RHDE Model with Hubble Horizon

• The paper introduces a novel framework where RHDE employs Rényi entropy and a Hubble horizon cutoff, resulting in a modified energy density scaling that differs from standard models.
• Phase space analysis reveals stable attractors that depend on both linear and non-linear interaction terms between dark energy and dark matter, with regimes spanning quintessence to phantom behavior.
• Numerical and analytical studies demonstrate that the model robustly accounts for late-time cosmic acceleration, offering a flexible alternative to traditional dark energy paradigms.

The RHDE (Rényi Holographic Dark Energy) model with Hubble horizon as the infrared (IR) cut-off is a theoretical framework that modifies standard holographic dark energy via the Rényi entropy formalism, introducing generalized thermodynamic effects at cosmological scales. By leveraging the Hubble horizon ($L = 1/H$) as the IR cut-off, the model links dark energy dynamics directly to the cosmic expansion rate, yielding a robust description of late-time acceleration, accommodating various interaction scenarios, and admitting stable (sometimes phantom-like) attractors. This model has recently been analyzed using phase space techniques, which illuminate its dynamical structure, its physical viability in both interacting and non-interacting regimes, and its implications for the evolution of the universe.

1. Construction of the Rényi HDE Model with Hubble Horizon

The RHDE model builds upon the holographic principle and the generalization of black hole entropy from the standard Bekenstein–Hawking area law to Rényi entropy. For a spatially flat FLRW Universe, the foundational entropy formula is

$S_R = \frac{1}{\delta} \ln(1 + \pi \delta L^2)$

where $L$ is the IR cut-off and $\delta$ parametrizes the deviation from extensivity. In the limit $\delta \to 0$, the standard area law $S = A/4$ is recovered.

Applying thermodynamic and holographic reasoning, the energy density of RHDE with $L = 1/H$ is

$$\rho_d = \frac{3 d^2}{8\pi} \frac{H^4}{H^2 + \pi \delta}$$

with $d$ a dimensionless constant and $H$ the Hubble parameter. This departs from the conventional HDE's $H^2$-scaling and introduces novel dynamical features, especially at late times.

Formulating the cosmological equations with normalized dimensionless density parameters: $$x = \frac{\rho_m}{3H^2}, ~~~ y = \frac{\rho_d}{3H^2}, ~~~ x + y = 1$$ allows recasting the system into an autonomous dynamical system suitable for phase space analysis (Das et al., 29 Jul 2025).

2. Phase Space Analysis and Critical Points

Rewriting the cosmological evolution equations in terms of $x$ and $y$ yields: $$\frac{dx}{dN} = 3xy\omega_d, ~~~ \frac{dy}{dN} = -3y(1-y)\omega_d$$ where $N=\ln a$ is the e-folding number and $\omega_d$ is the dynamical equation-of-state parameter. For the Hubble parameter evolution, a key function

$\lambda(x, y) = \frac{\dot{H}}{H^2 + \pi \delta}$

is introduced, with two primary ansatzes explored: linear, $\lambda = \alpha x + \beta y$, and exponential, $\lambda = \exp(\alpha x + \beta y)$.

Critical (fixed) points are identified by setting $dx/dN = 0$ and $dy/dN = 0$, leading, for instance, to full dark energy domination ($y=1$) or matter domination ($x=1$). The dynamical character (node, saddle, spiral) and stability (hyperbolic/non-hyperbolic) are determined by the eigenvalues of the Jacobian at these points:

• In the linear case, the fixed point "A" at $(x=0, y=1)$ is a hyperbolic stable node under $\beta > -3/2$, corresponding to a dark energy-dominated attractor.
• Exponential forms similarly yield late-time attractors, often with phantom behavior.
• Interacting scenarios (see below) generate new critical points related to the chosen interaction term form.

3. Interacting Scenarios: Linear and Non-Linear

The RHDE model admits both linear and non-linear interaction terms $Q$ between dark energy and dark matter, altering the continuity equations: $$\dot{\rho}_m + 3H \rho_m = Q, ~~~ \dot{\rho}_d + 3H(1+\omega_d)\rho_d = -Q$$ Explored forms include:

• $Q = 3H(\rho_m + \rho_d)$
• $Q = 3H\rho_d$
• $Q = 3H\rho_m$
• $Q = 3H\frac{\rho_d}{\rho_m + \rho_d}$ (nonlinear interaction)

Phase space structures in each case reflect the effect of energy exchange. For example, with $Q \propto \rho_m + \rho_d$, some critical points correspond to matter domination, while others (e.g., “D₁” and “D₂”) represent dark energy-dominated accelerated expansion for appropriate parameter choices. Nonlinear Q scenarios can yield unique attractors with quintessence-like or phantom-like evolution, depending on parameters $\alpha$, $\beta$ in the $\lambda(x, y)$ function.

4. Stability and Fluid Description

Stability of these fixed points is analyzed by linearizing the dynamical system around the critical points and computing the Jacobian's eigenvalues:

• Stable attractors require all eigenvalues to have negative real parts (hyperbolic node).
• For non-hyperbolic points (zero real part eigenvalues), further (nonlinear) stability analysis is required; some such points act as transition states.
• The fluid description of RHDE reveals that the equation-of-state parameter evolves as

$$\omega_d = \frac{1 + \frac{2}{3} \lambda(x, y)}{1 - 2y}$$

for linear $\lambda$. Numerical phase trajectories show that RHDE can interpolate between quintessence ($-1 < \omega_d < -1/3$) and phantom ($\omega_d < -1$) regimes depending on interaction and parameter choices.

Notably, RHDE models typically evolve toward $y=1$ (full dark energy domination) with $\omega_d$ evolving smoothly across the phantom divide or stabilizing in the quintessence regime.

5. Implications for Cosmic Evolution and Observations

The phase space analysis demonstrates that the RHDE model with Hubble horizon as IR cutoff generically leads to late-time acceleration: stable attractor solutions exist for a wide range of parameter values and interaction forms, with the system dynamically evolving toward a dark energy-dominated phase. The qualitative nature (phantom or quintessence) of the attractor regime can be tuned by model parameters.

Key mathematical expressions central to cosmological diagnostics include:

• RHDE energy density:

$$\rho_d = \frac{3 d^2}{8\pi} \frac{H^4}{H^2 + \pi \delta}$$

• Friedmann equation:

$3H^2 = \rho_m + \rho_d$

• Evolution equations for density parameters:

$x + y = 1, ~~~ \frac{dx}{dN} = 3xy \omega_d$

• EOS parameter for RHDE (with linear $\lambda$):

$$\omega_d = \frac{1 + (2/3)(\alpha x + \beta y)}{1 - 2y}$$

6. Cosmological Significance and Robustness

The RHDE model with Hubble horizon cutoff accommodates both non-interacting and a wide spectrum of interacting dark sector scenarios while maintaining observationally viable cosmic histories. In the non-interacting case, the system can move from matter-dominated to (phantom or quintessence) dark energy-dominated epochs. Introduction of dark sector interactions allows adjustment of the detailed expansion history, potentially alleviating cosmological tensions or mimicking various observational signatures beyond standard $\Lambda$CDM.

The robustness of late-time acceleration under this model is highlighted by the prevalence of stable attractors for a broad parameter range, the flexible fluid description of dark energy, and the capacity to realize (via interaction or entropy parameter tuning) both canonical and non-canonical (phantom) cosmic acceleration. Statefinder diagnostics and dynamical system theory confirm the physical consistency of RHDE scenarios, further strengthening the model’s viability as an alternative to standard dark energy paradigms (Das et al., 29 Jul 2025).