The slow-roll and quintessence inflation process of the universe in the context of the string theory-based D-BRANE KKLT potential and barrow entropy

In this thesis, the early inflationary phase of the universe is investigated within the framework of the D-brane KKLT inflation potential based on string theory and in the context of Barrow entropy, which incorporates the entropic reflections of quantum-gravitational effects. The proposed model, constructed on the basis of scalar field dynamics under the slow-roll condition, suggests that the inflationary process of the universe consists of two successive phases: the first phase corresponds to a quasi-de Sitter type expansion, while the second phase exhibits a dynamic structure characterized by quintessence-like dark energy. Barrow entropy quantitatively describes the fractal-entropic deformations that emerge at the high energy densities of the early universe. The effects of these deformations on the Friedmann equations are analyzed through modified field equations. The D-brane KKLT potential used in the study incorporates the dynamics of brane structures moving through the extra dimensions of string theory and is evaluated as a plateau-like potential compatible with observational data. Based on COBE normalization, it is shown that the model’s predictions for the scalar spectral index and tensor-to-scalar ratio are consistent with Planck data. By fixing the constant-roll parameter in accordance with observational constraints on the spectral index and the scalar-to-tensor ratio, the η-problem is resolved. The fixed value of the constant-roll parameter both addresses the η-problem and satisfies the slow-roll conditions. Furthermore, a theoretical continuity between the two phases of inflation is established through the constant-roll parameter, and the quintessence phase is shown to be interpretable within the scaling regime prior to reheating. The findings indicate that fractal entropic deformations not only leave imprints on the early universe but may also point to a dynamically active quintessence-like dark energy component influencing the evolution of the late-time universe. In this context, the study theoretically demonstrates the cosmological applicability of Barrow entropy and how string theory–based inflation models can construct a coherent cosmological framework within a quantum gravitational context.