Hierarchical cosmological constraints through strong lensing distance ratio

Strong gravitational lensing provides an independent and powerful probe of cosmic expansion by directly linking observables to cosmological distances. Upcoming surveys such as LSST will discover large number of galaxy-galaxy strong lensing systems, offering a new route to precise cosmological constraints. In this paper, we propose a Fisher-like sensitivity factor to map how the cosmological information of strong-lensing distances changes across the lens-source redshift plane. Applying such factor to the distance ratio $D_{ls}\mathrm{/}{D}_s$, the time-delay distance $D_{\mathrm{\Delta }t}$, and the double-source-plane ratio, we determine the ``sensitivity valleys’’ where an observable becomes insensitive to a given parameter. The realistically simulated LSST lens population, which largely lies outside the distance-ratio valleys, covers the most sensitive region for $(w_0,w_a)$ parameter space. We then develop a new hierarchical framework, which could calibrate the redshift evolution of lens mass-density slopes and constrain cosmological parameters simultaneously. Focusing on the LSST mock data, we demonstrate that ignoring mass-profile evolution can bias ${\mathrm{\Omega }}_m$ by up to $\sim 10\sigma$, while modeling the lens evolution could perfectly recovers the fiducial cosmology and yield stringent cosmological constraints (e.g., $\mathrm{\Delta }{\mathrm{\Omega }}_m\simeq 0.01$ and $\mathrm{\Delta }w\simeq 0.1$ for $\sim {10}^4$ lenses).