Third-order corrections to the slow-roll expansion: calculation and constraints with Planck, ACT, SPT, and BICEP/Keck

Abstract

We investigate the primordial power spectra (PPS) of scalar and tensor perturbations, derived through the slow-roll approximation. By solving the Mukhanov-Sasaki equation and the tensor perturbation equation with Green's function techniques, we extend the PPS calculations to third-order corrections, providing a comprehensive perturbative expansion in terms of slow-roll parameters. We investigate the accuracy of the analytic predictions with the numerical solutions of the perturbation equations for a selection of single-field slow-roll inflationary models. We derive the constraints on the Hubble flow functions $ε_i$ from Planck, ACT, SPT, and BICEP/Keck data. We find an upper bound $ε_1 \lesssim 0.002$ at 95\% CL dominated by BICEP/Keck data and robust to all the different combination of datasets. We derive the constraint $ε_2 \simeq 0.031 \pm 0.004$ at 68\% confidence level (CL) from the combination of Planck data and late-time probes such as baryon acoustic oscillations, redshift space distortions, and supernovae data at first order in the slow-roll expansion. The uncertainty on $ε_2$ gets larger including second- and third-order corrections, allowing for a non-vanishing running and running of the running respectively, leading to $ε_2 \simeq 0.034 \pm 0.007$ at 68\% CL. We find $ε_3 \simeq 0.1 \pm 0.4$ at 95\% CL both at second and at third order in the slow-roll expansion of the spectra. $ε_4$ remains always unconstrained. The combination of Planck and SPT data leads to slightly tighter constraints on $ε_2$ and $ε_3$. On the contrary, the combination of Planck data with ACT measurements, which point to higher values of the scalar spectral index compared to Planck findings, leads to shifts in the means and maximum likelihood values for $ε_2$ and $ε_3$.