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

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 S12 and show its constraints from recent surveys in comparison with Planck 2018. In WL alone, we find that the absolute distance convention correlates S12 with H0. Accounting for this correlation in the 3D space S12 x o iota m x h reproduces the usual levels of 2 similar to 3a-tension inferred from S8 x omega m. Additionally, we derive scaling relations in the S8 x h and S12 x h planes that are allowed by ACDM and extrapolate target scalings needed to solve the H0 and lensing-amplitude tensions jointly in a hypothetical beyond-ACDM model. As a test example, we quantify how the early dark energy scenario compares with these target scalings. Useful fitting formulas for S8 and S12 as a function of other cosmological parameters in ACDM are provided, with 1% precision.