Role of the Hubble scale in the weak lensing versus CMB tension
Here, 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 ×w m ×ℎ reproduces the usual levels of 2~3σ tension inferred from S 8 ×Ω m . Additionally, we derive scaling relations in the S 8 ×ℎ and S 12 ×ℎ planes that are allowed by ΛCDM and extrapolate target scalings needed to solve the H 0 and lensing-amplitude tensions jointly in a hypothetical beyond-ΛCDM 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 ΛCDM are provided, with 1% precision.