Revisiting the Hubble tension problem in the framework of holographic dark energy

This paper systematically analyzes six holographic dark energy models against multiple observational datasets, concluding that while models using the Hubble scale as an infrared cutoff fail to resolve the Hubble tension, those utilizing the future event horizon can partially alleviate it.

The Gist

The Great Cosmic Speed Limit Dispute: A Holographic Detective Story

Imagine the universe is a giant, expanding balloon. For decades, astronomers have been trying to measure exactly how fast this balloon is inflating right now. This speed is called the Hubble Constant ($H_0$).

Here is the problem: We have two very different ways of measuring this speed, and they don't agree. It's like asking two people to measure the speed of a car, and one says "60 mph" while the other says "73 mph." In the world of physics, a difference this big is a crisis. It's called the Hubble Tension.

• Team Early Universe (The "Planck" Team): They look at the baby photos of the universe (the Cosmic Microwave Background) and calculate the speed based on how the universe started. They get a slow speed: 67.4.
• Team Late Universe (The "SH0ES" Team): They look at the universe today, measuring the distance to nearby exploding stars (Supernovae). They get a fast speed: 73.2.

The gap between them is so wide that it's statistically impossible for them to be right at the same time under our current rules of physics. Something is missing from our rulebook.

The New Theory: The Holographic Universe

Enter the authors of this paper, Jun-Xian Li and Shuang Wang. They are testing a wild idea called Holographic Dark Energy (HDE).

The Analogy: Imagine a 3D movie projected onto a 2D screen. The "holographic principle" suggests that all the information in our 3D universe might actually be encoded on a 2D boundary, like a hologram. If this is true, the energy driving the universe's expansion (Dark Energy) isn't just random; it's determined by the size of this "screen" or boundary.

The big question is: What is the size of this screen? The paper tests six different theories about what this "screen" looks like.

The Six Suspects (The Models)

The researchers picked six different "detectives" (theoretical models) to see which one could solve the speed limit dispute. They split them into two teams based on how they define the "screen" (the boundary):

Team A: The "Current Speed" Team (Hubble Scale)

These models say the boundary is determined by the current speed of expansion (the Hubble scale).

• The Models: Generalized Ricci, Interacting HDE (Type 1), Tsallis.
• The Result: Total Failure.
  • The Metaphor: Imagine trying to fix a broken clock by looking at the hands moving right now. These models just ended up confirming the slow speed (67.4). They couldn't push the number up to match the fast measurements. They are stuck in the "slow lane."

Team B: The "Future Horizon" Team (Event Horizon)

These models say the boundary is the future event horizon. This is the farthest point we will ever be able to see in the future, even if we wait forever. It's a "look ahead" boundary.

• The Models: Original HDE, Barrow HDE, Interacting HDE (Type 2).
• The Result: Partial Success.
  • The Metaphor: These models are like looking at the horizon of a road trip. By looking further into the future, they allow the universe to expand a bit faster now.
  • They managed to push the calculated speed up from 67.4 to somewhere between 69 and 71.
  • Did they solve the mystery? Not completely. The gap shrank from a massive 5-sigma (a huge, screaming discrepancy) to a more manageable 1.7 to 3.2 sigma. It's like going from "The car is definitely speeding" to "The car might be speeding, but we're not 100% sure yet."

The Twist: New Data Makes it Harder

The researchers didn't just use old data. They used the DESI (Dark Energy Spectroscopic Instrument) data, which is the latest, most precise map of the universe we have.

• The Good News: The "Future Horizon" models still worked better than the standard theory (Lambda-CDM).
• The Bad News: Adding more data (like new supernova measurements) actually made the tension worse for everyone. It's like adding more witnesses to a crime scene; sometimes, the more you look, the more confusing the picture gets.

The Verdict: A Trade-Off

The paper concludes with a crucial insight: You can't have it all.

1. The "Hubble Scale" models fit the data perfectly but fail to fix the speed limit problem. They are boring but accurate.
2. The "Future Horizon" models fix the speed limit problem (partially) but are "ugly" statistically. They require more complex math and don't fit the other data points as neatly as the standard model.

The Simple Takeaway:
The universe is like a puzzle where the pieces don't quite fit. The authors tried six different ways to reshape the pieces using the "holographic" idea. They found that only the pieces that look into the future (the event horizon) can make the picture look a little less broken. However, even those pieces don't make the picture perfect.

The Hubble Tension is still a mystery, but this paper tells us that if we want to solve it, we probably need to stop looking at how fast the universe is expanding now and start thinking about where it is going next.

Technical Summary

1. Problem Statement

The paper addresses the Hubble tension, a significant discrepancy (>5$\sigma$) between the locally measured Hubble constant ($H_0 \approx 73.17$ km s$^{-1}$ Mpc$^{-1}$ from SH0ES) and the value inferred from early-universe Cosmic Microwave Background (CMB) data under the standard $\Lambda$CDM model ($H_0 \approx 67.4$ km s$^{-1}$ Mpc$^{-1}$ from Planck 2018).

While various dynamical Dark Energy (DE) models have been proposed to resolve this, Holographic Dark Energy (HDE) models have been studied individually in the past. The authors argue that a systematic comparison across all four theoretical categories of HDE is necessary to determine if the choice of the Infrared (IR) cutoff is the critical factor in alleviating the tension.

2. Methodology

Theoretical Framework

The study analyzes six representative HDE models selected from the four main categories of HDE theory:

1. Other Characteristic Length Scale: Original HDE (OHDE) using the future event horizon as the IR cutoff.
2. Extended Hubble Scale: Generalized Ricci HDE (GRDE) using a combination of the Hubble scale and its time derivative.
3. Dark Sector Interaction:
   • IHDE1: Interaction with the Hubble scale as the IR cutoff.
   • IHDE2: Interaction with the future event horizon as the IR cutoff.
4. Modified Black Hole Entropy:
   • Tsallis HDE (THDE): Uses the Hubble scale.
   • Barrow HDE (BHDE): Uses the future event horizon.

The $\Lambda$CDM model serves as the fiducial benchmark.

Observational Data

The authors utilize the latest cosmological datasets to constrain model parameters:

• Baryon Acoustic Oscillations (BAO):
  • Primary: DESI Data Release 2 (DR2) (covering BGS, LRG, ELG, QSO, and Ly$\alpha$ tracers).
  • Alternative: A compilation of previous BAO measurements (6dFGS, SDSS, BOSS, eBOSS, DESY3).
• Cosmic Microwave Background (CMB): Distance priors from Planck 2018 (shift parameter $R$ and acoustic scale $l_a$).
• Type Ia Supernovae (SN):
  • Primary: DESY5 compilation.
  • Alternatives: PantheonPlus and Union3 compilations.

Statistical Analysis

• Parameter Estimation: Markov Chain Monte Carlo (MCMC) simulations using the Cobaya code to derive posterior distributions for parameters ($\Omega_m$, $H_0$, and model-specific parameters like $C$, $\lambda$, $\beta$, etc.).
• Goodness of Fit: Evaluated using the $\chi^2$ statistic.
• Model Selection: Assessed via Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) relative to $\Lambda$CDM ($\Delta$AIC, $\Delta$BIC).
• Tension Metric: The significance of the Hubble tension is calculated as the difference between the model's inferred $H_0$ and the SH0ES measurement, expressed in standard deviations ($\sigma$).

3. Key Results

Impact of IR Cutoff Choice

The study reveals a stark dichotomy based on the choice of the IR cutoff:

• Hubble Scale Cutoffs (No Alleviation): Models using the Hubble scale or its extensions (GRDE, IHDE1, THDE) fail to resolve the tension. They yield $H_0$ values close to the $\Lambda$CDM result, resulting in tensions ranging from 5.12$\sigma$ to 6.31$\sigma$.
• Future Event Horizon Cutoffs (Partial Mitigation): Models using the future event horizon (OHDE, IHDE2, BHDE) successfully raise the inferred $H_0$ value, significantly reducing the tension.
  • DESI + CMB: Tensions reduced to 1.74$\sigma$ (OHDE), 3.19$\sigma$ (IHDE2), and 2.65$\sigma$ (BHDE).
  • DESI + CMB + DESY5: Tensions reduced to 2.49$\sigma$ (OHDE), 4.49$\sigma$ (IHDE2), and 3.12$\sigma$ (BHDE).

Robustness Across Datasets

The conclusion that "Future Event Horizon" models alleviate tension while "Hubble Scale" models do not holds true across different data combinations:

• Alternative BAO: Replacing DESI with non-DESI BAO data further reduces tensions for future-horizon models (e.g., OHDE drops to 0.95$\sigma$), though with larger error bars.
• Alternative SN: Adding PantheonPlus or Union3 SN data increases the tension compared to DESI+CMB alone (due to tighter constraints on $H_0$), but future-horizon models still outperform $\Lambda$CDM (tensions remain < 4.65$\sigma$ vs. > 5.4$\sigma$ for $\Lambda$CDM).

Statistical Trade-off (AIC/BIC)

While future-event-horizon models alleviate the tension, they suffer from poorer statistical performance compared to $\Lambda$CDM and Hubble-scale models.

• Models like OHDE and BHDE have high $\Delta$AIC and $\Delta$BIC values, indicating they are less favored by the data due to increased model complexity (more parameters) required to achieve the higher $H_0$.
• The improvement in $H_0$ is achieved at the cost of a worse fit to the overall cosmological dataset.

4. Physical Interpretation

The authors attribute the difference in performance to the evolution of the Dark Energy Equation of State ($w$):

• Hubble Scale Models: $w(z)$ evolves from $w > -1$ to $w \approx -1$ at late times, mimicking $\Lambda$CDM behavior and failing to shift $H_0$ significantly.
• Future Event Horizon Models: $w(z)$ evolves from $w > -1$ to $w < -1$ (phantom regime) at late times. This dynamical evolution introduces parameter degeneracies between $H_0$ and $w(z)$, allowing the model to accommodate a higher $H_0$ value consistent with local measurements, albeit with larger uncertainties.

5. Significance and Conclusion

• Systematic Validation: This is the first comprehensive study to test all four categories of HDE models against the latest DESI DR2 data.
• Cutoff Dependency: The study definitively establishes that the choice of IR cutoff is the determining factor for HDE's ability to address the Hubble tension. Only models utilizing the future event horizon show promise.
• Limitations: While future-horizon models reduce the tension, they do not fully resolve it (remaining at ~2.5–4.5$\sigma$) and are statistically disfavored compared to $\Lambda$CDM.
• Future Directions: The authors suggest that resolving the tension completely may require dynamical DE models that offer both a high $H_0$ and a good statistical fit, or the inclusion of new probes like standard sirens (gravitational waves) to break parameter degeneracies.

In summary, the paper concludes that while HDE models based on the future event horizon can partially mitigate the Hubble tension, they currently do so at the expense of statistical preference, leaving the problem unsolved within the current observational framework.