⚛️ general relativity

Kaniadakis Holographic Dark Energy with Particle Horizon as IR Cutoff

This paper proposes a Kaniadakis entropy-based holographic dark energy model using the particle horizon as an infrared cutoff, demonstrating that the Kaniadakis deformation alone enables late-time cosmic acceleration in the noninteracting case—a feature absent in standard Einstein gravity—while interactions only slightly shift the transition redshift.

Original authors: A. Asvar, M. Mohammadi, A. Sheykhi

Published 2026-09-10

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: A. Asvar, M. Mohammadi, A. Sheykhi

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 universe is expanding, and for decades, astronomers have watched this expansion speed up. This surprising discovery, made by observing distant exploding stars, suggests that something invisible is pushing galaxies apart. Scientists call this mysterious force "dark energy." To understand it, researchers often look to a concept called the holographic principle. Imagine a three-dimensional room where all the information needed to describe what happens inside is actually stored on the two-dimensional walls surrounding it. In cosmology, this idea suggests that the amount of energy in a region of space is limited by the size of its boundary, much like how the amount of information a hard drive can hold is limited by its surface area. When scientists apply this rule to the entire universe, they try to calculate how much dark energy should exist based on the size of the visible universe. However, a specific way of measuring this size, known as the particle horizon—which represents the furthest distance light could have traveled to reach us since the beginning of time—has historically failed to explain the current acceleration. In standard theories, using this specific boundary results in a model where the universe should be slowing down, not speeding up.

A team of researchers from Shiraz University and Persian Gulf University in Iran has revisited this problem by changing the rules of the game, not the boundary itself. They asked a simple but profound question: if we keep the same measurement of the universe's size but change the underlying mathematics that describe how information and energy behave, can we finally get the universe to accelerate? They turned to a newer statistical framework called Kaniadakis statistics. This approach offers a slight modification to the standard laws of thermodynamics, introducing a small adjustment parameter that accounts for relativistic effects in a way that standard physics does not. The researchers built a model where this new statistical rule changes both the amount of dark energy calculated and the way gravity behaves on the largest scales. They did not switch to a different boundary or add a mysterious energy transfer between dark matter and dark energy; they simply applied this new mathematical lens to the existing setup.

The results were striking. By incorporating this Kaniadakis correction, the team found that the model, which previously predicted a slowing universe, now naturally transitions into an accelerating one. Using a specific value for their adjustment parameter, they calculated that the universe would have started speeding up at a redshift of approximately 0.585. This transition point aligns well with what astronomers observe in the real world. The researchers also tested what happens if dark energy and dark matter exchange energy, a common idea in other theories. They found that adding this interaction changed the timing of the acceleration by only a tiny fraction, shifting the start of the speed-up to redshifts of 0.586 or 0.589. This small difference proves that the acceleration is not caused by the energy exchange itself, but is a direct result of the new statistical rules applied to the entropy of the universe.

To ensure their findings were physically sound, the team ran several checks. They examined the stability of their model, looking at whether the theoretical fluid representing dark energy would behave reasonably or collapse into chaos. They found that while the model successfully produces acceleration, the stability of this fluid is not perfect across all times; in certain periods, the model shows signs of instability, suggesting that while the background expansion works, the full picture might need further refinement. They also compared their model to the standard cosmological model using geometric tools designed to spot differences. Their model followed a unique path that distinguished it from the standard theory, confirming that the new statistical approach creates a genuinely different cosmic history.

Perhaps the most important part of this work is what happens when the researchers turn off their new correction. When they remove the Kaniadakis adjustment, the model instantly reverts to the old, failing version where the universe does not accelerate. This confirms that the acceleration is not an accident of the math or a result of the boundary choice, but a direct consequence of the new statistical deformation. The study demonstrates that the key to explaining the universe's speed-up might not lie in finding a new boundary or a new force, but in understanding that the fundamental rules of information and entropy might be slightly different than we thought. While the model is not yet a complete replacement for all existing theories, and questions about its stability remain, it offers a compelling new path forward. It shows that by tweaking the statistical foundation of how we count the universe's possibilities, we can recover the observed acceleration without abandoning the familiar particle horizon that has long been a stumbling block for other theories.

Technical Summary: Kaniadakis Holographic Dark Energy with Particle Horizon as IR Cutoff

Problem Statement
The paper addresses the inability of standard noninteracting Holographic Dark Energy (HDE) models, when using the particle horizon as the infrared (IR) cutoff within Einstein gravity, to produce the late-time accelerated expansion of the Universe. In the standard framework, the equation of state for noninteracting particle-horizon HDE is bounded by $w_{DE} > -1/3$ in the dark-energy-dominated limit, resulting in a decelerating universe. While interactions between dark matter and dark energy can alter this behavior, the authors seek to determine if the acceleration can be generated purely by modifying the underlying entropy and gravitational sectors, while retaining the particle horizon as the IR cutoff and initially assuming no interaction.

Methodology
The authors construct a cosmological model based on Kaniadakis entropy, a one-parameter relativistic generalization of Boltzmann-Gibbs statistics. The methodology involves a consistent modification of both the holographic energy density and the Friedmann background equations:

Entropy Deformation: The Kaniadakis entropy $S_K$ is applied to the horizon area. For small deformation parameters ( $K \ll 1$ ), the entropy expands to include a leading correction term proportional to $K^2 S_{BH}^3$ .
Holographic Density: The holographic bound $\rho_{DE} L^4 \lesssim S$ is applied using the Kaniadakis entropy. This yields a modified dark energy density containing the standard inverse-square term and a leading Kaniadakis correction proportional to $L^2$ :
$\rho_{DE}^{(H)} = 3c^2 M_p^2 L^{-2} + K^2 M_p^6 L^2$
where $L$ is the IR cutoff (particle horizon $R_P$ ).
Modified Friedmann Equations: Crucially, the authors do not merely modify the energy density; they also incorporate the Kaniadakis correction into the gravitational background equations derived from horizon thermodynamics. The modified Friedmann equations include a geometric correction term dependent on a parameter $\alpha \propto K^2$ .
Autonomous System: The authors derive a set of autonomous evolution equations for the dark energy density parameter $\Omega_{DE}$ and the equation of state $w_{DE}$ . They analyze both noninteracting ( $b_2=0$ ) and interacting ( $b_2 \neq 0$ ) scenarios, where energy exchange is governed by a coupling term $Q$ .
Numerical Analysis: The system is integrated numerically from the present epoch ( $z=0$ ) to the past ( $z=5$ ) using representative parameters: $\Omega_{DE0} = 0.7$ , $c^2 = 0.64$ , and a fixed deformation parameter $K = 1.90 \times 10^{-36}$ .
Diagnostics: The model is evaluated using the deceleration parameter ( $q$ ), the adiabatic squared sound speed ( $v_s^2$ ) for stability, and statefinder variables ( $r, s$ ) to distinguish the model from the $\Lambda$ CDM paradigm.

Key Contributions and Results

Acceleration without Interaction: The primary result is that the Kaniadakis deformation enables the particle-horizon HDE model to undergo a transition from deceleration to acceleration even in the noninteracting case. For the chosen parameters, the deceleration parameter $q$ changes sign at $z \simeq 0.585$ . This demonstrates that the acceleration is a direct consequence of the entropy deformation and the associated modification of the gravitational background, rather than a change in the IR cutoff or energy transfer.
Role of Interaction: When interaction strengths of $b_2 = 0.03$ and $b_2 = 0.06$ are introduced, the transition redshift shifts only marginally to $z \simeq 0.586$ and $z \simeq 0.589$ , respectively. The authors conclude that the interaction does not generate the acceleration; rather, the acceleration is already present in the noninteracting Kaniadakis model, and interaction only produces a slight shift in the timing of the transition.
Consistency with Standard Limits: The model continuously recovers standard Einstein-gravity particle-horizon HDE in the limit $K \to 0$ . In this limit, the geometric corrections vanish, the density reduces to the standard form, and the model reverts to the known nonaccelerating behavior, validating the physical consistency of the deformation.
Stability Analysis: The analysis of the adiabatic squared sound speed ( $v_s^2$ ) reveals that while the background solution accelerates, the effective fluid exhibits instability (negative $v_s^2$ ) over substantial portions of the redshift range, with singular behavior at intermediate redshifts. This instability becomes more pronounced with increasing interaction strength. The authors caution that this indicates the background solution alone is insufficient to guarantee physical viability without a full perturbation analysis.
Statefinder Diagnostics: The statefinder trajectories ( $s-r$ plane) show that the noninteracting Kaniadakis model does not pass through the $\Lambda$ CDM fixed point $\{1, 0\}$ . Interacting trajectories may approach or cross this point at intermediate redshifts, but this represents only an instantaneous geometric agreement, not a full dynamical degeneracy with $\Lambda$ CDM.

Significance and Claims
The paper claims to provide a "clean test" of the physical role of entropy deformation in holographic cosmology. By retaining the particle horizon (which fails to produce acceleration in standard gravity) and the noninteracting assumption, the authors isolate the Kaniadakis entropy deformation as the specific mechanism responsible for generating late-time acceleration.

The authors emphasize that their result distinguishes the Kaniadakis mechanism from models where acceleration is driven primarily by dark-sector energy transfer. However, the paper remains modest regarding the model's ultimate viability. It explicitly states that the observed background acceleration is accompanied by effective-fluid instabilities (negative sound speed) and that a complete assessment requires a full perturbation analysis and confrontation with observational data (Type-Ia supernovae, BAO, CMB), which are beyond the scope of the current work. The study serves as a theoretical demonstration that generalized entropy can fundamentally alter the cosmological dynamics of HDE models even when the IR cutoff remains unchanged.

Technical Summary: Kaniadakis Holographic Dark Energy with Particle Horizon as IR Cutoff

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