Reconstructing inflation in a generalized Rastall theory of gravity

TL;DR

The paper develops a reconstruction framework for inflation in standard and generalized Rastall gravity by expressing the scalar spectral index $n_s$ as a function of the number of $e$-folds $N$ and by treating the Rastall parameter as constant or dynamic. It derives the effective potential $V(N)$ and the corresponding $V(\phi)$, showing that $V(N)$ matches the GR result while Rastall effects modify $V(\phi)$; in the generalized case, an analytic form for $V(N)$ is obtained for a slowly varying $\lambda_{\text{Ras}}$, with GR recovered in the limit $\beta\to0$. Observational constraints from Planck are employed to bound the integration constants and model parameters, and a stability analysis demonstrates the absence of ghosts and gradient instabilities through positivity conditions on $Q_s$, $c_s^2$, and $Q_T$. The work lays groundwork for future MCMC analyses to tighten parameter constraints in both standard and generalized Rastall scenarios, potentially illuminating deviations from GR in the inflationary epoch.

Abstract

We investigate the reconstruction of standard and generalized Rastall gravity inflationary models, using the scalar spectral index and the Rastall parameter expressed as functions of the number of $e-$folds $N$. Within a general formalism, we derive the effective potential in terms of the relevant cosmological parameters and the Rastall parameter for these gravity frameworks. As a specific example, we analyze the attractor $n_s(N) - 1 \propto N^{-1}$, first by considering constant values of the Rastall parameter to reconstruct the inflationary stage in standard Rastall gravity, and then by assuming a linear dependence on the number of $e-$folds $N$ to reconstruct the inflationary model in generalized Rastall gravity. Thus, the reconstruction of the potential $V(φ)$ is obtained for both standard and generalized Rastall gravity inflationary models. In both frameworks, we constrain key parameters of the reconstructed models during inflation using the latest observational data from Planck.

Figures (4)

• Figure 1: The left panel illustrates the number of $e$-folds $N$ as a function of the shifted scalar field $\varphi=\phi- C$. The right panel shows the reconstructed effective potential as a function of the scalar field $\varphi$. In both cases, we have considered three distinct values of the Rastall parameter $\lambda_\text{Ras}$; 1.00 (blue curve), 1.05 (orange curve), and 1.10 (green curve).
• Figure 2: Dependence of the number of $e$-folds (upper panel), the reconstructed effective potential (central panel), and the Rastall parameter (lower panel) on the scalar field during the inflationary epoch. In these plots, we have considered three distinct values of the parameter $\beta$, together with the corresponding constraints on the parameters $A$ and $B$, as provided by Eqs.(\ref{['A2']}) and (\ref{['B2']}), respectively.
• Figure 3: The plot illustrates the comparison between the numerical solution of Eq.(\ref{['ddd']}) together with Eq.(\ref{['VN2']})(dashed line) and the approximate analytical solution of the reconstructed potential $V(\phi)$ provided by Eq.(\ref{['VV2']}) under the condition $\beta\ll 1$ (solid line). Here we have considered three different values of the parameter $\beta$.
• Figure 4: The upper left panel shows the evolution of the coefficient $Q_s$ related to the scalar perturbations in terms of the scalar field $\varphi=\phi-C$, in standard Rastall gravity for three different constant values of the parameter $\lambda_\text{Ras}$. The upper right panel presents the evolution of the same coefficient $Q_s$ but in generalized Rastall gravity as a function of $d(\phi)$ associated to scalar field, for three different values of the parameter $\beta$. The lower panel displays the evolution of the speed of sound in generalized Rastall gravity in terms of the function $d(\phi)$ for different values of the parameter $\beta$.