General Relativistic Entropic Acceleration at the perturbation level: a CLASS implementation and first Boltzmann-code constraints

Technical Summary: General Relativistic Entropic Acceleration at the perturbation level

Problem Statement
The origin of the late-time accelerated expansion of the Universe remains an open problem. The standard $\Lambda$CDM model, while successful, relies on a cosmological constant $\Lambda$ that suffers from severe fine-tuning and coincidence problems. Furthermore, observational tensions, particularly the Hubble constant ($H_0$) discrepancy and the amplitude of matter fluctuations ($S_8$), suggest potential deviations from $\Lambda$CDM. Previous phenomenological approaches often rely on arbitrary parametrizations of the dark energy equation of state, $w(z)$, or model-independent reconstructions that lack fundamental physical grounding. General Relativistic Entropic Acceleration (GREA) offers a theoretically motivated alternative, attributing acceleration to the entropy growth of the causal cosmological horizon without a cosmological constant. However, prior confrontations with data were limited to the background level, utilizing compressed distance priors rather than full likelihoods, and treating perturbations only semi-analytically. This prevented a rigorous test of the model against the full Cosmic Microwave Background (CMB) and matter power spectra.

Methodology
This work presents the first full implementation of GREA within an Einstein–Boltzmann solver (class). The methodology involves:

1. Background Integration: The GREA background dynamics, governed by a modified Friedmann equation where the entropic term arises from the Gibbons–Hawking–York boundary term, are integrated directly into the class background module. The model is characterized by a single $O(1)$ parameter, $\alpha$, defined as the ratio of the spatial curvature scale to the causal horizon today.
2. Perturbation Treatment: Since the entropic sector involves non-local quantities (perturbations of the light-cone integral) that are not yet derived from first principles, the entropic component is modeled at the linear level as an effective dark-energy fluid.
   • The equation of state, $w(a)$, is derived analytically from the background.
   • The fluid is evolved using the Parametrized-Post-Friedmann (PPF) scheme to ensure regularity across the transient phantom crossing ($w = -1$).
   • A smooth-fluid approximation is adopted ($c_s^2 = 1$) as a conservative baseline, with the understanding that sub-horizon growth is fixed by the background expansion.
3. Numerical Implementation: The code rescales the normalization to account for the fact that the physical present-day Hubble rate $H(z=0)$ is a derived quantity in GREA, distinct from the normalization parameter $H_0$ used in the conformal time definition.
4. Statistical Analysis: A Markov-chain Monte Carlo (MCMC) analysis is performed using cobaya against a comprehensive dataset:
   • CMB-SPA: A combination of Planck 2018, ACT DR6, and SPT-3G likelihoods (temperature, polarization, and lensing).
   • BAO: DESI Data Release 2 measurements.
   • Supernovae: Pantheon+ (PP), SH0ES-calibrated Pantheon+ (PPS), and DES Year-5 Dovekie (DD) compilations.

Key Contributions

• First Boltzmann-Level Implementation: The paper provides the first code capable of computing the full CMB temperature, polarization, and lensing spectra, as well as the linear matter power spectrum and growth rate $f\sigma_8(z)$, for the GREA model.
• Validation: The implementation is validated against semi-analytic results for the growth rate $f\sigma_8(z)$ and reproduces the expected background expansion history.
• Full Likelihood Constraints: Unlike previous studies, this work constrains the model using full primary-CMB likelihoods rather than compressed distance priors, allowing for a direct test of the model's perturbation sector.
• Public Release: The modified class code, validation harness, and run configurations are made publicly available.

Results

• Parameter Constraints: Across all dataset combinations, the coupling parameter $\alpha$ is tightly constrained to values close to unity ($\alpha \sim 1$). For instance, the full combination (CMB-SPA + PPS + DESI) yields $\alpha = 1.079 \pm 0.026$, and CMB-SPA + DD + DESI yields $\alpha = 1.004^{+0.030}_{-0.031}$. This is in excellent agreement with the theoretical prediction that $\alpha \sim 1$.
• Goodness of Fit: GREA provides a fit to the data comparable to $\Lambda$CDM. The difference in minimum $\chi^2$ is small ($|\Delta\chi^2| \lesssim 6$) despite the addition of a single free parameter. The Bayesian evidence is generally inconclusive or shows mild preference for GREA in late-time combinations, but remains statistically indistinguishable from $\Lambda$CDM in most cases.
• Expansion History and Equation of State: The reconstructed expansion history shows a characteristic deviation at intermediate redshifts ($0.2 \lesssim z \lesssim 0.6$) where the expansion rate is higher than in $\Lambda$CDM. The effective equation of state exhibits a transient phantom crossing at $z \lesssim 2$. Notably, the model predicts a second crossing of the phantom divide at $z \simeq 2$, returning to a quintessence-like regime, a feature not imposed by parametrization but arising naturally from the thermodynamic dynamics.
• Tensions:
  • Hubble Tension: When including the SH0ES-calibrated supernova data (PPS), GREA accommodates a high $H_0$ ($73.4 \pm 1.0$ km s$^{-1}$ Mpc$^{-1}$), but this combination remains in tension with CMB data. For consistent datasets (CMB + BAO + SNIa), GREA yields $H_0$ values similar to $\Lambda$CDM and does not alleviate the Hubble tension.
  • $S_8$ Tension: The full Boltzmann computation confirms that GREA enhances structure growth relative to $\Lambda$CDM at fixed primordial amplitude, leading to a higher $S_8$ ($\sim 0.82$). The paper notes that whether this worsens the $S_8$ tension cannot be definitively assessed without a dedicated weak-lensing analysis within the GREA framework, as current constraints assume $\Lambda$CDM.
• Robustness: The results are shown to be robust against the uncertainty in the effective sound speed of the entropic fluid; varying $c_s^2$ shifts observables by less than 1.6%, well below current sensitivity.

Significance and Claims
The paper claims a "striking vindication" of GREA's defining prediction: the data single out the specific theoretical value $\alpha \sim 1$ at the few-percent level without fine-tuning. The model successfully reproduces the observed expansion history and CMB spectra without a cosmological constant, matching $\Lambda$CDM's performance despite being a single-parameter, thermodynamically motivated theory.

The authors emphasize that the model's ability to match the data with a single parameter, while replacing a cosmological constant fine-tuned by $\sim 120$ orders of magnitude with a horizon-thermodynamics mechanism, is a strong statement in its favor. The paper concludes that the distinctive predictions of GREA—specifically the non-monotonic growth index, the second phantom crossing, and the specific ISW/lensing signatures—are now accessible to next-generation surveys (DESI, Euclid, Vera Rubin). The work establishes the necessary theoretical and computational framework to test whether the causal horizon drives the current acceleration and whether the Universe is destined for a Minkowski rather than a de Sitter fate.

The authors remain modest regarding the perturbation sector, acknowledging that the derivation of the perturbed entropic force tensor $\delta f_{\mu\nu}$ (specifically the non-local response) remains an open theoretical task. They posit that the current effective-fluid description is a conservative approximation, and a full derivation is required to uniquely fix the low-$\ell$ ISW and lensing responses.