New Holographic Dark Energy (NHDE)

• NHDE is a family of holographic dark-energy models that redefine the infrared cutoff through local curvature, action-based dynamics, or deformed entropy measures.
• It preserves the key ultraviolet–infrared relation while allowing varied cosmic evolutions such as quintessence-like, de Sitter, or phantom regimes.
• The approach addresses fundamental issues like causality and perturbative stability by replacing non-local event horizons with local geometric or thermodynamic inputs.

New Holographic Dark Energy (NHDE) denotes a class of holographically motivated dark-energy constructions in which the original holographic energy density is modified either by replacing the infrared cutoff with local curvature quantities, by deriving the cutoff dynamically from an action, or by deforming the horizon entropy entering the holographic bound. In the literature, the same label has been attached to the Granda–Oliveros local-cutoff density, the action-based model of Li and Miao, and several entropy-driven extensions based on Tsallis, Sharma–Mittal, DGP, AdS, and more recent two-parameter generalized entropies. A common aim is to preserve the holographic relation between ultraviolet and infrared physics while avoiding, reformulating, or generalizing the standard event-horizon prescription of holographic dark energy (Granda, 2011, Li et al., 2012, Saridakis et al., 2018, Jahromi et al., 2018, Sheykhi et al., 2015, Nakarachinda et al., 2022, Luciano et al., 11 Mar 2026).

1. Terminology and conceptual scope

The basic holographic starting point is the bound $L^{3}\Lambda^{4}\leq L M_{P}^{2}$, which, when saturated, gives the usual scaling $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$. NHDE models retain this ultraviolet–infrared logic but alter one of its structural inputs: the infrared length $L$, the entropy–area law, or the variational formulation of the cosmological system. In the original local-cutoff formulation, the infrared scale is replaced by a combination of $H$ and $\dot H$, yielding a dark-energy density that depends only on local geometric quantities and is therefore manifestly causal (Granda, 2011, Malekjani et al., 2010). In the action-based construction, the future event horizon is not imposed by hand but emerges from the equations of motion, together with an additional term interpreted as dark radiation (Li et al., 2012). In entropy-based NHDE, modified horizon entropies change the scaling of $\rho_{DE}$ with the cutoff and can make the Hubble horizon viable, something that fails in standard holographic dark energy without interaction (Sheykhi et al., 2015, Jahromi et al., 2018).

NHDE usage	Defining relation	Distinctive feature
Granda–Oliveros local cutoff	$\rho_D=3M_p^2(\alpha H^2+\beta \dot H)$ or $3[\lambda+\alpha H^2+\beta \dot H]$	Uses local curvature quantities
Action-principle NHDE	$$\rho_{DE}=\frac{1}{8\pi}\left(\frac{c}{a^2L^2}+\frac{\lambda}{2a^4}\right)$$	Future event horizon emerges dynamically
Entropy-based NHDE	Modified $S(A)$ gives modified $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$0	Deforms holographic scaling

This multiplicity of definitions means that NHDE is not a single model in the narrow sense. This suggests that the term functions as a family name for holographic dark-energy theories that depart from the standard Li prescription in a controlled way.

2. Local-curvature NHDE of Granda and Oliveros

In the Granda–Oliveros framework, the effective infrared cutoff is defined through local background quantities. One common parametrization is

$\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$1

while a closely related extension introduces a genuine constant term,

$\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$2

The constant $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$3 is interpreted as the first, vacuum-energy approximation to the infrared cutoff, whereas the $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$4 and $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$5 pieces provide dynamical corrections (Sharif et al., 2013, Granda, 2011).

A notable result of the constant-term version is an analytic Hubble law,

$\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$6

with $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$7. The term proportional to $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$8 behaves like an effective fluid even though no explicit matter sector is introduced. For $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$9, the model reproduces $L$0 exactly and hence recovers $L$1CDM, while small departures from $L$2 generate a dynamical equation of state (Granda, 2011).

The same local-cutoff structure has been studied in non-flat FRW backgrounds and in correspondences with Chaplygin-gas-type fluids. In that setting the background solution can be written as

$L$3

and the associated equation-of-state parameter $L$4 can evolve from $L$5 at early times to $L$6 at late times, depending on $L$7 (Malekjani et al., 2010). The flat-universe limit is more delicate: for $L$8, the model gives constant

$L$9

and the sound speed satisfies $H$0, which is negative for the quoted best-fit choice, leading that analysis to describe the flat limit as generically unstable (Malekjani et al., 2010).

The local-cutoff NHDE has also been reconstructed in $H$1 gravity. Using

$H$2

Sharif and Zubair obtained a third-order differential equation for $H$3 and found that $H$4 reproduces $H$5, whereas $H$6 yields phantom-like behavior and a big-rip future in their numerical reconstruction (Sharif et al., 2013).

3. Action-based and physically motivated reformulations

A distinct NHDE construction was proposed from an action principle in a Robertson–Walker background with auxiliary fields $H$7 and $H$8. The action contains the term

$H$9

and the constraint sector

$\dot H$0

Variation yields the equations

$\dot H$1

together with modified Friedmann equations (Li et al., 2012).

In this formulation the effective dark-energy density and pressure are

$\dot H$2

The key structural result is that the constraint integrates to

$\dot H$3

so the future event horizon appears as an output of the local equations of motion rather than as an external input. The extra $\dot H$4 term behaves as dark radiation, and the late-time behavior depends on $\dot H$5: $\dot H$6 gives quintessence-like expansion, $\dot H$7 yields de Sitter, and $\dot H$8 gives phantom behavior with a big-rip future (Li et al., 2012).

A separate physically motivated reformulation, due to Viaggiu, argues that the standard black-hole bound used in holographic dark energy is not appropriate in an expanding FLRW background because the non-formation of trapped surfaces depends on both $\dot H$9 and the background matter density $\rho_{DE}$0. This leads to

$\rho_{DE}$1

with a time-dependent saturation parameter $\rho_{DE}$2. For $\rho_{DE}$3, one finds

$\rho_{DE}$4

so that $\rho_{DE}$5 in the matter era and $\rho_{DE}$6 in the dark-energy-dominated future (Viaggiu, 2013). The same analysis also proposes a correction to Bekenstein–Hawking entropy in expanding universes,

$\rho_{DE}$7

linking NHDE to modified horizon thermodynamics (Viaggiu, 2013).

4. Entropic NHDE and generalized horizon thermodynamics

A major later direction redefines NHDE through generalized entropy functionals. In the Tsallis construction, the horizon entropy is

$\rho_{DE}$8

which, through the holographic principle, gives

$\rho_{DE}$9

With the future event horizon $\rho_D=3M_p^2(\alpha H^2+\beta \dot H)$0, the model becomes

$\rho_D=3M_p^2(\alpha H^2+\beta \dot H)$1

The parameter $\rho_D=3M_p^2(\alpha H^2+\beta \dot H)$2 measures the deviation from additive entropy; $\rho_D=3M_p^2(\alpha H^2+\beta \dot H)$3 reproduces standard holographic dark energy. The model yields the usual thermal history, allows quintessence-like, phantom-like, or phantom-divide-crossing behavior depending on $\rho_D=3M_p^2(\alpha H^2+\beta \dot H)$4, and a joint fit to 40 binned SNIa and 36 cosmic-chronometer $\rho_D=3M_p^2(\alpha H^2+\beta \dot H)$5 points gives $\rho_D=3M_p^2(\alpha H^2+\beta \dot H)$6, $\rho_D=3M_p^2(\alpha H^2+\beta \dot H)$7, $\rho_D=3M_p^2(\alpha H^2+\beta \dot H)$8, and $\rho_D=3M_p^2(\alpha H^2+\beta \dot H)$9 (Saridakis et al., 2018).

The Sharma–Mittal version uses a two-parameter entropy and takes the Hubble horizon $3[\lambda+\alpha H^2+\beta \dot H]$0 as cutoff. The corresponding density is

$3[\lambda+\alpha H^2+\beta \dot H]$1

Its central significance is that the Hubble horizon can now generate acceleration because the entropy deformation changes the simple $3[\lambda+\alpha H^2+\beta \dot H]$2 scaling that obstructs standard HDE. The model approaches a de Sitter phase at late times, and the reported stability analysis finds $3[\lambda+\alpha H^2+\beta \dot H]$3 at late times but typically $3[\lambda+\alpha H^2+\beta \dot H]$4 during deep matter domination (Jahromi et al., 2018).

Braneworld thermodynamics gives another entropy-based NHDE. In the DGP-inspired model, the modified apparent-horizon entropy implies

$3[\lambda+\alpha H^2+\beta \dot H]$5

For $3[\lambda+\alpha H^2+\beta \dot H]$6,

$3[\lambda+\alpha H^2+\beta \dot H]$7

which differs from standard HDE precisely by an extra-dimensional correction. This correction makes $3[\lambda+\alpha H^2+\beta \dot H]$8 dynamical, produces $3[\lambda+\alpha H^2+\beta \dot H]$9, and allows late-time acceleration even without dark-sector interaction; for $$\rho_{DE}=\frac{1}{8\pi}\left(\frac{c}{a^2L^2}+\frac{\lambda}{2a^4}\right)$$0, the quoted transition to $$\rho_{DE}=\frac{1}{8\pi}\left(\frac{c}{a^2L^2}+\frac{\lambda}{2a^4}\right)$$1 occurs at $$\rho_{DE}=\frac{1}{8\pi}\left(\frac{c}{a^2L^2}+\frac{\lambda}{2a^4}\right)$$2 (Sheykhi et al., 2015).

A related AdS-black-hole derivation replaces the Schwarzschild mass bound by the AdS one and obtains

$$\rho_{DE}=\frac{1}{8\pi}\left(\frac{c}{a^2L^2}+\frac{\lambda}{2a^4}\right)$$3

The constant term originates from the AdS cosmological constant and plays a cosmological-constant-like role. Both $$\rho_{DE}=\frac{1}{8\pi}\left(\frac{c}{a^2L^2}+\frac{\lambda}{2a^4}\right)$$4 and the particle horizon were analyzed, and both give late-time acceleration without future-horizon causality issues. The Hubble-radius version was fit to Pantheon + CC + BAO, yielding $$\rho_{DE}=\frac{1}{8\pi}\left(\frac{c}{a^2L^2}+\frac{\lambda}{2a^4}\right)$$5, $$\rho_{DE}=\frac{1}{8\pi}\left(\frac{c}{a^2L^2}+\frac{\lambda}{2a^4}\right)$$6, $$\rho_{DE}=\frac{1}{8\pi}\left(\frac{c}{a^2L^2}+\frac{\lambda}{2a^4}\right)$$7, with $$\rho_{DE}=\frac{1}{8\pi}\left(\frac{c}{a^2L^2}+\frac{\lambda}{2a^4}\right)$$8 relative to $$\rho_{DE}=\frac{1}{8\pi}\left(\frac{c}{a^2L^2}+\frac{\lambda}{2a^4}\right)$$9CDM (Nakarachinda et al., 2022).

More recent work extends NHDE to a microscopic two-parameter entropic functional,

$S(A)$0

leading to

$S(A)$1

This framework embeds standard HDE, Tsallis HDE, Barrow HDE, and $S(A)$2CDM as limiting cases. For $S(A)$3, the behavior is quintessence-like; for $S(A)$4, phantom regimes are possible; and near $S(A)$5, the model approaches $S(A)$6CDM (Luciano et al., 11 Mar 2026). Nojiri, Odintsov, and Faraoni developed another entropy-driven NHDE based on a three-parameter entropy $S(A)$7, where the cutoff-space energy density acquires a hypergeometric correction and the resulting cosmology admits multiple de Sitter points, including scales relevant to inflation and late-time acceleration (Nojiri et al., 2022).

5. Interacting sectors, modified gravity, and field reconstructions

NHDE has been embedded in several extended-gravity settings. In chameleon Brans–Dicke cosmology, the density is

$S(A)$8

with power-law ansätze $S(A)$9, $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$00, and $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$01. The resulting $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$02 is expressed in terms of incomplete gamma functions, and the model is reconstructed onto quintessence, DBI-essence, and tachyon fields. The phantom-divide crossing occurs when $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$03, and the quoted conformal-anomaly estimate gives $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$04, so the crossing is described as stable under the quantum correction. Stronger matter–chameleon coupling raises the reconstructed potentials (Chattopadhyay et al., 2014).

Interacting NHDE has also been studied in flat FRW cosmology with the non-linear coupling

$\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$05

For this model, the Hubble rate has an analytic trigonometric-exponential form, the equation of state shows quintom behavior, and the squared sound speed remains positive for all $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$06, so the model is classified there as classically stable. The ratio $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$07 stays of order unity over an extended interval, which is presented as an alleviation of the coincidence problem. For $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$08, $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$09–0.06, and $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$10, the transition redshift is $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$11 (Oliveros et al., 2014).

In fractal cosmology, the DGP-inspired NHDE density

$\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$12

is coupled to dark matter through either $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$13 or $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$14. The interacting case with $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$15 leads to $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$16, a deceleration parameter approaching $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$17, suppression of the CMB spectrum at low multipoles $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$18, and enhancement of the acoustic peaks in a modified CAMB analysis. The reported MCMC fit to SNIa, BAO, CMB, and OHD gives $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$19, $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$20, $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$21, and $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$22 (Sadri et al., 2018).

A large reconstruction literature maps NHDE backgrounds to effective fields and fluids. One-to-one correspondences have been constructed with standard, generalized, and modified Chaplygin gas models in non-flat FRW (Malekjani et al., 2010); with quintessence and tachyon fields in the non-linearly interacting model (Oliveros et al., 2014); with generalized, modified, modified variable, and viscous generalized Chaplygin gas, as well as DBI, Yang–Mills, and nonlinear-electrodynamics sectors, in later analytical reconstructions (Pasqua, 9 Sep 2025). These reconstructions do not define NHDE anew; rather, they interpret a given NHDE background as an effective scalar or fluid dynamics.

6. Phenomenology, observational status, and recurring issues

Across the literature, NHDE models are designed to reproduce the standard cosmic sequence from matter domination to dark-energy domination. The local-curvature unification model fits supernovae, CMB, and BAO with $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$23, $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$24, and $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$25, placing it very close to $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$26CDM while still allowing a dynamical $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$27 when $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$28 (Granda, 2011). The Tsallis model similarly preserves the standard thermal history and mildly prefers $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$29, though $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$30 remains allowed within $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$31 (Saridakis et al., 2018). The AdS-black-hole model is statistically compatible with $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$32CDM at $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$33, and the fractal interacting model was reported to be consistent with background and perturbation data (Nakarachinda et al., 2022, Sadri et al., 2018).

Three recurrent issues organize the debate around NHDE. The first is causality. The Granda–Oliveros local cutoff and several entropy-deformed Hubble-horizon models are explicitly constructed to avoid future-horizon non-locality, while the action-based NHDE solves the issue differently by deriving the future event horizon from local equations rather than postulating it (Granda, 2011, Jahromi et al., 2018, Sheykhi et al., 2015, Li et al., 2012). The second is perturbative stability. Some analyses report $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$34 and therefore classical stability, particularly in interacting or non-flat settings, whereas others find that the flat local-cutoff model has $\rho_{\Lambda}\sim M_{P}^{2}L^{-2}$35 and is generically unstable (Oliveros et al., 2014, Malekjani et al., 2010). The third is model identity. Because the same acronym labels local-cutoff, action-based, and entropy-based theories, statements about “NHDE” are often model-class statements rather than properties of a single Lagrangian or single energy density. This suggests that precision requires specifying the defining density, entropy, and infrared cutoff each time the term is used.

In that restricted but important sense, NHDE is best understood as a research program within holographic cosmology: a set of attempts to retain the holographic dark-energy paradigm while modifying its infrared, thermodynamic, or variational foundations. The program includes geometrical unification models, entropy-deformed models with viable Hubble-horizon cutoffs, action-based formulations with emergent future horizons, and a wide range of interacting and modified-gravity embeddings. The detailed late-time behavior—quintessence-like, de Sitter, phantom, or big-rip—depends on which NHDE realization is under consideration (Li et al., 2012, Sheykhi et al., 2015, Saridakis et al., 2018, Luciano et al., 11 Mar 2026).