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.
