  We study small field models of inflation, which, against previous
expectations, yield significant Gravitational Wave (GW) signal while
reproducing other measured observable quantities in the Cosmic Microwave
Background (CMB). We numerically study these, using previously published
analytic works as general guidelines. We first discuss the framework necessary
to understand model building, and some of its motivations. We review the
slow-roll paradigm, derive the slow-roll parameters, and discuss different
formulations thereof. We review the Lyth bound and its theoretical descendants,
we outline the small/large field taxonomy and their characterization in the
current nomenclature. We then present our models and the methods employed in
their building and examination. We employ MCMC simulations to evaluate model
likelihood and by process of marginalization extract the most probable
coefficients for these inflationary potentials. An additional method employed
is a multinomial fit, where we create a correspondence between coefficients and
observables. This allows the use of observable values directly to yield the
most likely coefficients. We compare the results of the two methods and
evaluate the level of tuning required for these models. We discuss an apparent
discrepancy between analytical approaches of evaluating Primordial Power
Spectrum (PPS) observables and the precise numerical results in our models. We
identify some of the sources of this discrepancy and remark on their meaning in
the age of precision cosmology. Finally, we present the results of our study,
for the most likely inflationary models with polynomial potentials of degree 5,
and 6. We demonstrate our ability to produce potentials that yield GW with a
tensor-to-scalar ratio r = 0.03. This is a realistic expectation of GW
detection sensitivity in the near future.
