Numerical Analysis of the Primordial Power Spectrum for (Small Field) Inflationary Potentials

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.