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arXiv · 2606.22431

An Interplay Between Fractional Calculus and Holographic Dark Energy

Ayush Bidlan
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Abstract

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 \(1<\alpha\leq2\). 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 \(\Lambda\)CDM.

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Ayush Bidlan. 2026-06-21. An Interplay Between Fractional Calculus and Holographic Dark Energy. https://arxiv.org/abs/2606.22431

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