This thesis studies the relationship between type Ia supernovae (SNe Ia) and their host galaxies. The sample consists of 527 SNe Ia with redshift z<0.09 discovered by the Palomar Transient Factory (PTF). We obtained high-quality photometric and spectroscopic data of the host galaxies and determined their stellar mass M_stellar, star formation rate (SFR), gas-phase/stellar metallicity, stellar age and SN offset. In the first part of the analysis, we compare the SN Ia photometric properties to the host parameters. Strong correlations between the SN Ia light-curve width (stretch) and the host age/mass/metallicity are found: fainter, faster-declining events tend to be hosted by older/massive/metal-rich galaxies. There is also some evidence that redder SNe Ia explode in higher metallicity galaxies. SNe Ia in higher-mass/metallicity galaxies also appear brighter after stretch/colour corrections than their counterparts in lower mass hosts, and the stronger correlation is with gas-phase metallicity suggesting this may be the more important variable. We also compare the host stellar mass distribution to that in galaxy targeted SN surveys and the high-redshift untargeted Supernova Legacy Survey (SNLS). The difference between each stellar mass distribution can be explained by an evolution in the galaxy stellar mass function, coupled with a SN delay-time distribution proportional to t^-1. Finally, we found no significant difference in the mass-metallicity relation of our SN Ia hosts compared to field galaxies, suggesting any metallicity effect on the SN Ia rate is small. In the second part of the analysis, we compare the SN spectral features to the host parameters. We find that SNe Ia with higher Si ii λ6355 velocities tend to explode in more massive galaxies. We study the strength of the high-velocity component of the Ca ii NIR absorption, and find that SNe Ia with a stronger high-velocity component are preferentially hosted by galaxies with a low M_stellar, a blue colour, and a high SFR, and are therefore likely to arise from the youngest progenitor systems. When combined with other studies, our results support the scenario that these high-velocity features are related to an interaction between the SN ejecta and a circumstellar medium (CSM) local to the SN.

Type Ia supernovae (SNe Ia) have proven to be extremely useful for measuring cosmological distances and were used for the discovery of the accelerated expansion of the universe. Although thousands of SNe Ia have been observed to date, many questions surrounding the physics of the explosions and the nature of their progenitor systems remain unanswered. An notable property of many SNe Ia is the relation between extinction due to dust and their colour. For example SN 2014J, the nearest SN Ia in recent years, has an extinction relation which would be very unusual to observe in the Milky Way. One possible explanation to the peculiar extinction could be the presence of circumstellar (CS) dust surrounding the explosions. Incidentally, some proposed progenitor models of SNe Ia suggest that the explosions are surrounded by shells of matter, which could account for the unusual extinction. CS gas would be ionised, if it is exposed to the intense ultraviolet (UV) radiation of a SN Ia. The research presented in this thesis focuses on the search for CS gas by observing the effects of photoionisation on absorption lines commonly detected in optical spectra. Simple models suggest that the frequently studied sodium doublet (Na I D) should significantly decrease or even disappear if the gas is in the CS environment. Conversely, the absence of variations implies that the absorbing gas clouds must be far from the explosion, in the interstellar medium (ISM). To date, few SNe Ia have been shown to have variable absorption lines, to which we have added another case with SN 2013gh. Yet, we have also shown that most observations searching for variable absorption lines have been taken at too late phases, when most CS gas will have already been ionised. Setting out to obtain the earliest possible coverage of a SN Ia with high-resolution spectra, we have been able to set strong limits on the presence of CS gas surrounding SN 2017cbv. Along with evidence from other observational methods, these results have shown that there is little matter in the CS environments of SNe Ia, suggesting that the peculiar extinction likely results from the dust properties of their host galaxy ISM. Although the progenitor question cannot be resolved by these observations, nondetections of CS gas point to models which do not deposit large amounts of matter in their surroundings. / <p>At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 5: Submitted.</p>