Addressing $H_0$ and $S_8$ tensions within $f(Q)$ cosmology

Technical Summary: Addressing $H_0$ and $S_8$ Tensions within $f(Q)$ Cosmology

Problem Statement
The Standard Model of Cosmology ($\Lambda$CDM), while highly effective, faces significant observational tensions, specifically the discrepancy in the Hubble constant ($H_0$) and the clustering amplitude ($S_8$). The $H_0$ tension arises from the $\sim 4.4\sigma$ difference between the Planck collaboration's estimation ($67.27 \pm 0.60$ km/s/Mpc) and direct local measurements by the SH0ES collaboration ($74.03 \pm 1.42$ km/s/Mpc). Similarly, the $S_8$ tension involves a discrepancy between matter clustering estimates derived from the Cosmic Microwave Background (CMB) and those from large-scale structure surveys (e.g., SDSS/BOSS). This study investigates whether $f(Q)$ gravity—a modified gravity theory based on non-metricity—can serve as a viable alternative framework to alleviate these tensions without introducing a cosmological constant.

Methodology
The authors analyze three representative non-linear $f(Q)$ models:

1. Model 1: $f_1(Q) = Q e^{\lambda_1 Q_0/Q}$
2. Model 2: $f_2(Q) = Q + Q_0 e^{-\lambda_2 Q_0/Q}$
3. Model 3: $f_3(Q) = Q + \lambda_3 Q_0 [1 - e^{-Q_0/Q}]$

These models are designed to reduce to General Relativity (GR) at early times ($Q \gg Q_0$) while generating late-time acceleration through geometric modifications. The study employs a comprehensive Bayesian analysis using the Cobaya MCMC sampler. The parameter space includes the Hubble constant ($H_0$), matter density ($\Omega_{m0}$), and baryon density ($\Omega_{b0}$).

The analysis confronts these models with five distinct combinations of cosmological datasets:

• Combination I: Cosmic Chronometers (CC), Type Ia Supernovae (SN), and Gamma-Ray Bursts (GRB).
• Combination II: Baryon Acoustic Oscillations (BAO).
• Combination III: CMB distance priors.
• Combination IV: BAO + CMB.
• Combination V: Full combination (CC + SN + GRB + BAO + CMB).

Model performance is evaluated using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) relative to the standard $\Lambda$CDM model. Theoretical predictions for the effective Newton's constant ($G_{\text{eff}}$) and the effective dark energy equation of state ($w_{\text{DE}}$) are derived to assess implications for structure growth and the $S_8$ tension.

Key Contributions and Results

• Alleviation of $H_0$ Tension: Models 1 and 3 successfully predict higher values of $H_0$ compared to $\Lambda$CDM when CMB data is included, bringing estimates closer to local measurements (e.g., $H_0 \approx 70.4 - 71.5$ km/s/Mpc in Combination V). This offers a partial alleviation of the $H_0$ tension. Conversely, Model 2 predicts a lower $H_0$ than $\Lambda$CDM, exacerbating the tension.
• Implications for $S_8$ Tension: The study identifies a trade-off between the two tensions. Model 2 is the only candidate satisfying $G_{\text{eff}} < G$, a condition typically associated with suppressed structure growth, making it a promising candidate for addressing the $S_8$ tension. However, Models 1 and 3 exhibit $G_{\text{eff}} > G$, which enhances clustering and is likely to worsen the $S_8$ tension.
• Internal Data Inconsistencies: A critical finding is the emergence of internal tensions between different dataset subsets within the $f(Q)$ framework.
  • Background-only probes (Combination I) favor higher $\Omega_{m0}$ for Models 1 and 3.
  • Early-universe and large-scale structure probes (BAO + CMB) drive these same models toward lower $\Omega_{m0}$.
  • Both Models 2 and 3 exhibit the strongest internal mismatch between BAO and CMB constraints, revealing a clear tension in the $\Omega_{m0} - H_0$ plane. While Model 2 shows the strongest overall discrepancy when comparing background data (Combination I) against BAO+CMB (Combination IV), the specific BAO vs. CMB tension is most pronounced in Models 2 and 3.
• Statistical Preference: While the models are statistically compatible with $\Lambda$CDM when analyzed against individual datasets (e.g., CMB alone or BAO alone), the combination of all datasets (Combination V) reveals significant internal inconsistencies. Consequently, $\Lambda$CDM is strongly favored over all three $f(Q)$ models in the global fit, as indicated by high $\Delta$AIC values (e.g., $\Delta$AIC $\approx 11.0$ for Model 1 and $37.1$ for Model 2 in Combination V).

Significance and Conclusions
The paper concludes that while $f(Q)$ gravity provides a flexible and theoretically motivated framework capable of addressing either the $H_0$ or the $S_8$ tension depending on the specific model chosen, it currently fails to resolve both simultaneously within the minimal formulations tested. The inability to reconcile background expansion data with large-scale structure constraints suggests that the absence of a cosmological constant term in these minimal models may be a limiting factor.

The authors assert that $f(Q)$ gravity remains a promising avenue for late-time cosmology but emphasize that future work must explore extended or hybrid models (e.g., including a cosmological constant, more general functional forms, or different connection choices) to fully reconcile all observational constraints. The study highlights that the specific geometric nature of non-metricity offers distinct phenomenological behaviors (phantom-like vs. quintessence-like dynamics) compared to other modified gravity theories, warranting further investigation.