Reconstructing inflation in Einstein-Gauss-Bonnet gravity in light of ACT data

TL;DR

The work tackles reconstructing the inflationary background in Einstein-Gauss-Bonnet gravity from observable attractors $n_s(N)$ and $r(N)$ under slow-roll. It derives general integral relations for $V(N)$ and the GB coupling $\xi(N)$ and prescribes a path to obtain $V(\phi)$ and $\xi(\phi)$ via $Q\,dN=d\phi$. A concrete example with $n_s=1-\gamma/N$ and $r=1/[N(1+\beta N)^p]$ yields analytic forms for $V(N)$ and $\xi(N)$, and, after solving for $N(\phi)$, explicit $V(\phi)$ and $\xi(\phi)$ for $p=1,2,3$, with ACT-compatible $\gamma=\tfrac{3}{2}$. The results show $V(\phi)\neq 1/\xi(\phi)$, emphasize the GB coupling's impact on inflationary predictions, and constrain constants such as $\alpha$ and $\beta$ from the power spectrum and $r$ data, revealing two inflationary branches and a path for exploring other observational parametrizations.

Abstract

During the inflationary epoch, we investigate the reconstruction of the background variables within the framework of Einstein-Gauss-Bonnet gravity, considering the scalar spectral index $n_s(N)$ and the tensor-to-scalar ratio $r(N)$, where $N$ denotes the number of $e-$folds. Under a general formalism, we determine the effective potential and the coupling function associated with the Gauss-Bonnet term as functions of the cosmological parameters $n_s(N)$ and $r(N)$, respectively. To implement the reconstruction methodology for the background variables, we study an example in which the attractors for the index $n_s$ and the ratio $r$ are in agreement with Atacama Cosmology Telescope (ACT) data. In this context, explicit expressions for the effective potential $V(φ)$ and the coupling parameter $ξ(φ)$ are reconstructed. Moreover, the reconstruction based on observational parameters shows that $V(φ)\not\propto 1/ξ(φ)$, in contrast to the assumption adopted in the literature for the study of the evolution of the universe in Einstein-Gauss-Bonnet gravity.

Figures (1)

• Figure 1: The upper-left panel shows the number of $e$-folds $N$ as a function of the scalar field $\varphi=\sqrt{2\beta}\phi+C_1$. The upper-right panel displays the reconstructed scalar potential $V(\varphi)$ in terms of $\varphi$. The lower panel shows the reconstructed coupling function associated with the Gauss-Bonnet term as a function of the field $\varphi$. In all panels we have considered the special case in which the parameter $p=1$. In addition, we have used the constant $C=0$ together with two different values of the parameter $\gamma$ related to the scalar spectral index; $\gamma=3/2$ and $\gamma=2$, respectively.