Tsallis Holographic Dark Energy

Updated 3 February 2026

Tsallis holographic dark energy is an extension of the standard HDE framework that replaces additive Bekenstein–Hawking entropy with a nonadditive power-law form.
The model modifies the relationship between the IR cutoff and vacuum energy density, leading to distinctive cosmic acceleration and a rich phase-space structure.
Stability and observational consistency are achieved for specific parameter choices (δ > 2, mild dark energy–dark matter interactions), offering a viable alternative to ΛCDM.

Tsallis holographic dark energy (THDE) is a cosmological model that extends the standard holographic @@@@1@@@@ energy paradigm by replacing the additive Bekenstein–Hawking entropy with the nonadditive Tsallis entropy. Motivated by nonextensive statistical mechanics and quantum gravity considerations, this proposal modifies the relationship between infrared (IR) cutoff scales and the energy density of the vacuum, resulting in significant differences for cosmic acceleration, dynamical stability, and the phase structure of the universe. The model is characterized by a nonadditivity parameter δ (or equivalently, γ in some notations), and can incorporate interactions between dark energy and dark matter, as well as generalizations to modified gravity, fractal spacetimes, and higher-dimensional brane scenarios.

1. Tsallis Entropy, Holographic Principle, and the Energy Density Law

Tsallis entropy replaces the additive area law with a power-law form: $S_\delta = \gamma\,A^\delta$ where $A$ is the horizon area, $\gamma$ a constant, and $\delta>0$ the nonadditivity parameter. For $\delta=1$ , the Bekenstein–Hawking scenario is recovered.

By saturating a holographic energy bound (the condition that the energy in a region of size $L$ does not exceed the corresponding black hole mass), the dark energy density is found to scale as: $\rho_D = B\,L^{2\delta-4}$ where $B$ is a (model-dependent) constant. For $\delta=1$ , this reduces to the standard holographic dark energy formula ( $\rho_D\propto L^{-2}$ ); for $\delta=2$ , dark energy becomes a constant, corresponding to ΛCDM. The specific choice of IR cutoff (such as the Hubble radius $L=H^{-1}$ , the future event horizon, or generalized scales) strongly impacts the cosmological evolution (Tavayef et al., 2018, Saridakis et al., 2018, Huang et al., 2022).

2. Cosmological Dynamics and Key Background Equations

In a spatially flat Friedmann–Robertson–Walker universe with matter, radiation, and THDE (possibly interacting), the background equations are: $H^2 = \frac{8\pi G}{3} (\rho_r + \rho_m + \rho_D)$

$\dot\rho_r + 4H\rho_r = 0, \quad \dot\rho_m + 3H\rho_m = Q, \quad \dot\rho_D + 3H(1+w_D)\rho_D = -Q$

with the phenomenological interaction term $Q = H(\alpha \rho_m + \beta \rho_D)$ , where α, β ≥ 0 (Huang et al., 2022, Mamon et al., 2020).

The dark energy equation of state is derived as: $w_D = \frac{[(\delta-2)(\Omega_m+\Omega_D)+5-4\delta]\Omega_D - \sigma}{3[(\delta-2)\Omega_D + 1]\Omega_D}$ where $\Omega_i = \frac{8\pi G\rho_i}{3H^2}$ and $\sigma = \alpha\Omega_m + \beta\Omega_D$ .

The dimensionless dynamical system for ( $\Omega_m,\, \Omega_D$ ) (with Ω_r given by the Friedmann constraint) yields autonomous evolution equations suitable for phase-space and fixed-point analysis.

3. Classical Stability and Sound Speed Analysis

Classical stability of the model is diagnosed via the squared adiabatic sound speed: $v_s^2 = \frac{dp_D}{d\rho_D} = \frac{\dot w_D + w_D \Gamma}{\Gamma} , \quad \Gamma = \frac{\dot\rho_D}{\rho_D}$ A sufficient condition for $v_s^2 > 0$ is met when $\delta > 2$ , $\alpha \geq 0$ , $\beta \geq 0$ , and $0 < \Omega_D < 1$ throughout cosmic history. Instabilities manifest for lower δ or for some alternative choices of interaction or cutoff, as examined for flat, fractal, brane, and higher-dimensional models (Huang et al., 2022, Mamon et al., 2020, Ghaffari et al., 2018, Mamon, 2020).

4. Dynamical Attractors and Cosmological Solutions

The THDE dynamical system exhibits a rich fixed-point structure:

P₁: (0,0) – radiation-dominated, unstable
P₂: (1,0) – matter-dominated, typically a saddle
P₃: $(\beta/(3-\alpha+\beta), (3-\alpha)/(3-\alpha+\beta))$ – dark-energy dominated, de Sitter with $q=-1$

For the special case $\beta=0$ , $P_3 \to (0,1)$ , reproducing the ΛCDM late-time behavior. The eigenvalues at $P_3$ confirm stability (attractor) when $0 \le \alpha < 3,\, \beta \ge 0,\, \delta > 1$ (Huang et al., 2022).

5. Equation of State Evolution and Cosmological Observables

The equation of state for THDE evolves from $w_D \to 0$ at early times ( $\Omega_D \to 0$ ), transits through the acceleration epoch, and settles to a constant at the attractor. In the limit $\beta \to 0$ , $w_D \to -1$ , matching ΛCDM.

The effective total EoS is $w_{\text{eff}} = w_D \Omega_D + (\Omega_r/3)$ . The deceleration parameter is: $q = -1 - \frac{\dot H}{H^2} = \frac{2 - (1+\alpha)\Omega_m - (\beta+2\delta)\Omega_D}{2(1 + (\delta-2)\Omega_D)}$ In viable parameter regions, the model predicts a transition from deceleration ( $q>0$ ) at high redshift to $q \approx -1$ at late times. The cosmic age formula,

$t_0 \approx \frac{2}{4 - (\Omega_m^0 + (1-3w_D^0)\Omega_D^0)}\, \frac{1}{H_0}$

gives ages in the range $t_0 \approx 14-17$ Gyr for $\delta \gtrsim 2.1, \alpha \sim 0.5-0.7, \beta \lesssim 0.1$ , and current density fractions. This prediction is compatible with Planck and stellar chronometry (Huang et al., 2022).

6. Comparison with Standard HDE, ΛCDM, and Alternatives

Standard HDE is recovered for $\delta=2$ and $\alpha=\beta=0$ , leading to $\rho_D \propto H^2$ , a model susceptible to IR problems and classical instabilities. Tsallis generalization ( $\delta \neq 2$ ) interpolates between quintessence, cosmological constant, and phantom-like behaviors depending on δ, interaction parameters, and the chosen IR cutoff.

THDE, particularly for $\delta > 2$ with mild interaction, achieves classical stability and a stable accelerated attractor, resolving issues endemic in standard HDE. At late times, with β=0, the model matches ΛCDM exactly (Huang et al., 2022, Mamon et al., 2020, Tavayef et al., 2018). Extensions to fractal cosmology, scalar–tensor, f(R), brane and higher-dimensional scenarios demonstrate the flexibility and physical robustness of the framework, though classical stability is sensitive to specific choices of δ and coupling (Mamon, 2020, Saridakis et al., 2018, Ens et al., 2020, Ghaffari et al., 2018, Ghaffari et al., 2018, Saha et al., 2020).

7. Observational and Phenomenological Implications

THDE models can fit existing cosmological expansion data (SNeIa, H(z), BAO, CMB, GRB) when the Tsallis parameter is modestly larger than unity ( $\delta \sim 1-2.1$ ), the present DE fraction is near $\Omega^0_D \sim 0.69$ , with a small positive matter–dark energy interaction ( $\alpha,\,\beta \ll 1$ ). For these values, the transition redshift to cosmic acceleration and the current deceleration parameter are consistent with Planck (z_t ~ 0.7, q_0 ~ -0.55), and the model can accommodate the measured cosmic age (Saridakis et al., 2018, Huang et al., 2022, Astashenok et al., 2024). In general, stability, realistic background history, and avoidance of unphysical values ( $\Omega_D > 1$ ) require careful calibration of all parameters, with the best-behaved models typically in the class of positive, mildly superarea Tsallis exponents and weak DE–DM interaction.