Observational constraints on fractional holographic dark energy in the light of DESI DR2

Based on the paper, here is the summary:

Based on the fractional entropy from fractional quantum mechanics, fractional holographic dark energy (FHDE) has been proposed with the Hubble horizon as the IR cutoff (FHDEH). This paper extends this framework by adopting the future event horizon and the particle horizon as the IR cutoff, proposing the FHDEF and FHDEP models. Using the SN+OHD+DESI DR2 dataset to constrain these models, the authors find that all three models provide a marginally lower χ2min compared to ΛCDM but without significant preference according to AIC and BIC. When CMB distance priors are included, the FHDEH and FHDEP models are strongly ruled out. The authors further analyze the cosmological evolution for these models, and find that only the FHDEF model predicts nearly identical evolutions of omegam and omegade to those of the ΛCDM model across cosmic history, but its deceleration parameter q deviate from the ΛCDM model in the future, indicating richer late time dynamics beyond the standard ΛCDM cosmology.

Specifically, when the dataset SN+OHD+DESI DR2 is used to constrain the FHDE models, the minimum χ2min values for FHDEH, FHDEF, and FHDEP are 1426.7, 1426.8, and 1430.9, respectively, all lower than that of the ΛCDM model (1431.5). The ∆AIC values for FHDEH, FHDEF, and FHDEP are −2.8, −0.7, and 3.4, respectively, with the AIC of FHDEH and FHDEF slightly lower than that of the ΛCDM model, indicating a marginal preference according to the AIC criterion. The ∆BIC values for FHDEH, FHDEF, and FHDEP are 2.6, 10.1, and 14.2, respectively, all higher than that of the ΛCDM model, reflecting a penalty for the extra parameters α and C. These results indicate that the BIC strongly favors the ΛCDM model over FHDEF and FHDEP models, and only slightly disfavors the FHDEH model.

When the dataset SN+OHD+DESI DR2+CMB, which includes the CMB distance priors, is used, the FHDEH and FHDEP models are strongly disfavored by the data, as the values of ∆χ2min exceed 200 (200.2 and 250.5, respectively). For the FHDEF model, the values of ∆χ2min, ∆AIC, and ∆BIC are −0.8, 3.2, and 14, respectively, suggesting that the FHDEF model yields a slightly lower χ2min than ΛCDM model, but it is strongly disfavored by the BIC due to the penalty for extra parameters. The FHDEF model remains the only FHDE model that survives the inclusion of CMB distance priors.

Using the mean values obtained from the dataset SN+OHD+DESI DR2+CMB, the authors solve the corresponding dynamical equation and plot the evolutionary curves of cosmological parameters for the FHDE models. They find that the FHDEF model predicts evolutions of omegade and omegam that are nearly identical to those of ΛCDM across the entire cosmic history, but q deviate from ΛCDM in the future, while the FHDEH and FHDEP models exhibit significant deviations. For the FHDEF model, ωde behaves as quintessence in the past, has recently crossed the phantom divide, and currently lies in the phantom regime, indicating a quintom-like behavior that emerged at very late times. This phantom behavior becomes increasingly severe in the future, leading to a big rip singularity, which is qualitatively different from the ΛCDM model.

To further investigate the evolution of the universe in the FHDEF model, the authors plot its phase space trajectories in the (omegam, omegade) plane and the (omegam, omegade, F) space using the mean parameter values from the dataset SN+OHD+DESI DR2+CMB. The evolution of the universe in the FHDEF model can be summarized as follows: the universe originates from the radiation dominated epoch P1, then passes through the pressureless matter dominated epoch P2 and the dark energy dominated epoch P4 successively, and eventually evolves parallel to the F axis with omegam = 0 and omegade = 1. The results show that, after passing through the ΛCDM-like evolutionary stages, the universe does not converge to P3 but continues to evolve further, revealing a richer dynamical behavior beyond the standard ΛCDM cosmology. The eigenvalue analysis reveals that P1 is an unstable node, P2 and P4 are saddle points, while P3 is a stable attractor, indicating that the FHDEF model admits a stable late time attractor P3, consistent with the observed accelerated expansion, whereas the cosmological constant like point P4 is only a saddle point and does not serve as the final evolutionary state of the universe.

Improvements to AI Systems Based on This Paper:

1. Enhanced Bayesian Model Comparison in Cosmology:

The AI can be improved to automatically compute and report ΔAIC, ΔBIC, and Δχ2 min alongside parameter constraints. This allows the system to flag models that are statistically indistinguishable from ΛCDM (e.g., FHDEF) versus those strongly ruled out (e.g., FHDEH, FHDEP with CMB priors), preventing overinterpretation of marginal χ2 improvements.

2. Dynamic System Stability Analysis for Dark Energy Models:

The AI can integrate phase-space trajectory plotting and eigenvalue analysis (unstable nodes, saddle points, stable attractors) for any given dark energy model. This enables automatic classification of late-time attractors (e.g., P3 vs. P4) and prediction of future singularities (e.g., big rip) without manual derivation.

3. Cross-Dataset Robustness Scoring:

The AI can be trained to evaluate model viability across multiple observational datasets (e.g., SN+OHD+DESI DR2 vs. adding CMB distance priors). It can output a robustness score indicating whether a model's preference persists or collapses when new data are included—useful for flagging models that are only favored by incomplete datasets.

4. Automated Phantom Divide Detection and Quintom Classification:

The AI can analyze the equation-of-state parameter ω de(z) to automatically detect phantom crossing events and classify the dark energy behavior (quintessence, phantom, quintom). This would allow real-time monitoring of observational data for signs of evolving dark energy beyond ΛCDM.

5. Predictive Extrapolation of Cosmic Evolution:

Using fitted model parameters, the AI can generate future evolution curves (e.g., q, ω de, omega m) and automatically compare them to ΛCDM predictions. It can then issue alerts for qualitative differences (e.g., future deceleration, big rip) that are not present in the standard model.

6. Model Selection with Parameter Penalty Awareness:

The AI can be enhanced to explicitly separate goodness-of-fit (χ2 min) from model complexity penalties (BIC). This prevents the system from recommending models that merely overfit, as seen where FHDEF has lower χ2 but is strongly disfavored by BIC due to extra parameters (α, C).

7. Phase-Space Trajectory Interpretation for Non-Experts:

The AI can translate complex phase-space plots (e.g., in (omega m, omega de, F) space) into plain-language summaries, such as the universe passes through radiation, matter, and dark-energy-dominated epochs, then approaches a stable attractor rather than a cosmological-constant-like saddle point. This improves interpretability for researchers.

8. Data-Driven Prior Sensitivity Analysis:

The AI can automatically test how inclusion of specific datasets (e.g., CMB distance priors) changes parameter constraints and model rankings. It can highlight which data points drive the rejection of certain models, aiding in identifying systematic tensions.