An Interplay Between Fractional Calculus and Holographic Dark Energy

This dissertation aims to put forth a systematic construction of a fractional-calculus extension of holographic dark energy (HDE). We show that linking late-time cosmic acceleration to non-local or memory effects encoded in a fractional (Riesz) derivative within black hole thermodynamics produces deviations from standard HDE and can address some challenges of the Hubble cutoff. In particular, a Riesz fractional spatial derivative is introduced into the Hamiltonian constraint of a Schwarzschild black hole in quantum geometrodynamics, leading to a Fractional Wheeler--DeWitt equation whose solutions yield fractionally corrected thermodynamic quantities, notably fractional Bekenstein--Hawking entropy governed by the L'evy index alpha, with 1< alpha at most2. Using this entropy with Cohen's inequality, a new dark energy density is constructed, defining the Fractional Holographic Dark Energy (FHDE) framework. The cosmological implications of FHDE are then investigated. Within the Hubble cutoff, its late-time evolution is analysed through cosmological observables, and the model is reconstructed using effective field descriptions with spin- 0 and spin- 1 candidates, allowing kinetic and potential terms to be extracted as functions of redshift z and alpha. The framework is then extended to BD, DGP, EPN, and Horndeski theories to derive the equation-of-state and deceleration parameters in terms of z and alpha. In addition, the fate of the Universe is studied through late-time singularities, namely the big, little, and pseudo-rip, within the Granda--Oliveros FHDE setting. In short, this dissertation proposes FHDE as a theoretically motivated extension of HDE, bridging non-locality in quantum gravity with the late-time dynamics of the Universe, and offering a route toward understanding cosmic acceleration beyond CDM.