Bayesian and Machine-Learning Analyses of Nonminimal $f(Q)$ Gravity and $H_0$ Tension

TL;DR

This work investigates nonminimal couplings between the nonmetricity scalar $Q$ and matter within the metric–affine, symmetric teleparallel framework, formulating a two-function action $S=\int d^4x\,\sqrt{-g}\left[\tfrac{1}{2}f_1(Q)+f_2(Q)\mathcal{L}_M\right]$ and deriving second-order field equations. By adopting $f_1(Q)=-Q+\alpha Q^2$ and $f_2(Q)=1+\beta Q$ in a flat FLRW background, the authors obtain modified Friedmann equations and define effective density and pressure, enabling a direct confrontation with late-time data. A comprehensive MCMC analysis using cosmic chronometers, DESI DR2 BAO, and multiple SNe datasets shows that the model achieves a partial alleviation of the $H_0$ tension, with $H_0$ typically in the range $67$–$69$ km s$^{-1}$ Mpc$^{-1}$ and $r_d\approx147$ Mpc, while remaining consistent with $\Lambda$CDM in many observables. Complementary machine-learning approaches (linear regression, SVR, and random forests) demonstrate that SVR with an RBF kernel most accurately reproduces the theoretical $H(z)$ evolution, reinforcing the robustness of the background reconstruction. Overall, the study positions nonminimal $f(Q)$ gravity as a flexible and viable framework for late-time cosmology with potential extensions to further address cosmological tensions.

Abstract

In this study, the cosmological implications of nonminimally coupled $f(Q)$ gravity are examined within the metric-affine formalism, in which the nonmetricity scalar $Q$ couples directly to the matter Lagrangian. Within the symmetric teleparallel framework, a representative $f(Q)$ model is constructed, and the corresponding background cosmological equations are derived. The analysis aims to test whether this geometric formulation yields more consistent realizations of nonminimal matter-geometry couplings. A comprehensive statistical MCMC analysis is performed using cosmic chronometers, DESI BAO DR2, and Type Ia supernovae from the Pantheon+, DESY5, and Union3 samples. To complement the statistical study, we employ machine learning methods, such as linear regression, support vector regression (SVR), and random forest algorithms, to evaluate the predictive performance and robustness of the data. The results indicate that a partial alleviation of the $H_0$ tension can be achieved for a broad range of parameter choices. Nonetheless, $f(Q)$ gravity emerges as a promising and flexible framework for late-time cosmology, motivating further exploration of extended models consistent with all observations.

Figures (8)

• Figure 1: Two-dimensional contours of the parameter space for the $f(Q)$ model using different observational data.
• Figure 2: The behavior of cosmological parameters using best-fit parameters from observational data. The top panel shows $H(z)$ compared with cosmic chronometer data, and the bottom panel shows the distance modulus $\mu(z)$ against Pantheon$+$ and Union3 supernova samples.
• Figure 3: Evolution of the deceleration parameter $q(z)$ and the effective equation-of-state parameter $w(z)$ using best-fit parameters.
• Figure 4: The panel displays the cosmological parameters plotted alongside DESI BAO data. The first plot shows the behavior of $D_H/rd$, the second presents the $D_M/rd$, and the last illustrates the $D_V/rd$ profile.
• Figure 5: Evolution of the $Om(z)$ diagnostic using best-fit parameters from the considered observational data.