New modified cosmology from a new generalized entropy

1. Problem Statement

The standard cosmological model ($\Lambda$CDM) relies on General Relativity (GR) and the Boltzmann-Gibbs-Shannon (BGS) entropy. While successful, it faces challenges such as the $H_0$ tension (discrepancy in Hubble constant measurements) and the theoretical nature of Dark Energy (DE).

• Thermodynamic Limitations: Standard BGS entropy assumes the "separability" axiom (K4), which holds for non-interacting systems but fails for complex, interacting systems with long-range correlations or memory effects.
• Cosmological Motivation: Extending thermodynamics to include generalized entropies offers a way to modify cosmological dynamics without altering the fundamental laws of gravity (keeping GR intact). Previous attempts using single-parameter generalized entropies (like Tsallis or Barrow) have been explored, but a more flexible framework is needed to capture richer phenomenological behaviors, such as phantom-divide crossing and dynamic equation-of-state evolution.

2. Methodology

The authors employ the gravity-thermodynamics conjecture, which posits that the Friedmann equations governing the Universe's expansion can be derived from the first law of thermodynamics applied to the apparent horizon of the Universe.

Step 1: Construction of a New Generalized Entropy ($S_{\delta,\epsilon}$)

• Starting from the general definition of entropy $S_f[p] = \sum f(p_i)$, the authors violate the separability axiom (K4) to account for interacting systems.
• They propose a generalized microstate scaling $W_{\delta,\epsilon}$ that combines two distinct scaling behaviors.
• This leads to a new entropic functional composed of two terms:
  $$S_{\delta,\epsilon} = \eta_\delta (\log W)^\delta + \eta_\epsilon (\log W)^\epsilon$$
• In the context of holographic thermodynamics (systems with boundaries), this translates to a generalized area-law scaling with two exponents ($\delta$ and $\epsilon$):
  $$S_{\delta,\epsilon} = \gamma_\delta A^\delta + \gamma_\epsilon A^\epsilon$$
  where $A$ is the horizon area. This expression generalizes both the standard Bekenstein-Hawking entropy (recovered when $\delta=\epsilon=1$) and previous models like Tsallis-Cirto ($\delta=\epsilon$) or Barrow entropy.

Step 2: Derivation of Modified Friedmann Equations

• The authors apply the first law of thermodynamics ($-dQ = T_h dS + W dV$) to the apparent horizon of a flat FRW universe.
• They retain the standard definitions for horizon temperature ($T_h$) and work density ($W$) but substitute the standard Bekenstein-Hawking entropy ($S \propto A$) with the new $S_{\delta,\epsilon}$.
• This procedure yields modified Friedmann equations containing additional terms dependent on the Hubble parameter $H$ and the exponents $\delta, \epsilon$.

Step 3: Cosmological Analysis

• The additional terms in the Friedmann equations are interpreted as an effective Dark Energy sector ($\rho_{DE}, p_{DE}$) of entropic origin.
• The authors derive analytical expressions for the dark energy density parameter $\Omega_{DE}$ and the equation-of-state (EoS) parameter $w_{DE}$ as functions of redshift $z$.
• They analyze two scenarios:
  1. General Case: Including a cosmological constant ($\Lambda$) as an integration constant.
  2. Radical Case ($\Lambda = 0$): Where the effective DE arises solely from the entropic modifications.

3. Key Contributions

• Novel Entropic Functional: Introduction of a two-parameter generalized entropy ($S_{\delta,\epsilon}$) derived from first principles (violation of separability) rather than being postulated ad-hoc at the area-law level.
• Unified Framework: The model naturally encompasses standard $\Lambda$CDM, Tsallis cosmology, and Barrow cosmology as specific limiting cases.
• Dynamic Dark Energy: The model generates a dynamic DE sector that can transition between different regimes (quintessence, phantom, and cosmological constant) depending on the values of $\delta$ and $\epsilon$.
• Phantom-Divide Crossing: The model explicitly demonstrates the ability to cross the phantom divide ($w_{DE} = -1$), a feature difficult to achieve in standard scalar field models without instabilities.

4. Results

Cosmological Evolution:

• History: The Universe follows the standard thermal history, transitioning from a matter-dominated era to a dark-energy-dominated era.
• Asymptotic Behavior: Regardless of the initial values of $\delta$ and $\epsilon$, the effective dark energy stabilizes at $w_{DE} \to -1$ in the far future ($z \to -1$), driving the Universe toward a de Sitter phase. This ensures the model avoids catastrophic fates like the "Big Rip."

Equation of State ($w_{DE}$):

• $\delta, \epsilon > 1$: The model exhibits phantom-like behavior ($w_{DE} < -1$) at present times, potentially evolving from quintessence ($w_{DE} > -1$) to phantom. This phantom behavior can help alleviate the $H_0$ tension by enhancing late-time acceleration.
• $\delta, \epsilon < 1$: The model exhibits quintessence-like behavior ($w_{DE} > -1$).
• Intermediate Regime: When one exponent is $<1$ and the other $>1$, the corrections nearly cancel, mimicking $\Lambda$CDM.

Observational Consistency:

• Deceleration Parameter: The predicted transition redshift ($z_{tr} \approx 0.67$) and present deceleration parameter ($q_0 \approx -0.55$) align well with combined observational data (Hubble parameter, Supernovae, BAO).
• Supernovae Type Ia (SN Ia): The theoretical distance modulus $\mu(z)$ derived from the model shows agreement with SN Ia data, with some parameter combinations ($\delta, \epsilon < 1$) showing slightly better fit at high redshifts than standard $\Lambda$CDM.
• $\Lambda = 0$ Case: Even without an explicit cosmological constant, the model can reproduce the observed matter-DE sequence and late-time acceleration by tuning $\delta$ and $\epsilon$, though it does not strictly recover $\Lambda$CDM as a limit.

5. Significance

• Theoretical Robustness: The work provides a microscopic foundation for modified cosmology rooted in statistical mechanics, moving beyond phenomenological area-law modifications.
• Phenomenological Richness: The two-parameter structure offers greater flexibility than single-parameter models (like Barrow or Tsallis), allowing for a wider range of cosmological behaviors, including the crucial phantom-divide crossing.
• Hubble Tension: The ability of the model to naturally produce a phantom-like EoS at present times offers a potential theoretical mechanism to resolve the $H_0$ tension without invoking new physics in the early universe.
• Future Outlook: The authors suggest that a full statistical analysis using CMB, BAO, and Hubble data, along with information criteria (AIC/BIC), is necessary to constrain the parameters $\delta$ and $\epsilon$ precisely. The model also opens avenues for investigating the connection between this entropic gravity and other extended gravity theories.

In summary, Luciano and Saridakis present a compelling modified cosmological scenario where the complexity of the Universe's microstates (via a generalized entropy) naturally generates an effective dark energy sector, reproducing standard cosmology in limits while offering richer dynamics to address current observational tensions.