Starobinsky’s $R+\alpha R^2$ 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 $R+\alpha R^2-\beta W^2$, where the coupling constant $\alpha$ 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 $\beta$, at the linear order in $\beta /\alpha$. 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 $c_{\mathrm{t}}/c_{\mathrm{s}}\simeq 1+\frac{\beta }{6\alpha }$ 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 $r\simeq 3(1-\frac{\beta }{6\alpha })(n_{\mathrm{s}}-1{)}^2$, the tensor tilt $n_{\mathrm{t}}\simeq -\frac{r}{8}$, and the running of the scalar tilt ${\mathfrak{a}}_{\mathrm{s}}\simeq -\frac{1}{2}(n_{\mathrm{s}}-1{)}^2$, all expressed in terms of the observed scalar tilt $n_{\mathrm{s}}$. We also provide the corresponding corrections up to N3LO, $\mathcal{O}((n_{\mathrm{s}}-1{)}^3)$.

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• Alternative gravity theories
• Inflation
• Quantum gravity