Cosmology from weak lensing alone and implications for the Hubble tension

We investigate the origin of $\Lambda$CDM parameter constraints in weak lensing, with a focus on the Hubble constant. We explain why current cosmic shear data are sensitive to the parameter combination $S_8 \propto \sigma_8 \Omega_m^{0.5}$, improving upon previous studies through use of the halo model. Motivated by the ongoing discrepancy in measurements of the Hubble constant from high and low redshift, we explain why cosmic shear provides almost no constraint on $H_0$ by showing how the lensing angular power spectrum depends on physical length scales in the dark matter distribution. We derive parameter constraints from galaxy lensing in KiDS and cosmic microwave background weak lensing from Planck and SPTpol, separately and jointly, showing how degeneracies between $\sigma_8$ and $\Omega_m$ can be broken. Using lensing and Big Bang Nucleosynthesis to calibrate the sound horizon measured in projection by baryon acoustic oscillations gives $H_0 = 67.4 \pm 0.9 \; \mathrm{km} \, \mathrm{s}^{-1} \, \mathrm{Mpc}^{-1}$, consistent with previous results from Planck and the Dark Energy Survey. We find that a toy Euclid-like lensing survey provides only weak constraints on the Hubble constant due to degeneracies with other parameters that affect the shape of the lensing correlation functions. If external priors on $n_s$, the baryon density, and the amplitude of baryon feedback are available then sub-percent $H_0$ constraints are achievable with forthcoming lensing surveys.