Slow-rolling down the curvature: a reassessment of the Planck constraints on $ϕ^2$ inflation in a closed universe

We revisit the Cosmic Microwave Background (CMB) constraints on the spatial curvature of the Universe, assessing how they change when the curvature parameter and the primordial inflationary scalar spectrum are treated consistently within theoretically motivated frameworks. Instead of relying on the phenomenological parametrisation commonly used to capture curvature effects at the largest scales, we present a case study based on closed quadratic inflation, where the primordial spectrum is derived in full generality and in a gauge-invariant manner. Within this framework, we analyze both the $\texttt{plik}$ PR3 and $\texttt{CamSpec}$ PR4 Planck CMB likelihoods and find that the constraints on $Ω_{\mathcal{K}}$ shift towards spatial flatness. In $\texttt{plik}$ the preference for $Ω_{\mathcal{K}}<0$ decreases from $\gtrsim 3.5σ$ to $\sim 2.5σ$, while in $\texttt{CamSpec}$ it reduces to $\sim 2σ$. At large angular scales ($\ell < 10$), our model explains the low-$\ell$ power suppression anomaly, notably improving the fit to the quadrupole. However, the reduced preference for highly negative values of $Ω_{\mathcal{K}}$ only partially accounts for the lensing anomaly at high multipoles, worsening the fit to the $\texttt{plik}$ spectrum at small scales. By contrast, in the $\texttt{CamSpec}$ PR4 spectrum, where the lensing anomaly is less pronounced, the model yields an overall improvement. Our analysis highlights a key conceptual point: closed-inflation models tie the curvature parameter to the inflationary dynamics and the primordial spectrum, enforcing consistency conditions that do not necessarily allow for the large deviations from flatness seen in phenomenological parametrisations. In the case of quadratic inflation, these restrictions reduce the apparent evidence for negative curvature reported by earlier analyses, while allowing for a mildly closed geometry.