Holographic Dark-Energy Densities

• Holographic dark-energy densities are theoretical constructs that connect quantum-gravity entropy bounds to cosmic acceleration through boundary information.
• They are realized via various infrared cutoff choices and modified entropy-area prescriptions, underpinning models that unify horizon thermodynamics and effective field theory.
• Recent variants such as Tsallis, Barrow, and Polynomial HDE offer improved observational fits by incorporating dark sector interactions and modified gravity corrections.

Holographic dark-energy densities are theoretical constructs that generalize quantum-gravity–motivated energy bounds to cosmological settings. They link the dark-energy content of the universe to boundary information, as formulated by the holographic principle, and are realized through various choices of infrared (IR) cutoffs and entropy-area prescriptions. These densities underpin a broad class of cosmological models unifying horizon thermodynamics, effective field theory, and late-time cosmic acceleration.

1. Fundamental Principle and Standard Formulation

The holographic principle constrains the vacuum energy in a region of typical size $L$ to not exceed the mass of a black hole with the same size. In quantum field theory with Planck mass $M_p$, IR cutoff $L$, and UV cutoff $\Lambda$, this yields $L^3\Lambda^4\lesssim L M_p^2$, leading to the standard holographic dark-energy (HDE) density \cite{(Wang et al., 2016, Zapata et al., 29 Jul 2025)}: $\rho_{\rm HDE} = 3c^2 M_p^2 L^{-2}$ where $c$ is a dimensionless parameter. In most models, the IR cutoff $L$ is chosen as the future event horizon,

$L = a\int_a^\infty \frac{da'}{H(a') a'^2}$

leading to a dynamical DE density that can support late-time cosmic acceleration, in contrast to alternatives such as $L=H^{-1}$ or the particle horizon, which fail to produce acceleration in standard HDE \cite{(Wang et al., 2016)}.

2. Variants and Generalizations: Modified Entropy and Cutoff Choices

Recent extensions of holographic DE densities follow from adopting non-standard black-hole entropy–area relations or alternative cosmological cutoffs:

• Tsallis HDE. Generalizes $S\sim A$ to $S_\delta\propto A^\delta$, yielding $\rho_{DE}=B L^{2\delta-4}$. For $\delta=1$, this reduces to standard HDE; $\delta>1$ (non-extensive entropy) produces phenomenologically viable models, with distinctive EoS features such as quintessence, phantom, or crossing behavior depending on $\delta$ \cite{(Saridakis et al., 2018)}.
• Barrow HDE. Quantum-gravitational corrections deform the area law to $S_B\propto(A/A_0)^{1+\Delta/2}$, leading to $\rho_{DE}\propto L^{\Delta-2}$ and allowing a fractal deformation parameter $\Delta\in[0,1]$. The resulting EoS generically interpolates between quintessence and phantom \cite{(Saridakis, 2020)}.
• Fractional HDE. Fractional calculus applied to horizon entropy yields $S_h\propto A^{(2+\alpha)/(2\alpha)}$. The FHDE density becomes $\rho_{\rm FHDE}\propto H^{(3\alpha-2)/\alpha}$, continuously connecting to the standard $H^2$ scaling as $\alpha\to2$. This allows the Hubble cutoff ($L=H^{-1}$) to yield accelerating solutions, unattainable in ordinary HDE \cite{(Trivedi et al., 2024)}.
• Polynomial HDE. Inspired by quantum gravity corrections, polynomial expansions in $H$ such as $\rho_{DE} = \alpha H^2 + \beta H^4 + \gamma H^6$ capture additional running effects, leading to phase phenomena and transient phantom behaviour closely tracking $\Lambda$CDM at low redshift \cite{(Cruz et al., 29 Oct 2025)}.
• Modified Ricci and Granda–Oliveros Cutoffs. Take $L^{-2}\propto \alpha H^2 + \beta\dot H$, so that $\rho_{DE}=3(\alpha H^2+\beta\dot H)$ \cite{(Oliveros et al., 2014)}. The Ricci scalar $R=6(\dot H+2H^2)$ underpins these models, producing a density scaling as a fixed fraction $\sim0.25–0.27$ of $R$ and fitting cosmological acceleration without fine-tuned cosmological constants \cite{(Forte et al., 2012)}.

Model	Entropy/Scale Law	HDE Density Formula	Distinctive Parameter
Standard HDE	$S\sim A$	$3c^2 M_p^2 L^{-2}$	$c$
Tsallis HDE	$S_\delta\sim A^{\delta}$	$B L^{2\delta-4}$	$\delta$
Barrow HDE	$S_B\sim A^{1+\Delta/2}$	$C L^{\Delta-2}$	$\Delta$
Fractional HDE	$S\sim A^{(2+\alpha)/(2\alpha)}$	$3c^2H^{(3\alpha-2)/\alpha}$	$\alpha$
Polynomial HDE	N/A	$\alpha H^2+\beta H^4+\gamma H^6$	$\alpha$, $\beta$, $\gamma$

3. Interacting Holographic Dark Energy

A major development consists in coupling the HDE sector to dark matter via non-gravitational interaction terms. In EFT or scalar-tensor frameworks, the coupling arises either phenomenologically ($Q\propto H \rho_i$ or its variants) or from scalar field–matter interactions in the action \cite{(Farajollahi et al., 2011, Rozas-Fernández et al., 2010)}. The continuity equations for matter and HDE become

$$\begin{align*} \dot\rho_m + 3H\rho_m &= +Q \ \dot\rho_{DE} + 3H(1+w)\rho_{DE} &= -Q \end{align*}$$

where $Q>0$ transfers energy from DE to DM and can alleviate the coincidence problem.

• Chameleon–tachyon scenarios: Here, the action is $$S = \int d^4x \sqrt{-g}[\frac12M_p^2R - V(\phi)\sqrt{1-\partial_\mu\phi\partial^\mu\phi} + f(\phi)\mathcal{L}_m]$$, and variation induces an interaction $Q \propto \rho_m \dot f(\phi)$, with field-dependent coupling, yielding cosmic histories in agreement with data for suitable parameter ranges \cite{(Farajollahi et al., 2011)}.
• Ricci HDE with interaction: Models of the form $\rho_{DE} = 3\alpha(\dot H+2H^2)$ including $Q=3bH\rho_i$ ($\rho_i$ being $\rho_{DE}$, $\rho_m$, or total), offer analytic solutions for $H(z)$ and $w(z)$ and are strongly favoured by BAO+SNe+CMB data relative to noninteracting Ricci-type models \cite{(Fu et al., 2011)}.
• Nonlinear interactions: Generalizations such as $Q \propto H \rho_{DE}^2/(\rho_m+\rho_{DE})$ (arising in “new holographic” schemes) can flatten the energy density ratio curve $\rho_m/\rho_{DE}$ over an extended period, further mitigating the coincidence issue and generating stable models, as evinced by positive adiabatic sound speed $c_s^2>0$ across cosmic history \cite{(Oliveros et al., 2014)}.

4. Dynamical System Analysis and Phenomenology

The dynamical behaviour, critical points, and cosmic attractors are accessed by recasting evolution equations as autonomous systems in $\Omega_{DE}$ or related quantities \cite{(Mahata et al., 2015)}. For standard HDE with future event horizon cutoff, the EoS is $$w_{\rm HDE}=-\frac13-\frac{2}{3c}\sqrt{\Omega_{DE}}$$ and the evolution

$$\frac{d\Omega_{DE}}{dz} = -\frac{\Omega_{DE}(1-\Omega_{DE})}{1+z}\left[1+\frac{2}{c}\sqrt{\Omega_{DE}}\right]$$

which yields a quintessence-like EoS for $c>1$, phantom for $c<1$. A line of non-hyperbolic (saddle-type) fixed points with $w<-1$ exists when Q is included, partially alleviating the coincidence problem but precluding hyperbolic attractors at observationally viable $\Omega_{DE}\sim0.7$ unless extra ingredients are added \cite{(Mahata et al., 2015)}.

In all phenomenologically acceptable models, matching observed $H(z)$, transition redshift $z_{\rm acc}\sim 0.6–0.9$, and $w(z=0)\sim-1$ is achievable for specific parameter sets: e.g., Ricci DE with $\alpha=4/3$, $\beta<0.1$ gives $\rho_{DE}\sim 0.26R$, $w_s\simeq-0.84$, $z_{\rm acc}\simeq0.89$ \cite{(Forte et al., 2012, Oliveros et al., 2014)}.

5. Modified Gravity and Entropic Approaches

Holographic energy densities extend to modified-gravity frameworks:

• Lovelock gravity: Black-hole (or apparent-horizon) thermodynamics in higher-order gravity naturally contains a holographic term $\rho_\Lambda(r_h) = 3/(16\pi G r_h^2)$ alongside topological-density corrections proportional to higher curvature terms. This topological mass structure provides a geometric origin for HDE, and the corresponding EoS has a stable $w_\Lambda=-1$ late-time attractor, linking higher-curvature quantum gravity to cosmic acceleration \cite{(Bousder et al., 2023)}.
• Braneworld (DGP) modifications: The DGP-induced area-entropy correction alters the HDE formula to $\rho_D=3c^2 M_p^2 L^{-2}(1 - \epsilon L/(3r_c))$; with $L=H^{-1}$ even non-interacting HDE solutions can produce acceleration due to the bulk correction $f(r_c,H)=-\epsilon/(3r_c H)$ \cite{(Sheykhi et al., 2015)}.

6. Model Reconstruction and Observational Constraints

Recent works employ non-parametric or nodal-spline methods to reconstruct the functional dependence of the HDE entropy exponent as a function of redshift, $\rho_{DE}\propto L^{f(a)}$, directly from data, leading to the following findings \cite{(Zapata et al., 29 Jul 2025)}:

• Standard HDE (fixed $f=-2$), as well as $\Lambda$CDM ($f=0$), are both statistically disfavored relative to reconstructed $f(z)$ with 3 nodes, yielding $\Delta\chi^2\sim10$–$12$ improvements in fits to BAO+SNe+$H_0$.
• The data favor moderate, $z$-dependent deviations from the area law, with $f(z)$ transitioning from Barrow/Tsallis-like at high $z$ to nearly $\Lambda$CDM at low $z$, and an EoS evolution from quintessence to mildly phantom at late times.

Large-scale structure, SNe, and BAO data all constrain HDE models to tight parameter ranges:

• Flat HDE: $c=0.7$–$0.8\pm0.1$, $\Omega_{m0}=0.27\pm0.02$, $H_0=67$ km/s/Mpc \cite{(Wang et al., 2016)}
• Ricci/cutoff models: $\alpha \simeq 0.44$, $b=0.02$–$0.05$ best fit, compatible with Planck SNe$H(z)$ measurements \cite{(Fu et al., 2011)}
• Tsallis/Barrow: non-extensive index near unity, $\delta=1.03_{-0.10}^{+0.12}$, $\Delta\lesssim0.5$ \cite{(Saridakis et al., 2018, Saridakis, 2020)}

7. Physical Implications and Theoretical Significance

Holographic dark-energy densities establish a deep connection between quantum-gravity–driven entropy bounds, cosmic information, and late-universe dynamics. In all viable models:

• Dynamical equations of state $w(z)$ interpolate between matter-like, quintessence, and sometimes trans-phantom values.
• Cosmic acceleration is tied to the crossover from subdominant to dominant HDE, with $q(z)$ tracking observationally required values and the age of the universe, CMB, and BAO-compatible expansions.
• Modifications to area law (Barrow, Tsallis, fractional, DGP, Ricci/Granda–Oliveros, polynomial) enable finer fits to data, alleviate the coincidence problem, and can embed the cosmological constant as a limiting case.

The extension to three-component systems or generalized entropy scaling necessitates going beyond the $L^{-2}$ ansatz, introducing higher-order derivatives (the “jerk” term), which are required for consistency in the presence of general dark-sector interactions \cite{(Forte, 2018)}.

These approaches continue to motivate both phenomenological studies and efforts to ground dark energy in fundamental quantum-gravity principles. Research avenues include systematic MCMC constraints, dynamical systems analysis, non-parametric function reconstruction, and embedding in higher-curvature or modified-gravity theories.

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References

• (Zapata et al., 29 Jul 2025) How Holographic is the Dark Energy? A Spline Nodal reconstruction approach
• (Trivedi et al., 2024) Fractional Holographic Dark Energy
• (Wang et al., 2016) Holographic Dark Energy
• (Farajollahi et al., 2011) Interacting Holographic dark energy in chameleon tachyon cosmology
• (Oliveros et al., 2014) New holographic dark energy model with non-linear interaction
• (Fu et al., 2011) Holographic Ricci dark energy: Interacting model and cosmological constraints
• (Mahata et al., 2015) A Dynamical System Analysis of Holographic Dark Energy Models with Different IR Cutoff
• (Saridakis et al., 2018) Holographic dark energy through Tsallis entropy
• (Cruz et al., 29 Oct 2025) Holographic Dark Energy from a Polynomial Expansion in the Hubble Parameter
• (Saridakis, 2020) Barrow holographic dark energy
• (Bousder et al., 2023) Holographic dark energy satisfying the energy conditions in Lovelock gravity
• (Sheykhi et al., 2015) New holographic dark energy model inspired by the DGP braneworld
• (Chimento et al., 2012) Holographic dark energy linearly interacting with dark matter
• (Forte et al., 2012) Holographic dark energy interacting with dark matter
• (Sinha et al., 2019) Density perturbation in an interacting holographic dark energy model