After the epoch of reionization, Lyman continuum radiation permeate the universe. Immediately after reionization (redshifts $z\approx3-7$), the influence of the cosmic Ultraviolet (UV) background on the Intergalactic Medium (IGM) and the Circumgalactic Medium (CGM) is important to understand. We focus on building a 1D radiative transfer algorithm using the Accelerated Lambda Iteration scheme and the Lambda Iteration scheme to solve the time-independent radiative transfer equation. This approach captures the dynamics of radiation fields, and the convergence, accuracy, and mathematical limitations of these schemes are thoroughly tested. We then expand the algorithm to include a 4th order Runge-Kutta scheme to trace the fraction of Hydrogen atoms between the UV background and the center of a massive galaxy. In our calculations, the UV background dominates the ionization of the IGM and CGM near our example star-forming galaxy.

Understanding the evolution of the intergalactic medium (IGM) and how large-scale structures in the Universe develop over cosmic history represents one of the fundamental goals of modern astrophysics. The filamentary network of dark matter and gas, known as the "cosmic web" encodes information about the relative abundance of baryons and dark matter, the evolution of the radiation emitted by galaxies and quasars, the expansion history of the Universe, the nature andproperties of the dark matter particle and the mass of neutrinos, among other relevant physics. The Lyman-alpha forest originates from the absorption signature that cosmic neutral hydrogen imprints on the spectra of distant quasars. Therefore, the forest provides a blueprint of the cosmic web, making it a primary probe of the properties of the IGM. The promise of the Lyman-alpha forest for constraining the nature of dark matter and dark energy has in part motivated the construction of the Dark Energy Spectroscopic Instrument (DESI), which will measure absorption line spectra from nearly a million distant quasars.

To extract the physics encoded in the observations of the Lyman-alpha forest, one requires sophisticated numerical simulations. For my thesis, I have extended the GPU-native hydrodynamical solver Cholla to run cosmological simulations. Using Cholla on the largest supercomputers in the world, we have been able to run over 1500 high-resolution simulations that vary the physical models that shape the structure of the forest, allowing us to study the properties of the IGM with unprecedented detail.

This thesis presents an analysis of the impact on the IGM from different models for the photoionization and photoheating due to the metagalactic UVB radiation emitted by early galaxies and quasars. When comparing the Lyman-alpha forest from simulations that apply current models for the UVB to observations, we find that the models fail to reproduce the evolution of the statistical properties from the observational measurements. From a Bayesian approach where we compared the simulations to observations of the Lyman-alpha forest, we were able to improve the current models and provide a better fit to the observations. Our best-fit model provides an inference of the thermal and ionization history of the IGM post-reionization consistent with other independent determinations.

Additionally, this thesis presents the results from our comparison of high-resolution observations of the Lyman-alpha forest power spectrum to a massive grid of 1080 simulations that simultaneously vary the mass of a possible WDM candidate and the thermal history of the IGM. Interestingly, we find that a WDM particle mass of ~4.5 keV provides a formally better fit to the small-scale power spectrum than Lambda-CDM, which motivates continued observational experiments of the Lyman-alpha forest.