Cosmic discordance: are Planck CMB and CFHTLenS weak lensing measurements out of tune?

We examine the level of agreement between low-redshift weak lensing data and the cosmic microwave background using measurements from the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS) and Planck+Wilkinson Microwave Anisotropy Probe (WMAP) polarization. We perform an independent analysis of the CFHTLenS six bin tomography results of Heymans et al. We extend their systematics treatment and find the cosmological constraints to be relatively robust to the choice of non-linear modelling, extension to the intrinsic alignment model and inclusion of baryons. We find that when marginalized in the Omega(m)-sigma(8) plane, the 95 per cent confidence contours of CFHTLenS and Planck+WMAP only just touch, but the discrepancy is less significant in the full six-dimensional parameter space of Lambda cold dark matter (Lambda CDM). Allowing a massive active neutrino or tensor modes does not significantly resolve the tension in the full n-dimensional parameter space. Our results differ from some in the literature because we use the full tomographic information in the weak lensing data and marginalize over systematics. We note that adding a sterile neutrino to Lambda CDM brings the 2D marginalized contours into greater overlap, mainly due to the extra effective number of neutrino species, which we find to be 0.88 +/- 0.43 (68 per cent) greater than standard on combining the data sets. We discuss why this is not a completely satisfactory resolution, leaving open the possibility of other new physics or observational systematics as contributing factors. We provide updated cosmology fitting functions for the CFHTLenS constraints and discuss the differences from ones used in the literature.