In this dissertation, I present the work I have done using simulations to understand cosmology and astrophysics.

I developed a novel method for detecting the presence of cosmic strings, which are early universe topological defects, via their effect on cosmic filaments. The method was developed by inserting the effects of cosmic strings into N-body simulations, and determining what effects are most observable. I applied this method to filamentary structure from observations of galaxies. While there was no indication of the presence of cosmic strings in current observations, future prospects are promising.

I then turned my attention to smaller scales and lower redshift phenomena, determining how simulations of the Lyman-$\alpha$ forest are affected by including species specific transfer functions in the initialization of simulations. The effect on the Lyman-$\alpha$ forest flux power spectrum is particularly important, as it is widely used, and will be measured to increasingly high precision by future observations.

Continuing on that topic, I developed a machine learning approach to interpolating Lyman-$\alpha$ forest summary statistics from simulations. This used Gaussian processes to construct an emulator for the Lyman-$\alpha$ forest flux power spectrum. Two suites of simulations were used to train the emulator: a small sample of high resolution simulations, and a large sample of low resolution simulations. This combination allowed an emulator that predicts high resolution simulation outputs, but costs a fraction of previously used methods. This is the multi-fidelity emulator model.

Finally, I present inference using a multi-fidelity emulator, trained on large volume $(120 \text{Mpc h}^{-1})^3$ and high resolution ($1536^3$ and $3072^2$ gas particles) simulations. The volume and resolution of these simulations allow a more robust modeling of the scales important to the Lyman-$\alpha$ forest. By combining the Lyman-$\alpha$ forest flux power spectrum with the mean temperature of the intergalactic medium, I constrain several cosmological and astrophysical parameters.