Measuring the Stellar-to-Halo Mass Relation at $\sim10^{10}$ Solar masses, using forthcoming space-based imaging of galaxy-galaxy strong lenses

The stellar-to-halo mass relation (SHMR) is central to understanding the co-evolution of galaxies and their host dark matter haloes, yet it remains weakly constrained for dwarf galaxies owing to their faintness, especially beyond the Local Group. Strong gravitational lensing offers a unique probe of the SHMR at sub-galactic scales and cosmological distances, as the masses of subhalos within the main lens can be inferred from the perturbations they imprint on lensed images. Anticipating the discovery of $\sim10^5$ galaxy--galaxy strong lenses by forthcoming facilities such as \textit{Euclid}, we perform an end-to-end simulation to forecast \textit{Euclid}'s constraints on the SHMR at the halo mass scale of $\sim10^{10}\,\mathrm{M}_\odot$. We generate mock \textit{Euclid} VIS images of lens systems hosting a fiducial $3\times10^{10}\,\mathrm{M}_\odot$ subhalo and vary its properties to assess the robustness of mass inference. We find that \textit{Euclid}'s angular resolution cannot break the intrinsic mass--concentration degeneracy of subhaloes, nor deblend the light of satellite galaxies (when present) associated with them, leading to biased inferred halo masses. These limitations are overcome with high-resolution follow-up imaging from facilities such as the \textit{Hubble Space Telescope}, enabling accurate halo-mass measurements. We forecast that a statistical sample of $\sim100$ such systems, combining lensing-derived halo masses with stellar masses from photometric SED fitting, can constrain the SHMR at dwarf-galaxy scales with a precision of $\sim0.05$~dex in halo mass and $\sim0.03$~dex in stellar mass, enabling powerful tests of galaxy formation theories.