  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.
