  We consider a phenomenological model of inflation where the inflaton is the
phase of a complex scalar field $\Phi$. Planck-suppressed operators of
$\mathcal O(f^5/M_\mathrm{pl})$ modify the geometry of the vev $\langle \Phi
\rangle$ at first order in the decay constant $f$, which adds a first order
periodic term to the definition of the canonically normalized inflaton $\phi$.
This correction to the inflaton induces a fixed number of extra oscillatory
terms in the potential $V \sim \theta^p$. We derive the same result in a toy
scenario where the vacuum $\langle \Phi \rangle$ is an ellipse with an
arbitrarily large eccentricity. These extra oscillations change the form of the
power spectrum as a function of scale $k$ and provide a possible mechanism for
differentiating EFT-motivated inflation from models where the angular shift
symmetry is a gauge symmetry.

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology;High Energy Physics - Theory