Tension between the power spectrum of density perturbations measured on large and small scales

There is a tension between measurements of the amplitude of the power spectrum of density perturbations inferred using the Cosmic Microwave Background (CMB) and directly measured by Large-Scale Structure (LSS) on smaller scales. We show that this tension exists, and is robust, for a range of LSS indicators including clusters, lensing and redshift space distortions and using CMB data from either $Planck$ or WMAP+SPT/ACT. One obvious way to try to reconcile this is the inclusion of a massive neutrino which could be either active or sterile. Using $Planck$ and a combination of all the LSS data we find that (i) for an active neutrino $\sum m_{\nu}= (0.357\pm0.099)\,{\rm eV}$ and (ii) for a sterile neutrino $m_{\rm sterile}^{\rm eff}= (0.67\pm0.18)\,{\rm eV}$ and $\Delta N_{\rm eff}= 0.32\pm0.20$. This is, however, at the expense of a degraded fit to $Planck$ temperature data, and we quantify the residual tension at $2.5\sigma$ and $1.6 \sigma$ for massive and sterile neutrinos respectively. We also consider alternative explanations including a lower redshift for reionization that would be in conflict with polarisation measurements made by WMAP and $ad$-$hoc$ modifications to primordial power spectrum.