We present posterior likelihoods and Bayesian model selection analysis for generalized cosmological models where the primordial perturbations include correlated adiabatic and cold dark matter isocurvature components .
We perform nested sampling with flat and , for the first time , curved spatial geometries of the Universe , using data from the cosmic microwave background ( CMB ) anisotropies , the Union supernovae ( SN ) sample and a combined measurement of the integrated Sachs–Wolfe ( ISW ) effect .
The CMB alone favors a 3 \% ( positively correlated ) isocurvature contribution in both the flat and curved cases .
The non-adiabatic contribution to the observed CMB temperature variance is 0 < \alpha _ { T } < 7 \% at 98 % CL in the curved case .
In the flat case , combining the CMB with SN data artificially biases the result towards the pure adiabatic \Lambda CDM concordance model , whereas in the curved case the favored level of non-adiabaticity stays at 3 % level with all combinations of data .
However , the ratio of Bayes factors , or \Delta \ln ( \mathrm { evidence } ) , is more than 5 points in favor of the flat adiabatic \Lambda CDM model , which suggests that the inclusion of the 5 extra parameters of the curved isocurvature model is not supported by the current data .
The results are very sensitive to the second and third acoustic peak regions in the CMB temperature angular power : therefore a careful calibration of these data will be required before drawing decisive conclusions on the nature of primordial perturbations .
Finally , we point out that the odds for the flat non-adiabatic model are 1:3 compared to the curved adiabatic model .
This may suggest that it is not much less motivated to extend the concordance model with 4 isocurvature degrees of freedom than it is to study the spatially curved adiabatic model , though at the moment the model selection disfavors both of these models .