Constraints on primordial isocurvature perturbations and spatial curvature by Bayesian model selection

Abstract

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 LCDM 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(evidence), is more than 5 points in favor of the flat adiabatic LCDM 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.

Figures (5)

• Figure 1: Left: Posterior likelihood distributions for the model parameters assuming mixed initial conditions and flat spatial geometry of the Universe. Right: The same for curved spatial geometry. 'ALL' refers to the CMB&SN&ISW data.
• Figure 2: Illustrative 95% CL regions when using the SN or ISW data alone. The dotted black curves indicate the angular diameter distance $D_A$ to last scattering in units of Mpc. Two of them are highlighted --- in the mixed models the CMB data favor $D_A \simeq 13\, 000\,$Mpc, whereas in the pure adiabatic models $D_A \simeq 14\, 000\,$Mpc is favored. The thin black dashed line indicates flat models. The thin red line indicates in which part of the parameter space the expansion of the Universe is accelerating or decelerating today.
• Figure 3: Selected 2d posterior likelihoods for the mixed model when allowing for a spatially curved geometry of the Universe. The inner contours indicate 68% CL, and the outer ones 95% CL regions.
• Figure 4: Posterior likelihoods with the CMB&SN data for selected model parameters in the flat and curved case assuming either pure adiabatic or mixed initial conditions.
• Figure 5: Posterior likelihoods with the CMB&SN data for selected model parameters in the curved mixed model. 'Amp. par. $k_0=0.01$' indicates the results reported in this paper, obtained assuming flat priors for the amplitudes $\alpha_{1,2}$ and $\gamma_{1,2}$ at scales $k_1=0.002\,$Mpc$^{-1}$ and $k_2=0.05\,$Mpc$^{-1}$, and converted to spectral indices and amplitudes at the pivot scale $k_0=0.01\,$Mpc$^{-1}$. 'n-par. $k_0=0.002$' indicates what the results would be if we assumed flat priors in the spectral index parametrization and chose the pivot scale $k_0=0.002\,$Mpc$^{-1}$. 'n-par. $k_0=0.01$' and 'n-par. $k_0=0.05$' are is the same as the previous one, but choosing $k_0=0.01\,$Mpc$^{-1}$ or $k_0=0.05\,$Mpc$^{-1}$, respectively. The raggedness of the $k_0=0.05$ curves is due to a small amount of well-fitting samples, after reweighting by the Jacobian, Eq. (\ref{['eqn:jacobian']}).