Type Ia supernovae (SNe Ia) are currently the best probe of the expansion history of the universe. Their usefulness is due chiefly to their uniformity between supernovae (SNe). However, there are some slight variations amongst SNe that have yet to be understood and accounted for. The goal of this work is to uncover relationships between the spectral features and the light curve decline rate, [delta]m₁₅. Wavelet decomposition has been used to develop a new spectral index to measure spectral line strengths independent of the continuum and easily corrected for noise. This new method yields consistent results without the arbitrary uncertainties introduced by current methods and is particularly useful for spectra which do not have a clearly defined continuum. These techniques are applied to SN Ia spectra and correlations are found between the spectral features and light curve decline rate. The wavelet spectral indexes are used to measure the evolution of spectral features which are characterized by 3 or 4 parameters for the most complicated evolution. The three absorption features studied here are associated with sulfur and silicon and all show a transition in strength between 1 to 2 weeks after B-band maximum. Pearson correlation coefficients between spectral features and [delta]m₁₅ are found to be significant within a week of maximum brightness and 3 to 4 weeks post-maximum. These correlations are used to determine the principal components at each epoch among the set of SN spectra in this work. The variation contained in the first principal component (PC1) is found to be greater than 60% to 70% for most epochs and reaching as high as 80% to 90% for epochs with the highest correlations. The same first principal component can be used to relate spectral feature strengths to the decline rate. These relations were used to estimate a SN light curve decline rate from a set of spectra taken over the course of the explosion, from a single spectrum, or from even a single spectral feature. These relationships could be used for future surveys to estimate spectral characteristics from light curve data, such as photometric redshift.

Type Ia supernovae (SNe Ia) are excellent tools in cosmology. Their intrinsic luminosities are found to vary systematically with the light-curve widths, providing an empirical calibration. This property, called the width-luminosity relation (WLR), is the basis of modern SN Ia cosmology and led to the unexpected discovery of the current accelerated rate of cosmic expansion. By examining the spectroscopic diversity of SNe Ia, this thesis aims to improve both the use of SNe Ia in cosmology and the physical understanding of the observed properties. Spectra of SNe Ia contain a wealth of information, but are difficult to organize. In this thesis, new methods are developed to consistently quantify and analyze the spectral features of supernovae. The efficacy of the methods is tested on a large library of observed spectra encompassing a wide range of properties. The spectroscopic diversity of SNe Ia enters cosmology through K-correction calculations. Before this work, K-correction was a major contributor of the systematic errors in cosmology. It is shown here that the systematic errors can be largely diminished by carefully quantifying the mean spectroscopic properties of SNe Ia. The remaining statistical errors are also quantified and shown to be redshift dependent. With the aid of principal component analysis (PCA), the multidimensional spectral information is reduced to a few components describing the largest variations in the spectral library. Using this tool, it is shown here that SN Ia intrinsic luminosity is the main driver of the spectroscopic diversity at maximum light, for every spectral feature from the ultraviolet to the near-infrared. These spectroscopic sequences can potentially account for a large fraction of the K-correction statistical errors and even enable the use of SN Ia spectra as independent indicators of intrinsic luminosity and colors. The established relations will also disentangle the effects of demographic shift and true evolution in high-redshift SN Ia spectra. The temporal evolution of the spectral features is shown to exhibit the persistence of the spectroscopic sequences throughout other epochs. The effect is attributed to the more rapid spectroscopic temporal evolution of fainter SNe Ia. This conclusion supports the theory that WLR is primarily a spectroscopic effect, rather than a bolometric one.