Preheating constraints in $\alpha$-attractor inflation and Gravitational Waves production

We propose a scenario where preheating occurs for a specific duration that is parametrized by an e-folds number $N_{pre}$, our results suggest a direct correlation between the preheating duration and the density of gravitational waves (GWs) produced during this phase. Moreover, we investigate the consequences of the inflationary parameters on the $\alpha$-attractor E model in the small $\alpha$ limits. In this framework, we perform investigations on the preheating parameters involving the number of e-folds $N_{pre}$, and the temperature of reheating $T_{re}$, then we show that the parameter $n$ associated with the E model of $\alpha$-attractor inflation has a negligible effect on the preheating duration, and we demonstrate that gravitational wave generation during preheating satisfies the restrictions from Planck's recent data.