Comment on: “Third-order corrections to the slow-roll expansion: Calculation and constraints with Planck, ACT, SPT, and BICEP/Keck [2025 PDU 47 101813]”

Author(s)

Auclair, Pierre, Ringeval, Christophe

Abstract

We point out that Ballardini et al. [41] adopt approximations that do not reproduce the exact result originally presented by Auclair and Ringeval [39]. In our original work, all terms at all orders have been derived exactly and any difference between two expansions performed at the same pivot wavenumber signals a problem. As we show in this comment, the simplifications performed by Ballardini et al. [41] fail to reproduce the finite part of some definite three-dimensional integrals. We explain in details the discrepancy between the approximate results of Ballardini et al. [41] and the exact result of Auclair and Ringeval [39], which can be summarized as the difference between integrating a truncated Taylor expansion and Taylor expanding an integral. Our claim is backed-up with a Monte-Carlo numerical integration of the incriminated three-dimensional integrals, which, unsurprisingly, matches the analytical value derived in Auclair and Ringeval [39].

Figures

Caption

: Real part.

Caption

: Imaginary part.