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.
