Primordial power spectrum at N3LO in effective theories of inflation

We develop a systematic framework to compute the primordial power spectrum up to next-to-next-to-next to leading order (N3LO) in the Hubble-flow parameters for a large class of effective theories of inflation. We assume that the quadratic action for perturbations is characterized by two functions of time, the kinetic amplitude and the speed of sound, that are independent of the Fourier mode $k$. Using the Green's function method introduced by Stewart and Gong and extended by Auclair and Ringeval, we determine the primordial power spectrum fully expanded around a pivot scale up to N3LO, starting from a given generic action for perturbations. As a check, we reproduce the state-of-the-art results for scalar and the tensor power spectra of the simplest "vanilla" models of single-field inflation. The framework applies to Weinberg's effective field theory of inflation (with the condition of no parity violation) and to the effective theory of spontaneous de Sitter-symmetry breaking. As a concrete application, we provide the expression for the N3LO power spectrum of $R+R^2$ Starobinsky inflation in metric variables. All expressions are provided in terms of an expansion in one single parameter, the number of inflationary e-foldings $N_\ast$. Surprisingly, we find that, compared to previous leading-order calculations, for $N_\ast = 55$ the N3LO correction results in a $7\%$ decrease of the predicted tensor-to-scalar ratio, in addition to a deviation from the consistency relation and a prediction of a negative running $\alpha_\mathrm{s}=-\frac{1}{2}(n_\mathrm{s}-1)^2+\ldots$ of the scalar tilt. These results provide precise theoretical predictions for the next generation of CMB observations.