Recently, the Planck satellite found a larger and most precise value of the matter energy density, that impacts on the present values of other cosmological parameters such as the Hubble constant, the present cluster abundances and the age of the Universe. The existing tension between Planck determination of these parameters in the frame of the base LambdaCDM model and their direct measurements generated lively discussions and several interpretations. In this paper we quantify this tension by exploring several extensions of the base LambdaCDM model that include the leptonic asymmetry. We set bounds on the radiation content of the Universe and neutrino properties by using the latest cosmological measurements, imposing also self-consistent BBN constraints on the primordial helium abundance. For all cosmological asymmetric models we find the preference of cosmological data for smaller values of active and sterile neutrino masses. This increases the tension between cosmological and short baseline neutrino oscillation data that favor a sterile neutrino with the mass of around 1 eV. For the case of degenerate massive neutrinos, we find that the discrepancies with direct determinations of the Hubble constant, the present cluster abundances and the age of the Universe are alleviated at ~ 1.3 sigma for all leptonic asymmetric models. We also find ~2 sigma statistical evidence of the preference of cosmological data for the normal neutrino hierarchy. This is more evident for the case of cosmological models involving leptonic asymmetry and three massive neutrino species. We conclude that the current cosmological data favor the leptonic asymmetric extension of the base LambdaCDM model and normal neutrino mass hierarchy over the models with additional sterile neutrino species and/or inverted neutrino mass hierarchy.
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