On the renormalization of ultraviolet divergences in the inflationary angular power spectrum

We revise the role that ultraviolet divergences of quantum fields play in slow-roll inflation, and discuss the renormalization of cosmological observables from a space-time perspective, namely the angular power spectrum. We first derive an explicit expression for the multipole coefficients $C_{\mathrm{\ell }}$ in the Sachs-Wolfe regime in terms of the two-point function of primordial perturbations. We then analyze the ultraviolet behavior, and point out that the standard result in the literature is equivalent to a renormalization of $C_{\mathrm{\ell }}$ at zero adiabatic'' order. We further argue that renormalization at second adiabatic’’ order may be more appropriate from the viewpoint of standard quantum field theory. This may change significantly the predictions for $C_{\mathrm{\ell }}$, while maintaining scale invariance.