⚛️ general relativity

An Interplay Between Fractional Calculus and Holographic Dark Energy

This dissertation proposes a Fractional Holographic Dark Energy (FHDE) framework by integrating Riesz fractional derivatives into black hole thermodynamics to derive a corrected entropy, which is then used to construct a new dark energy model that addresses Hubble cutoff challenges and explains late-time cosmic acceleration across various gravitational theories.

Original authors: Ayush Bidlan

Published 2026-06-23

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ayush Bidlan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Why is the Universe Speeding Up?

Imagine the universe is a giant balloon. For a long time, scientists thought this balloon was expanding at a steady pace, or perhaps even slowing down because gravity was pulling everything together. But in the late 1990s, we discovered something shocking: the balloon isn't just expanding; it's speeding up.

Something invisible is pushing the balloon apart. We call this "Dark Energy." The standard theory (called $\Lambda$ CDM) says this push comes from a constant "Cosmological Constant" (a fixed energy of empty space). However, this theory has a major problem: it doesn't explain why the push is the size it is, and recent data suggests the push might actually be changing over time, not staying constant.

This paper proposes a new idea to fix these problems. It suggests that Dark Energy isn't a simple, constant push, but a complex force that remembers its past.

The Core Idea: The Universe Has a "Memory"

To understand the author's solution, we need two main ingredients: Holography and Fractional Calculus.

1. The Hologram (The Shadow on the Wall)

Imagine you have a 3D object, like a potato. If you shine a light on it, you get a 2D shadow on the wall. The "Holographic Principle" in physics suggests that all the information inside a 3D volume (like the universe) is actually encoded on its 2D boundary (like the surface of a sphere).

In this paper, the author uses this idea to calculate how much "Dark Energy" exists. Usually, scientists assume the boundary is the "Hubble Horizon" (the distance light has traveled since the Big Bang). But in standard theories, using this specific boundary leads to a boring, unchanging result that doesn't match what we see.

2. Fractional Calculus (The "Swirly" Math)

This is the paper's secret sauce.

Normal Math (Integer Order): Think of a car driving on a straight road. If you look at its speed, you look at where it is right now compared to where it was a split second ago. It's a simple, local step.
Fractional Math (Non-Integer Order): Now, imagine the car is driving through thick mud. Its movement right now depends not just on where it was a split second ago, but on where it was a minute ago, an hour ago, and even yesterday. The mud creates friction and memory. The car "remembers" its past path.

In mathematics, this is called Fractional Calculus. It allows us to describe systems that have "memory" or "non-local" effects. The author introduces a specific type of fractional math (using something called a Riesz derivative) into the equations that describe the universe.

The "Fractional Swirl": How It Works

The author takes the standard rules for how Black Holes work (specifically, how they store information and entropy) and applies this "fractional math" to them.

The Black Hole Connection: In standard physics, a black hole's entropy (a measure of its information) is proportional to its surface area. The author calculates what happens if the math describing the black hole's interior is "fractional" (swirly, with memory).
The New Formula: This calculation changes the formula for the black hole's entropy. Instead of a simple square relationship, it becomes a power law involving a new number called $\alpha$ (the Lévy index).
- If $\alpha = 2$ , the math is normal (standard physics).
- If $1 < \alpha < 2$ , the math is "fractional" (it has memory).
The Resulting Dark Energy: When the author plugs this new "fractional entropy" back into the Holographic Dark Energy formula, something magical happens. The "Hubble Horizon" cutoff (which usually fails) suddenly starts working! It produces a Dark Energy that changes over time, exactly like the universe seems to be behaving.

What the Paper Found

The author didn't just write equations; they tested this new "Fractional Holographic Dark Energy" (FHDE) model against real-world data.

The "Memory" Parameter ( $\alpha$ ): The model depends on a number $\alpha$ . The paper finds that if $\alpha$ is close to 2, the universe behaves like the standard model. But if $\alpha$ is smaller (around 1.1 to 1.5), the model perfectly matches recent observations (like data from the DESI telescope) that suggest Dark Energy is evolving.
Reconstructing the Engine: The author tried to see what kind of "engine" could drive this universe. They tested various theoretical engines (like scalar fields, string theory concepts, and gauge fields). They found that for the model to work, these engines need to behave in a specific way that aligns with the "fractional" math.
The Future of the Universe (The "Rips"): The paper asks: Will the universe tear itself apart?
- Big Rip: A scenario where the universe tears apart in a finite time.
- Little Rip / Pseudo Rip: Slower, more gradual tearing.
- The Finding: The "fractional memory" changes the outcome. Depending on how the Dark Energy interacts with matter, the universe might avoid a catastrophic "Big Rip" or head toward a "Little Rip." The math shows that the "memory" effect can either suppress or encourage these disasters.

The "So What?"

Think of the universe as a song.

Standard Theory ( $\Lambda$ CDM): The song is a single, unchanging note played forever. It's simple, but it doesn't match the melody we are hearing in the data.
This Paper (FHDE): The song has a "reverb" or an echo. The note you hear now is influenced by the notes you heard before. This "echo" (the fractional memory) makes the song dynamic and changing.

The paper claims that by adding this "echo" to the math of the universe, we can explain why the universe is accelerating in a way that fits our new, more precise measurements. It bridges the gap between the tiny world of quantum mechanics (where things are fuzzy and non-local) and the huge world of cosmology (where the universe expands).

Summary of Claims

The Problem: Standard Dark Energy models struggle to explain recent data, especially when using the "Hubble Horizon" as a boundary.
The Solution: Introduce "Fractional Calculus" (math with memory) into the thermodynamics of black holes.
The Outcome: This creates a new type of Dark Energy (FHDE) that naturally evolves over time.
The Evidence: When tested against data, this model works well if the "memory" parameter ( $\alpha$ ) is slightly less than 2.
The Future: The model predicts different endings for the universe (Big Rip vs. Little Rip) depending on how the "memory" interacts with matter, and it suggests that the universe might be classically unstable in the far future (a technical detail about how the energy fluctuates).

The paper does not claim to have found a new particle, nor does it claim to solve the mystery of Dark Energy completely. It proposes a specific mathematical framework that makes the existing "Holographic" idea work better with current data.

Technical Summary: An Interplay Between Fractional Calculus and Holographic Dark Energy

Problem Statement
The standard $\Lambda$ CDM model, while observationally successful, faces significant theoretical challenges, most notably the fine-tuning and cosmic coincidence problems associated with the cosmological constant ( $\Lambda$ ). Holographic Dark Energy (HDE) offers a dynamical alternative motivated by the Holographic Principle (HP) and Black Hole Thermodynamics (BHT). However, standard HDE models often struggle with specific Infrared (IR) cutoffs; notably, the Hubble cutoff ( $L_{IR} = H^{-1}$ ) typically fails to produce a viable accelerating universe within the standard framework, often yielding an equation of state (EoS) parameter $w \approx 0$ rather than the required negative pressure. Furthermore, recent observational data (e.g., from DESI) suggest a time-evolving dark energy sector, motivating extensions beyond the static cosmological constant. This dissertation addresses the need for a theoretically motivated extension of HDE that incorporates non-local effects and memory, potentially resolving the limitations of the Hubble cutoff and providing a link between quantum gravity and late-time cosmology.

Methodology
The research constructs a Fractional Holographic Dark Energy (FHDE) framework by integrating Fractional Calculus (FC), specifically the Riesz fractional derivative, into the quantum geometrodynamics of black holes. The methodology proceeds through the following steps:

Fractional Quantum Mechanics (FQM) and Black Hole Thermodynamics: The study begins by applying space-fractional quantum mechanics to the canonical quantization of a Schwarzschild black hole. By replacing the standard Laplacian in the Wheeler–DeWitt (WDW) equation with a Riesz fractional derivative (characterized by the Lévy index $\alpha$ , where $1 < \alpha \leq 2$ ), a Fractional WDW (FWDW) equation is derived. This yields a fractionally corrected Bekenstein–Hawking entropy, $\tilde{S}_{BH} \propto A^{\frac{2+\alpha}{2\alpha}}$ , where $A$ is the horizon area.
Derivation of FHDE Density: Using Cohen's inequality (which relates the UV cutoff of a local quantum field theory to the mass of a black hole to avoid collapse) and the modified fractional entropy, a new dark energy density is derived:
$\rho_{DE}^{(fr)} = 3c^2 M_{pl}^2 L_{IR}^{\frac{2-3\alpha}{\alpha}}$
where $L_{IR}$ is the IR cutoff. The standard HDE density is recovered in the limit $\alpha \to 2$ .
Cosmological Evolution and Reconstruction: The framework adopts the Hubble horizon as the IR cutoff ( $L_{IR} = H^{-1}$ ). The cosmological evolution is analyzed by deriving the EoS parameter ( $w_{DE}$ ), deceleration parameter ( $q$ ), and density parameters ( $\Omega_{DE}, \Omega_{DM}$ ) as functions of redshift $z$ and the fractional parameter $\alpha$ .
Effective Field Reconstruction: To provide a physical interpretation, the FHDE model is reconstructed using various effective field theories (EFTs). The study establishes correspondences between the FHDE EoS and the dynamics of:
- Spin-0 fields: Quintessence, K-essence, Dilaton, DBI-essence, and Tachyon fields.
- Spin-1 fields: Yang–Mills Condensate (YMC).
  For each, the kinetic terms and potentials are reconstructed as functions of $z$ and $\alpha$ .
Extension to Modified Gravity and Singularities: The framework is extended to modified gravity theories (Brans–Dicke, Post-GW170817 Horndeski, Extended Proca–Nuevo, and Dvali–Gabadadze–Porrati braneworlds) to derive EoS and deceleration parameters. Additionally, the occurrence of future "rip" singularities (Big Rip, Little Rip, Pseudo Rip) is investigated using both the Granda–Oliveros (GO) and Hubble cutoffs, employing ansatz-based approaches for the Hubble parameter.

Key Contributions and Results

Rehabilitation of the Hubble Cutoff: A primary result is that the FHDE model successfully generates a viable late-time accelerating universe even when using the Hubble cutoff, a choice that typically fails in standard HDE. The fractional parameter $\alpha$ introduces non-local memory effects that allow for dynamical evolution.
Observational Viability: Numerical analysis indicates that smaller values of the Lévy index (e.g., $\alpha \approx 1.1$ to $1.5$) yield cosmological evolution consistent with current observational constraints (including DESI data). Specifically, the transition from deceleration to acceleration occurs at redshifts ( $z_T \approx 0.38$ for $\alpha=1.1$ ) compatible with observations, and the EoS parameter remains within the quintessence regime ( $-1 < w < -1/3$ ) for a significant portion of cosmic history, approaching $w \to -1$ in the far future.
Field Reconstruction Patterns: The reconstruction of effective fields reveals distinct behaviors:
- For Quintessence, K-essence, DBI-essence, Tachyon, and YMC, smaller $\alpha$ values generally produce smoother kinetic evolution and potentials that decay asymptotically, supporting a stable dark energy phase.
- The Dilaton model exhibits an exception, where larger $\alpha$ values are preferred for the kinetic term to ensure slow evolution, though the potential behavior favors smaller $\alpha$ .
- Across most models, the EoS parameter asymptotically approaches the $\Lambda$ CDM limit ( $w = -1$ ) as $z \to -1$ , particularly for smaller $\alpha$ .
Singularity Analysis:
- Granda–Oliveros Cutoff: The analysis shows that a Big Rip singularity is a generic outcome for fractional values of $\alpha$ in the range $1 < \alpha \leq 2$ , driven by linear interaction terms. Pseudo Rip scenarios require $\alpha > 2$ , which is outside the physically admissible domain of the model. Little Rip scenarios are not generic and require highly specific, unmotivated IR cutoffs.
- Hubble Cutoff: Under the Hubble cutoff, the occurrence of singularities depends heavily on the interaction term. Linear interactions tend to drive the system toward a Big Rip (finite-time divergence) for small $\alpha$ . Non-linear interactions, however, allow for genuine Little Rip and Pseudo Rip scenarios across the full range of $\alpha$ , where the EoS approaches $-1$ from below without finite-time divergence.
- Stability: The squared sound speed ( $v_s^2$ ) is found to be negative in the late-time limit for rip scenarios, indicating classical instability (gradient instability) in the perturbative regime, a common feature in phantom-dominated models.
Modified Gravity Extensions: The thesis derives explicit expressions for the EoS and deceleration parameters in Brans–Dicke, Horndeski, EPN, and DGP theories. These expressions explicitly depend on $\alpha$ , demonstrating how fractional non-locality can be embedded into diverse gravitational frameworks.

Significance and Claims
The paper claims that FHDE represents a theoretically motivated extension of HDE that bridges the gap between non-local quantum gravity effects (via fractional calculus and black hole thermodynamics) and late-time cosmic acceleration. Its significance lies in:

Conceptual Coherence: It provides a coherent chain of reasoning from microscopic fractional quantum dynamics to macroscopic cosmology, avoiding arbitrary parameterizations. The modification to the entropy-area relation is derived from the FWDW equation rather than postulated phenomenologically.
Viability of the Hubble Cutoff: It demonstrates that the Hubble cutoff, often dismissed in standard HDE, becomes a viable option when the underlying entropy is corrected by fractional effects. This suggests that the failure of the Hubble cutoff in standard models may reflect an incomplete entropy prescription rather than a fundamental flaw in the cutoff itself.
Phenomenological Flexibility: The model offers a single parameter ( $\alpha$ ) to control the strength of non-local/memory effects, allowing the model to interpolate between standard HDE ( $\alpha=2$ ) and more dynamical regimes ( $\alpha < 2$ ). This flexibility allows the model to accommodate recent observational hints of evolving dark energy.
Distinction from Other Models: While structurally similar to Barrow and Tsallis HDE (due to power-law entropy corrections), the paper emphasizes that FHDE is distinct in its physical motivation, arising specifically from fractional quantum mechanics and the Riesz derivative, rather than non-extensive thermodynamics or horizon deformations.

The work concludes that FHDE is a promising, though not yet complete, framework. It successfully addresses the dynamical nature of dark energy and the Hubble cutoff issue but acknowledges remaining challenges, such as classical instability in the far future and the need for a quantum cosmological treatment to resolve potential singularities. The paper positions FHDE as a competitive candidate for dynamical dark energy that warrants further observational constraints and theoretical development.