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 ∑m_ν=(0.357±0.099) eV and (ii) for a sterile neutrino mₛₜₑᵣᵢₗₑᵉᶠᶠ=(0.67±0.18) eV and ΔNeff=0.32±0.20. This is, however, at the expense of a degraded fit to Planck temperature data, and we quantify the residual tension at 2.5σ and 1.6σ for massive and sterile neutrinos, respectively. We also consider alternative explanations including a lower redshift for reionization that would be in conflict with polarization measurements made by WMAP and ad hoc modifications to the primordial power spectrum.