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Accurate and precise measurements of the Hubble constant are critical fortesting our current standard cosmological model and revealing possibly newphysics. With Hubble Space Telescope (HST) imaging, each strong gravitationallens system with measured time delays can allow one to determine the Hubbleconstant with an uncertainty of $\sim 7\%$. Since HST will not last forever, weexplore adaptive-optics (AO) imaging as an alternative that can provide higherangular resolution than HST imaging but has a less stable point spread function(PSF) due to atmospheric distortion. To make AO imaging useful fortime-delay-lens cosmography, we develop a method to extract the unknown PSFdirectly from the imaging of strongly lensed quasars. In a blind test with twomock data sets created with different PSFs, we are able to recover theimportant cosmological parameters (time-delay distance, external shear, lensmass profile slope, and total Einstein radius). Our analysis of the Keck AOimage of the strong lens system RXJ1131-1231 shows that the importantparameters for cosmography agree with those based on HST imaging and modelingwithin 1-$\sigma$ uncertainties. Most importantly, the constraint on the modeltime-delay distance by using AO imaging with $0.045"$resolution is tighter by$\sim 50\%$ than the constraint of time-delay distance by using HST imagingwith $0.09"$when a power-law mass distribution for the lens system is adopted.Our PSF reconstruction technique is generic and applicable to data sets thathave multiple nearby point sources, enabling scientific studies that requirehigh-precision models of the PSF.

SHARP - III: First Use Of Adaptive Optics Imaging To Constrain Cosmology With Gravitational Lens Time Delays