One of the foundational goals of modern cosmology is the precise and accurate measurement of the expansion rate of the universe, which is denoted by the Hubble constant ($H_0$). In recent years, the results of the most mature methods available for making this measurement have diverged to such an extent that some have called it a ``crisis in cosmology." The ultimate resolution of the crisis is not yet clear, but there is no doubt that additional methods for constraining $H_0$ can shed light on this underlying problem.
Time-delay cosmography is one such method for constraining the expansion rate. It involves careful measurements and modeling of strong lensing systems where the luminosity of the background object is variable. This variability appears at different times to the observer, and the length of this ``time delay" depends on the underlying cosmology in addition to the geometry and mass distribution of the lens system itself. Chapter \ref{chapter:1} of this dissertation provides an introduction to modern physical cosmology, and further introduction to the theory of gravitational lensing and time-delay cosmography.
The environment of the lensing system introduces perturbations that affect the results of the cosmographic analysis, generally at the percent level. Accounting for this bias is crucial for accurately measuring the underlying cosmological parameters. While some structures can be accounted for in the primary mass model itself, this requires high quality data and significant modeling time, and is generally not possible for the majority of objects in the field. The cumulative effect of the remaining perturbers is known as the ``external convergence" ($\kappa_{\rm{ext}}$) and must be handled statistically.
This dissertation explores one technique for quantifying the collective impact of these remaining objects. The technique involves comparing a summary statistics from a line of sight of interest to summary statistics from large number of similar lines of sight in a suitable reference survey. This provides an empirical estimate of the mass density in the field as compared to the Universe as a whole, which can be compared to cosmological simulations to estimate the value of $\kappa_{\rm{ext}}$. This quantity is used directly in the final cosmological inference as a correction factor on the time-delay distance, which is inversely proportional to $H_0$. Chapter \ref{chapter:2} provides an introduction to these techniques and an application to a time-delay lens.
Currently, there are only around a dozen time-delay lenses that have been fully analyzed to put constraints on cosmological parameters. However the number of known systems is rising fast, and the Legacy Survey of Space and Time (LSST) is expected to increase this number by around a factor of 10. Working with these systems requires the development of tools that are capable of operating at larger scale, and statistical models that can combine information from many lens systems into a single estimate of cosmological parameters. Chapter \ref{chapter:3} introduces a framework for estimating the population distribution of $\kappa_{\rm{ext}}$ for a sample of lenses. I use this framework to show that a sample of 25 lenses from the Strong Lensing Legacy Survey (SL2S) fall in preferentially overdense lines of sight. This finding is expected based on previous work, but this dissertation represents the first time this overdensity has been quantified in a way that is useful for time-delay cosmography. Additionally, I present a new statistical model which may shed light into underlying mass distribution in our lines of sight and provide a path forward for improving the measurement.
The analyses presented in this dissertation where accompanied by a significant amount of work developing high-quality software to perform the analyses and enable future ones. The software produced is capable of scaling to many more systems than are currently available for analysis and adapting to significant changes in the underlying techniques without being rewritten. This software, and some of the philosophy behind its development, is presented in detail in Chapter \ref{chapter:4}
Finally in Chapter \ref{chapter:conclusions} I draw some conclusions based on this work and look forward to the future of the $\kappa_{\rm{ext}}$ measurement and cosmological software.
