In theories with multiple vacua, reheating to a temperature greater than the height of a barrier can stimulate transitions from a desirable metastable vacuum to a lower energy state. We discuss the constraints this places on various theories and demonstrate that in a class of supersymmetric models this transition does not occur even for arbitrarily high reheating temperature.

In this thesis, we study some possible extensions of the Standard Model and the insight they provide into treating the Standard Model as an effective field theory. Following an introduction highlighting the uses and difficulties of effective field theory in study of the Standard Model and physics beyond the Standard Model (BSM), we first focus on Loryons, particles which acquire most of their mass from the Higgs. We explain how BSM Loryons necessitate the use of HEFT as an effective field theory description and examine the experimental viability of scalars and vector-like fermions. We then investigate how BSM Loryons could affect the electroweak phase transition and gravitational wave background. Finally, we analyze the effective field theory of the two Higgs doublet model, demonstrating the importance of choosing an appropriate flavor basis for the two doublets and introducing a new basis particularly suited to effective field theory analysis.

It's an exciting time in particle theory. Fantastic experimental effort has revealed that the major ideas of the past few decades of particle theory are increasingly in tension with empirical data---underscoring the need for new ideas and strategies. In this thesis we focus on the application of this philosophy toward the hierarchy problem. We begin with a pedagogical introduction aimed at grounding the hierarchy problem with a clear understanding of renormalization and a comprehensive discussion of the problem while debunking common confusions. We survey classic approaches and argue that the LHC data has qualitatively changed the problem. We discuss the Twin Higgs as the avatar of the `neutral naturalness' strategy, and its exotic signatures which shift the signals of naturalness away from the energy frontier. Finally, we suggest a new trail for the naturalness paradigm: violations of effective field theory---as demanded by quantum gravity---challenge the assumptions of the space of solutions and suggest qualitatively new avenues of pursuit.

As experimental efforts to uncover the nature of physics beyond the Standard Model continue to push the boundaries of energy and sensitivity, theoretical predictions of well-motivated physics models will need to commensurately increase in precision so that we may reliably distinguish signal from background. A prime example may be found in the stochastic gravitational wave background that will soon be within reach of observatories and which could contain imprints from a variety of ultraviolet phenomena, including first order cosmological phase transitions and topological defects. We begin this thesis by discussing scenarios in which the gravitational wave spectrum due to a phase transition can be substantially altered by particle reflection off of relativistic bubble walls, an effect which has been largely ignored in the literature thus far. We then move on to discussing a particular class of parity-based solutions to the strong CP problem which also features a potential gravitational wave signal, this time due to domain wall topological defects. In addition, these models provide testable predictions for near-future colliders and tabletop experiments. Finally, we point out an exciting new computational avenue for discriminating between signal and background in particle collider data using machine learning coupled with a physically motivated metric on the space of collider events.

In theories with multiple vacua, reheating to a temperature greater than the height of a barrier can stimulate transitions from a desirable metastable vacuum to a lower energy state. We discuss the constraints this places on various theories and demonstrate that in a class of supersymmetric models this transition does not occur even for arbitrarily high reheating temperature.

The interplay between particle physics models and their cosmological implicationsprovides a crucial set of constraints for model-builders. As upcoming experiments set out to determine the particle nature of dark matter, detect primordial gravitational waves (GWs), and map primordial non-Gaussianities (PNGs), a unique opportunity arises to search for physics beyond the Standard Model (BSM) at scales that may be as high as 10^13 GeV. This thesis explores these three examples of cosmological aspects of model- building. We begin with a review of the problems that motivate the search for BSM physics, touching on both the naturalness problems and the inflationary paradigm. Then, we introduce a parity-based solution to the strong CP problem, motivating the question of whether a model containing a heavy mirror copy of the SM gauge group may contain a viable candidate for dark matter. We then turn to GWs, introducing a new source of primordial GWs from stochastic fluctuations of scalar fields in the early universe. Finally, we turn to effective field theories (EFTs) in de-Sitter space, introducing the problem in the context of cosmological collider physics and initiating a systematic understanding of EFT construction in inflationary spacetimes. In turn, we illustrate the promise of near-future cosmological probes to provide insights into the underlying physics of our universe.

High energy physics, like many other scientific disciplines, has entered an exciting new era of big data, where both particle accelerators at the \emph{energy frontier} and astrophysical surveys at the \emph{cosmic frontier} are producing an enormous amount of data which may hold the very key to the most fundamental questions about nature. Mining such gold inevitably calls for revolutionary designs of ever more powerful and efficient statistical analysis frameworks, while at the same time scientific rigorousness places an additional requirement on the interpretability of any novel model proposed. Among a plethora of available modern machine learning techniques, the theory of \emph{optimal transport} stands out as a distinct approach that is both high performing and mathematically well grounded. By equipping the space of data represented as distributions with a suitable metric, optimal transport replaces \emph{ad hoc} notions of similarity with a well-defined distance, opening up a range of new applications with profound theoretical implications.

This thesis introduces the theory of optimal transport with an eye towards its usage in physics. Special emphasis is put on two particular optimal transport distances which enjoy unique geometric properties. Utilizing their geometric structure, we develop a computationally efficient linearization framework for the two distances and highlight their approximations for discrete distributions encountered in practice. We then showcase the power of this linearized optimal transport framework by applying it to two use cases---one in collider physics at the energy frontier and the other in dark matter astrophysics at the cosmic frontier. As the adoption of optimal transport in high energy physics is still in its early stage, the present thesis invites the readers to think of other potential applications for their own research.