Reionization optical depth and CMB-BAO tension in punctuated inflation

Within the standard six-parameter Lambda cold dark matter ($\Lambda$CDM) model, a $2$-$3\sigma$ tension persists between baryon acoustic oscillation (BAO) measurements from the Dark Energy Spectroscopic Instrument (DESI) and observations of the cosmic microwave background (CMB). Although this tension has often been interpreted as evidence for dynamical dark energy or a sum of neutrino masses below the established minimum, recent studies suggest it may instead originate from an underestimation of the reionization optical depth, particularly when inferred from large-scale CMB polarization. Jhaveri et al. propose that a suppression of large-scale primordial curvature power could partially cancel the contribution of $\tau$ to the CMB low-$\ell$ polarization power spectrum, leading to a biased low $\tau$ measurement in standard analyses. In this work, we investigate whether punctuated inflation - which generates a suppression of primordial power on large scales through a transient fast-roll phase - can raise the inferred $\tau$ value and thereby reconcile the consistency between CMB and BAO. For simple models with step-like features in the inflaton potential, we find that the constraint on $\tau$ and the CMB-BAO tension remain nearly identical to those in the standard six-parameter $\Lambda$CDM model. We provide a physical explanation for this negative result.