With the Vera C. Rubin Observatory Legacy Survey of Space and Time, and the Nancy Grace Roman Space Telescope both on the horizon, Type Ia supernova (SN Ia) cosmology is about to undergo a paradigm shift. Whilst this new era has the potential to bring unprecedented constraints on the nature of dark energy, this will only be possible if we are able to overcome the considerable challenge of controlling the astrophysical sources of systematic uncertainty that will dominate the error budget of these future experiments.

In this thesis, I develop and deploy a robust new hierarchical Bayesian framework for modelling the spectral energy distributions (SEDs) of SNe Ia in the optical and near-infrared (NIR). I apply this framework to detailed studies of the dust in SN Ia host galaxies – particularly the distribution of the dust law R_V. This model is able to leverage the optical and NIR in a principled manner, whilst coherently marginalising over all sources of uncertainty.

I use this model to analyse the Pan-STARRS grid light curves of 157 low-redshift SNe Ia from the Foundation Supernova Survey. I use these data to robustly model the dust laws in SN Ia host galaxies, and to place constraints on the extent to which R_V differs between low- and high-mass hosts. I do not find statistically significant evidence for strongly different R_V between low- and high-mass hosts, and am able to place upper limits on the R_V population variance. My results indicate that observed correlations between SNe Ia and host-galaxy mass cannot be fully explained by dust.

I follow this with an analysis of optical and NIR (B–H band; ≈3500–18000 Å) data of 86 SNe Ia from the Carnegie Supernova Project (CSP). I use this independent sample to uphold my previous results, place tighter constraints on the R_V population variance, and to investigate the dust laws for more-highly reddened SNe Ia (beyond the apparent B–V≤0.3 colour cut typically used in cosmological analyses). I also argue for the critical importance of combining optical and NIR data to break degeneracies when investigating the effect of dust on SNe Ia, and present a simulation-based demonstration of the challenges inherent in estimating R_V and its population distribution. I show that hierarchical Bayes is suited to overcoming these challenges.

Finally, I analyse high-redshift data from the RAISIN (SN IA in the IR) Survey. These unique data include Hubble Space Telescope F125W and F160W observations for 37 SNe Ia with 0.22≤z≤0.61, providing the first sample of rest-frame NIR (YJ-band) observations at such high redshifts. I use these data to place constraints on R_V in the host galaxies of high-z SNe Ia. By combining the high-z RAISIN data with a low-z CSP companion sample, I am able to place limits on the possible redshift evolution of SN Ia host galaxy dust laws.