In this thesis, I will discuss two separate topics which are related to gravitational wave production in the early universe. The first part will focus on the tensor power spectrum from inflation, derived using the Ashtekar variables of loop quantum gravity. This formalism is different from the ordinary approach in that it uses a complex connection as the central gravitational variable instead of the metric. Although the choice of variables should not affect any classical results, it becomes vital when considering quantum mechanical quantities like vacuum fluctuations. We will find that in this formalism, the tensor power spectrum is chiral, which would lead to a non-zero TB correlator in the CMB. Obtaining the full TB power spectrum would enable us to probe this chirality and provide clues about the nature of gravity. In the second part, I will consider gravitational waves produced from massless preheating, during which the inflaton transfers energy to a scalar field chi. If chi is light, it acquires a scale invariant spectrum of perturbations from inflation. At the time of preheating, the field will therefore have fluctuations on superhorizon scales and take a different value in different parts of the observable universe. I will study GW production for different initial values of chi numerically using 3d lattice simulations. The GW amplitude strongly depends on this initial value, leading to a GW background that is anisotropic today, with relative fluctuations of order 1%. In general, anisotropies will occur in any model of preheating with a light scalar field, and the characteristics should strongly depend on the model parameters. If a GW background from preheating was measured in the future, it would provide a novel way to distinguish between different inflationary scenarios.