Inflation in $F(R)$ gravity models revisited after ACT

The $F(R)$ gravity models of inflation are revisited in light of the recent observations of cosmic microwave background radiation by Atacama Cosmology Telescope (ACT) and DESI Collaboration. A detailed study of the evolution equations in the Jordan frame is given and a new description of the slow-roll approximation in the $F(R)$-gravity-based models of inflation is proposed. It is found that all those models of inflation are significantly constrained by demanding a higher (than the Planck Telescope value) cosmological tilt $n_s$ of scalar perturbations and a positive running index $α_s$ favored by ACT. It is not difficult to meet the ACT constraints on the scalar tilt $n_s$ by modifying the existing models of inflation, but simultaneously demanding a positive running $α_s$ would rule out many of them. Using the proposed slow-roll approximation in the Jordan frame, we provide a new modification of the Starobinsky inflation model in the framework of $F(R)$ gravity, which satisfies all ACT constraints. An extension of our ACT-consistent inflation model to the unified $F(R)$-gravity description of Starobinsky-like inflation and production of primordial black holes on a smaller scale is also proposed.

💡 Research Summary

The paper revisits inflationary models based on (F(R)) gravity in light of the latest cosmic microwave background (CMB) measurements from the Atacama Cosmology Telescope (ACT) and the DESI collaboration. While the classic Starobinsky model (F(R)=R+R^{2}/(6m^{2})) has long been compatible with Planck data, ACT prefers a slightly higher scalar spectral index (n_{s}) (and a positive running (\alpha_{s})) than Planck, thereby tightening constraints on any viable (F(R)) inflationary scenario.

The authors begin by deriving the field equations for a generic (F(R)) action in the Jordan frame, emphasizing the necessity of the stability conditions (F_{R}>0) and (F_{RR}>0). They then introduce the e‑fold number (N) as the evolution variable and rewrite the Friedmann‑type equations in terms of (H(N)) and the Ricci scalar (R(N)). A central technical development is the definition of a new quantity (Y\equiv M_{\rm Pl}\sqrt{F_{R}},H), which they prove coincides with the Einstein‑frame Hubble parameter (H_{E}) when expressed in Jordan‑frame time. This identification allows them to formulate a slow‑roll (SR) approximation directly in the Jordan frame, introducing SR parameters (\epsilon_{1},\zeta_{1},\zeta_{2},\dots) that capture the dynamics of (H) and (F(R)).

Through a systematic expansion they derive exact relations between the Jordan‑frame SR parameters and the usual Einstein‑frame SR parameters (\epsilon^{(E)}) and (\eta^{(E)}). Keeping terms up to second order, they obtain expressions such as
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