Holographic dark energy in modified Kaniadakis cosmology

This paper proposes a Kaniadakis holographic dark energy model within modified Kaniadakis cosmology using the Hubble radius as an IR cutoff, demonstrating that it can explain current cosmic acceleration without interaction, mimics a cosmological constant in a dark energy-dominated era, and exhibits phantom behavior and distinct statefinder evolution when interaction with dark matter is considered.

The Gist

Imagine the universe as a giant, expanding balloon. For a long time, scientists thought this balloon was slowing down as it grew, like a car running out of gas. But about 25 years ago, we discovered something shocking: the balloon isn't just growing; it's speeding up. Something invisible is pushing it faster and faster. Scientists call this invisible pusher "Dark Energy."

This paper by Sheykhi, Asvar, and Ebrahimi proposes a new way to understand this invisible pusher. They combine two big ideas: Holographic Dark Energy and Kaniadakis Cosmology.

Here is the breakdown of their idea, using simple analogies:

1. The "Holographic" Rule (The Energy Budget)

Think of the universe like a giant hologram. In this theory, the amount of energy (Dark Energy) the universe can hold depends on the size of its "screen" (the horizon).

• The Old Rule: In standard physics, if you calculate the energy based on the size of the universe, you get a specific number. But when scientists tried to use the most obvious size (the Hubble radius, or how far light has traveled since the Big Bang), the math said the universe should be slowing down, not speeding up. It was like a recipe that said "add sugar" but resulted in a sour cake.
• The Fix: The authors say, "Let's change the recipe." They use a modified version of the "entropy" (a measure of disorder or information) called Kaniadakis entropy. Think of this as a new, slightly different ruler for measuring the universe's information.

2. The "Kaniadakis" Twist (The New Ruler)

Standard physics uses a "flat" ruler. Kaniadakis entropy is like a ruler that has a tiny, subtle curve to it (controlled by a parameter called $K$).

• The authors argue that because the universe is so vast and relativistic, this "curved ruler" is more accurate than the flat one.
• When they apply this curved ruler to the energy budget, something magical happens: The math suddenly works. Even without any extra help, the universe starts to speed up naturally.

3. The Big Discovery: No "Hand-Holding" Needed

In many previous models, to get the universe to speed up, scientists had to assume Dark Energy and Dark Matter (another invisible substance) were constantly trading energy back and forth, like two people passing a ball to keep a game going.

• The Paper's Claim: This new model is special because it doesn't need that "ball passing." Even if Dark Energy and Dark Matter are completely separate and don't talk to each other, the universe still accelerates. The "curved ruler" (Kaniadakis entropy) does all the heavy lifting.
• The Cosmological Constant: The authors also found that in a universe dominated only by this Dark Energy, it behaves exactly like a "Cosmological Constant" (a steady, unchanging push). This suggests that the mysterious "Lambda" ($\Lambda$) in our equations might actually come from this new way of measuring entropy.

4. What Happens When They Do Interact?

The authors also looked at what happens if Dark Energy and Dark Matter do interact (pass the ball).

• The "Phantom" Crossing: They found that with interaction, the "push" from Dark Energy can become so strong that it crosses a dangerous threshold called the "phantom divide." Imagine a car accelerating so hard it breaks the sound barrier; this model allows the universe to do something similar, entering a "phantom" state where the push gets even stronger.
• Stability Issues: They checked if this new universe is stable. They found that if the interaction is too strong, the universe becomes "wobbly" (unstable), like a house of cards.

5. The "Statefinder" Test (The Fingerprint)

To prove this model is different from the standard "Lambda-CDM" model (the current gold standard of cosmology), they used a diagnostic tool called the Statefinder.

• Think of the standard model as having a fingerprint of {1, 0}.
• The authors show that their new model has a different fingerprint. As you look further back in time (higher redshift), the fingerprint moves away from the standard one, proving this is a distinct theory. However, in the very far future, their model's fingerprint eventually drifts back to match the standard one.

Summary

The paper suggests that the reason our universe is speeding up might not be because of a mysterious new force or a constant push, but because our old way of measuring the universe's "information" (entropy) was slightly off. By using a more advanced, "curved" ruler (Kaniadakis entropy), the math naturally explains the acceleration without needing to force the pieces together. It offers a potential explanation for why the "Cosmological Constant" exists in the first place.

Technical Summary

Technical Summary: Holographic Dark Energy in Modified Kaniadakis Cosmology

Problem Statement
The standard $\Lambda$CDM model faces significant challenges, including the fine-tuning and coincidence problems, as well as observational tensions regarding the Hubble constant ($H_0$) and the amplitude of matter fluctuations ($\sigma_8$). While Holographic Dark Energy (HDE) offers a dynamical alternative based on the holographic principle, previous studies utilizing modified entropies (such as Tsallis or Barrow) typically modified only the HDE energy density expression while retaining the standard Friedmann field equations. This approach contradicts the thermodynamics-gravity correspondence, which posits that modifications to the horizon entropy must induce modifications to the gravitational field equations. Specifically, for Kaniadakis entropy, previous works did not consistently apply the resulting modified Friedmann equations to the cosmological evolution.

Methodology
The authors propose a Kaniadakis Holographic Dark Energy (KHDE) model within the framework of modified Kaniadakis cosmology. The methodology involves:

1. Entropy Modification: Utilizing the Kaniadakis entropy ($S_K$), a one-parameter generalization of Boltzmann-Gibbs-Shannon entropy characterized by the parameter $K$. For small $K$, the entropy is expanded as $S_K \approx S_{BH} + \frac{K^2}{6}S_{BH}^3$.
2. Field Equations: Applying the thermodynamics-gravity correspondence to derive modified Friedmann equations that incorporate the Kaniadakis correction term. The energy density of KHDE is derived as $\rho_{DE} = 3c^2 M_p^2 H^2 + 3\alpha M_p^2 H^{-2}$, where $\alpha \propto K^2$.
3. IR Cutoff: The Hubble radius ($L = H^{-1}$) is chosen as the infrared (IR) cutoff.
4. Scenarios: The model is analyzed under three distinct scenarios:
   • A Dark Energy (DE) dominated universe.
   • A universe containing both DE and Dark Matter (DM) evolving separately (non-interacting).
   • A universe with an interaction between DE and DM, modeled by an energy transfer term $Q = 3b^2 H(1+r)\rho_{DE}$.
5. Diagnostics: The authors analyze the equation of state (EoS) parameters ($w_{DE}$ and $w_{tot}$), the deceleration parameter ($q$), the squared speed of sound ($v_s^2$) for stability, and the statefinder parameters ($r, s$) to distinguish the model from $\Lambda$CDM.

Key Contributions and Results

• DE Dominated Universe: In a DE-dominated epoch, the model yields a constant Hubble parameter ($H = \text{const}$) and an EoS parameter $w_{DE} = -1$. The authors demonstrate that the KHDE mimics a cosmological constant ($\Lambda$). Crucially, they derive a theoretical expression for $\Lambda$ in terms of the Kaniadakis parameter $K$ ($\Lambda \sim K$), suggesting a thermodynamic origin for the cosmological constant within this framework.
• Non-Interacting Case (Novel Result): In standard HDE cosmology, choosing the Hubble radius as the IR cutoff without interaction fails to produce accelerated expansion ($w_{DE} = 0$). However, the authors find that in modified Kaniadakis cosmology, the inclusion of the entropy correction term allows the model to explain the current accelerated expansion ($w_{DE} < -1/3$) even without any interaction between DE and DM. This is a significant departure from standard HDE results.
• Interacting Case: When interaction is introduced, the total EoS parameter ($w_{tot}$) and the DE EoS ($w_{DE}$) decrease as the coupling constant $b^2$ increases. The model allows $w_{DE}$ to cross the phantom divide ($w_{DE} < -1$) at the present epoch, consistent with certain observational constraints.
• Cosmic Evolution: The transition from a decelerated to an accelerated phase occurs at a redshift $z \approx 0.4$ for the non-interacting case, which is later than the $\Lambda$CDM prediction ($z \approx 0.64$). Increasing the interaction parameter $b^2$ shifts this transition to higher redshifts.
• Stability Analysis: The analysis of the squared speed of sound ($v_s^2$) indicates that the model suffers from instability in the presence of interaction ($v_s^2 < 0$ for interacting KHDE), although it remains stable in the non-interacting case at the present epoch.
• Statefinder Diagnostic: The statefinder diagram ($r, s$) shows that the model is distinct from the $\Lambda$CDM point $\{1, 0\}$ at the present time. The trajectory moves away from $\{1, 0\}$ as the interaction parameter increases, but the model converges to the $\Lambda$CDM point in the far future.

Significance and Claims
The paper claims that its primary significance lies in the consistent application of the thermodynamics-gravity conjecture to Kaniadakis entropy. By modifying both the energy density and the background Friedmann equations, the authors demonstrate that:

1. The Kaniadakis correction provides a theoretical mechanism for the cosmological constant ($\Lambda$) to emerge from entropy considerations.
2. The model successfully resolves the limitation of standard HDE with a Hubble cutoff, achieving accelerated expansion without requiring an interaction between dark sectors.
3. The model offers rich cosmological phenomenology, including phantom-divide crossing and distinct evolutionary paths compared to $\Lambda$CDM, while remaining consistent with current observational bounds on the EoS parameter.

The authors conclude that the KHDE model in modified Kaniadakis cosmology provides a viable alternative to $\Lambda$CDM that addresses the coincidence problem and offers a potential thermodynamic origin for dark energy, though it faces stability challenges when interactions are present.