Constraints on Rastall gravity from current observational data

Technical Summary: Constraints on Rastall Gravity from Current Observational Data

Problem Statement
While the standard $\Lambda$CDM model within General Relativity (GR) successfully describes cosmic evolution, it faces significant theoretical and observational challenges. Most notably, there is a $\approx 6\sigma$ tension between direct local measurements of the Hubble constant ($H_0$) and those inferred from Cosmic Microwave Background (CMB) data under the $\Lambda$CDM assumption. Additionally, a mild discrepancy exists regarding the matter clustering parameter $S_8$, where local observations suggest less structure growth than predicted by Planck data. These tensions motivate the exploration of physics beyond GR. This paper investigates Rastall gravity, a phenomenological modification where the covariant conservation of the energy-momentum tensor is relaxed to $\nabla_\mu T^\mu_\nu = \lambda \nabla_\nu R$, introducing a non-minimal coupling between matter and geometry.

Methodology
The authors conduct a comprehensive cosmological analysis of Rastall gravity by:

1. Deriving Field Equations: They formulate the modified field equations for Rastall gravity at both the background level (FLRW metric) and the linear perturbation level (in both synchronous and conformal Newtonian gauges). The model is parameterized by a dimensionless Rastall parameter $\beta = \kappa\lambda$.
2. Numerical Implementation: The authors modified the Boltzmann code CLASS (Cosmic Linear Anisotropy Solving System) to incorporate the derived Rastall field equations, specifically focusing on the evolution of cosmological observables when dark energy is treated as a cosmological constant.
3. Observational Constraints: They performed a Markov Chain Monte Carlo (MCMC) analysis using the Monte Python package. The analysis utilized a combined dataset comprising:
   • Planck 2018 CMB data (high-$l$ TT, TE, EE; low-$l$ TT, EE; lensing).
   • Planck-SZ (Sunyaev-Zeldovich effect).
   • CFHTLenS (weak lensing).
   • Pantheon+ (Type Ia supernovae).
   • BAO (Baryon Acoustic Oscillations) and BAORSD (Redshift-Space Distortions).
4. Model Comparison: The study compares the best-fit parameters and derived constraints of Rastall gravity against the standard $\Lambda$CDM model, utilizing the Akaike Information Criterion (AIC) to assess model preference.

Key Contributions and Results

• Suppressed Structure Growth: The numerical analysis reveals that Rastall gravity predicts a lower growth rate for cosmic structures compared to $\Lambda$CDM. This is evidenced by the matter power spectrum, matter density contrast, and Newtonian potential diagrams. This suppression aligns with low-redshift large-scale structure observations that report lower $S_8$ values than Planck.
• Matter Acoustic Oscillations: The non-minimal matter-geometry coupling inherent in Rastall gravity induces matter acoustic oscillations. These are visible in the matter power spectrum, density perturbations, and velocity perturbations, a feature not present in standard GR.
• Hubble Tension: The study finds that Rastall gravity with a cosmological constant as the dark energy component does not resolve the Hubble tension; rather, the tension becomes more severe in the Rastall model. The best-fit value for $H_0$ in the Rastall model is lower ($68.42$ km/s/Mpc) than in $\Lambda$CDM, increasing the discrepancy with local $H_0$ measurements. The $68\%$ confidence level uncertainty is $0.37$ km/s/Mpc, with a $95\%$ confidence level uncertainty of $0.75$ km/s/Mpc.
• Parameter Constraints: The MCMC analysis constrains the Rastall parameter to $\beta \approx -3.27 \times 10^{-7}$, indicating a deviation from standard GR ($\beta=0$) at greater than $3\sigma$ significance. The derived universe age in Rastall gravity ($t_0 = 13.74$ Gyr) is slightly larger than in $\Lambda$CDM ($13.72$ Gyr), potentially offering better agreement with the ages of the oldest astrophysical objects.
• Model Preference: The Akaike Information Criterion (AIC) analysis yields $\Delta \text{AIC} = 29.24$ in favor of Rastall gravity over $\Lambda$CDM, suggesting the Rastall model provides a statistically superior fit to the combined observational data despite the added parameter.
• Theoretical Similarity: The authors note numerical similarities between Rastall gravity and $f(R, T)$ gravity, where both models exhibit comparable evolutionary behaviors for cosmological observables despite arising from different theoretical foundations.

Significance
The paper concludes that while Rastall gravity successfully addresses the $S_8$ tension by predicting suppressed structure growth consistent with local probes, it fails to alleviate the $H_0$ tension and actually worsens it under the assumption of a cosmological constant. The strong statistical preference for the Rastall model based on the AIC suggests that the non-minimal coupling between matter and geometry is a viable avenue for describing cosmic evolution. The authors assert that these results merit further investigation using more precise and reliable observational datasets to fully characterize the viability of Rastall gravity as an alternative to standard cosmology.