This dissertation presents a series of my contributions to research in theoretical cosmology, focusing on aspects of the very large scale universe, particularly dark energy, cosmic acceleration, modified gravity, and cosmic variance. Following an overview of the current understanding of the standard cosmological model in chapter 1, three pertinent topics are discussed in detail. A common theme among all chapters is the desire to explain the properties of the universe on the largest scales. One of the biggest mysteries on large scales is the need for dark energy to explain the observed accelerated expansion of the late universe. The unsatisfying explanation offered by the standard cosmological model and the associated enormous fine tuning problem have driven considerable interest in infrared (long-distance) modifications of general relativity. In this work, we consider a particularly well motivated modified theory, massive gravity, in which the modification is to simply assume that the particle mediating the gravitational force has a non-zero mass. For a mass on the order of the Hubble constant, this theory offers an alternative explanation of the accelerated cosmic expansion. Chapter 2 lays the theoretical groundwork for massive gravity, summarizing its history and formalism. A fundamental challenge for any modified gravity theory is sequestering the modification to large enough distance scales, so that the predictions match general relativity on solar system scales where it has been tested to high precision. Chapter 3 provides a detailed analysis of massive gravity's ability to screen its extra degrees of freedom, allowing for continuity with general relativity on short distance scales. Further, in chapter 4, we explore the cosmological production and propagation of gravitational waves in an extension of massive gravity, bigravity, determining whether there may be any testable deviations from general relativity. Understanding these predictions is crucial, as there is now a vigorous observational program to probe possible deviations from our standard model. As rapid progress in observational cosmology unfolds, not only is it paramount to construct viable modified gravity theories to test against general relativity, it is necessary to explore which observational methods will be the most powerful for constraining them. This dissertation contains progress on both of these fronts: analyzing potential modified gravity theories, and analyzing potential novel observational probes of the large scale universe. Chapter 5 provides the theoretical framework for one such novel probe, the large scale kinetic Sunyaev-Zeldovich effect. This effect is particularly intriguing because of its ability to overcome cosmic variance, and thus help us unlock the secrets of the universe on the largest scales.