Precision predictions of Starobinsky inflation with self-consistent Weyl-squared corrections

Starobinsky's $R+\ensuremath{\alpha}{R}^{2}$ inflation provides a compelling one-parameter inflationary model that is supported by current cosmological observations. However, at the same order in spacetime derivatives as the ${R}^{2}$ term, an effective theory of spacetime geometry must also include the Weyl-squared curvature invariant ${W}^{2}$. In this paper, we study the inflationary predictions of the gravitational theory with action of the form $R+\ensuremath{\alpha}{R}^{2}\ensuremath{-}\ensuremath{\beta}{W}^{2}$, where the coupling constant $\ensuremath{\alpha}$ sets the scale of inflation, and corrections due to the ${W}^{2}$ term are treated self-consistently via reduction of order in an expansion in the coupling constant $\ensuremath{\beta}$, at the linear order in $\ensuremath{\beta}/\ensuremath{\alpha}$. Cosmological perturbations are found to be described by an effective action with a nontrivial speed of sound ${c}_{\mathrm{s}}$ for scalar and ${c}_{\mathrm{t}}$ for tensor modes, satisfying the relation ${c}_{\mathrm{t}}/{c}_{\mathrm{s}}\ensuremath{\simeq}1+\frac{\ensuremath{\beta}}{6\ensuremath{\alpha}}$ during the inflationary phase. Within this self-consistent framework, we compute several primordial observables up to the next-to-next-to-next-to leading order (N3LO). We find the tensor-to-scalar ratio $r\ensuremath{\simeq}3(1\ensuremath{-}\frac{\ensuremath{\beta}}{6\ensuremath{\alpha}})({n}_{\mathrm{s}}\ensuremath{-}1{)}^{2}$, the tensor tilt ${n}_{\mathrm{t}}\ensuremath{\simeq}\ensuremath{-}\frac{r}{8}$, and the running of the scalar tilt ${\mathfrak{a}}_{\mathrm{s}}\ensuremath{\simeq}\ensuremath{-}\frac{1}{2}({n}_{\mathrm{s}}\ensuremath{-}1{)}^{2}$, all expressed in terms of the observed scalar tilt ${n}_{\mathrm{s}}$. We also provide the corresponding corrections up to N3LO, $\mathcal{O}(({n}_{\mathrm{s}}\ensuremath{-}1{)}^{3})$.