Neutrino constraints: what large-scale structure and CMB data are telling us?

(Abridged) We discuss the reliability of neutrino mass constraints, either active or sterile, from the combination of WMAP 9-year or Planck CMB data with BAO measurements from BOSS DR11, galaxy shear measurements from CFHTLenS, SDSS Ly-$\alpha$ forest constraints and galaxy cluster mass function from Chandra observations. To avoid model dependence of the constraints we perform a full likelihood analysis for all the datasets employed. As for the cluster data analysis we rely on to the most recent calibration of massive neutrino effects in the halo mass function and we explore the impact of the uncertainty in the mass bias and re-calibration of the halo mass function due to baryonic feedback processes on cosmological parameters. We find that none of the low redshift probes alone provide evidence for massive neutrinos in combination with CMB measurements, while a larger than $2\sigma$ detection of non zero neutrino mass, either active or sterile, is achieved combining cluster or shear data with CMB and BAO measurements. The preference for massive neutrino is larger in the sterile neutrino scenario, and for the combination of Planck, BAO, shear and cluster datasets we find that the vanilla $\Lambda$CDM model is rejected at more than $3\sigma$ and a sterile neutrino mass as motivated by accelerator anomaly is within the $2\sigma$ errors. Finally, results from the full data combination reflect the tension between the $\sigma_8$ constraints obtained from cluster and shear data and that inferred from Ly-$\alpha$ forest measurements; in the active neutrino scenario for both CMB datasets employed, the full data combination yields only an upper limits on $\sum m_\nu$, while assuming an extra sterile neutrino we still get preference for non-vanishing mass, $m_s^{\rm eff}=0.26^{+0.22}_{-0.24}$ eV, and dark contribution to the radiation content, $\Delta N_{\rm eff}=0.82\pm0.55$.