In this dissertation, I used early-type galaxies (ETGs) at intermediate redshifts (0.25 < ? < 0.75) to study both galaxy evolution and cosmology. I assembled a novel sample of 180 galaxies with integral field unit (IFU) data within this redshift range. Utilizing the extracted kinematic maps and photometric data, I measured the structural, kinematic, and stellar population properties of these galaxies. By comparing this sample with a subset of ETGs from the MaNGA survey, I demonstrated that the degree of rotational support in ETGs has decreased at lower redshifts. Furthermore, I investigated the discrepancies between observational results from lensing studies and predictions from cosmological simulations concerning the evolution of the total density slope of ETGs, which present conflicting views. Using Jeans anisotropic modeling, I modeled the observed resolved kinematics and measured the total density slope of this sample. My findings suggest that the slope in the ? − ? plane is generally negative, indicating an increase in the total density slope with decreasing redshift. However, due to the high level of uncertainty, I cannot conclusively dismiss the possibility of a zero or positive slope. In addition to studying these non-lensing ETGs, I also worked with ETGs which act as strong gravitational lenses, to measure the Hubble constant using the time-delay cosmography method. A crucial factor in accurately measuring the Hubble constant is the stellar kinematics of the lensing galaxies, which is challenging to measure due to the spatial blending of light caused by atmospheric seeing in ground-based telescopes. To address this, I developed a forward modeling method to extract high signal-to-noise ratio (SNR) one-dimensional spectra, enabling reliable kinematic measurements. The results presented in this dissertation contribute to our understanding of galaxy evolution and provide insights that could help resolve the Hubble tension, i.e., the discrepancy between independent inferences of the current expansion rate of the Universe. By improving our knowledge of ETG properties across redshifts and refining methods to measure the Hubble constant, this work advances our comprehension of the fundamental processes governing the Universe.

Dark matter makes up roughly 85 percent of the mass budget of the Universe. Though we cannot see it, hence the name, the formation and behavior of structures from the largest cosmological scales down to sub-galactic scales betray the existence of dark matter via its gravitational interactions. The widely accepted view of cosmological history, informed by observations of hierarchical structure formation, posits non-relativistic dark matter particles that only interact gravitationally, known as cold dark matter (CDM). However, alternative theories of dark matter may describe small-scale observations as well or better. It is essential to test variations on CDM and differentiate between models that agree on larger scales. We can observe even completely invisible dark matter structures from the way they deflect and magnify light. This phenomenon, gravitational lensing, has been used to infer some of the tightest constraints on alternative dark matter models and grows stronger as telescopes improve and sample sizes increase.

This dissertation is focused on strong gravitational lensing, when multiple images of the light source appear on the sky. Particularly, I demonstrate and evaluate a method called flux-ratio analysis, which uses bright, compact sources like quasars that are lensed into four distinct images. In these systems, the main lens is an elliptical galaxy, the mass distribution of which determines the positions and relative brightnesses of the images. Low-mass dark matter structures, called halos, associated with the main lens or along the line of sight are detectable in the additional magnification or demagnification they introduce to individual images. Discrepancies between observed flux ratios and those predicted by a smooth model of the main lens indicate the presence of perturbing halos. With a sample of many lensed quasars, we can characterize the population statistics of low-mass halos. These statistics are determined by the underlying dark matter model, thus they enable us to infer constraints on it.

In Chapter 1 of this work, I provide general background information on dark matter and cosmology and detail the predictions of CDM. I then discuss the use of gravitational lensing to detect dark matter and review the history of flux-ratio analysis to put my work in the context of the field.

Chapter 2 of this work contains a detailed description of my flux-ratio analysis procedure as well as constraints on warm dark matter (WDM) inferred from a sample of 14 lensed quasars. I provide background on WDM theory and describe the process of simulating low-mass halo populations that align with theoretical predictions. Then I walk through my Bayesian inference procedure and its application to infer the strongest gravitational lensing constraints on WDM to date. Lastly, I explore the context of my analysis in relation to other works that used subsets of this lens sample. I highlight the differences in between our analysis procedures and potential sources of systematic error.

In Chapter 3, I investigate the addition of general third- and fourth-order multipoles to the mass model. I find that multipoles with realistic amplitudes, at least when compared to the isophotes of elliptical galaxies, can perturb flux ratios as significantly as low-mass halos. If their orientation angles are left to vary freely, joint third- and fourth-order multipoles are completely degenerate with CDM halo populations when modeling quadruply-imaged quasars. This work calls into question all previous dark matter constraints from flux-ratio analysis, including those presented in Chapter 2. However, those results are still critical to the field moving forward. Until this degeneracy is mitigated, the method and degree of inclusion of multipoles into an inference procedure will strongly impact the result. Chapter 2 explores one extreme end of this spectrum -- the exclusion of multipoles.

In Chapter 4, I summarize and provide a framework for reevaluating past dark matter constraints from flux-ratio analysis. I explain how comparison between observations of lens galaxy isophotes and mass models from associated lensed arcs can provide more informative priors on multipoles in future lens models. Additionally, I forecast ways that high-resolution simulations can provide complementary prior information both regarding multipoles that may be present in galactic dark matter halos and their connection to the distribution of baryons in the galaxy. Finally, I give an overview of the increase in the sample size of strong lens systems that is expected from JWST and Euclid. Though data from these telescopes holds great potential to help us understand the nature of dark matter, we must first untangle the degeneracy between smooth model complexity and low-mass halos.

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.