Strong Lensing Tomography: Double and pseudo multi-source plane strong gravitational lensing to constrain dark energy

Tomographic measurements of gravitational lensing with different lens and source redshift distributions contain crucial information about the universe's relative expansion rate, and hence dark energy. While this technique is well-established in weak lensing, its application to strong lensing has traditionally focused on Double Source Plane Lenses (DSPLs). However, DSPLs are exceedingly rare and fundamentally limited by the Mass-Sheet Degeneracy (MSD), a systematic uncertainty underexplored in previous literature. To overcome these challenges, we introduce Pseudo Double-Source Plane Lenses (PDSPLs): pairs of independent single-source plane lenses with self-similar deflectors. This generalizes the DSPL formalism to the $\sim 10^5$ galaxy-galaxy lenses expected from upcoming surveys like LSST, Euclid, and Roman. Unlike true DSPLs, PDSPLs are free from the intermediate source mass problem by construction, eliminating the associated secondary MSD and the need for multi-plane ray tracing. We incorporate the deflector galaxy's MSD into a hierarchical forecasting framework, demonstrating that this degeneracy severely degrades constraints from small DSPL samples, thus motivating our PDSPL statistical approach. We forecast constraints on the dark energy equation of state under a Flat $w_0w_a$CDM cosmology. The LSST 10-year photometric sample alone achieves $\sigma(w_0) \sim 0.45$, while simultaneously constraining the MSD parameter and deflector power-law slope to $\sim 2\%$. Adding a prior $\mathcal{N}(0.3, 0.05)$ on $\Omega_{\rm m}$ -- simulating combination with external probes like CMB, BAO, or SNe Ia -- tightens this to $\sigma(w_0) \sim 0.29$, competitive with current Stage III weak lensing analyses. Notably, this massive photometric sample outperforms smaller subsets with precise spectroscopic follow-up (e.g., from 4MOST), confirming statistical volume dominates over per-pair precision.