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, $\Delta N_{\rm eff}=0.82\pm0.55$.
