Expectations for Inflationary Observables: Simple or Natural?

We describe the general inflationary dynamics that can arise with a single, canonically coupled field where the inflaton potential is a 4-th order polynomial. This scenario yields a wide range of combinations of the empirical spectral observables, $n_s$, $r$ and $\alpha_s$. However, not all combinations are possible and next-generation cosmological experiments have the ability to rule out all inflationary scenarios based on this potential. Further, we construct inflationary priors for this potential based on physically motivated choices for its free parameters. These can be used to determine the degree of tuning associated with different combinations of $n_s$, $r$ and $\alpha_s$ and will facilitate treatments of the inflationary model selection problem. Finally, we comment on the implications of these results for the naturalness of the overall inflationary paradigm. We argue that ruling out all simple potentials would not necessarily imply that the inflationary paradigm itself was unnatural, but that this eventuality would increase the importance of building inflationary scenarios in the context of broader paradigms of ultra-high energy physics.