A Map Between Primordial Power Spectra and the Effective Field Theory of Inflation

Abstract

We have developed a precise dictionary between the spectrum of primordial density fluctuations and the parameters of the effective field theory (EFT) of inflation that determine the primordial power spectrum (PPS). At lowest order the EFT contains two parameters: the slow-roll parameter $\epsilon$, which acts as an order parameter, and the speed of sound $c_s$. Applying second-order perturbation theory, we provide maps from the PPS to the EFT parameters that are precise up to the cube of the fractional change in the PPS $(\Delta \mathcal{P}/\mathcal{P})^3$, or less than $1\%$ for spectral features that modulate the PPS by $20\%$. While such features are not required when the underlying cosmological model is assumed to be $\Lambda$CDM they are necessary for alternative models that have no cosmological constant/dark energy. We verify the dictionary numerically and find those excursions in the slow-roll parameter that reproduce the PPS needed to fit Planck data for both $\Lambda$ and no-$\Lambda$ cosmological models.