No Evidence for Extensions to the Standard Cosmological Model

We compute the Bayesian Evidence for models considered in the main analysis of Planck cosmic microwave background data. By utilising carefully-defined nearest-neighbour distances in parameter space, we reuse the Monte Carlo Markov Chains already produced for parameter inference to compute Bayes factors $B$ for many different model-dataset combinations. Standard 6-parameter flat $Lambda$CDM model is favoured over all other models considered, with curvature being mildly favoured only when CMB lensing is not included. Many alternative models are strongly disfavoured by the data, including primordial correlated isocurvature models ($ln B-7.8$), non-zero scalar-to-tensor ratio ($ln B-4.3$), running of the spectral index ($ln B = -4.7$), curvature ($ln B-3.6$), non-standard numbers of neutrinos ($ln B-3.1$), non-standard neutrino masses ($ln B-3.2$), non-standard lensing potential ($ln B-4.6$), evolving dark energy ($ln B-3.2$), sterile neutrinos ($ln B-6.9$), and extra sterile neutrinos with a non-zero scalar-to-tensor ratio ($ln B-10.8$). Other models are less strongly disfavoured with respect to flat $Lambda$CDM. As with all analyses based on Bayesian Evidence, the final numbers depend on the widths of the parameter priors. We adopt the priors used in the Planck analysis, while performing a prior sensitivity analysis. Our quantitative conclusion is that extensions beyond the standard cosmological model are disfavoured by Planck data. Only when newer Hubble constant measurements are included does $Lambda$CDM become disfavoured, and only mildly, compared with a dynamical dark energy model ($ln Bsim +2$).