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Third-order corrections to the slow-roll expansion: calculation and constraints with Planck, ACT, SPT, and BICEP/Keck
Mario Ballardini, Alessandro Davoli, Salvatore Samuele Sirletti
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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 ϵᵢ from Planck, ACT, SPT, and BICEP/Keck data. We find an upper bound ϵ₁ ≲0.002 at 95% CL dominated by BICEP/Keck data and robust to all the different combination of datasets. We derive the constraint ϵ₂ ≃0.031 ±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 ϵ₂ gets larger including second- and third-order corrections, allowing for a non-vanishing running and running of the running respectively, leading to ϵ₂ ≃0.034 ±0.007 at 68% CL. We find ϵ₃ ≃0.1 ±0.4 at 95% CL both at second and at third order in the slow-roll expansion of the spectra. ϵ₄ remains always unconstrained. The combination of Planck and SPT data leads to slightly tighter constraints on ϵ₂ and ϵ₃. 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 ϵ₂ and ϵ₃.
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