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

TL;DR

This work analyzes inflation driven by the geometric action $R+\alpha R^2-\beta W^2$ with $\beta\ll\alpha$, treating the Weyl-squared term as a self-consistent effective correction via reduction of order. The authors derive a modified background dynamics that yields a quasi-de Sitter phase and compute the full set of cosmological perturbations, showing that both scalar and tensor modes acquire time-dependent kinetic amplitudes and speeds of sound related by $c_t/c_s \approx 1+{\beta}/{(6\alpha)}$. Using a Green’s-function approach for the quasi-Bunch-Davies vacuum, they obtain inflationary observables up to N3LO, including a precise relation $r \simeq 3\left(1-{\beta}/{(6\alpha)}\right)(n_s-1)^2$ and $n_t \simeq -r/8$, all expressed in terms of the observed scalar tilt $n_s$. The framework yields a robust, frame-aware pathway to connect measurements of $n_s$ and the amplitude $\mathcal{A}_s$ to the fundamental couplings $\alpha$ and $\beta$, enabling stringent tests of Weyl corrections to gravity with future CMB and gravitational-wave observations, and it discusses implications for reheating and potential preinflationary imprints.

Abstract

Starobinsky's $R+αR^2$ 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 $R+αR^2 - βW^2$, where the coupling constant $α$ 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 $β$, at the linear order in $β/α$. Cosmological perturbations are found to be described by an effective action with a nontrivial speed of sound $c_{\textrm{s}}$ for scalar and $c_{\textrm{t}}$ for tensor modes, satisfying the relation $c_{\textrm{t}}/c_{\textrm{s}} \simeq 1+ \fracβ{6\, α}$ 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 $r \simeq 3(1-\fracβ{6α})(n_\textrm{s}-1)^2$, the tensor tilt $n_{\textrm{t}}\simeq-\frac{r}{8}$ and the running of the scalar tilt $\mathfrak{a}_{\textrm{s}}\simeq-\frac{1}{2} (n_{\textrm{s}} - 1)^2$, all expressed in terms of the observed scalar tilt $n_{\textrm{s}}$. We also provide the corresponding corrections up to N3LO, $\mathcal{O}((n_{\textrm{s}} - 1)^3)$.

Figures (3)

• Figure 1: A concrete example based on the action \ref{['eq:Action-Oscillator-Acceleration']} which illustrates the three necessary conditions for a self-consistent solution from reduction of order. The numerical solution refers to \ref{['eq:App_EoM_toymodel1']}. Panel a): All three conditions are satisfied and the exact physical solution \ref{['eq:Reduction-Physical-Exact']} matches almost perfectly the effective solution \ref{['eq:Reduction-Effective-Solution']}. Panel b): Condition $C_3$ is not satisfied, as $\beta \omega_0^2\approx 0.2$, which goes slightly beyond the regime of validity of the lowest order truncation. Panel c): Condition $C_2$ is not satisfied, as initial conditions are chosen arbitrarily, and both frequencies $\Lambda_{+}$ and $\Lambda_{-}$ play a role in the dynamics, introducing a nonanalytic dependence on $\beta$. Panel d): Condition $C_1$ is not satisfied, as $\beta<0$ corresponds to an instability, and the characteristic runaway solution quickly becomes dominant.
• Figure 2: Inflationary solution obtained from the modified Friedmann equation \ref{['eq:Friedmann_Modified']}. In the geometric framework, a quasi--de Sitter inflationary phase (solid blue) arises naturally from the nontrivial dynamics of the Starobinsky action, followed by an oscillatory period of reheating (solid tan). The red dot indicates the end of the inflationary phase, defined as $\ddot{a}(t_{\mathrm{end}})=0$ or equivalently by $\epsilon_{1H}(t_{\mathrm{end}})=1$, while the gray star denotes a typical freezing time used to define a pivot scale. At leading order, we have $\dot{H}\approx-1/36\alpha$, as shown in the green dashed line. The dotted gray line indicates the characteristic scale at the end of inflation, $H_{\mathrm{end}}^2 \approx 1/36\,\alpha$.
• Figure 3: Marginalized joint 68% and 95% C.L. regions for $n_s$ and $r$ at $k = 0.05$ Mpc$^{-1}$ as reported by the Atacama Cosmology Telescope (ACT) Collaboration in their ACT Data Release 6 ACT:2025tim, where constraints from Planck, BICEP2/Keck and CMB Lensing+Baryonic Acoustic Oscillations are also included. The results of Table \ref{['Tab:Results_Summaryv2']} for Starobinsky inflation up to N3LO are shown in the solid green line, while the red shaded region indicates the possible values on the plane for a range $0\leq \beta/\alpha \leq 1$. For $\beta>0$, the predicted values of $r$ are lower than the ones for Starobinsky inflation. As argued in Zharov:2025evb, larger values of $N_\ast$ can be taken into account in reheating constraints, allowing the model to remain consistent with the latest observational data.