Holographic dark energy from generalized entropies and extended infrared cut-offs

The gist The study investigates holographic dark energy models constructed from generalized entropies and extended infrared cut-offs against joint DESI DR2 BAO, Pantheon+ and Planck compressed CMB likelihoods to determine observationally viable combinations

Holographic Dark Energy Construction

The holographic dark energy (HDE) models are based on the principle that the vacuum energy inside a region of size L cannot exceed that of a black hole of the same size, leading to an energy density relation such as ρDE ∝ M 2PlL-2 The choice of infrared cut-off L is crucial, with various possibilities including the Hubble scale (L = 1/H), the particle horizon Lp, and the future event horizon Lf Any HDE represents a particular case of the Nojiri–Odintsov (NO) generalized HDE The cosmological evolution is determined by combining the definition of the fractional dark energy density omegaDE(z) = ρDE(z)/3M 2PlH 2(z) with the Friedmann equation

Generalized Entropies and Cut-offs

The framework combines generalized entropy functionals, such as Barrow entropy [40], with local infrared cut-offs to create nine models The three entropies studied are Barrow, S3, and S5, each combined with three local IR cut-offs: the Granda–Oliveros (GO) scale, cutoff 2 (adding a γH-2 term), and cutoff 3 (adding a γ H/H¨ term) The GO cut-off is attractive because it depends only on local, present-epoch quantities and avoids causality issues associated with the future event horizon

Model Analysis and Constraints

The analysis constrained all nine models against a joint dataset combining DESI DR2 BAO measurements [8], Pantheon+ Type Ia supernova compilation [56], and a Planck 2018 compressed CMB likelihood [3, 57] The aims were to determine which entropy–cut-off combinations are observationally viable, assess whether the extended cut-offs provide genuine dynamical leverage beyond GO or merely add unconstrained parameters, and extract physical lessons from parameter degeneracies

1. The Hubble constant remains stable across all models, H0 ≃ 67.0–68.1 km s−1 Mpc−1

2. Cut-off 2 (γ H-2 term) is unconstrained in every case, as it becomes relevant only in the far future, and never improves on the GO cut-off

3. The H/H¨ term in cut-off 3 is tightly constrained for Barrow and S5, where it improves the fit substantially but not for S3

4. The best fit is obtained for Barrow entropy with the H/H¨ cut-off, yet the Barrow exponent remains compatible with zero in all cases

Sector-Specific Findings

(Barrow Entropy)

With the GO cut-off, the Barrow exponent is tightly bounded to ∆ < 0.00432 (68% C.L.), consistent with the Bekenstein–Hawking limit ∆ = 0 Moving to cut-off 3 produces a qualitatively different result, where the shooting parameter omega′DE(0) is tightly constrained to −0.594+0.017−0.016, demonstrating that the data are sensitive to the second-order dynamics

(S3 Entropy)

The GO cut-off yields the best fit in this sector and the second-best overall (∆AIC = 3.09) Cut-off 2 fails to improve the fit, as γr is prior-dominated, confirming that the γ H-2 correction becomes relevant only in the asymptotic future

(S5 Entropy)

S5 + cut-off 3 is the only model in this sector for which the extended cut-off makes a material difference, with both cut-off parameters tightly constrained and achieving the best fit in the S5 sector (∆AIC = 17.61)

Cross-cutting Patterns and Conclusion

Cut-off 2 never improves the fit across all three entropy cases, as the parameter multiplying the H-2 term is entirely prior-dominated, confirming that this term only becomes dynamically relevant in the asymptotic future Cut-off 3 shows a sharp dichotomy: for Barrow and S5, the second-order dynamics is genuinely probed by the data, while for S3, the shooting parameter remains prior-dominated and γr is driven to zero The overall best fit is Barrow + cut-off 3, followed by S3 + GO (∆AIC = 3.09) and S5 + cut-off 3 (∆AIC = 17.61) The data favour a higher-order infrared cut-off in two of the three entropy sectors, while showing no significant preference for a departure from the Bekenstein–Hawking area law in the Barrow sector

The paper concludes that meaningful extensions of the GO cut-off must act at intermediate redshifts where BAO and supernovae carry constraining power, rather than in asymptotic regimes The data favour such a higher-order cut-off in two of the three entropy sectors, while showing no significant preference for a departure from the Bekenstein–Hawking area law in the Barrow sector. This systematic baseline provides a reference against which more refined tests of holographic dark energy can be measured The study spans a 3 × 3 grid of nine models—Barrow, S3 and S5 entropies, each combined with the GO cut-off and cut-offs 2 and 3—with the Bekenstein–Hawking case recovered in the ∆ → 0 limit of the Barrow models. The BIC confirms Barrow + cut-off 3 as the preferred model (∆BIC = 24.61 for S3 + GO and 60.66 for S5 + cut-off 3) The Barrow exponent stays near the classical limit, with only Barrow + cut-off 3 allowing a non-zero central value compatible with zero at about 1.4σ. The paper is organized as follows. Section II introduces the HDE construction based on Bekenstein–Hawking entropy and derives the evolution equations for the GO cutoff and cut-offs 2 and 3 Sections III, IV and V extend the framework to the Barrow, S3 and S5 entropies, highlighting the qualitative change of structure—most notably the appearance of a second-order (cut-off 3) or higher-order system—that arises for each Section VI describes the observational datasets and likelihood construction, and details the MCMC methodology and priors Section VII presents the constraints and model comparison for all nine combinations, organized by entropy sector, together with the cross-cutting patterns Our conclusions are summarized in Section VIII. The full posterior distributions are collected in Appendix B; here we summarize the main findings, organized by entropy sector The parameter space and uniform priors adopted for each entropy–cut-off combination are summarized in Table I Table II, III, and IV list the marginalised constraints (68% C.L.) and the information criteria ∆AIC and ∆BIC [73–75] The full parameter space for a given model is the union of the base parameters, the cutoff-specific block, and the entropy-specific block The Hubble constant is in units of km s−1 Mpc−1.

1. textbf Rewriting Dark Energy Model Selection Logic for AI Training: The improved system can now distinguish between physically relevant extensions and unconstrained parameters by prioritizing models with high information-criteria penalties, as shown in Section VII: The overall best fit is Barrow + cut-off 3, followed by S3 + GO (∆AIC = 3.09) and S5 + cut-off 3 (∆AIC = 17.61), while all other models, including Barrow + GO, are disfavoured by ∆AIC ≳ 140.

2. textbf Dynamical System Sensitivity Analysis: The AI can now perform a second-order dynamics sensitivity check by analyzing the constraints on the shooting parameter, noting that for S3 and S5: For S3, the shooting parameter remains prior-dominated and γr is driven to zero, whereas for Barrow and S5: the second-order dynamics is genuinely probed by the data.

3. textbf Model Hierarchy Mapping: The system can map the observational preference hierarchy directly from the results, identifying that Cut-off 3 shows a sharp dichotomy where it probes the second-order dynamics, contrasting this with Cut-off 2, which is characterized by constraints being entirely prior-dominated.

4. textbf Parameter Degeneracy Identification: The AI can quantify parameter degeneracies by identifying cases where the data do not distinguish between alternatives; for instance, in S5 + GO, the results show broad, mutually correlated posteriors, indicating that the five-parameter entropy functional is over-parameterized for the current dataset.