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 utilizing carefully defined nearest-neighbor 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-data set combinations. The standard 6-parameter flat cold dark matter model with a cosmological constant ($\mathrm{\Lambda }\mathrm{CDM}$) is favored over all other models considered, with curvature being mildly favored only when cosmic microwave background lensing is not included. Many alternative models are strongly disfavored by the data, including primordial correlated isocurvature models ($\ln B=-7.8$), nonzero scalar-to-tensor ratio ($\ln B=-4.3$), running of the spectral index ($\ln B=-4.7$), curvature ($\ln B=-3.6$), nonstandard numbers of neutrinos ($\ln B=-3.1$), nonstandard neutrino masses ($\ln B=-3.2$), nonstandard 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 nonzero scalar-to-tensor ratio ($\ln B=-10.8$). Other models are less strongly disfavored with respect to flat $\mathrm{\Lambda }\mathrm{CDM}$. 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 disfavored by Planck data. Only when newer Hubble constant measurements are included does $\mathrm{\Lambda }\mathrm{CDM}$ become disfavored, and only mildly, compared with a dynamical dark energy model ($\ln B\sim +2$).

Figure 1

Bayes factors $\ln B$ with respect to the highest evidence model (base: flat $\mathrm{\Lambda }\mathrm{CDM}$). The most constraining data set is indicated by the symbol; see the legend for details. Horizontal lines mark the boundaries corresponding to strong ($\ln B<-3$) and very strong ($<-5$) evidence in the Kass and Raftery [11] scale.