Current and future constraints on the expansion history of the GREA model

1. Problem Statement

The paper addresses the theoretical and observational challenges facing the standard $\Lambda$CDM cosmological model, specifically the cosmological constant problem (the discrepancy between the observed value of $\Lambda$ and quantum field theory predictions) and the Hubble tension (the discrepancy between early-universe $H_0$ inferred from CMB and late-universe $H_0$ measured locally).

The authors investigate the General Relativistic Entropic Acceleration (GREA) framework as a viable alternative. Unlike $\Lambda$CDM, which attributes acceleration to a fundamental dark energy fluid, GREA posits that late-time acceleration is an emergent phenomenon arising from entropy production associated with the cosmological horizon. The study aims to:

1. Test the GREA model against current observational data.
2. Introduce and analyze a modified GREA theory that includes an additional dark energy component ($\Omega_{de}$) to allow for a continuous interpolation between entropic acceleration and $\Lambda$CDM.
3. Forecast the ability of future surveys (SKAO, LSST, Einstein Telescope) to discriminate between these models using Bayesian model comparison.

2. Methodology

Theoretical Framework

• GREA Theory: Based on the thermodynamics of the cosmological horizon, the Einstein field equations are modified by an entropic force tensor $f_{\mu\nu}$. The acceleration arises from the horizon's entropy growth. The model is governed by a specific geometric parameter, $\sqrt{-k}\eta_0$ (related to the rescaled conformal time and curvature), which determines the late-time expansion history.
  • Key Distinction: In GREA, the physical expansion rate today, $H(0)$, is a derived quantity from the numerical solution of the modified Friedmann equations and is not necessarily equal to the parameter $H_0$ used for normalization.
• Modified GREA: An empirical extension adding a constant dark energy term ($\Omega_{de}$) to the Friedmann equation. This allows the model to recover the $\Lambda$CDM limit when the entropic contribution becomes negligible or when $\Omega_{de}$ dominates.

Data and Simulations

• Current Data: The authors combined four complementary datasets:
  • BAO: DESI DR2 data (7 redshift bins, $z=0.3-2.33$).
  • SNIa: Pantheon+ catalogue (1701 light curves, $z=0.001-2.26$).
  • CC: Cosmic Chronometers (direct $H(z)$ measurements).
  • CMB: Compressed Planck+ACT (P-ACT) data ($100\theta_*, \omega_b, \omega_{cb}$).
• Mock Data (Forecasts): Realistic mock datasets were generated for next-generation surveys:
  • SKAO: BAO measurements (21-cm line, $z=0.2-2.0$).
  • LSST: Type Ia Supernovae ($z=0.1-1.0$).
  • ET (Einstein Telescope): Gravitational Wave (GW) standard sirens from binary neutron star mergers ($N=20,000$ events).
  • Fiducial Cosmologies: Mocks were generated assuming both GREA and $\Lambda$CDM as the true underlying universe to test model recovery and bias.

Statistical Analysis

• Inference: Bayesian parameter estimation using the Nautilus nested sampling algorithm (via the Cobaya framework).
• Model Comparison: Computation of Bayesian Evidence ($Z$) and Bayes Factors ($B_{12}$) relative to $\Lambda$CDM (or GREA for forecasts), interpreted via the Jeffreys scale to determine statistical preference.
• Priors: Flat priors were used for geometric parameters ($\sqrt{-k}\eta_0$) and $\Omega_{de}$. A Gaussian prior on $M_B$ (absolute magnitude) was used to break degeneracies with $H_0$.

3. Key Contributions

1. Implementation and Validation: The authors developed a custom code to numerically integrate the GREA Friedmann equations and validated it against previous literature, confirming consistency within $1\sigma$.
2. Phenomenological Extension: They introduced the "Modified GREA" model, creating a bridge between pure entropic acceleration and standard $\Lambda$CDM, allowing for a rigorous test of whether an explicit dark energy component is required by data.
3. GW Standard Sirens as Discriminators: The study highlights the unique power of future GW observations to break degeneracies between entropic and dark-energy-driven expansion, as GWs provide absolute distance measurements independent of the cosmic distance ladder.
4. Comprehensive Forecasting: The paper provides the first detailed forecast analysis of GREA using the combined power of SKAO, LSST, and ET, assessing both parameter recovery and model selection capabilities.

4. Results

Current Data Constraints

• $\Lambda$CDM Preference: When compressed CMB data is included, $\Lambda$CDM is strongly preferred over GREA.
  • For DESI+P-ACT+CC+Pantheon+, $\Delta \log Z = -9.36$ (Strong evidence against GREA).
  • The Modified GREA performs better ($\Delta \log Z = -3.88$) but is still disfavored compared to $\Lambda$CDM.
• Low-Redshift Competitiveness: Without CMB data (e.g., DESI+CC+Pantheon+), GREA becomes competitive ($\Delta \log Z = -2.76$), suggesting that current low-redshift tensions might be accommodated by GREA.
• Parameter Shifts: GREA prefers a higher present-day expansion rate, $H(0)$, than $\Lambda$CDM for the same datasets, potentially alleviating the Hubble tension by shifting $H(0)$ closer to local probe values.
• Degeneracies: In the Modified GREA, $\Omega_{de}$ exhibits strong degeneracy with the geometric parameter $\sqrt{-k}\eta_0$. A large prior range for $\sqrt{-k}\eta_0$ was necessary to properly explore the parameter space and allow the $\Lambda$CDM limit.

Forecast Results (Mock Data)

• Recovery of True Cosmology:
  • If GREA is true: Future data (especially with GWs) successfully recovers GREA parameters and strongly disfavors $\Lambda$CDM ($\Delta \log Z = -7.95$). The Modified GREA also disfavors the unnecessary $\Omega_{de}$ term, correctly converging to $\Omega_{de} \approx 0$.
  • If $\Lambda$CDM is true: GREA fails to reproduce the expansion history without significant systematic shifts in $H(0)$ and $\Omega_m$. The Modified GREA can reproduce the data via a non-zero $\Omega_{de}$, but the model is penalized by Bayesian evidence due to the extra free parameter ($\Delta \log Z \approx -0.9$).
• Role of GWs: The inclusion of Einstein Telescope (ET) data significantly tightens constraints and enhances the statistical power to distinguish between models. GW standard sirens are particularly effective at breaking the degeneracy between entropic effects and dark energy because they measure luminosity distance directly.

5. Significance and Conclusion

The paper concludes that while $\Lambda$CDM remains the statistically preferred model given current data (driven largely by CMB constraints), the GREA framework is a viable and competitive alternative for low-redshift cosmology.

The study demonstrates that future multi-messenger observations, particularly Gravitational Wave standard sirens, will be decisive in testing the nature of cosmic acceleration. They will be able to:

1. Distinguish between horizon-thermodynamics-driven acceleration and fundamental dark energy.
2. Identify whether the observed expansion history requires an explicit cosmological constant or if it can be explained by entropic effects.
3. Resolve the Hubble tension by determining if the true expansion rate aligns with local probes (favored by GREA) or CMB inferences (favored by $\Lambda$CDM).

The authors suggest that future work should incorporate linear perturbation theory and full CMB likelihoods to further test the growth of structure within the GREA framework.