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 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 MSD into a hierarchical population model to forecast dark energy constraints in Flat $w_0w_a$CDM, showing that MSD severely degrades constraints from small DSPL samples and motivates our PDSPL statistical approach. Assuming unbiased Gaussian photo-z errors of $σ_z/(1+z)=0.03$, the mock LSST Y10 photometric sample yields $σ(w_0)\sim0.45$, constraining the population-mean MSD parameter and deflector power-law slope to $\sim2\%$. A prior $Ω_m\sim\mathcal{N}(0.3,0.05)$, representing combination with external cosmological probes such as CMB, BAO, or SNe Ia, improves this to $σ(w_0)\sim0.29$, comparable to Stage III weak lensing. The statistical advantage over smaller spectroscopic subsets (e.g., 4MOST) assumes unbiased photo-zs. Real-data use requires spectroscopic photo-z calibration and validation of how paired deflectors' mass-profile differences affect Einstein-radius ratios.