Preheating and Reheating Constraints in Supersymmetric Braneworld Inflation

We study the evolution of the Universe at early stages, we discuss also preheating in the framework of hybrid braneworld inflation by setting conditions on the coupling constants $\lambda$ and $g$\ for effective production of $\chi$-particles. Considering the phase between the time observable CMB scales crossed the horizon and the present time, we write reheating and preheating parameters $N_{re}$, $T_{re}$ and $N_{pre}$ in terms of the scalar spectral index $n_{s}$, and prove that, unlike the reheating case, the preheating duration does not depend on the values of the equation of state $\omega ^{\ast }$. We apply the slow-roll approximation in the high energy limit to constrain the parameters of D-term hybrid potential. We show also that some inflationary parameters, in particular, the spectral index $n_{s}$ demand that the potential parameter $\alpha$ is bounded as $\alpha \geq 1$ to be consistent with $Planck$'s data, while the ratio $r$ is in agreement with observation for $\alpha \leq 1$ considering high inflationary e-folds. We also propose an investigation of the brane tension effect on the reheating temperature. Comparing our results to recent CMB measurements, we study preheating and reheating parameters $N_{re}$, $T_{re}$ and $N_{pre}$ in the Hybrid D-term inflation model in the range $0.8\leq \alpha\leq 1.1$\, and conclude that $T_{re}$ and $N_{re}$ require $\alpha \leq 1$, while for $N_{pre}$ the condition $\alpha \leq 0.9$ must be satisfied, to be compatible with $Planck$'s results.