Quasar lensing: the time-delay distance determines $H_0$

Fix the speed of light $c>0$, a matter density parameter $\Omega_m \ge 0$ of a flat ΛCDM universe, and lens and source redshifts with $0 < z_{\rm lens} < z_s$. Then there is a constant $K>0$, depending only on $c$, $\Omega_m$, $z_{\rm lens}$, $z_s$, such that for every Hubble constant $H_0>0$ the time-delay distance (Eq. (1)) satisfies

In particular $H_0 = K/D_{\Delta t}$ is uniquely recovered from a measured time-delay distance once the redshifts and $\Omega_m$ are fixed. This is the mathematical content of the statement in Section 3.1 that "with observed time delay and lens mass model, $H_0$ can be inferred" and that the time-delay distance is "primarily sensitive to $H_0$".

Formalization Note The time-delay distance is built from the angular diameter distances of Eqs. (2)–(3) (with the standard integrand $1/E(z')$). The constraint $\Omega_m \le 1$ (non-negative $\Omega_\Lambda$) is not needed and is omitted.