What Can Cosmic Microwave Background Observations Already Say about Cosmological Parameters in Open and Critical‐Density Cold Dark Matter Models?

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Abstract
We use a combination of the most recent cosmic microwave background (CMB) flat-band power measurements to place constraints on Hubble's constant h and the total density of the universe Ω0 in the context of inflation-based cold dark matter (CDM) models with no cosmological constant. We use χ2 minimization to explore the four-dimensional parameter space having as free parameters, h, Ω0, the power-spectrum slope n, and the power-spectrum normalization at ℓ = 10. Conditioning on Ω0 = 1, we obtain h = 0.33 ± 0.08. Allowing Ω0 to be a free parameter reduces the ability of the CMB data to constrain h, and we obtain 0.26 < h < 0.97 with a best-fit value at h = 0.40. We obtain Ω0 = 0.85 and set a lower limit Ω0 > 0.53. A strong correlation between acceptable h and Ω0 values leads to a new constraint Ω0h1/2 = 0.55 ± 0.10. We quote Δχ2 = 1 contours as error bars; however, because of nonlinearities of the models, these may be only crude approximations to 1 σ confidence limits. A favored open model with Ω0 = 0.3 and h = 0.70 is more than ~4 σ from the CMB data best-fit model and is rejected by goodness-of-fit statistics at the 99% confidence level. High baryonic models (Ωbh2 ~ 0.026) yield the best CMB χ2 fits and are more consistent with other cosmological constraints. The best-fit model has n = 0.91+ 0.29−0.09 and Q10 = 18.0+ 1.2−1.5 μK. Conditioning on n = 1, we obtain h = 0.55+ 0.13−0.19, Ω0 = 0.70 with a lower limit Ω0 > 0.58, and Q10 = 18.0+ 1.4−1.5 μK. The amplitude and position of the dominant peak in the best-fit power spectrum are Apeak = 76+ 3−7 μK and ℓpeak = 260+ 30−20. Unlike the Ω0 = 1 case we considered previously, CMB h results are now consistent with the higher values favored by local measurements of h but only if 0.55 Ω0 0.85. Using an approximate joint likelihood to combine our CMB constraint on Ω0h1/2 with other cosmological constraints, we obtain h = 0.58 ± 0.11 and Ω0 = 0.65+ 0.16−0.15.