Beyond six parameters: extending $\Lambda$CDM

Cosmological constraints are usually derived under the assumption of a $6$ parameters $\Lambda$-CDM theoretical framework or simple one-parameter extensions. In this paper we present, for the first time, cosmological constraints in a significantly extended scenario, varying up to $12$ cosmological parameters simultaneously, including the sum of neutrino masses, the neutrino effective number, the dark energy equation of state, the gravitational waves background and the running of the spectral index of primordial perturbations. Using the latest Planck 2015 data release (with polarization) we found no significant indication for extensions to the standard $\Lambda$-CDM scenario, with the notable exception of the angular power spectrum lensing amplitude, $A_{\rm lens}$ that is larger than the expected value at more than two standard deviations even when combining the Planck data with BAO and supernovae type Ia external datasets. In our extended cosmological framework, we find that a combined Planck+BAO analysis constrains the value of the r.m.s. density fluctuation parameter to $\sigma_8=0.781_{-0.063}^{+0.065}$ at $95 \%$ c.l., helping to relieve the possible tensions with the CFHTlenS cosmic shear survey. We also find a lower value for the reionization optical depth $\tau=0.058_{-0.043}^{+0.040}$ at $95$ \% c.l. respect to the one derived under the assumption of $\Lambda$-CDM. The scalar spectral index $n_S$ is now compatible with a Harrison-Zeldovich spectrum to within $2.5$ standard deviations. Combining the Planck dataset with the HST prior on the Hubble constant provides a value for the equation of state $w < -1$ at more than two standard deviations while the neutrino effective number is fully compatible with the expectations of the standard three neutrino framework.