -19.255^{+0.040}_{-0.039} $$	$- 1 . 19_{- 0.70}^{+ 1.14}$ $Mathematical equation: $ -1.19^{+1.14}_{-0.70} $$	$0 . 032_{- 0.029}^{+ 0.065}$ $Mathematical equation: $ 0.032^{+0.065}_{-0.029} $$	$1 . 36_{- 1.23}^{+ 2.72}$ $Mathematical equation: $ 1.36^{+2.72}_{-1.23} $$	1462.96	2.72

Base 1	66.1 ± 1.3	$0 . 21_{- 0.12}^{+ 0.13}$ $Mathematical equation: $ 0.21^{+0.13}_{-0.12} $$	$0 . 021_{- 0.017}^{+ 0.018}$ $Mathematical equation: $ 0.021^{+0.018}_{-0.017} $$	$0 . 119_{- 0.015}^{+ 0.016}$ $Mathematical equation: $ 0.119^{+0.016}_{-0.015} $$	$- 19 . 474_{- 0.036}^{+ 0.037}$ $Mathematical equation: $ -19.474^{+0.037}_{-0.036} $$	$- 0 . 41_{- 1.04}^{+ 0.95}$ $Mathematical equation: $ -0.41^{+0.95}_{-1.04} $$	$0 . 210_{- 0.180}^{+ 0.230}$ $Mathematical equation: $ 0.210^{+0.230}_{-0.180} $$	$8 . 03_{- 6.89}^{+ 8.80}$ $Mathematical equation: $ 8.03^{+8.80}_{-6.89} $$	1416.83	10.53
Base 1 + CC	66.2 ± 1.2	$0 . 20_{- 0.11}^{+ 0.12}$ $Mathematical equation: $ 0.20^{+0.12}_{-0.11} $$	0.021 ± 0.012	0.119 ± 0.012	$- 19 . 470_{- 0.033}^{+ 0.034}$ $Mathematical equation: $ -19.470^{+0.034}_{-0.033} $$	$- 0 . 45_{- 1.03}^{+ 0.95}$ $Mathematical equation: $ -0.45^{+0.95}_{-1.03} $$	$0 . 180_{- 0.160}^{+ 0.230}$ $Mathematical equation: $ 0.180^{+0.230}_{-0.160} $$	$6 . 90_{- 6.13}^{+ 8.81}$ $Mathematical equation: $ 6.90^{+8.81}_{-6.13} $$	1431.98	10.55
Base 1 + CC + R22	$69 . 58_{- 0.86}^{+ 0.90}$ $Mathematical equation: $ 69.58^{+0.90}_{-0.86} $$	$0 . 026_{- 0.082}^{+ 0.087}$ $Mathematical equation: $ 0.026^{+0.087}_{-0.082} $$	$0 . 026_{- 0.082}^{+ 0.087}$ $Mathematical equation: $ 0.026^{+0.087}_{-0.082} $$	$0 . 122_{- 0.011}^{+ 0.012}$ $Mathematical equation: $ 0.122^{+0.012}_{-0.011} $$	$- 19 . 380_{- 0.024}^{+ 0.023}$ $Mathematical equation: $ -19.380^{+0.023}_{-0.024} $$	$0 . 29_{- 1.31}^{+ 0.92}$ $Mathematical equation: $ 0.29^{+0.92}_{-1.31} $$	$0 . 059_{- 0.057}^{+ 0.110}$ $Mathematical equation: $ 0.059^{+0.110}_{-0.057} $$	$2 . 38_{- 2.30}^{+ 4.43}$ $Mathematical equation: $ 2.38^{+4.43}_{-2.30} $$	1448.63	8.12

Base 2	$70 . 14_{- 0.91}^{+ 0.93}$ $Mathematical equation: $ 70.14^{+0.93}_{-0.91} $$	$0 . 007_{- 0.077}^{+ 0.079}$ $Mathematical equation: $ 0.007^{+0.079}_{-0.077} $$	0.026 ± 0.012	0.124 ± 0.012	$- 19 . 363_{- 0.023}^{+ 0.024}$ $Mathematical equation: $ -19.363^{+0.024}_{-0.023} $$	$0 . 47_{- 1.06}^{+ 0.91}$ $Mathematical equation: $ 0.47^{+0.91}_{-1.06} $$	$0 . 049_{- 0.047}^{+ 0.089}$ $Mathematical equation: $ 0.049^{+0.089}_{-0.047} $$	$1 . 99_{- 1.91}^{+ 3.61}$ $Mathematical equation: $ 1.99^{+3.61}_{-1.91} $$	1489.50	9.02
Base 2 + CC	$70 . 07_{- 0.90}^{+ 0.88}$ $Mathematical equation: $ 70.07^{+0.88}_{-0.90} $$	$0 . 008_{- 0.081}^{+ 0.078}$ $Mathematical equation: $ 0.008^{+0.078}_{-0.081} $$	0.026 ± 0.012	0.124 ± 0.012	−19.365 ± 0.023	$0 . 49_{- 1.14}^{+ 0.92}$ $Mathematical equation: $ 0.49^{+0.92}_{-1.14} $$	$0 . 047_{- 0.045}^{+ 0.090}$ $Mathematical equation: $ 0.047^{+0.090}_{-0.045} $$	$1 . 91_{- 1.82}^{+ 3.65}$ $Mathematical equation: $ 1.91^{+3.65}_{-1.82} $$	1504.71	9.27
Base 2 + CC + R22	$71 . 07_{- 0.73}^{+ 0.74}$ $Mathematical equation: $ 71.07^{+0.74}_{-0.73} $$	$- 0 . 026_{- 0.074}^{+ 0.072}$ $Mathematical equation: $ -0.026^{+0.072}_{-0.074} $$	0.027 ± 0.012	0.125 ± 0.012	−19.340 ± 0.019	$0 . 51_{- 1.21}^{+ 0.94}$ $Mathematical equation: $ 0.51^{+0.94}_{-1.21} $$	$0 . 041_{- 0.039}^{+ 0.073}$ $Mathematical equation: $ 0.041^{+0.073}_{-0.039} $$	$1 . 69_{- 1.60}^{+ 3.00}$ $Mathematical equation: $ 1.69^{+3.00}_{-1.60} $$	1508.26	10.50

Base 3	$69 . 53_{- 0.72}^{+ 0.69}$ $Mathematical equation: $ 69.53^{+0.69}_{-0.72} $$	$0 . 008_{- 0.035}^{+ 0.037}$ $Mathematical equation: $ 0.008^{+0.037}_{-0.035} $$	0.022 ± 0.002	0.122 ± 0.009	−19.380 ± 0.019	$- 0 . 99_{- 0.80}^{+ 1.16}$ $Mathematical equation: $ -0.99^{+1.16}_{-0.80} $$	$0 . 036_{- 0.034}^{+ 0.063}$ $Mathematical equation: $ 0.036^{+0.063}_{-0.034} $$	$1 . 45_{- 1.37}^{+ 2.34}$ $Mathematical equation: $ 1.45^{+2.34}_{-1.37} $$	1511.75	1.39
Base 3 + CC	$69 . 49_{- 0.69}^{+ 0.68}$ $Mathematical equation: $ 69.49^{+0.68}_{-0.69} $$	$0 . 010_{- 0.035}^{+ 0.036}$ $Mathematical equation: $ 0.010^{+0.036}_{-0.035} $$	0.022 ± 0.002	$0 . 122_{- 0.008}^{+ 0.009}$ $Mathematical equation: $ 0.122^{+0.009}_{-0.008} $$	−19.380 ± 0.019	$- 1 . 01_{- 0.78}^{+ 1.14}$ $Mathematical equation: $ -1.01^{+1.14}_{-0.78} $$	$0 . 037_{- 0.035}^{+ 0.065}$ $Mathematical equation: $ 0.037^{+0.065}_{-0.035} $$	$1 . 49_{- 1.41}^{+ 2.61}$ $Mathematical equation: $ 1.49^{+2.61}_{-1.41} $$	1526.52	1.57
Base 3 + CC + R22	$70 . 36_{- 0.57}^{+ 0.58}$ $Mathematical equation: $ 70.36^{+0.58}_{-0.57} $$	−0.009 ± 0.032	0.022 ± 0.002	0.124 ± 0.008	−19.358 ± 0.016	$- 1 . 31_{- 0.62}^{+ 1.09}$ $Mathematical equation: $ -1.31^{+1.09}_{-0.62} $$	$0 . 032_{- 0.030}^{+ 0.067}$ $Mathematical equation: $ 0.032^{+0.067}_{-0.030} $$	$1 . 30_{- 1.22}^{+ 2.73}$ $Mathematical equation: $ 1.30^{+2.73}_{-1.22} $$	1535.44	1.73

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. We calculate ζ₀ using Eq. (15).

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