Background solutions to the Hubble tension in $f(Q)$ gravity and consistency with BAO measurements

TL;DR

This work tests whether symmetric teleparallel $f(Q)$ gravity can resolve the Hubble tension while remaining consistent with DESI DR2 BAO measurements. By examining three functional forms—Log, Exp, and Tanh—along with variants including a cosmological constant and a flexible phenomenological model, the authors constrain late-time background evolution against multiple datasets, employing Bayesian model comparison. They find that the Log and Tanh models struggle with BAO and Bayes factors, while the Exp model can alleviate the Hubble tension but remains in mild tension with BAO; including a Λ term improves BAO compatibility at the expense of reduced theoretical motivation. Phenomenological models fit BAO data well and can reproduce high local $H_0$ values, but their lack of theoretical grounding and presence of EoS pathologies limits their physical viability. Overall, the study suggests that fully solving both the Hubble tension and BAO anomaly at the background level remains challenging, with best results arising from phenomenology, while theoretically motivated $f(Q)$ models face BAO constraints.

Abstract

We test whether $f(Q)$ symmetric teleparallel theories of gravity are capable of solving the Hubble tension while producing consistency with DESI DR2 BAO. We consider three different forms for the $f(Q)$ function: logarithmic, exponential, and hyperbolic tangent. We also consider extensions of these models by adding the Cosmological constant, and compare to phenomenological models with a flexible double exponential addition to the standard $Λ$CDM $H(z)$. We test these models against DESI DR2 BAO, Planck 2018, Local $H_0$, and Cosmic Chronometers data. The logarithmic and hyperbolic tangent models do not provide an adequate solution, while the exponential model gives a reasonable solution, though it faces mild tension with the BAO data. The models assisted by the cosmological constant perform slightly better than the exponential, at the cost of reduced theoretical motivation. This highlights the difficulties of finding a theoretically motivated solution to the Hubble tension while producing BAO consistency. The phenomenological models are able to achieve a good fit to all data, giving guidelines for what a background solution to the Hubble tension would look like in general.

Figures (7)

• Figure 1: $1\sigma$ and $2\sigma$ C.L. contours for the cosmological parameters $\Theta = \{ \Omega_m, \Omega_\Lambda, H_0$}. Each panel shows results for a particular $f(Q)$ gravity models, the same model assisted by the cosmological constant $\Lambda$, and $\Lambda$CDM. We show (a) the logarithmic form $f(Q) = Q/(8 \pi G) - \alpha \ln(Q/Q_0)$, (b) the exponential form $f(Q) = Q/(8\pi G) \exp(\lambda Q_0/Q)$, and (c) the hyperbolic tangent form $f(Q) = Q/(8\pi G) + \alpha \tanh (Q_0/Q)$.
• Figure 2: $1\sigma$ and $2\sigma$ C.L. contours for the cosmological parameters $\Theta = \{ \Omega_m, H_0, A, B, z_1, k \}$ in the phenomenological models where $H(z) = H_{\Lambda CDM}(z) + A \exp(-z/z_1) + B (\exp(-z/z_1)-\exp(-z/(k z_1))))$. We present the results for $k=2$, $k=3$, and with $k$ as a free parameter. We also show the $\Lambda$CDM model for comparison.
• Figure 3: Predicted $\dot{a}$ as a function of redshift for (a) the $f(Q)$ models, (b) the $f(Q)$ models assisted with the cosmological constant, and (c) the phenomenological models. The curves correspond to the mean values of the cosmological parameters given in tables \ref{['tab:ResultsfQ']} and \ref{['tab:ResultsPhen']}. We also plot the $\Lambda$CDM case for comparison. We include the residuals with respect to $\Lambda$CDM in the lower panels. The $1\sigma$ C.L. error bars are not visible here due to the very tight Planck 2018 constraints.
• Figure 4: Ratio between the isotropically averaged comoving BAO scale $D_V$ to the $\Lambda$CDM case for (a) the $f(Q)$ models, (b) the $f(Q)$ models plus cosmological constant, and (c) the phenomenological models. We take the best fits for all models with respect to the full dataset ($H_0$+Planck 2018+CC+DESI DR2 BAO), including the $\Lambda$CDM case. We use the standard Planck sound horizon $r_d = 147.05 \, \pm \, 0.30$ Mpc. We also plot the BAO data from the last 20 years for comparison.
• Figure 5: Equation of state (EoS) for the effective dark energy of (a) the $f(Q)$ models, (b) the $f(Q)$ models assisted with the cosmological constant, and (c) the phenomenological models. The $1\sigma$ C.L. error bars are invisibly small due to the very tight Planck 2018 constraints.