Type Ia supernovae (SNe Ia) are the final brilliant explosion of a carbon-oxygen white dwarf accreting mass from a companion star. At peak brightness, a SN Ia can outshine an entire galaxy of billions of stars. Most SNe Ia have a standardizable luminosity, ideal for use as an extragalactic distance indicator. Measurements of a large sample of SNe Ia over a range of distances enables the estimate of cosmological parameters to help determine the mass-energy content of the Universe (Riess et al. 1998; Perlmutter et al. 1999; Riess et al. 2004; Astier et al. 2006; Riess et al. 2007; Wood-Vasey et al. 2008; Kowalski et al. 2008; Hicken et al. 2009a; Amanullah et al. 2010; Sullivan et al. 2011a; Suzuki et al. 2012).

The cosmological application of SNe Ia is predicated upon relationships between the intrinsic luminosity and light-curve properties. Despite the successful measurement of cosmological parameters using SNe Ia, our understanding of SNe Ia themselves is surprisingly lacking. The SN Ia progenitor system has never been directly observed, making it unclear how many different channels exist to make a SN Ia. The physical nature of the relationship between light-curve parameters and luminosity is also not well understood, and it remains to be seen whether other correlations exist to improve SN Ia distance estimates.

The goal of this dissertation is to shed light on the physics of SNe Ia and search for new correlations to improve distance estimates to SNe Ia by analyzing a large sample of well-observed, high-quality SN Ia light curves. I have collected, reduced, and analyzed optical photometric data for 165 nearby SNe Ia as part of the Lick Observatory Supernova Search (LOSS). These data represent a significant contribution to the existing sample of nearby SN Ia light curves.

After giving a general overview of SNe Ia in Chapter 1, I present the methods used to obtain and reduce the LOSS data in Chapter 2. In Chapter 3, I use the LOSS data in an analysis of the earliest photometry epochs to understand the explosion physics governing the initial rise of the SN Ia light curve. These early data points also provide a means of testing models to constrain the nature of the binary companion star. I do not find evidence for interaction between SN ejecta and a companion star, ruling out theories requiring a red giant as a companion in most cases. In Chapter 4, I combine the data presented in this thesis with other samples in the literature to place constraints on cosmological parameters. I reject a non-accelerating Universe with 99.999% confidence. In Chapter 5, I present a study of an individual peculiar SN Ia that is unlike any previously published object, bucking the relationships normally observed in SNe Ia. Studying extreme SNe Ia may provide insights into understanding the physics of normal SNe Ia.

The explosive deaths of stars, known as a supernovae (SNe), have been critical to our understanding of the Universe for centuries. From the first evidence of a changing Universe beyond the Moon (Brahe1573) to the first evidence of the accelerating expansion of the Universe (Riess et al. 1998; Perlmutter et al. 1999), SNe - and often a specific subclass of SNe called Type Ia SNe (SNe Ia) - have been integral to astronomical research. An introduction to SNe, their importance in astronomy, and how we observe them is given in Chapter 1. How SNe Ia explode, what progenitor systems give rise to them, and how different initial conditions affect the observed outcomes of these objects are understood only at a relatively basic level. In other words, a detailed understanding of the physics behind SNe Ia is still lacking. One way astronomers can begin to solve these problems, and others involving SNe Ia, is to obtain and analyze a large, self-consistent dataset of SN Ia observations. This is the goal of the Berkeley SN Ia Program (BSNIP) which comprises the majority of this Thesis.

In the second Chapter, I present the full BSNIP sample which consists of 1298 low-redshift (z ≤ 0.2) optical spectra of 582 SNe Ia observed from 1989 through the end of 2008. Many of the SNe have well-calibrated light curves with measured distances as well as spectra which have been corrected for host-galaxy contamination. Most of the data were obtained using the Kast double spectrograph mounted on the Shane 3 m telescope at Lick Observatory with typical wavelength coverage of 3300-10400 Å, which is significantly larger than that of most previously published SN Ia spectral datasets. I also present the BSNIP observing and reduction procedures used during the two decades over which the data were collected. In addition, I describe our spectral classification scheme (using the SuperNova IDentification code, SNID; Blondin & Tonry 2007), utilizing my newly constructed set of SNID spectral templates. These templates allow me to accurately spectroscopically classify the entire BSNIP dataset, and by doing so I am able to reclassify a handful of objects as bona fide SNe Ia and a few other objects as members of some of the peculiar SN Ia subtypes. In fact, the BSNIP dataset includes spectra of nearly 90 spectroscopically peculiar SNe Ia. I also present spectroscopic host-galaxy redshifts of some SNe Ia where these values were previously unknown. The sheer size of the BSNIP dataset and the consistency of the observation and reduction methods makes this sample unique among all other published SN Ia datasets and is complementary in many ways to the large, low-redshift SN Ia spectra presented by Matheson et al. 2008 and Blondin et al. 2011.

I present measurements of spectral features of 432 low-redshift (z < 0.1) optical spectra within 20 d of maximum brightness of 261 SNe Ia from the BSNIP sample in the third Chapter. I describe in detail my method of automated, robust spectral feature definition and measurement which expands upon similar previous studies. Using this procedure, I attempt to measure expansion velocities, (pseudo-)equivalent widths (pEWs), spectral feature depths, and fluxes at the center and endpoints of each of nine major spectral feature complexes. A sanity check of the consistency of the measurements is performed using the BSNIP data (as well as a separate spectral dataset). I investigate how velocity and pEW evolve with time and how they correlate with each other. Various spectral classification schemes are employed and quantitative spectral differences among the subclasses are investigated. Several ratios of pEW values are calculated and studied. Furthermore, SNe Ia that show strong evidence for interaction with circumstellar material or an aspherical explosion are found to have the largest near-maximum expansion velocities and pEWs, possibly linking extreme values of spectral observables with specific progenitor or explosion scenarios. A discussion of the relative merits of various classification schemes is presented and I find that purely spectroscopic classification schemes are useful in identifying the most peculiar SNe Ia. However, in almost all spectral parameters investigated the full sample of objects spans a nearly continuous range of values. Comparisons to previously published theoretical models of SNe Ia are made and some of the predictions of these models match the observations presented here. I conclude with a brief discussion of how these measurements and the possible correlations presented will be crucial to future SN surveys.

The fourth Chapter of this Thesis presents comparisons of spectral feature measurements to photometric properties of 115 low-redshift (z < 0.1) SNe Ia with optical spectra within 5 d of maximum brightness. The spectral data come from the BSNIP sample described in Chapter 2, and the photometric data come mainly from the Lick Observatory Supernova Search (LOSS) and are published by Ganeshalingam et al. (2010). The spectral measurements come from BSNIP II (Chapter 3 of this Thesis) and the light-curve fits and photometric parameters can be found in Ganeshalingam et al. (in preparation). A variety of previously proposed correlations between spectral and photometric parameters are investigated using the large and self-consistent BSNIP dataset. We also use a combination of light-curve parameters (specifically the SALT2 stretch and color parameters x₁ and c) and spectral measurements to calculate distance moduli. The residuals from these models is then compared to the standard model which only uses light-curve stretch and color. The pEW of Si II λ4000 is found to be a good indicator of light-curve width and the pEWs of the Mg II and Fe II complexes are relatively good proxies for color. However, a distance model only using these spectroscopic measurements performs worse than the standard model which uses only light-curve parameters. When using a distance model which combines the ratio of fluxes near ~3600 Å and ~4300 Å with both x₁ and c, the Hubble residuals are decreased by 12%, which is found to be significant at the 2.4σ level. The weighted root-mean square of the residuals using this model is 0.130 ± 0.019 mag (as compared to 0.146 ± 0.019 mag when using the same sample with the standard model). This Hubble diagram fit has one of the smallest scatters ever published and at the highest significance ever seen in such a study. Finally, these results are discussed with regard to how they can improve the cosmological accuracy of future, large-scale SN Ia surveys.

Finally, I conclude this Thesis with an in-depth study of a quite peculiar SN Ia, not included in the BSNIP sample. Chapter 5 presents and analyzes optical photometry and spectra of the extremely luminous and slowly evolving Type Ia SN 2009dc, and offers evidence that it is a super-Chandrasekhar mass (SC) SN Ia and thus had a SC white dwarf (WD) progenitor. Optical spectra of SN 2007if, a similar object, are also shown. SN 2009dc had one of the most slowly evolving light curves ever observed for a SN Ia, with a rise time of ~23 d and Δm₁₅(B) = 0.72 mag. I calculate a lower limit to the peak bolometric luminosity of ~2.4×10⁴³ erg s^-1, though the actual value is likely almost 40% larger. Optical spectra of SNe 2009dc and 2007if obtained near maximum brightness exhibit strong C II features (indicative of a significant amount of unburned material), and the post-maximum spectra are dominated by iron-group elements. All of the spectra of SNe 2009dc and 2007if also show low expansion velocities. However, I see no strong evidence in SN 2009dc for a velocity "plateau" near maximum light like the one seen in SN 2007if (Scalzo et al. 2010). The high luminosity and low expansion velocities of SN 2009dc lead to a derived WD progenitor mass of more than 2 M_Sun and a ⁵⁶Ni mass of about 1.4-1.7 M_Sun. I propose that the host galaxy of SN 2009dc underwent a gravitational interaction with a neighboring galaxy in the relatively recent past. This may have led to a sudden burst of star formation which could have produced the SC WD progenitor of SN 2009dc and likely turned the neighboring galaxy into a "post-starburst galaxy." No published model seems to match the extreme values observed in SN 2009dc, but simulations do show that such massive progenitors can exist (likely as a result of the merger of two WDs) and can possibly explode as SC SNe Ia.

Astrophysical transient events, particularly Type Ia supernovae (SNe Ia) and tidal disruption events (TDEs), provide critical insights into stellar systems and black hole activity. This thesis studies polarimetry of SNe Ia and TDEs in an attempt to better understand the geometry and physical processes behind these explosive events.

For SN 2019ein, spectropolarimetric data from approximately 11 days before to 10 days after peak brightness revealed a continuum polarization of 0.0--0.3\%. The constant polarization angle indicates axial symmetry in the explosion, consistent across inner and outer ejecta regions. Notably, polarization at the Si II and Ca II lines reached about 1\%, suggesting a largely spherical explosion with localized clumping of intermediate-mass elements. These observations largely disfavor merger-induced models for SN 2019ein. A larger sample of SNe Ia spectropolarimetry observed at Lick Observatory is presented and described.

In the study of the TDE AT\,2019qiz, spectropolarimetry showed negligible continuum polarization at peak brightness, suggesting a nearly spherical, optically thick electron scattering photosphere around the black hole. By day 29, polarization increased to around 1\%, indicating a more aspherical interior as the fallback rate decreased and the photosphere receded. This marks the first detailed spectropolarimetric evolution observed in a TDE, and supports the reprocessing model for the source of optical emission in TDEs. The study of AT\,2019qiz is followed by imaging polarimetry of several TDEs from Keck Observatory in an attempt to build a more comprehensive picture of gas geometry in TDEs.

Additionally, the thesis explores the newly discovered X-ray quasi-periodic eruptions (QPEs), analyzing archival {\it Hubble Space Telescope} images of the QPE source GSN 069. The data revealed a compact [O III] emission region within a 35 pc radius of the nucleus, along with extended emission up to 2 kpc away. Simulations with the photoionization code \textsc{Cloudy} suggest this emission comes from dense gas, likely ionized by X-rays from a young accretion disk, offering a unique glimpse into the dynamic processes at play in the host galaxies of QPEs.

Type Ia supernovae (SNe Ia) are magnificent explosions in the Cosmos that are thought to result from the thermonuclear runaway of white dwarf stars in multistar systems (see, e.g., Jha et al. 2019, for a recent review). Though the exact details of the progenitor system(s) and explosion mechanism(s) remain elusive, SNe Ia have proven themselves to be immensely valuable in shaping our understanding of the physical laws that govern the evolution of the Universe (i.e., physical cosmology). This value is manifested chiefly in two empirical facts: (i) SNe Ia are incredibly luminous (reaching the equivalent of several billion Suns), and (ii) the relatively similar peak luminosities that all "normal" SNe Ia reach can be further homogenized by exploiting a correlation with the rate of photometric evolution (e.g., Phillips 1993). Together, these facts make SNe Ia excellent extragalactic distance indicators, and their use as such led to the discovery of the accelerating expansion of the Universe (Riess et al. 1998; Perlmutter et al. 1999). Through this, the current cosmological paradigm came into favor — the so-called ΛCDM model, where the Universe consists primarily of repulsive dark energy (of which a leading candidate is Einstein’s cosmological constant, Λ) and cold dark matter (CDM).

In this thesis, I present a comprehensive study that follows the entire SN Ia cosmology lifecyle, from data acquisition to cosmological analysis (albeit of a different flavor than those mentioned above). While these “bookends” provide natural segmentation points in this thesis, there is a third, intermediate segment which serves to present a complementary method for SN Ia distance measurement that is far less data intensive than conventional approaches. In this way, the segments are hierarchical, each depending on its predecessor and enabling its successor.

After appropriately setting the stage in Chapter 1, I delve into the first segment (data acquisition) with Chapter 2, a data release and analysis of 93 multipassband SN Ia light curves collected between 2005 and 2018, and Chapter 3, a complementary release of 637 low-redshift SN Ia optical spectra from a similar time interval. In both, I describe open-source software I developed for data processing and analysis purposes, and make — in addition to the data themselves — useful, value-added data products (e.g., fitted parameters from light curves) available to the community. When combined with prior releases, the Berkeley SN Ia sample now reaches nearly 2000 optical spectra and more than 250 multiband light curves, all observed and processed with the utmost care for quality and internal consistency.

This large, homogeneous sample proves critical for the second segment of this thesis, in which I ultimately develop and validate the aforementioned technique — the snapshot distance method (SDM) — for estimating the distance to an SN Ia from sparse observations. As a prerequisite to the SDM, I develop, in Chapter 4, an open-source software package called deepSIP that is capable of determining the phase and light-curve shape of an SN Ia — both of which conventionally require a well-sampled light curve — from a single optical spectrum. At its heart, deepSIP consists of a set of three convolutional neural networks trained on a significant fraction of all publicly available SN Ia optical data (including those presented in the first segment of this thesis), with judicious augmentation steps included to promote telescope agnosticism and model robustness. The impressive performance of deepSIP enables the SDM, which, as I demonstrate in Chapter 5, is capable of deriving an SN Ia distance estimate from as little as one optical spectrum and one epoch of 2+ passband photometry with notable precision over a wide range of SN Ia parameters.

This leads, finally, into the last segment of this thesis (cosmological analysis), where I use the SDM to turn trash (i.e., SN Ia observations that were previously unusable owing to data sparsity) into treasure (i.e., reliable distance estimates to be used in a cosmological study). In particular, in Chapter 6, I combine a novel sample of 137 SDM-resurrected SNe Ia with a large literature sample of SNe Ia and SNe II to measure peculiar velocities and set leading (from an SN-only perspective) constraints on the cosmological parameter combination fσ8 and the nature of bulk flows in the local Universe. Moreover, the methods by which I perform this analysis establish a reproducible and extensible blueprint for future such analyses as large-scale surveys come online and unleash an unprecedented data volume.