Cosmography with the Double Source Plane Strong Gravitational Lens AGEL150745+052256

Strong gravitational lenses with two background sources at widely separated redshifts are a powerful and independent probe of cosmological parameters. We can use these systems, known as Double-Source-Plane Lenses (DSPLs), to measure the ratio ($\beta$) of angular-diameter distances of the sources, which is sensitive to the matter density ($\Omega_m$) and the equation-of-state parameter for dark-energy ($w$). However, DSPLs are rare and require high-resolution imaging and spectroscopy for detection, lens modeling, and measuring $\beta$. Here we report only the second DSPL ever used to measure cosmological parameters. We model the DSPL AGEL150745+052256 from the ASTRO 3D Galaxy Evolution with Lenses (AGEL) survey using HST/WFC3 imaging and Keck/KCWI spectroscopy. The spectroscopic redshifts for the deflector and two sources in AGEL1507 are $z_{\rm defl}=0.594$, $z_{\rm S1}=2.163$, and $z_{\rm S2}=2.591$. We measure a stellar velocity dispersion of $\sigma_{\rm obs}=109 \pm 27$ km s$^{-1}$ for the nearer source. Using $\sigma_{\rm obs}$ for the main deflector (from literature) and S1, we test the robustness of our DSPL model. We measure $\beta=0.953^{+0.008}_{-0.010}$ for AGEL1507 and infer $\Omega_{\rm m}=0.33^{+0.38}_{-0.23}$ for $\Lambda$CDM cosmology. Combining AGEL1507 with the published model of the Jackpot lens improves the precision on $\Omega_{\rm m}$ ($\Lambda$CDM) and w (wCDM) by $\sim 10 \%$. The inclusion of DSPLs significantly improves the constraints when combined with Plancks cosmic microwave background observations, enhancing precision on w by $30 \%$. This paper demonstrates the constraining power of DSPLs and their complementarity to other standard cosmological probes. Tighter future constraints from larger DSPL samples discovered from ongoing and forthcoming large-area sky surveys would provide insights into the nature of dark energy.