The phenomena of strong gravitational lensing is a powerful probe of the expansion history and scale of the

Universe. The Vera C. Rubin Observatory, and its Legacy Survey of Space and Time (LSST) is going to

discover tens of thousands of strong gravitational lenses, more than 200 lensed quasars, and about one hun-

dred multiply imaged lensed supernovae per year. These numbers correspond to several e-foldings of lenses

being used compared to current state-of-the-art studies. One key limitation of using strong gravitational

lensing is the mass-sheet degeneracy (MSD). We propose to use galaxy-galaxy weak gravitational lensing

measurements of the deflector galaxies to contain the MSD. This proposal aims to test and deploy an end-to-

end workflow from the LSST DR1 & DR2 data products combining strong and weak lensing measurements

to provide a precise and accurate measurement of dark energy and the Hubble constant.

A measurement of time delays from a strongly-lensed multiply-imaged time-variable sources is an absolute

distance indicator. The change in Einstein radius as a function of source redshift is a relative distance

indicator. This proposal addresses two key limiting factors of such an analysis: (i) The radial density profile

of the strong lensing deflector needs to be known precisely to accurately predict the time delays, while the

strong lensing observables do not have access to this information due to the MSD. A stacked galaxy-galaxy

weak lensing measurements from small to large angular scales with LSST data is able to break the MSD

and contain the uncertainty on the time delays to∼ 1% (Khadka et al. 2024). (ii) Strong lenses are rare

and inherently a result of a selection effect in the tail of the large-scale structure mass distribution. It is

crucial to understand the selection effects in the lens discovery channel to apply the needed corrections in

the inference. The overarching methodology of this proposal is to integrate the lens search, analysis, weak

lensing deflector selection, and cosmological inference components in one integrated pipeline.

We will perform a lens search and modeling analysis based on LSST DR1 &DR2 data. All our analysis

choices and population-level parameters are propagated through an end-to-end population-level simulation

and compared with the discovered lenses in the LSST data set, effectively correcting for population biases.

In parallel, we will select a larger sample of self-similar deflector galaxies to perform a galaxy-galaxy weak

lensing mass density profile measurement from LSST data. The relative selection function between strong

and weak lensing measurements will be accounted for with the same full population-level simulation tools.

Finally, the bias-corrected sampling over lensing deflector population hyper-parameters and cosmological

parameters will be conducted. The time-domain data allows to measure time delays to constrain an absolute

distance, the Einstein radii distribution allows for a relative distance tomographic analysis, and finally the

weak lensing data enables to constrain the MSD to the percent level.