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

TL;DR

This work tests non-metricity-based $f(Q)$ gravity as a way to address the $H_0$ and $S_8$ tensions, analyzing three representative models against a broad suite of cosmological data. Using a Bayesian MCMC framework with background probes and CMB distance priors, it finds that Models 1 and 3 can raise $H_0$ toward local measurements, while Model 2 can yield $G_{\mathrm{eff}}<G$ and potentially ease growth-related tensions, though not simultaneously. The results show internal data tensions when combining datasets, with ΛCDM often favored in the full data combination, suggesting that minimal $f(Q)$ models may need extensions (e.g., a cosmological constant) to fully reconcile observations. Overall, $f(Q)$ gravity remains a flexible late-time framework with the potential to address individual cosmological tensions, warranting further exploration and data-driven extensions.

Abstract

We investigate the viability of $f(Q)$ gravity as an alternative framework to address the $H_0$ and $S_8$ tensions in cosmology. Focusing on three representative $f(Q)$ models, we perform a comprehensive Bayesian analysis using a combination of cosmological observations, including cosmic chronometers, Type Ia supernovae, gamma-ray bursts, baryon acoustic oscillations, and CMB distance priors. Our results demonstrate that most of these models can yield higher values of $H_0$ than those predicted by $Λ$CDM, offering a partial alleviation of the tension. In addition, one model satisfies the condition $G_{\mathrm{eff}} < G$, making it a promising candidate for addressing the $S_8$ tension. However, these improvements are accompanied by mild internal inconsistencies between different subsets of data, which limit the overall statistical preference relative to $Λ$CDM. Despite this, $f(Q)$ gravity remains a promising and flexible framework for late-time cosmology, and our results motivate further exploration of extended or hybrid models that may reconcile all observational constraints.

Figures (5)

• Figure 1: Cosmological background and effective properties of the $f(Q)$ models compared to $\Lambda$CDM. Top-left: Total effective equation of state $w_{\mathrm{tot}}(z)$, showing the transition from radiation to matter domination and the late-time accelerated regime. Top-right: Effective dark energy equation of state $w_{\mathrm{DE}}(z)$ compared to $\Lambda$CDM ($w=-1$). Bottom: Effective Newton’s constant $G_{\mathrm{eff}}/G$ as a function of redshift. Models 1 and 3 exhibit $G_{\mathrm{eff}}>G$, whereas Model 2 shows $G_{\mathrm{eff}}<G$, with distinct implications for structure formation. The horizontal line $G_{\mathrm{eff}}/G=1$ marks the GR limit. All curves are obtained using the best-fit cosmological parameters corresponding to Combination V, which will be defined in the results section.
• Figure 2: Two-dimensional posterior distributions for the $f(Q)$ models and $\Lambda$CDM scenario, using Combination V (CC + SN + GRB + BAO + CMB). The contours correspond to the 68% and 95% confidence levels (C.L.). This figure summarises the full parameter constraints, including $\Omega_{b0}$. It illustrates how Models 1 and 3 accommodate higher values of $H_0$ in contrast to Model 2, which yields a lower $H_0$ compared to $\Lambda$CDM.
• Figure 3: Reconstruction of the effective dark energy equation of state $w_{\mathrm{DE}}(z)$ for the three $f(Q)$ models using the full dataset combination (Combination V). Solid curves correspond to the best-fit reconstruction, while shaded regions represent the 68% (dark) and 95% (light) confidence levels obtained from the MCMC analysis. The three panels show: Model 1 (top-left), Model 2 (top-right), and Model 3 (bottom). Compared to Fig. \ref{['fig:theory_motivation']}, which displayed only the best-fit behaviours, here we explicitly include the confidence regions, demonstrating that the constraints on $w_{\mathrm{DE}}$ are consistently tight across all three models.
• Figure 4: Comparison of the two-dimensional posterior distributions obtained from Combination II (BAO) and Combination III (CMB) for each model separately. The contours correspond to the 68% and 95% confidence levels (C.L.). The top-left panel shows the results for $\Lambda$CDM scenario, which displays excellent agreement between BAO and CMB. The remaining panels correspond to Models 1 (top-right), 2 (bottom-left), and 3 (bottom-right). A clear tension between BAO and CMB in the $\Omega_{m0}-H_0$ plane appears in Models 2 and 3.
• Figure 5: Comparison of the two-dimensional posterior distributions obtained from Combination I (CC + SN + GRB) and Combination IV (BAO + CMB) for each model separately. The contours correspond to the 68% and 95% confidence levels (C.L.). The top-left panel shows the results for $\Lambda$CDM scenario, which displays excellent agreement between the dataset combinations. The remaining panels correspond to Models 1 (top-right), 2 (bottom-left), and 3 (bottom-right), where a clear tension between the combinations emerges in the $\Omega_{m0}-H_0$ plane. These internal inconsistencies contribute to the poorer global fits obtained by the $f(Q)$ models when all datasets are combined.