Ricci-Cubic Holographic Dark Energy

• RCHDE is a holographic dark energy model defined by local curvature invariants, combining the Ricci scalar and a cubic invariant for its infrared cutoff.
• It interpolates between traditional Ricci holographic dark energy and higher-curvature models, exhibiting dynamic regimes such as quintessence, phantom crossing, and potential Big Rip scenarios.
• Empirical analyses using MCMC and machine-learning regression validate its background expansion history while highlighting the need for perturbative stability studies.

Ricci-Cubic Holographic Dark Energy (RCHDE) is a holographic dark-energy model in which the infrared cutoff is defined by a local combination of the Ricci scalar and a cubic curvature invariant, rather than by a nonlocal horizon such as the future event horizon. In its defining form,

$$\frac{1}{L^2}=-\alpha R+\lambda P^{1/3}, \qquad \rho_{DE}=\frac{3}{\kappa^2\left(-\alpha R+\lambda P^{1/3}\right)},$$

so the dark-energy density is determined by local spacetime curvature. The construction was proposed as a higher-curvature extension of Ricci holographic dark energy and later confronted with background observational data through MCMC inference and auxiliary machine-learning regression (Rudra, 2022, Sanyal et al., 27 Aug 2025).

1. Geometric definition and invariant content

The defining novelty of RCHDE is the replacement of the usual horizon-based infrared cutoff by a local invariant combination of the Ricci scalar $R$ and a cubic curvature scalar $P$. The original proposal motivates this choice by two linked considerations: the evolution then does not depend on the past or future features of the universe, but completely on its present features; and the use of invariants is presented as making the theory more fundamental in nature (Rudra, 2022).

The cubic invariant is introduced as a general scalar built from cubic contractions of curvature tensors,

$P=\beta_{1}R_{\mu~~\nu}^{~~\rho~~\sigma}R_{\rho~~\sigma}^{~~\gamma~~\delta}R_{\gamma~~\delta}^{~~\mu~~\nu} +\beta_{2}R_{\mu\nu}^{\rho\sigma}R_{\rho\sigma}^{\gamma\delta}R_{\gamma\delta}^{\mu\nu} +\beta_{3}R^{\sigma\gamma}R_{\mu\nu\rho\sigma}{R^{\mu\nu\rho}_{\gamma} +\beta_{4}RR_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} +\beta_{5}R_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma} +\beta_{6}R^{\nu}_{\mu}R^{\rho}_{\nu}R^{\mu}_{\rho} +\beta_{7}R_{\mu\nu}R^{\mu\nu}R +\beta_{8}R^{3}.$

For theories with the same spectrum as GR, the coefficients satisfy

$$\beta_{7}=\frac{1}{12}\left(3\beta_{1}-24\beta_{2}-16\beta_{3}-48\beta_{4}-5\beta_{5}-9\beta_{6}\right),$$

$$\beta_{8}=\frac{1}{72}\left(-6\beta_{1}+36\beta_{2}+22\beta_{3}+64\beta_{4}+3\beta_{5}+9\beta_{6}\right).$$

To ensure second-order FLRW equations, the cosmological construction further imposes

$$\beta_{6}=4\beta_{2}+2\beta_{3}+8\beta_{4}+\beta_{5},$$

and introduces the effective combination

$$\tilde{\beta}\equiv -\beta_1+4\beta_2+2\beta_3+8\beta_4.$$

At the homogeneous FLRW level, the cubic sector therefore collapses to the single parameter $\tilde{\beta}$ (Rudra, 2022).

Within this framework, RCHDE interpolates between lower- and higher-curvature holographic models. The limit $\lambda=0$ gives ordinary Ricci holographic dark energy, while $R$0 gives a pure cubic HDE model. The proposal is also explicitly positioned relative to Ricci–Gauss–Bonnet HDE: the dimensional logic is analogous, except that the quadratic invariant sector is replaced by a cubic invariant sector through $R$1 (Rudra, 2022).

2. Flat-FLRW reduction and governing equations

The cosmological analysis is formulated in FLRW spacetime and then specialized to the spatially flat case $R$2. The first Friedmann equation is

$R$3

and for dust matter,

$R$4

The density parameters are

$R$5

This yields

$R$6

Using $R$7 and primes for $R$8,

$R$9

These relations are the starting point for rewriting the curvature invariants entirely in terms of $P$0 and its derivatives (Rudra, 2022).

In flat FLRW, the Ricci scalar and the cubic invariant reduce to

$P$1

Equivalently, after substituting the $P$2 relations,

$P$3

$P$4

The dark-energy density therefore becomes

$P$5

Substituting this into the definition of $P$6 yields the central nonlinear first-order equation,

$P$7

This equation governs the flat dust-filled RCHDE cosmology (Rudra, 2022).

The original construction emphasizes that this master equation generally has no exact analytic solution. The cubic-root term is treated through a binomial expansion, keeping only terms linear in $P$8 and $P$9, which yields an approximate analytic solution for $P=\beta_{1}R_{\mu~~\nu}^{~~\rho~~\sigma}R_{\rho~~\sigma}^{~~\gamma~~\delta}R_{\gamma~~\delta}^{~~\mu~~\nu} +\beta_{2}R_{\mu\nu}^{\rho\sigma}R_{\rho\sigma}^{\gamma\delta}R_{\gamma\delta}^{\mu\nu} +\beta_{3}R^{\sigma\gamma}R_{\mu\nu\rho\sigma}{R^{\mu\nu\rho}_{\gamma} +\beta_{4}RR_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} +\beta_{5}R_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma} +\beta_{6}R^{\nu}_{\mu}R^{\rho}_{\nu}R^{\mu}_{\rho} +\beta_{7}R_{\mu\nu}R^{\mu\nu}R +\beta_{8}R^{3}.$0. In that solution, the combination

$P=\beta_{1}R_{\mu~~\nu}^{~~\rho~~\sigma}R_{\rho~~\sigma}^{~~\gamma~~\delta}R_{\gamma~~\delta}^{~~\mu~~\nu} +\beta_{2}R_{\mu\nu}^{\rho\sigma}R_{\rho\sigma}^{\gamma\delta}R_{\gamma\delta}^{\mu\nu} +\beta_{3}R^{\sigma\gamma}R_{\mu\nu\rho\sigma}{R^{\mu\nu\rho}_{\gamma} +\beta_{4}RR_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} +\beta_{5}R_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma} +\beta_{6}R^{\nu}_{\mu}R^{\rho}_{\nu}R^{\mu}_{\rho} +\beta_{7}R_{\mu\nu}R^{\mu\nu}R +\beta_{8}R^{3}.$1

appears, and the integration constant $P=\beta_{1}R_{\mu~~\nu}^{~~\rho~~\sigma}R_{\rho~~\sigma}^{~~\gamma~~\delta}R_{\gamma~~\delta}^{~~\mu~~\nu} +\beta_{2}R_{\mu\nu}^{\rho\sigma}R_{\rho\sigma}^{\gamma\delta}R_{\gamma\delta}^{\mu\nu} +\beta_{3}R^{\sigma\gamma}R_{\mu\nu\rho\sigma}{R^{\mu\nu\rho}_{\gamma} +\beta_{4}RR_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} +\beta_{5}R_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma} +\beta_{6}R^{\nu}_{\mu}R^{\rho}_{\nu}R^{\mu}_{\rho} +\beta_{7}R_{\mu\nu}R^{\mu\nu}R +\beta_{8}R^{3}.$2 is fixed by the present condition $P=\beta_{1}R_{\mu~~\nu}^{~~\rho~~\sigma}R_{\rho~~\sigma}^{~~\gamma~~\delta}R_{\gamma~~\delta}^{~~\mu~~\nu} +\beta_{2}R_{\mu\nu}^{\rho\sigma}R_{\rho\sigma}^{\gamma\delta}R_{\gamma\delta}^{\mu\nu} +\beta_{3}R^{\sigma\gamma}R_{\mu\nu\rho\sigma}{R^{\mu\nu\rho}_{\gamma} +\beta_{4}RR_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} +\beta_{5}R_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma} +\beta_{6}R^{\nu}_{\mu}R^{\rho}_{\nu}R^{\mu}_{\rho} +\beta_{7}R_{\mu\nu}R^{\mu\nu}R +\beta_{8}R^{3}.$3 (Rudra, 2022).

3. Cosmological sectors, extensions, and asymptotic behavior

At the background level, RCHDE was constructed to support a local-curvature description of late acceleration without event-horizon nonlocality. In the noninteracting dust case, the dark-energy equation of state is obtained from

$P=\beta_{1}R_{\mu~~\nu}^{~~\rho~~\sigma}R_{\rho~~\sigma}^{~~\gamma~~\delta}R_{\gamma~~\delta}^{~~\mu~~\nu} +\beta_{2}R_{\mu\nu}^{\rho\sigma}R_{\rho\sigma}^{\gamma\delta}R_{\gamma\delta}^{\mu\nu} +\beta_{3}R^{\sigma\gamma}R_{\mu\nu\rho\sigma}{R^{\mu\nu\rho}_{\gamma} +\beta_{4}RR_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} +\beta_{5}R_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma} +\beta_{6}R^{\nu}_{\mu}R^{\rho}_{\nu}R^{\mu}_{\rho} +\beta_{7}R_{\mu\nu}R^{\mu\nu}R +\beta_{8}R^{3}.$4

and the paper derives an explicit redshift-dependent expression for $P=\beta_{1}R_{\mu~~\nu}^{~~\rho~~\sigma}R_{\rho~~\sigma}^{~~\gamma~~\delta}R_{\gamma~~\delta}^{~~\mu~~\nu} +\beta_{2}R_{\mu\nu}^{\rho\sigma}R_{\rho\sigma}^{\gamma\delta}R_{\gamma\delta}^{\mu\nu} +\beta_{3}R^{\sigma\gamma}R_{\mu\nu\rho\sigma}{R^{\mu\nu\rho}_{\gamma} +\beta_{4}RR_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} +\beta_{5}R_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma} +\beta_{6}R^{\nu}_{\mu}R^{\rho}_{\nu}R^{\mu}_{\rho} +\beta_{7}R_{\mu\nu}R^{\mu\nu}R +\beta_{8}R^{3}.$5 in terms of $P=\beta_{1}R_{\mu~~\nu}^{~~\rho~~\sigma}R_{\rho~~\sigma}^{~~\gamma~~\delta}R_{\gamma~~\delta}^{~~\mu~~\nu} +\beta_{2}R_{\mu\nu}^{\rho\sigma}R_{\rho\sigma}^{\gamma\delta}R_{\gamma\delta}^{\mu\nu} +\beta_{3}R^{\sigma\gamma}R_{\mu\nu\rho\sigma}{R^{\mu\nu\rho}_{\gamma} +\beta_{4}RR_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} +\beta_{5}R_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma} +\beta_{6}R^{\nu}_{\mu}R^{\rho}_{\nu}R^{\mu}_{\rho} +\beta_{7}R_{\mu\nu}R^{\mu\nu}R +\beta_{8}R^{3}.$6, $P=\beta_{1}R_{\mu~~\nu}^{~~\rho~~\sigma}R_{\rho~~\sigma}^{~~\gamma~~\delta}R_{\gamma~~\delta}^{~~\mu~~\nu} +\beta_{2}R_{\mu\nu}^{\rho\sigma}R_{\rho\sigma}^{\gamma\delta}R_{\gamma\delta}^{\mu\nu} +\beta_{3}R^{\sigma\gamma}R_{\mu\nu\rho\sigma}{R^{\mu\nu\rho}_{\gamma} +\beta_{4}RR_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} +\beta_{5}R_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma} +\beta_{6}R^{\nu}_{\mu}R^{\rho}_{\nu}R^{\mu}_{\rho} +\beta_{7}R_{\mu\nu}R^{\mu\nu}R +\beta_{8}R^{3}.$7, and $P=\beta_{1}R_{\mu~~\nu}^{~~\rho~~\sigma}R_{\rho~~\sigma}^{~~\gamma~~\delta}R_{\gamma~~\delta}^{~~\mu~~\nu} +\beta_{2}R_{\mu\nu}^{\rho\sigma}R_{\rho\sigma}^{\gamma\delta}R_{\gamma\delta}^{\mu\nu} +\beta_{3}R^{\sigma\gamma}R_{\mu\nu\rho\sigma}{R^{\mu\nu\rho}_{\gamma} +\beta_{4}RR_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} +\beta_{5}R_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma} +\beta_{6}R^{\nu}_{\mu}R^{\rho}_{\nu}R^{\mu}_{\rho} +\beta_{7}R_{\mu\nu}R^{\mu\nu}R +\beta_{8}R^{3}.$8. The deceleration parameter is written as

$P=\beta_{1}R_{\mu~~\nu}^{~~\rho~~\sigma}R_{\rho~~\sigma}^{~~\gamma~~\delta}R_{\gamma~~\delta}^{~~\mu~~\nu} +\beta_{2}R_{\mu\nu}^{\rho\sigma}R_{\rho\sigma}^{\gamma\delta}R_{\gamma\delta}^{\mu\nu} +\beta_{3}R^{\sigma\gamma}R_{\mu\nu\rho\sigma}{R^{\mu\nu\rho}_{\gamma} +\beta_{4}RR_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} +\beta_{5}R_{\mu\nu\rho\sigma}R^{\mu\rho}R^{\nu\sigma} +\beta_{6}R^{\nu}_{\mu}R^{\rho}_{\nu}R^{\mu}_{\rho} +\beta_{7}R_{\mu\nu}R^{\mu\nu}R +\beta_{8}R^{3}.$9

for dust $$\beta_{7}=\frac{1}{12}\left(3\beta_{1}-24\beta_{2}-16\beta_{3}-48\beta_{4}-5\beta_{5}-9\beta_{6}\right),$$0 (Rudra, 2022).

The model admits several qualitatively distinct regimes. The 2022 construction reports that RCHDE can exhibit quintessence-like behavior, phantom-divide crossing, and pure phantom behavior depending on the parameter space $$\beta_{7}=\frac{1}{12}\left(3\beta_{1}-24\beta_{2}-16\beta_{3}-48\beta_{4}-5\beta_{5}-9\beta_{6}\right),$$1. Increasing $$\beta_{7}=\frac{1}{12}\left(3\beta_{1}-24\beta_{2}-16\beta_{3}-48\beta_{4}-5\beta_{5}-9\beta_{6}\right),$$2, which controls the cubic sector, lowers both present and future values of $$\beta_{7}=\frac{1}{12}\left(3\beta_{1}-24\beta_{2}-16\beta_{3}-48\beta_{4}-5\beta_{5}-9\beta_{6}\right),$$3, with the effect strongest in the future epoch $$\beta_{7}=\frac{1}{12}\left(3\beta_{1}-24\beta_{2}-16\beta_{3}-48\beta_{4}-5\beta_{5}-9\beta_{6}\right),$$4. Increasing $$\beta_{7}=\frac{1}{12}\left(3\beta_{1}-24\beta_{2}-16\beta_{3}-48\beta_{4}-5\beta_{5}-9\beta_{6}\right),$$5, which controls the Ricci part, also decreases $$\beta_{7}=\frac{1}{12}\left(3\beta_{1}-24\beta_{2}-16\beta_{3}-48\beta_{4}-5\beta_{5}-9\beta_{6}\right),$$6, again significantly affecting future evolution. In the plotted examples, the transition from deceleration to acceleration occurs around

$$\beta_{7}=\frac{1}{12}\left(3\beta_{1}-24\beta_{2}-16\beta_{3}-48\beta_{4}-5\beta_{5}-9\beta_{6}\right),$$7

and the model can reproduce the expected radiation $$\beta_{7}=\frac{1}{12}\left(3\beta_{1}-24\beta_{2}-16\beta_{3}-48\beta_{4}-5\beta_{5}-9\beta_{6}\right),$$8 matter $$\beta_{7}=\frac{1}{12}\left(3\beta_{1}-24\beta_{2}-16\beta_{3}-48\beta_{4}-5\beta_{5}-9\beta_{6}\right),$$9 dark-energy sequence when radiation is included (Rudra, 2022).

The formalism is extended in two directions. First, an interacting case is introduced through

$$\beta_{8}=\frac{1}{72}\left(-6\beta_{1}+36\beta_{2}+22\beta_{3}+64\beta_{4}+3\beta_{5}+9\beta_{6}\right).$$0

with

$$\beta_{8}=\frac{1}{72}\left(-6\beta_{1}+36\beta_{2}+22\beta_{3}+64\beta_{4}+3\beta_{5}+9\beta_{6}\right).$$1

For dust,

$$\beta_{8}=\frac{1}{72}\left(-6\beta_{1}+36\beta_{2}+22\beta_{3}+64\beta_{4}+3\beta_{5}+9\beta_{6}\right).$$2

so the Hubble function becomes

$$\beta_{8}=\frac{1}{72}\left(-6\beta_{1}+36\beta_{2}+22\beta_{3}+64\beta_{4}+3\beta_{5}+9\beta_{6}\right).$$3

The paper states that $$\beta_{8}=\frac{1}{72}\left(-6\beta_{1}+36\beta_{2}+22\beta_{3}+64\beta_{4}+3\beta_{5}+9\beta_{6}\right).$$4 gives earlier dark-energy domination, while $$\beta_{8}=\frac{1}{72}\left(-6\beta_{1}+36\beta_{2}+22\beta_{3}+64\beta_{4}+3\beta_{5}+9\beta_{6}\right).$$5 delays it (Rudra, 2022).

Second, a radiation-inclusive case imposes

$$\beta_{8}=\frac{1}{72}\left(-6\beta_{1}+36\beta_{2}+22\beta_{3}+64\beta_{4}+3\beta_{5}+9\beta_{6}\right).$$6

and the model is then solved numerically. This is the setting in which the sequential radiation-dominated, matter-dominated, and dark-energy-dominated epochs are displayed. In one representative example, the parameters are taken as

$$\beta_{8}=\frac{1}{72}\left(-6\beta_{1}+36\beta_{2}+22\beta_{3}+64\beta_{4}+3\beta_{5}+9\beta_{6}\right).$$7

with present values

$$\beta_{8}=\frac{1}{72}\left(-6\beta_{1}+36\beta_{2}+22\beta_{3}+64\beta_{4}+3\beta_{5}+9\beta_{6}\right).$$8

The original paper does not perform statistical fitting in this sector; these are illustrative parameter choices (Rudra, 2022).

The late-time fate is also parameter-dependent. A de Sitter asymptotic solution is possible if

$$\beta_{8}=\frac{1}{72}\left(-6\beta_{1}+36\beta_{2}+22\beta_{3}+64\beta_{4}+3\beta_{5}+9\beta_{6}\right).$$9

This follows from the constant-$$\beta_{6}=4\beta_{2}+2\beta_{3}+8\beta_{4}+\beta_{5},$$0 limit, where

$$\beta_{6}=4\beta_{2}+2\beta_{3}+8\beta_{4}+\beta_{5},$$1

If instead the parameters drive $$\beta_{6}=4\beta_{2}+2\beta_{3}+8\beta_{4}+\beta_{5},$$2 asymptotically, the model can approach a Big Rip. The original proposal therefore supports multiple late-time fates: asymptotic de Sitter, persistent quintessence, phantom evolution, and possible Big Rip behavior (Rudra, 2022).

4. Empirical constraints and data-driven reconstruction

The first direct background-level confrontation of RCHDE with data uses Hubble data, cosmic chronometers, BAO, GRB data, and a joint Hubble+CC analysis, with MCMC sampling and Bayesian inference for the parameter triplet $$\beta_{6}=4\beta_{2}+2\beta_{3}+8\beta_{4}+\beta_{5},$$3. In the main parameter-estimation table, the analysis fixes

$$\beta_{6}=4\beta_{2}+2\beta_{3}+8\beta_{4}+\beta_{5},$$4

and uses the generic Hubble-type chi-square

$$\beta_{6}=4\beta_{2}+2\beta_{3}+8\beta_{4}+\beta_{5},$$5

The paper states that the combined datasets tighten the confidence regions relative to single-probe constraints, and that the joint Hubble+CC case gives the tightest reported bounds (Sanyal et al., 27 Aug 2025).

The reported best-fit values are

$$\beta_{6}=4\beta_{2}+2\beta_{3}+8\beta_{4}+\beta_{5},$$6

for BAO;

$$\beta_{6}=4\beta_{2}+2\beta_{3}+8\beta_{4}+\beta_{5},$$7

for Hubble;

$$\beta_{6}=4\beta_{2}+2\beta_{3}+8\beta_{4}+\beta_{5},$$8

for cosmic chronometers;

$$\beta_{6}=4\beta_{2}+2\beta_{3}+8\beta_{4}+\beta_{5},$$9

for GRB; and

$$\tilde{\beta}\equiv -\beta_1+4\beta_2+2\beta_3+8\beta_4.$$0

for the joint Hubble+CC fit (Sanyal et al., 27 Aug 2025).

The observational interpretation in that study is explicitly background-oriented. At low redshift ($$\tilde{\beta}\equiv -\beta_1+4\beta_2+2\beta_3+8\beta_4.$$1), RCHDE is said to closely mimic $$\tilde{\beta}\equiv -\beta_1+4\beta_2+2\beta_3+8\beta_4.$$2CDM. At higher redshift ($$\tilde{\beta}\equiv -\beta_1+4\beta_2+2\beta_3+8\beta_4.$$3), the model shows small deviations attributed to higher-curvature effects. The near-unity values of $$\tilde{\beta}\equiv -\beta_1+4\beta_2+2\beta_3+8\beta_4.$$4 in several fits are interpreted as suggesting Ricci-term dominance at late times, while $$\tilde{\beta}\equiv -\beta_1+4\beta_2+2\beta_3+8\beta_4.$$5 indicates a non-negligible cubic sector (Sanyal et al., 27 Aug 2025).

The same paper adds an auxiliary machine-learning analysis based on 30 observed $$\tilde{\beta}\equiv -\beta_1+4\beta_2+2\beta_3+8\beta_4.$$6 points over

$$\tilde{\beta}\equiv -\beta_1+4\beta_2+2\beta_3+8\beta_4.$$7

split into 20 training points and 10 test points. Six regressors are used: Enhanced Linear Regression, Physics-Informed Linear Regression, Enhanced Artificial Neural Network, Enhanced Support Vector Regression, Enhanced Random Forest Regression, and Gradient Boosting Regression. The best-performing model is reported to be Enhanced SVR, with

$$\tilde{\beta}\equiv -\beta_1+4\beta_2+2\beta_3+8\beta_4.$$8

The stated role of this analysis is validation of the model’s predictive power for $$\tilde{\beta}\equiv -\beta_1+4\beta_2+2\beta_3+8\beta_4.$$9, rather than primary parameter inference (Sanyal et al., 27 Aug 2025).

That observational study also contains several explicit caveats. It does not provide full prior ranges, sampler choice, chain length, burn-in, or convergence diagnostics. It does not write dedicated BAO or GRB likelihoods in the text, and it does not report AIC, BIC, or Bayes factors. It also quotes, in the machine-learning section, a separate best fit

$\tilde{\beta}$0

with

$\tilde{\beta}$1

which should be treated as a separate fit/reporting stream rather than as the same parameter set used in the main MCMC table (Sanyal et al., 27 Aug 2025).

5. Relation to Ricci-scale holography and inherited benchmark problems

RCHDE is structurally continuous with earlier Ricci-based holographic dark-energy models. In the standard flat-FLRW Ricci construction,

$\tilde{\beta}$2

so the infrared cutoff is effectively fixed by the Ricci curvature scale. RCHDE preserves this local-curvature logic but replaces the pure Ricci cutoff by the mixed invariant $\tilde{\beta}$3, with the Ricci model recovered when $\tilde{\beta}$4 (Belkacemi et al., 2013).

The Ricci-scale literature defines the immediate comparison class. Interacting Ricci-cutoff models in flat FRW already exhibit late acceleration, variable $\tilde{\beta}$5, possible phantom crossing, and partial alleviation of the coincidence problem, especially when the matter-to-dark-energy ratio tends to a finite asymptotic value under a dark-sector interaction [(Duran et al., 2010); (Durán et al., 2010)]. This means that not all late-time phenomenology seen in RCHDE is unique to the cubic sector; some of it is already present in linear Ricci HDE.

At the same time, Ricci-based models also expose the pathologies that any Ricci-cubic extension must confront. In standard four-dimensional holographic Ricci dark energy, the future is asymptotically de Sitter for $\tilde{\beta}$6 and ends in a Big Rip for $\tilde{\beta}$7. In a DGP braneworld with Gauss–Bonnet curvature corrections, the original Big Rip can be replaced by either de Sitter evolution or a Big Freeze singularity, depending on parameters. This shows that higher-curvature structures can change the kind of doomsday rather than simply eliminate it (Belkacemi et al., 2013). A plausible implication is that RCHDE should also be analyzed for critical parameter surfaces separating de Sitter-like, phantom, and singular futures.

Perturbative viability is an equally important benchmark. A gauge-invariant perturbation analysis of noninteracting Ricci dark energy found that background fits alone are insufficient: the perturbation dynamics is plagued by instabilities that exclude phantom-type equations of state, and the only stable configuration is singled out by

$\tilde{\beta}$8

a relation only marginally consistent with the observationally preferred background values in that model (Campo et al., 2013). This is not a direct result for RCHDE, but it sharply limits the interpretive value of background-only fits and suggests that any Ricci-cubic model must be checked beyond the homogeneous expansion history.

A separate Ricci-based line treats holographic Ricci dark energy as a running vacuum. In one such construction, the addition of a constant vacuum term regularizes the future evolution, removes the dangerous homogeneous mode, and drives the universe to a de Sitter asymptote for

$\tilde{\beta}$9

This establishes another benchmark lesson: local-curvature holographic densities may require additive vacuum structure or conservation-law restrictions to avoid pathological late-time behavior (George et al., 2015). For RCHDE, this suggests that cubic corrections alone need not guarantee regularity.

6. Theoretical status and open directions

RCHDE is presently best understood as a geometrically motivated phenomenological model rather than as a fully derived modified-gravity theory. The 2022 proposal is explicit on this point: the construction is inspired by higher-curvature invariants, but it is not derived from a full action principle for dark energy, the master evolution equation generally has no exact analytic solution, the analytic treatment relies on a linearized/binomial approximation, and no full dynamical-system analysis, observational parameter constraints, or perturbative stability analysis are carried out there (Rudra, 2022).

The 2025 observational follow-up materially advances the background phenomenology, but it remains limited to expansion-history tests. No perturbation-level constraints are imposed, no information criteria are reported for formal comparison with $\lambda=0$0CDM, and no explicit reconstruction of $\lambda=0$1, $\lambda=0$2, the transition redshift, or the age of the universe is given in that paper’s statistical section (Sanyal et al., 27 Aug 2025). As a result, current evidence supports RCHDE as observationally viable at the background level, but not as decisively superior to $\lambda=0$3CDM.

The surrounding 2025 generalized Ricci literature also clarifies what RCHDE is not. Generalized holographic and generalized Ricci models can interpolate analytically between $\lambda=0$4-based and $\lambda=0$5-based cutoffs, and they support extensive diagnostics and scalar-field correspondences, but they do not introduce cubic curvature invariants. They therefore function as adjacent Ricci-type baselines rather than as alternative formulations of RCHDE proper (Pasqua, 21 Sep 2025).

The present state of the subject therefore has a clear structure. RCHDE has a well-defined local invariant cutoff, a flat-FLRW reduction, a parameter-dependent late-time phenomenology, and initial background constraints. What remains open are the stability of perturbations, the singularity structure outside the illustrative sectors already explored, the robustness of statistical inference under fully specified likelihoods and priors, and the extent to which the cubic sector provides genuinely new explanatory power beyond what is already achievable in lower-order Ricci holography.