Role of the Hubble scale in the weak lensing versus CMB tension

Abstract

We explore a reparametrization of the lensing amplitude tension between weak lensing (WL) and cosmic microwave background (CMB) data and its implications for a joint resolution with the Hubble tension. Specifically, we focus on the lensing amplitude over a scale of 12 Mpc in absolute distance units using a derived parameter ${S}_{12}$ and show its constraints from recent surveys in comparison with Planck 2018. In WL alone, we find that the absolute distance convention correlates ${S}_{12}$ with ${H}_{0}$. Accounting for this correlation in the 3D space ${S}_{12}\ifmmode\times\else\texttimes\fi{}{\ensuremath{\omega}}_{m}\ifmmode\times\else\texttimes\fi{}h$ reproduces the usual levels of $2\ensuremath{\sim}3\ensuremath{\sigma}$ tension inferred from ${S}_{8}\ifmmode\times\else\texttimes\fi{}{\mathrm{\ensuremath{\Omega}}}_{m}$. Additionally, we derive scaling relations in the ${S}_{8}\ifmmode\times\else\texttimes\fi{}h$ and ${S}_{12}\ifmmode\times\else\texttimes\fi{}h$ planes that are allowed by $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ and extrapolate target scalings needed to solve the ${H}_{0}$ and lensing-amplitude tensions jointly in a hypothetical beyond-$\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model. As a test example, we quantify how the early dark energy scenario compares with these target scalings. Useful fitting formulas for ${S}_{8}$ and ${S}_{12}$ as a function of other cosmological parameters in $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ are provided, with 1% precision.