This thesis is focused on two topics in which relativistic gravitational fields play an important role, namely early Universe cosmology and black hole physics. The theory of cosmic inflation has emerged as the most successful theory of the very early Universe with concrete and verifiable predictions for the properties of anisotropies of the cosmic microwave background radiation and large scale structure. Coalescences of black hole binaries have recently been detected by the Laser Interferometer Gravitational Wave Observatory (LIGO), opening a new arena for observationally testing the dynamics of gravity. In part I of this thesis we explore some modifications to the standard theory of inflation. The main predictions of single field slow-roll inflation have been largely consistent with cosmological observations. However, there remain some aspects of the theory that are not presently well understood. Among these are the somewhat interrelated issues of the choice of initial state for perturbations and the potential imprints of pre-inflationary dynamics. It is well known that a key prediction of the standard theory of inflation, namely the Gaussianity of perturbations, is a consequence of choosing a natural vacuum initial state. In chapter 3, we study the generation and detectability of non-Gaussianities in inflationary scalar perturbations that originate from more general choices of initial state. After that, in chapter 4, we study a simple but predictive model of pre-inflationary dynamics in an attempt to test the robustness of inflationary predictions. We find that significant deviations from the standard predictions are unlikely to result from models in which the inflaton field decouples from the pre-inflationary degrees of freedom prior to freeze-out of the observable modes.

In part II we turn to a study of an aspect of the thermodynamics of black holes, a subject which has led to important advances in our understanding of quantum gravity. For objects which collapse to form black holes, we examine a conjectured relationship between the objects' entropy, the collapse timescale, and the mass of the final black hole. This relationship is relevant for understanding the nature of generic quantum mechanical states of black hole interiors.
In chapter 6 we construct a counter-example to a weak version of the conjectured relation.