Next generation modeling techniques for strong gravitational lensing and measuring the Hubble constant

For many years, observations of the Universe suggest a series problems with our theoretical models, particularly its dark energy and dark matter components. Fortunately, the Universe also provides us with a tool to solve these problems, called strong gravitational lensing. This natural phenomenon, observed as multiple images of a distant background source, distorted by the mass of a foreground galaxy, offers a unique opportunity to detect the otherwise invisible total mass of that galaxy. Yet, this detection is only possible if both background and foreground objects are accurately modeled, a task of increasing difficulty because our telescopes reveal more and more of the intrinsic complexity of galaxies. The primary goal of this thesis is to demonstrate how novel and well-motivated techniques can go beyond the current simplifying assumptions to improve the modeling of luminous and dark components of galaxies. Gravitational lens modeling is an under-constrained problem that has several possible solutions. The new techniques I introduce here are based on two key concepts to overcome this difficulty: sparsity and the wavelet transform. Sparsity selects the solution that contains the fewest parameters, namely the least complex one that best fits the observation. The wavelet transform separates the various spatial scales of the solution and enables the reconstruction of the small-scale compact features up to the larger, smoother variations, which are all found in real galaxies. Since the complexity and size of data sets are dramatically increasing, modeling techniques must also be fast and scalable. I address these requirements using differentiable programming to enable unprecedented gains in computation time. The proposed modeling framework allows us to effortlessly combine simple and more complex techniques together, if required by the observations. In this work, I demonstrate that sparsity and wavelets can address the limitations of current methods for modeling the full complexity of gravitational lenses. Compared to the many methods based on smoothness assumptions, I show how multi-scale modeling techniques significantly improve the reconstruction of lensed galaxies at high resolution. Moreover, I demonstrate that those same techniques are well-suited to characterize the invisible mass distributions of galaxies, notably when it deviates from the widely used smooth elliptical profiles. These results offer exciting possibilities to measure better the properties of galaxies via gravitational lensing, including their dark matter content, ultimately improving our understanding of galaxy evolution. Additionally, I take part to the long-standing debate regarding the role of dark energy in the expansion of the Universe using the method of time-delay cosmography. Based on the gravitational lensing of distant quasars, this method plays a central role in this context because it can measure the expansion rate of the Universe (the Hubble constant) independently of all other methods. The results I present are various: searching for systematic errors in past measurements, testing modeling techniques on a blind challenge, and modeling recent Hubble Space Telescope observations of lensed quasars to measure their absolute distance. I also describe the new approach introduced within the TDCOSMO collaboration, based on relaxing most assumptions on the mass distribution of galaxies and replacing those with observations of stellar kinematics