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Cosmic inflation is an epoch of quasi-de Sitter expansion that is hypothesized to have occurred before the Hot Big Bang, that provides a mechanism for the production of primordial curvature perturbations with a nearly scale-invariant power spectrum, explaining the super-horizon correlations observed in the Cosmic Microwave Background. This early period of rapid expansion in the universe offers a unique observational window into the fundamental structure of spacetime, as the principles of quantum mechanics greatly influence geometry at the shortest scales. Hence, understanding this interplay not only provides insights into the origins of the universe but also helps in extract valuable information to compare with theories of quantum gravity.
As we enter an era of precision cosmology, future observations will improve constraints on many primordial observables, and the computation of precise theoretical predictions is essential for next-generation surveys. In the work Primordial power spectrum at N3LO in effective theories of inflation, we derived an explicit expression of fully expanded primordial power spectrum of cosmological perturbations up to next-to-next-to-next-to-leading order in the Hubble-flow expansion. For effective theories of inflation where a given Scalar-Vector-Tensor mode $\psi$ contribute to the quadratic action as:
\[S_{2}[\psi] = \frac{1}{2} \int \mathrm{d}^4 x \, a(t)^3\,Z_{\psi}(t)\Big( \dot{\psi}^2- \frac{c_{\psi}(t)^2}{a(t)^2} (\partial_i \psi)^2 \Big) \, ,\]we find a power spectrum of the form:
\[\mathcal{P}_0^{(\psi)}(k) = \frac{\hbar H_\ast^2}{4\pi^2 c_\ast^3 Z_\ast} \Bigg[ 1 + p_{0\ast} + p_{1\ast} \ln\Big(\frac{k}{k_\ast}\Big) +\, p_{2\ast} \ln\Big(\frac{k}{k_\ast}\Big)^2 + p_{3\ast} \ln\Big(\frac{k}{k_\ast}\Big)^3 \Bigg] \,\]where the coefficients are given in Tabs. III, IV, V, and VI of the paper.