Generalized Holographic Dark Energy Model

• Generalized Holographic Dark Energy Model is a framework where the dark energy density is defined via a versatile infrared cutoff linking horizon distances, curvature invariants, and their derivatives.
• It unifies various dark energy and modified gravity theories by accommodating analytic solutions, diagnostic tools, and structural dualities with models like scalar fields and Chaplygin gas.
• The model integrates entropic formalisms and interaction terms to reproduce cosmic acceleration and stability constraints observed in current cosmological data.

The Generalized Holographic Dark Energy (GHDE) Model is a broad class of phenomenological constructions in which the dark energy (DE) energy density is defined via the holographic principle but with an infrared (IR) cutoff that is a highly flexible functional of horizon and curvature invariants, their derivatives, and cosmological parameters. This framework unifies and subsumes a wide spectrum of DE and modified-gravity models, including those based on entropic dark energy, Ricci curvature, scalar fields, and generalized entropy formalisms. Generalized HDE models admit analytic background solutions, have well-developed diagnostic tools, and display deep structural dualities with models of covariant modified gravity and effective perfect fluids.

1. Covariant Definition and Model Space

In the generalized holographic framework, the DE energy density is given by

$$\rho_{\mathrm{HDE}} = \frac{3c^2}{\kappa^2\,L_{\mathrm{IR}}^2}$$

where $c^2$ is an order-unity free parameter and the IR cutoff $L_{\mathrm{IR}}$ is a highly general functional of the cosmological background: $$L_{\mathrm{IR}} = L_{\mathrm{IR}}\bigl(L_p, \dot L_p, \ddot L_p, \ldots; L_f, \dot L_f, \ddot L_f, \ldots; H, \dot H, \ldots; a; \Lambda; t_s; \ldots\bigr)$$ Here, $L_p$ is the particle horizon, $L_f$ the future/event horizon, $H$ the Hubble parameter, $a$ the scale factor, and $t_s$ a potential future singularity time. The holographic fluid evolves according to

$$\dot\rho_{\mathrm{HDE}} + 3H(\rho_{\mathrm{HDE}} + p_{\mathrm{HDE}}) = 0$$

yielding the effective equation of state (EoS) parameter

$c^2$0

The flexibility in $c^2$1 allows the model to encompass classical HDE (e.g., $c^2$2 or $c^2$3), as well as non-local, higher-derivative, or curvature-invariant cutoffs.

2. Generalized Ricci and Curvature-based GHDE Models

A key subclass are models in which $c^2$4 is a local function of $c^2$5 and its derivatives: $c^2$6 where $c^2$7 is the Ricci scalar and $c^2$8 parametrizes the interpolation between pure Hubble ($c^2$9) and Ricci ($L_{\mathrm{IR}}$0) cutoffs (Pasqua, 21 Sep 2025). This form emerges in the Granda–Oliveros cutoff and its extensions, yielding equations of motion for $L_{\mathrm{IR}}$1 that can be solved analytically or semi-analytically for a range of backgrounds, including spatial curvature and dark-sector interactions. The effective DE EoS, density, pressure, deceleration $L_{\mathrm{IR}}$2, and statefinder diagnostics can all be derived explicitly (Enkhili et al., 2024, Pasqua, 21 Sep 2025, Yu et al., 2010).

3. Entropic and Nonextensive Generalized HDE

Many generalized HDE models exploit entropy-area relations motivated by nonadditive (Tsallis, Rényi, Sharma–Mittal, Barrow) entropy formalisms. The DE density can be constructed as

$L_{\mathrm{IR}}$3

with $L_{\mathrm{IR}}$4 the relevant entropy. For instance, Tsallis $L_{\mathrm{IR}}$5 and Rényi $L_{\mathrm{IR}}$6 (Nojiri et al., 2021, Jahromi et al., 2018). Explicitly, with Sharma–Mittal entropy,

$L_{\mathrm{IR}}$7

with $L_{\mathrm{IR}}$8 the Bekenstein–Hawking entropy, and the associated $L_{\mathrm{IR}}$9 features a nontrivial $$L_{\mathrm{IR}} = L_{\mathrm{IR}}\bigl(L_p, \dot L_p, \ddot L_p, \ldots; L_f, \dot L_f, \ddot L_f, \ldots; H, \dot H, \ldots; a; \Lambda; t_s; \ldots\bigr)$$0-dependence, reducing to standard HDE for suitable limits (Jahromi et al., 2018). Such models can admit solvable background equations and pass cosmic-acceleration and stability constraints for appropriate parameter choices.

Importantly, one can construct explicit one-to-one mappings between entropic DE models (with either constant or running exponents) and generalized HDE forms by expressing the relevant cutoff $$L_{\mathrm{IR}} = L_{\mathrm{IR}}\bigl(L_p, \dot L_p, \ddot L_p, \ldots; L_f, \dot L_f, \ddot L_f, \ldots; H, \dot H, \ldots; a; \Lambda; t_s; \ldots\bigr)$$1 in terms of either horizon and its derivatives (Nojiri et al., 2021, Nojiri et al., 2021).

4. Scalar-field, Chaplygin Gas, and Modified Gravity Dualities

Generalized HDE models admit precise correspondences with scalar-field models (quintessence, k-essence, tachyon, dilaton, DBI), Chaplygin gas variants, and Yang–Mills or nonlinear electrodynamics condensate DE. The mapping equates energy density and EoS in both sectors, reconstructing scalar-field potentials and kinetic terms as functionals of the GHDE background (Pasqua, 21 Sep 2025, Pasqua, 9 Sep 2025):

• Quintessence: $$L_{\mathrm{IR}} = L_{\mathrm{IR}}\bigl(L_p, \dot L_p, \ddot L_p, \ldots; L_f, \dot L_f, \ddot L_f, \ldots; H, \dot H, \ldots; a; \Lambda; t_s; \ldots\bigr)$$2, $$L_{\mathrm{IR}} = L_{\mathrm{IR}}\bigl(L_p, \dot L_p, \ddot L_p, \ldots; L_f, \dot L_f, \ddot L_f, \ldots; H, \dot H, \ldots; a; \Lambda; t_s; \ldots\bigr)$$3.
• k-essence: $$L_{\mathrm{IR}} = L_{\mathrm{IR}}\bigl(L_p, \dot L_p, \ddot L_p, \ldots; L_f, \dot L_f, \ddot L_f, \ldots; H, \dot H, \ldots; a; \Lambda; t_s; \ldots\bigr)$$4, $$L_{\mathrm{IR}} = L_{\mathrm{IR}}\bigl(L_p, \dot L_p, \ddot L_p, \ldots; L_f, \dot L_f, \ddot L_f, \ldots; H, \dot H, \ldots; a; \Lambda; t_s; \ldots\bigr)$$5 reconstructed from HDE EoS.
• Chaplygin gas mappings use $$L_{\mathrm{IR}} = L_{\mathrm{IR}}\bigl(L_p, \dot L_p, \ddot L_p, \ldots; L_f, \dot L_f, \ddot L_f, \ldots; H, \dot H, \ldots; a; \Lambda; t_s; \ldots\bigr)$$6, establishing correspondence via the expansion history.

Similarly, any FLRW background induced by HDE with a suitable $$L_{\mathrm{IR}} = L_{\mathrm{IR}}\bigl(L_p, \dot L_p, \ddot L_p, \ldots; L_f, \dot L_f, \ddot L_f, \ldots; H, \dot H, \ldots; a; \Lambda; t_s; \ldots\bigr)$$7 can be mapped to modified gravity (notably $$L_{\mathrm{IR}} = L_{\mathrm{IR}}\bigl(L_p, \dot L_p, \ddot L_p, \ldots; L_f, \dot L_f, \ddot L_f, \ldots; H, \dot H, \ldots; a; \Lambda; t_s; \ldots\bigr)$$8 or $$L_{\mathrm{IR}} = L_{\mathrm{IR}}\bigl(L_p, \dot L_p, \ddot L_p, \ldots; L_f, \dot L_f, \ddot L_f, \ldots; H, \dot H, \ldots; a; \Lambda; t_s; \ldots\bigr)$$9) by identifying the HDE energy density with the geometric sector, and the resulting cutoff is expressible as a function of curvature invariants and their derivatives (Nojiri et al., 2017, Chattopadhyay, 7 Oct 2025).

5. Interacting and Time-varying GHDE Models

Generalized HDE allows inclusion of interaction terms between dark energy and dark matter: $L_p$0 yielding a coupled system of continuity equations. In such models, analytical background solutions for the Hubble rate, density parameters, and EoS are available. Diagnostics such as statefinder $L_p$1, Om$L_p$2, hierarchy parameters, and growth factors provide means to distinguish interaction strength and cutoff parameters from $L_p$3CDM (Enkhili et al., 2024, Yu et al., 2010, Zhang et al., 2012).

Allowance for a time- or redshift-dependent holographic parameter $L_p$4 (e.g., $L_p$5 in $L_p$6) or a running entropy exponent substantially broadens phenomenology. Observational constraints from SNIa, BAO, and CMB data indicate flexibility in matching the data and reproducing acceleration and phantom–quintessence transitions (Zhang et al., 2012, Lu et al., 2012, Majeed et al., 2014).

6. Thermodynamics, Stability, and Observational Diagnostics

GHDE models possess well-defined thermodynamic properties, with the first and generalized second laws holding under specific conditions. On the apparent horizon the first law $L_p$7 and generalized second law (GSL) $L_p$8 are always satisfied, but these fail generically on particle or event horizons except for specific parameter domains (Bhattacharya et al., 2011). Analytical evaluation of the speed of sound $L_p$9 and stability against classical perturbations reveals constraints on the allowed parameter space for $L_f$0 (Enkhili et al., 2024, Cruz et al., 29 Oct 2025).

A suite of cosmological diagnostics is established:

• Statefinder parameters $L_f$1, $L_f$2, and statefinder hierarchy.
• Cosmographic parameters ($L_f$3, $L_f$4, $L_f$5, $L_f$6, $L_f$7), with present-day values consistent with observational bounds.
• Growth rate of matter perturbations $L_f$8 and composite null-diagnostics.
• $L_f$9–$H$0 plane trajectories to distinguish freezing/thawing and departures from $H$1CDM (Enkhili et al., 2024, Pasqua, 21 Sep 2025).

Comparison to data shows that with appropriate parameter choices, generalized HDE models smoothly interpolate between matter-dominated, quintessence, phantom, and de Sitter final states, reproducing the observed transition redshift $H$2–$H$3, $H$4 in [$H$5,$H$6], and $H$7 (Lu et al., 2012, Zhang et al., 2012, Enkhili et al., 2024).

7. Unification and Symmetry Structure

A central insight is the demonstrated symmetry or duality: any FLRW model in which the background evolution $H$8 is a function of $H$9, its derivatives, horizons, or their derivatives, can be recast as a generalized HDE model with a suitable $a$0 (Nojiri et al., 2021, Nojiri et al., 2017). Conversely, any generalized HDE admits reinterpretation as an entropic DE, scalar field, or curvature-based theory—modulo smoothness and energy conservation. This umbrella property shows that the GHDE class is not merely a phenomenological extension, but a structural unification, encompassing a wide landscape of dark energy and modified gravity phenomenology.

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Selected References:

• "Diagnostic Approaches for Interacting generalized holographic Ricci Dark Energy Models" (Enkhili et al., 2024)
• "Generalized Holographic and Ricci Dark Energy: Cosmological Diagnostics and Scalar Field Realizations" (Pasqua, 21 Sep 2025)
• "Different faces of generalized holographic dark energy" (Nojiri et al., 2021)
• "Covariant Generalized Holographic Dark Energy and Accelerating Universe" (Nojiri et al., 2017)
• "Statefinder Description in Generalized Holographic and Ricci Dark Energy Models" (Khatua et al., 2011)
• "Generalized entropy formalism and a new holographic dark energy model" (Jahromi et al., 2018)
• "Generalized holographic dark energy model described at the Hubble length" (Malekjani, 2012)
• "Holographic Dark Energy with Time Varying n^2 Parameter in Non-Flat Universe" (Majeed et al., 2014)
• "Scalar Field Reconstructions of Holographic Dark Energy Models with Applications to Chaplygin Gas, DBI, Yang-Mills, and NLED Frameworks" (Pasqua, 9 Sep 2025)
• "Barrow entropic dark energy: A member of generalized holographic dark energy family" (Nojiri et al., 2021)