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 $M_p$0 to $M_p$1, yielding $M_p$2. For $M_p$3, this reduces to standard HDE; $M_p$4 (non-extensive entropy) produces phenomenologically viable models, with distinctive EoS features such as quintessence, phantom, or crossing behavior depending on $M_p$5 \cite{(Saridakis et al., 2018)}.
• Barrow HDE. Quantum-gravitational corrections deform the area law to $M_p$6, leading to $M_p$7 and allowing a fractal deformation parameter $M_p$8. The resulting EoS generically interpolates between quintessence and phantom \cite{(Saridakis, 2020)}.
• Fractional HDE. Fractional calculus applied to horizon entropy yields $M_p$9. The FHDE density becomes $L$0, continuously connecting to the standard $L$1 scaling as $L$2. This allows the Hubble cutoff ($L$3) to yield accelerating solutions, unattainable in ordinary HDE \cite{(Trivedi et al., 2024)}.
• Polynomial HDE. Inspired by quantum gravity corrections, polynomial expansions in $L$4 such as $L$5 capture additional running effects, leading to phase phenomena and transient phantom behaviour closely tracking $L$6CDM at low redshift \cite{(Cruz et al., 29 Oct 2025)}.
• Modified Ricci and Granda–Oliveros Cutoffs. Take $L$7, so that $L$8 \cite{(Oliveros et al., 2014)}. The Ricci scalar $L$9 underpins these models, producing a density scaling as a fixed fraction $\Lambda$0 of $\Lambda$1 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	$\Lambda$2	$\Lambda$3	$\Lambda$4
Tsallis HDE	$\Lambda$5	$\Lambda$6	$\Lambda$7
Barrow HDE	$\Lambda$8	$\Lambda$9	$L^3\Lambda^4\lesssim L M_p^2$0
Fractional HDE	$L^3\Lambda^4\lesssim L M_p^2$1	$L^3\Lambda^4\lesssim L M_p^2$2	$L^3\Lambda^4\lesssim L M_p^2$3
Polynomial HDE	N/A	$L^3\Lambda^4\lesssim L M_p^2$4	$L^3\Lambda^4\lesssim L M_p^2$5, $L^3\Lambda^4\lesssim L M_p^2$6, $L^3\Lambda^4\lesssim L M_p^2$7

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 ($L^3\Lambda^4\lesssim L M_p^2$8 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

$L^3\Lambda^4\lesssim L M_p^2$9

where $\rho_{\rm HDE} = 3c^2 M_p^2 L^{-2}$0 transfers energy from DE to DM and can alleviate the coincidence problem.

• Chameleon–tachyon scenarios: Here, the action is $\rho_{\rm HDE} = 3c^2 M_p^2 L^{-2}$1, and variation induces an interaction $\rho_{\rm HDE} = 3c^2 M_p^2 L^{-2}$2, 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_{\rm HDE} = 3c^2 M_p^2 L^{-2}$3 including $\rho_{\rm HDE} = 3c^2 M_p^2 L^{-2}$4 ($\rho_{\rm HDE} = 3c^2 M_p^2 L^{-2}$5 being $\rho_{\rm HDE} = 3c^2 M_p^2 L^{-2}$6, $\rho_{\rm HDE} = 3c^2 M_p^2 L^{-2}$7, or total), offer analytic solutions for $\rho_{\rm HDE} = 3c^2 M_p^2 L^{-2}$8 and $\rho_{\rm HDE} = 3c^2 M_p^2 L^{-2}$9 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 $c$0 (arising in “new holographic” schemes) can flatten the energy density ratio curve $c$1 over an extended period, further mitigating the coincidence issue and generating stable models, as evinced by positive adiabatic sound speed $c$2 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 $c$3 or related quantities \cite{(Mahata et al., 2015)}. For standard HDE with future event horizon cutoff, the EoS is $c$4 and the evolution

$c$5

which yields a quintessence-like EoS for $c$6, phantom for $c$7. A line of non-hyperbolic (saddle-type) fixed points with $c$8 exists when Q is included, partially alleviating the coincidence problem but precluding hyperbolic attractors at observationally viable $c$9 unless extra ingredients are added \cite{(Mahata et al., 2015)}.

In all phenomenologically acceptable models, matching observed $L$0, transition redshift $L$1, and $L$2 is achievable for specific parameter sets: e.g., Ricci DE with $L$3, $L$4 gives $L$5, $L$6, $L$7 \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 $L$8 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 $L$9 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 $L = a\int_a^\infty \frac{da'}{H(a') a'^2}$0; with $L = a\int_a^\infty \frac{da'}{H(a') a'^2}$1 even non-interacting HDE solutions can produce acceleration due to the bulk correction $L = a\int_a^\infty \frac{da'}{H(a') a'^2}$2 \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, $L = a\int_a^\infty \frac{da'}{H(a') a'^2}$3, directly from data, leading to the following findings \cite{(Zapata et al., 29 Jul 2025)}:

• Standard HDE (fixed $L = a\int_a^\infty \frac{da'}{H(a') a'^2}$4), as well as $L = a\int_a^\infty \frac{da'}{H(a') a'^2}$5CDM ($L = a\int_a^\infty \frac{da'}{H(a') a'^2}$6), are both statistically disfavored relative to reconstructed $L = a\int_a^\infty \frac{da'}{H(a') a'^2}$7 with 3 nodes, yielding $L = a\int_a^\infty \frac{da'}{H(a') a'^2}$8–$L = a\int_a^\infty \frac{da'}{H(a') a'^2}$9 improvements in fits to BAO+SNe+$L=H^{-1}$0.
• The data favor moderate, $L=H^{-1}$1-dependent deviations from the area law, with $L=H^{-1}$2 transitioning from Barrow/Tsallis-like at high $L=H^{-1}$3 to nearly $L=H^{-1}$4CDM at low $L=H^{-1}$5, 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: $L=H^{-1}$6–$L=H^{-1}$7, $L=H^{-1}$8, $L=H^{-1}$9 km/s/Mpc \cite{(Wang et al., 2016)}
• Ricci/cutoff models: $M_p$00, $M_p$01–$M_p$02 best fit, compatible with Planck SNe$M_p$03 measurements \cite{(Fu et al., 2011)}
• Tsallis/Barrow: non-extensive index near unity, $M_p$04, $M_p$05 \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 $M_p$06 interpolate between matter-like, quintessence, and sometimes trans-phantom values.
• Cosmic acceleration is tied to the crossover from subdominant to dominant HDE, with $M_p$07 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 $M_p$08 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