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The primordial power spectrum informs the possible inflationary histories ofour universe. Given a power spectrum, the ensuing cosmic microwave backgroundis calculated and compared to the observed one. Thus, one focus of moderncosmology is building well-motivated inflationary models that predict theprimordial power spectrum observables. The common practice uses analytic termsfor the scalar spectral index $n_s$ and the index running $\alpha$, forgoingthe effort required to evaluate the model numerically. However, the validity ofthese terms has never been rigorously probed and relies on perturbativemethods, which may lose their efficacy for large perturbations. The requirementfor more accurate theoretical predictions becomes crucial with the advent ofhighly sensitive measuring instruments. This paper probes the limits of theperturbative treatment that connects inflationary potential parameters toprimordial power spectrum observables. We show that the validity of analyticapproximations of the scalar index roughly respects the large-field/small-fielddichotomy. We supply an easily calculated measure for relative perturbationamplitude and show that, for large field models, the validity of analyticalterms extends to $\sim 3\%$ perturbation relative to a power-law inflationmodel. Conversely, the analytical treatment loses its validity for small-fieldmodels with as little as $0.1\%$ perturbation relative to the small-fieldtest-case. By employing the most general artificial neural networks andmultinomial functions up to the twentieth degree and demonstrating theirshortcomings, we show that no reasonable analytic expressions correlating smallfield models to the observables the yield exists. Finally, we discuss thepossible implications of this work and supply the validity heuristic for largeand small field models.

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