Slow-rolling and quintessence inflation with the D-brane KKLT potential in Barrow entropy

In this study, we investigate the brane inflation scenario by modifying the field equations of motion based on Barrow's entropy proposal, while also considering the constant-roll condition. We address the -problem by fixing the constant-roll parameter at , consistent with observational constraints on the spectral index and the scalar-to-tensor ratio $\textit{r}$. The fixed value of the constant-roll parameter ensures that the -problem is addressed while satisfying the slow-roll condition. A correlation is found between the parameter of the D-brane KKLT potential and the parameter present in Barrow entropy. Another parameter of the D-brane KKLT potential is determined using the COBE formalism, which describes the energy scale of inflation. The quintessential behavior of the D-brane KKLT potential is examined within a scaling regime. The parameters established during the slow-rolling phase significantly influence the emergence of the quintessence phase. This scenario demonstrates the presence of a quintessence-type fluid following the slow-rolling phase, as approximated by the effective equation of state (EoS) parameter derived from the scaling solutions of the model. The early inflation of the universe is shown to emerge from two phases via the D-brane KKLT potential.