Generalized Holographic Dark Energy

• Generalized holographic dark energy (GHDE) is a dark energy framework that extends standard HDE by allowing a dynamic parameter c(z), generalized infrared cutoffs, and modified entropy-area relations.
• GHDE employs various parametrizations, such as CPL, JBP, and Wetterich types, to reconcile early matter domination with late-time acceleration.
• The framework integrates generalized Ricci and entropic models, enabling effective field reconstructions and improved alignment with observational dark energy diagnostics.

Generalized holographic dark energy (GHDE) denotes a class of dark-energy models obtained by relaxing one or more defining ingredients of standard holographic dark energy, whose basic density is $\rho_{de}=3c^2 M_{pl}^2 L^{-2}$. In the literature, the term is used both for the direct replacement $c\to c(z)$ and for broader extensions in which the infrared cutoff $L$, the entropy-area relation, or both are generalized. Across these constructions, the common objective is to preserve the holographic scaling while removing the pathological behavior of oversimplified implementations—especially the Hubble-cutoff case with constant $c$, which behaves like pressureless matter—and to obtain viable cosmic evolution with matter domination at early times and dark-energy domination at late times (Malekjani, 2012, Zhang et al., 2012, Nojiri et al., 2017, Luciano et al., 11 Mar 2026).

1. Foundational structure and scope of the generalization

The starting point of holographic dark energy is the holographic bound

$L^3 \rho_d \leq L\,m_p^2,$

whose saturation yields

$\rho_d = 3c^2 m_p^2 L^{-2}.$

Here $L$ is an infrared cutoff and $c$ is a dimensionless parameter of order unity in the standard formulation. In this form, HDE is a horizon-based realization of the idea that vacuum energy in a region of size $L$ must not exceed the mass of a black hole of the same size (Malekjani, 2012).

The main ambiguity is the choice of $L$. Standard constructions have used the Hubble scale $c\to c(z)$0, the particle horizon

$c\to c(z)$1

or the future event horizon

$c\to c(z)$2

The literature repeatedly emphasizes that these choices are not equivalent: the Hubble-scale model with constant $c\to c(z)$3 gives $c\to c(z)$4, the particle-horizon choice does not yield acceleration, and the future-event-horizon choice can generate acceleration but raises causality concerns in some formulations (Yu et al., 2010, Nojiri et al., 2017).

Within this context, GHDE emerged as a controlled enlargement of the HDE ansatz. One branch promotes the normalization $c\to c(z)$5 to a time- or redshift-dependent function, while another enlarges the cutoff itself to a functional

$c\to c(z)$6

or replaces the Bekenstein–Hawking entropy by generalized entropies. A broad synthesis of this viewpoint is that many dark-energy models can be represented as members of a generalized holographic family once the cutoff is allowed to depend on horizon data and derivatives (Nojiri et al., 2021, Nojiri et al., 2021).

2. Variable $c\to c(z)$7 models and the original GHDE program

A canonical GHDE construction replaces the constant holographic parameter by a redshift-dependent function,

$c\to c(z)$8

with $c\to c(z)$9 the future event horizon. In this formulation, the cosmology depends not only on the magnitude of $L$0 but also on its derivative $L$1; the equation of state acquires an additional term proportional to $L$2, and the evolution equation for $L$3 is modified accordingly (Zhang et al., 2012).

Four parametrizations of $L$4 were fitted in an early observational study: CPL-type,

$L$5

JBP-type,

$L$6

Wetterich-type,

$L$7

and Ma–Zhang-type,

$L$8

Using SNLS3, BAO, CMB, and $L$9 data, all GHDE variants fit at least as well as the original HDE model, and the JBP-type case reduced $c$0 by $c$1 relative to constant-$c$2 HDE (Zhang et al., 2012).

A particularly influential special case is the Hubble-cutoff GHDE,

$c$3

in a spatially flat FRW universe with separately conserved pressureless matter and dark energy. Because

$c$4

the generalized holographic function is directly the dark-energy density fraction. The model imposes a slow-variation condition,

$c$5

and yields the central relations

$c$6

with prime denoting derivative with respect to $c$7. In the constant-$c$8 limit, one recovers $c$9 and $L^3 \rho_d \leq L\,m_p^2,$0, which is precisely the failure mode of standard HDE at Hubble cutoff. With the Wetterich-type choice for $L^3 \rho_d \leq L\,m_p^2,$1, the model gives $L^3 \rho_d \leq L\,m_p^2,$2 at early times, a later dark-energy-dominated phase, a transition from decelerated to accelerated expansion, and a crossing of the phantom divide from quintessence regime to phantom regime (Malekjani, 2012).

The physical consequence of the $L^3 \rho_d \leq L\,m_p^2,$3 program is that the future fate of the universe is no longer fixed by a single constant. In the constant-$L^3 \rho_d \leq L\,m_p^2,$4 event-horizon model, $L^3 \rho_d \leq L\,m_p^2,$5, $L^3 \rho_d \leq L\,m_p^2,$6, and $L^3 \rho_d \leq L\,m_p^2,$7 correspond respectively to quintessence-like, cosmological-constant-like, and phantom-like futures. GHDE broadens this outcome space: depending on the form of $L^3 \rho_d \leq L\,m_p^2,$8, the future can be quintessence-like, cosmological-constant-like, or phantom-like (Zhang et al., 2012).

3. Generalized cutoffs, Ricci-type extensions, and covariant formulations

A second major strand of GHDE generalizes the infrared cutoff rather than only the parameter $L^3 \rho_d \leq L\,m_p^2,$9. In the extended Hubble-scale class, the cutoff is replaced by a local combination of background quantities,

$\rho_d = 3c^2 m_p^2 L^{-2}.$0

which leads to

$\rho_d = 3c^2 m_p^2 L^{-2}.$1

This generalizes Ricci dark energy, for which $\rho_d = 3c^2 m_p^2 L^{-2}.$2, and is often denoted generalized Ricci dark energy (GRDE). In recent numerical comparisons, this class was treated as the representative generalized-HDE realization based on an extended Hubble scale (Li et al., 2024).

An interacting generalization of the Granda–Oliveros framework uses

$\rho_d = 3c^2 m_p^2 L^{-2}.$3

together with the coupling

$\rho_d = 3c^2 m_p^2 L^{-2}.$4

In that model, analytic expressions were obtained for $\rho_d = 3c^2 m_p^2 L^{-2}.$5, $\rho_d = 3c^2 m_p^2 L^{-2}.$6, $\rho_d = 3c^2 m_p^2 L^{-2}.$7, and $\rho_d = 3c^2 m_p^2 L^{-2}.$8, and the framework was presented as a local alternative to future-event-horizon HDE, with the Ricci model recovered by $\rho_d = 3c^2 m_p^2 L^{-2}.$9 (Yu et al., 2010).

A more expansive formulation is the covariant generalized HDE program, in which the cutoff is promoted to a functional of horizon scales, curvature, lifetime, and derivatives,

$L$0

Within this framework, suitable choices of $L$1 can reproduce viable dark energy, unified inflation-plus-dark-energy cosmologies, $L$2 gravity, and general perfect-fluid cosmologies at the background level. The paper giving this construction explicitly showed how Starobinsky inflation with $L$3 can be recast in covariant generalized-HDE language (Nojiri et al., 2017).

A closely related development is the claim that generalized HDE possesses multiple “faces”: Tsallis, Rényi, and Sharma–Mittal entropic dark energy, quintessence, and Ricci dark energy can each be represented as generalized holographic models with suitable cutoffs written either in terms of the particle horizon and its derivatives or the future horizon and its derivatives. In this classification, entropic models typically require only first derivatives of $L$4 or $L$5, whereas quintessence and Ricci constructions require second derivatives (Nojiri et al., 2021). Barrow entropic dark energy was later shown to fit into this same scheme with cutoffs depending only on $L$6 or $L$7, even when the Barrow exponent is allowed to run with the cosmological expansion (Nojiri et al., 2021).

4. Entropic and nonextensive generalizations

A third major line of development generalizes HDE through entropy rather than through $L$8 or the geometric cutoff alone. In entropy-corrected HDE motivated by loop quantum gravity, the Bekenstein–Hawking law is replaced by a logarithmically corrected entropy, leading to

$L$9

The $c$0 terms are interpreted as quantum corrections, important when $c$1 is small, and the model has been embedded in interacting dark-sector cosmologies and mapped to modified variable, new modified, and viscous generalized Chaplygin gas constructions (Jamil et al., 2010, Farooq et al., 2010).

Rényi-based HDE modifies both the dark-energy density and the Friedmann equations. In a flat FLRW universe with apparent-horizon cutoff $c$2, the generalized HDE density becomes

$c$3

and the modified Friedmann equation contains an extra purely geometric term proportional to

$c$4

That term is treated as an additional dark-energy-like contribution, so the total dark-energy sector is the sum of generalized HDE and a geometric component (2002.04097).

A more recent microscopic construction introduced a two-parameter entropy

$c$5

which yields the generalized holographic density

$c$6

This is explicitly a two-sector holographic density. Standard HDE is recovered in the Bekenstein–Hawking limit, while $c$7CDM appears when the generalized density becomes constant, for example when $c$8 or $c$9 in the relevant one-sector limits, or when both exponents equal $L$0 (Luciano et al., 11 Mar 2026).

The same unifying tendency appears in four-parameter entropy models. A generalized entropy

$L$1

was used to reconstruct a “most generalized” entropic HDE family that contains generalized HDE with Nojiri–Odintsov cutoff, Barrow entropic HDE with particle horizon cutoff, and Tsallis entropic HDE with future event horizon cutoff as particular cases. This framework was analyzed in a viscous interacting setting and tested against the generalized second law using Bekenstein, logarithmic, and power-law entropy corrections (Saha et al., 2024).

At an even more abstract level, a general nonextensive formalism defines

$L$2

so that the choice of horizon entropy $L$3 directly determines the dark-energy density. This formalism has been applied to Bekenstein–Hawking, Barrow, Tsallis-Cirto, Rényi, Sharma-Mittal, and Kaniadakis entropies, with the future event horizon used in the data analysis though the authors emphasize that other horizons can be treated as well (Cimdiker et al., 23 Mar 2025). This suggests that, in contemporary usage, GHDE often functions as a container concept for horizon-entropy deformations.

5. Dynamical behavior, diagnostics, and effective field correspondences

The principal cosmological motivation for GHDE is dynamical flexibility. In the Hubble-cutoff $L$4 model, the evolution of $L$5, $L$6, and $L$7 shows a dust-like early universe, a later dark-energy-dominated epoch, and a deceleration-to-acceleration transition, with genuine crossing of the phantom line from quintessence regime to phantom regime (Malekjani, 2012). In the future-event-horizon $L$8 program, the derivative term $L$9 is the new dynamical ingredient responsible for richer background evolution and multiple possible future asymptotics (Zhang et al., 2012).

Generalized entropic models broaden the possibilities further. The two-parameter entropy model can remain entirely quintessence-like or entirely phantom-like in the Hubble-horizon case, depending on $L$0, while still reproducing the matter-to-dark-energy transition and the standard thermal history (Luciano et al., 11 Mar 2026). In the four-parameter entropic framework, the reported qualitative behavior differs by subfamily: the generalized four-parameter GHDE is phantom-like, Nojiri–Odintsov HDE crosses from quintessence-like to phantom, Tsallis HDE crosses the phantom divide early and later becomes quintessence-like, and Barrow HDE remains quintessence-like while its effective EoS can become phantom-like at late times (Saha et al., 2024).

The generalized-HDE literature also makes extensive use of dynamical diagnostics. Recent generalized holographic and generalized Ricci models were analyzed through the deceleration parameter $L$1, statefinder parameters $L$2, the $L$3 diagnostic, cosmography, and the squared sound speed $L$4. In that setting, the models were formulated for four cases—flat, curved, interacting, and curved-plus-interacting—and explicit formulas were obtained for $L$5, $L$6, $L$7, $L$8, and $L$9 in each case (Pasqua, 21 Sep 2025). A hybrid length-time cutoff combining HDE and NADE similarly employed $c\to c(z)$00, $c\to c(z)$01, statefinder trajectories, and $c\to c(z)$02, finding classical stability only in the interacting case (Maity et al., 19 Jun 2026).

A recurrent feature of GHDE research is reconstruction into effective field variables. Barrow-entropic generalized HDE has been mapped to quintessence and dilaton dark-energy models (Garg et al., 2023). More broadly, generalized holographic and Ricci models have been matched to tachyon, k-essence, dilaton, quintessence, DBI, Yang–Mills, and nonlinear electrodynamics fields by equating $c\to c(z)$03 and $c\to c(z)$04 with the corresponding scalar- or gauge-field energy densities and pressures (Pasqua, 21 Sep 2025). These reconstructions do not replace the holographic interpretation; rather, they provide alternative effective descriptions of the same background dynamics.

6. Observational constraints, reconstruction strategies, and present status

The earliest dedicated observational study of GHDE found that allowing $c\to c(z)$05 to vary with redshift improves the fit relative to original HDE. Using SNLS3, WMAP7 distance priors, SDSS DR7 BAO, and HST $c\to c(z)$06, the best-fit JBP-type model achieved $c\to c(z)$07, compared with $c\to c(z)$08 for constant-$c\to c(z)$09 HDE, while the original model remained inside the $c\to c(z)$10 regions of the GHDE parametrizations, indicating that then-current data did not determine the form of $c\to c(z)$11 uniquely (Zhang et al., 2012).

More recent broad comparisons are less favorable to simple HDE variants. A comprehensive 2024 numerical study organized the literature into four categories—other characteristic length scale, extended Hubble scale, dark sector interaction, and modified black-hole entropy—and found that $c\to c(z)$12CDM remains the most competitive model, Ricci dark energy is ruled out, and interacting HDE models perform worst across the four categories. The original HDE model performed better in BAO+CMB, whereas entropy-modified HDE models performed better in BAO+CMB+SN (Li et al., 2024).

Generalized nonextensive entropy models have also been subjected to direct Bayesian comparison with $c\to c(z)$13CDM using Pantheon+, cosmic chronometers, GRBs, CMB distance priors, and BAO. In that analysis, all HDE models under study were statistically disfavored relative to $c\to c(z)$14CDM. Standard HDE, Rényi, Sharma-Mittal, and Kaniadakis cases were very strongly disfavored, Barrow HDE was moderately disfavored, and Tsallis-Cirto HDE was the closest to $c\to c(z)$15CDM, with the fit pushing the model toward its $c\to c(z)$16CDM-like limit (Cimdiker et al., 23 Mar 2025).

At the same time, reconstruction approaches indicate that the decisive issue may be rigidity rather than holography itself. A spline-nodal reconstruction of generalized HDE introduced a scale-factor-dependent function $c\to c(z)$17 through

$c\to c(z)$18

with $c\to c(z)$19 the future event horizon and $c\to c(z)$20 reconstructed by 0-, 1-, 2-, and 3-node interpolations. The standard HDE case corresponds to $c\to c(z)$21, while the model approaches $c\to c(z)$22CDM when $c\to c(z)$23. In that analysis, the 3-node reconstruction improved the fit over standard HDE by $c\to c(z)$24 and over $c\to c(z)$25CDM by $c\to c(z)$26, although Bayesian evidence still left $c\to c(z)$27CDM competitive or mildly preferred in some dataset combinations because of complexity penalties (Zapata et al., 29 Jul 2025).

The present observational status is therefore mixed. Simple or rigid generalized constructions are often not preferred over $c\to c(z)$28CDM, but models with enough functional freedom to depart from fixed-$c\to c(z)$29, fixed-entropy, or fixed-cutoff assumptions can fit late-time data substantially better than standard HDE. A plausible implication is that current observations favor flexibility in the generalized sector while strongly penalizing unnecessary parameter volume.