More accurate slow-roll approximations for inflation in scalar-tensor theories

TL;DR

The paper tackles the accuracy limitations of slow-roll approximations in inflationary models with a nonminimally coupled scalar field in the Jordan frame. It derives new slow-roll expressions for key observables by formulating and solving in terms of the scalar field and by introducing three new approximations (I, II, III) that improve upon the standard and previously known methods. Through the induced gravity term with a quartic potential, including Higgs-driven inflation, the authors show that approximations I and III provide significantly better estimates for the tensor-to-scalar ratio $r$ and the scalar amplitude $A_s$ than the traditional approaches, while II may fail in certain parameter regimes. The results demonstrate that careful handling of the nonminimal coupling yields more reliable inflationary predictions and extend the slow-roll framework to a broader class of gravity theories, with potential applications to generalized gravity models in the Einstein frame or beyond.

Abstract

We propose new versions of the slow-roll approximation for inflationary models with nonminimally coupled scalar fields. We derive more precise expressions for the standard slow-roll parameters as functions of the scalar field. To verify the accuracy of the proposed approximations, we consider inflationary models with the induced gravity term and the fourth-order monomial potential. For specific values of the model parameters, this model is the well-known Higgs-driven inflationary model. We investigate the inflationary dynamics in the Jordan frame and come to the conclusion that the proposed versions of the slow-roll approximation are not only more accurate at the end of inflation, but also give essentially more precise estimations for the tensor-to-scalar ratio $r$ and the amplitude of scalar perturbations $A_s$.

Figures (10)

• Figure 1: The functions $\phi(N)$ (left), $H^2(N)$ (center), and $\varepsilon_1(N)$ (right) for the model with the potential $V(\phi)=0.0125\phi^4$ and the coupling function $F(\phi)=1+17367\phi^2$. The curves are obtained by the numerical integration of the system (\ref{['PhiSYSN']}) with the initial data $\phi(N_0)=0.077$, $\chi(N_0)=0$
• Figure 2: The functions $\varepsilon_1(\phi)$ (left) and $\zeta_1(\phi)$ (right) for the model with the potential $V(\phi)=0.0125\phi^4$ and the coupling function $F(\phi)=1+17367\phi^2$. The black lines are the result of the numerical integration of the system (\ref{['PhiSYSN']}), the blue curves are obtained in the known approximation, the red curves are obtained in the approximation I, the magenta curve is obtained in the approximation II and the green curves are obtained in the approximation III.
• Figure 3: The function $r(\phi)$ for the model with the potential $V(\phi)=0.0125\phi^4$ and the coupling function $F(\phi)=1+17367\phi^2$. The black lines are the result of the numerical integration of the system (\ref{['PhiSYSN']}), the blue curves are obtained in the known approximation, the red curves are obtained in the approximation I and the green curves are obtained in the approximation III.
• Figure 4: The functions $A_s(\phi)$ (the left and center pictures) and $n_s(\phi)$ (the right picture) for the model with the potential $V(\phi)=0.0125\phi^4$ and the coupling function $F(\phi)=1+17367\phi^2$. The black lines are the result of the numerical integration of the system (\ref{['PhiSYSN']}), the blue curves are obtained in the known approximation, the red curves are obtained in the approximation I, and the green curves are obtained in the approximation III.
• Figure 5: The functions $\varepsilon_1(\phi)$ (left) and $\zeta_1(\phi)$ (right) for the model with the potential $V(\phi)=5\cdot 10^{-11}\phi^4$ and the coupling function $F(\phi)=1+\phi^2$. The black lines are the result of the numerical integration of the system (\ref{['PhiSYSN']}) with the initial data $\phi(N_0)=10$, $\chi(N_0)=0$ at $N_0=-27$, the blue curves are obtained in the known approximation, the red curves are obtained in the approximation I, and the green curves are obtained in the approximation III.