Inflationary dynamics of non-minimally coupled $f(R)$ matter-curvature theories

TL;DR

This paper analyzes inflation in $f(R)$ gravity with a non-minimal coupling between matter and curvature, considering both positive and negative corrections to the minimal coupling. A robust numerical method evolves the metric and inflaton beyond slow-roll, revealing that negative-type (−) corrections support a stable de Sitter attractor, while positive-type (+) corrections are unstable during slow-roll. By confronting the resulting scalar and tensor observables with Planck, BK18, and ACT data, the study finds that the non-minimal coupling scale must be at least around $M\gtrsim 10^{13}$ GeV, limiting the effects of the coupling to near-perturbative levels. The analysis shows that the inflaton matter Lagrangian choice does not affect pivot-scale observables, and across several potentials the data still favor GR, though certain parameter regions in the NMC models remain viable and distinguishable with future measurements. Overall, the work tightly bounds matter-curvature couplings in the early universe and demonstrates a concrete, testable framework for non-minimally coupled inflation, while ruling out the graceful exit rescue for old inflation within this class of theories.

Abstract

This study examines how inflationary dynamics are affected by $f(R)$ theories with a non-minimal coupling between matter and curvature. Both positive and negative corrections to the minimal coupling of General Relativity are considered, and a robust numerical method is developed that evolves the metric and the inflaton field in this modified theory beyond slow-roll. Through a stability analysis, we find that positive models are inherently unstable during slow-roll, whereas negative ones can accommodate a stable attractor de Sitter solution. Using the amplitude of the scalar power spectrum from the latest data releases, we constrain the scale of the non-minimal coupling to be above $10^{13}$ GeV. In light of the 2018 Planck, BICEP/Keck and the recent Atacama Cosmology Telescope data for the scalar spectral index and tensor-to-scalar ratio, strong constraints on the coupling strength force the effects of these modified theories to be, at most, slightly above the perturbative level. Furthermore, we determine that the choice of the perfect fluid matter Lagrangian does not impact the inflationary observables at the pivot scale. Finally, we present the predicted observables for different inflationary potentials and show that even though classical gravity is still preferred by the data, there are areas of the parameter space that are viable for non-minimally coupled inflationary models.

Figures (8)

• Figure 1: The slow-roll solutions for $\rho( H^2)$ under the $\dot H=0$ assumption in the $f_2^+$(left) and $f_2^-$ (right) models. The GR solution $\rho=3H^2$ is shown for comparison.
• Figure 2: The maximum values of the inflationary expansion rate $H_{\text{inf}}$ and inflaton density $\rho_{\text{inf}}$ in the $(\pm)$-type models as functions of the power-law exponent $n$. The maximum density is unconstrained in $(+)$-type models and the corresponding maximum Hubble parameter is achieved asymptotically for large density values.
• Figure 3: Eigenvalues of the dynamical system (\ref{['eq:system']}) for the slow-roll solution $\dot H=0$ in the $(+)$-type (left) and $(-)$-type (right) power-law NMC models up to the corresponding maximum values of $H_{\text{inf}}$.
• Figure 4: The ratio between the two spectral indices in $(\pm)$-type NMC models as a function of the density.
• Figure 5: Predictions for the scalar-to-tensor ratio and the scalar spectral index with a quartic symmetry-breaking potential, where $\gamma$ decreases from left to right in the range $[0.05,0.001]$, and the boundaries of the shaded region correspond to inflation lasting $N_*=50-60$$e$-folds. The $n=2$ NMC model matches GR, but both $n=2,3$ are over $2\sigma$ away from the observational measurements.