Analytical approximations for curved primordial tensor spectra

Abstract

We build upon previous analytical treatments of scalar perturbations in curved inflationary universes to obtain analytical templates for the primordial tensor power spectrum in models with non-zero primordial spatial curvature. These templates are derived without assuming a particular inflaton potential, within a background history consisting of an initial kinetically dominated phase followed by an ultra-slow-roll inflationary regime, thereby isolating the universal imprints of curvature on tensor modes. Our results predict characteristic large-scale features -- including low-$\ell$ cut-offs and oscillatory patterns -- that are consistent with numerical solutions and provide a clear physical interpretation of how curvature modifies the underlying dynamics. In particular, we show that curvature effects manifest mathematically as systematic shifts in the dynamically relevant wavevectors, mirroring the behaviour previously identified in the scalar power spectrum. These features translate into distinctive signatures in the large-angle $B$-mode polarisation spectrum, offering a potential discriminant for spatial curvature in forthcoming CMB observations.