Analytic Correlation of Inflationary Potential to Power Spectrum Shape: Limits of Validity, and `No-Go' for Small Field Model Analytics

The primordial power spectrum informs the possible inflationary histories of our universe. Given a power spectrum, the ensuing cosmic microwave background is calculated and compared to the observed one. Thus, one focus of modern cosmology is building well-motivated inflationary models that predict the primordial power spectrum observables. The common practice uses analytic terms for the scalar spectral index $n_s$ and the index running $\alpha$, forgoing the effort required to evaluate the model numerically. However, the validity of these terms has never been rigorously probed and relies on perturbative methods, which may lose their efficacy for large perturbations. The requirement for more accurate theoretical predictions becomes crucial with the advent of highly sensitive measuring instruments. This paper probes the limits of the perturbative treatment that connects inflationary potential parameters to primordial power spectrum observables. We show that the validity of analytic approximations of the scalar index roughly respects the large-field/small-field dichotomy. We supply an easily calculated measure for relative perturbation amplitude and show that, for large field models, the validity of analytical terms extends to $\sim 3\%$ perturbation relative to a power-law inflation model. Conversely, the analytical treatment loses its validity for small-field models with as little as $0.1\%$ perturbation relative to the small-field test-case. By employing the most general artificial neural networks and multinomial functions up to the twentieth degree and demonstrating their shortcomings, we show that no reasonable analytic expressions correlating small field models to the observables the yield exists. Finally, we discuss the possible implications of this work and supply the validity heuristic for large and small field models.