Slow-roll approximations for Gauss-Bonnet inflation revisited

TL;DR

This work probes the validity of slow-roll approximations for Gauss-Bonnet inflation with growing scalar-GB couplings, contrasting them with prior decaying-coupling scenarios. By formulating the full GB-inflation dynamics and three competing slow-roll schemes, the authors show that the commonly used standard slow-roll approximation frequently remains the most accurate, while the more complex approximations offer limited or even negative gains due to term cancellations and stability issues. Numerical experiments with both quadratic and asymptotically flat potentials reveal regime-dependent behavior: in many cases the old approximation outperforms the new ones, though the new schemes can outperform in isolated parameter regions. The study highlights the importance of numerical cross-validation when applying slow-roll techniques to modified gravity inflation models and motivates exploring broader coupling forms.

Abstract

In our paper we consider the validity of slow-roll approximations for Gauss-Bonnet inflation introduced in [1]. In contrast to the cited paper where the coupling function before the Gauss-Bonnet term have been chosen as a decaying function of the scalar field, here we consider growing coupling functions. We have found that while in [1] new slow-roll approximations work considerably better, now they do not increase the precision. Moreover, we identify some cases where more involved approximations work worse than the standard one. Corresponding explanations of such a situation are given.

Figures (8)

• Figure 1: The plot depicts the evolution of the inflationary parameters $r$ (left), $A_s$ (middle), $n_s$ (right) at the onset of inflation, including the region around 65 e-folds. The black line represents the numerical solution of system \ref{['eq:numerical-1']}, blue --- classical slow-roll approximation \ref{['eq:classical-approx']}, red --- new Approximation I \ref{['eq:first-approx']}, green --- new Approximation II \ref{['eq:second-approx']}. The two plots in the first row illustrate the case described by the set $V=V_0\phi^2$; $\xi=\phi^2/(\phi_0^2+(a\phi)^2)$; $V_0=1.5\cdot 10^{-11}$; $\phi_0=0.003$; $a=0$.
• Figure 2: On left panel: The evolution of the slow-roll parameter $\varepsilon_1(\phi)$ at the end of inflation for the Example 1. The black line represents the numerical solution of system (\ref{['eq:numerical-1']}), blue --- classical slow-roll approximation \ref{['eq:classical-approx']}, red --- new Approximation I (\ref{['eq:first-approx']}), green --- new Approximation II (\ref{['eq:second-approx']}). For better visibility of the curves, the $\phi$-axis display range was reduced from $[0; 20]$ to $[1; 4]$. On right panel: The evolution of the slow-roll parameter $\varepsilon_1(\phi)$ at the beginning of inflation for the Example 1. For comparative purposes, the plot indicates different numbers of e-foldings: $N = 65$ (triangles) and $N = 60$ (squares). For visual clarity, the blue curve representing the standard slow-roll approximation is shown as a dashed curve. The inflaton field $\phi$ is given in units of the Planck mass $M_{\text{Pl}}$.
• Figure 3: On left panel: The evolution of the slow-roll parameter $\delta_1(\phi)$ at the end of inflation for the Example 1. On right panel: The plot of the tensor-to-scalar ratio $r$ (\ref{['eq:r']}) at the start of inflation, including the region around 65 e-foldings for the Example 1.
• Figure 4: The plot of the inflation parameter $A_s(\phi)$ (\ref{['eq:a_s']}) (left) and $n_s(\phi)$ (\ref{['eq:n_s']}) (right) at the beginning of inflation near 65 e-foldings fot the Example 1. As the plots show, the old approximation performs better than the new ones.
• Figure 5: The numerical evolution of the different terms of dynamical equation (\ref{['eq:2.4a']}) as a functions of the number of e-foldings $N$ for the Example 1, where $-12UH^2$ is plotted in gray, ${\dot\phi}^2$ --- cyan, $2V$ --- orange, $24\xi_{,\phi}\dot\phi H^3$ --- magenta. In the left plot we see main terms, in the right plot --- corrections.