Non-Metricity Corrections Approach to Alleviate $H _0$ Tension: The Logarithmic and Nonlinear $f(Q)$ Models

Abstract: The persistent discrepancy between early-time and late-Universe measurements of the Hubble constant commonly known as the $H_0$ tension remains one of the most pressing open questions in modern cosmology. In this work, we explore whether modifications to the gravitational sector, specifically within the framework of symmetric teleparallel gravity, can offer a viable pathway toward alleviating this tension. We consider two functional forms of $f(Q)$ gravity: a logarithmic model and a nonlinear saturation model, both of which introduce geometric corrections to the standard expansion history without invoking a cosmological constant. Constraining these models through a Bayesian MCMC analysis against a comprehensive suite of observational data, including cosmic chronometers, Type Ia supernova compilations (Pantheon, Pantheon$+$SH0ES, and DES SN5YR), and BAO measurements from SDSS and DESI, we find that both models remain statistically competitive with $Λ$CDM. The logarithmic model, in particular, consistently infers intermediate values of $H_0$ between the \textit{Planck} and SH0ES benchmarks across all dataset combinations, and carries lower AIC and BIC penalties, establishing it as the more promising candidate for partially easing the $H_0$ tension within a modified gravity framework.

• The paper demonstrates that nonmetricity corrections in f(Q) gravity, especially the logarithmic model, can reconcile discrepancies between local and early-Universe H₀ measurements.
• The study employs Bayesian MCMC with diverse datasets—including cosmic chronometers, Type Ia supernovae, and BAO—to robustly constrain model parameters and assess performance.
• The findings imply that logarithmic f(Q) models offer a geometrically driven alternative to dark energy, reducing model-selection penalties compared to ΛCDM.

Non-Metricity Corrections Approach to Alleviate $H_0$ Tension: The Logarithmic and Nonlinear $f(Q)$ Models

Context and Motivation

The persistent discrepancy between early- and late-Universe measurements of the Hubble constant ($H_0$), currently in excess of $5\sigma$, presents a robust challenge to the standard $\Lambda$CDM framework. Measurements calibrated via local distance ladders (e.g., SH0ES, JWST) consistently yield $H_0 > 73$ km s$^{-1}$ Mpc$^{-1}$, whereas large-scale structure and BAO analyses (e.g., DESI DR2) favor $H_0 \approx 68.5$ km s$^{-1}$ Mpc$f(Q)$0. Traditional attempts to account for this discrepancy by invoking dynamical dark energy, or modifying the expansion history at low $f(Q)$1, have encountered model-selection penalties or required exotic new fields. An alternative paradigm is to modify the gravitational sector itself while retaining geometric consistency at early times.

Symmetric teleparallel gravity, constructed via a nonmetricity scalar $f(Q)$2 and generalized to $f(Q)$3 gravity, is a promising candidate. Within this framework, the gravitational action is a function of $f(Q)$4 (distinct from curvature and torsion), introducing new geometric degrees of freedom. These can alter late-time expansion without impacting the high-$f(Q)$5 regime, offering controlled mechanisms to address $f(Q)$6 tension.

$f(Q)$7 Gravity: Theoretical Structure

The action for $f(Q)$8 gravity is

$f(Q)$9

with field equations derived via metric and affine connection variation. Under FLRW dynamics and the coincident gauge, $H_0$0 and the modified Friedmann equations are

$H_0$1

With $H_0$2, geometric corrections $H_0$3 act as alternatives to dark energy, impacting late-time expansion.

Observational Dataset and Inference

The models are constrained by a comprehensive set of late-time cosmological probes:

• Cosmic Chronometers (CC): 31 direct $H_0$4 measurements using passively evolving galaxies.
• Type Ia Supernovae: Pantheon, Pantheon+SH0ES, and DES SN5YR compilations spanning extensive $H_0$5 precision and robust host distance calibration.
• BAO: Measurements from SDSS and DESI DR2, with careful treatment of degeneracies between $H_0$6 and $H_0$7.

Parameter estimation relies on Bayesian MCMC, with explicit chi-square construction and priors over $H_0$8, as appropriate. Model comparison is performed via AIC and BIC, penalizing unnecessary parameter proliferation.

Logarithmic and Nonlinear $H_0$9 Models

Model I: Logarithmic Correction

Functional form:

$5\sigma$0

with normalization $5\sigma$1 fixed by present Friedmann constraints.

The normalized Hubble parameter, $5\sigma$2, satisfies a nonlinear equation accommodating geometric deviations at low $5\sigma$3. Model reduces to $5\sigma$4CDM in the limit $5\sigma$5.

Confidence parameter contours and posterior distributions for Model I, across dataset combinations, consistently yield intermediate $5\sigma$6 values between Planck and SH0ES benchmarks, with improved precision as BAO data shift from SDSS to DESI.

Figure 1: Contour plot for the combined dataset for CC, Pantheon, Pantheon+SHOES, BAO(SDSS), DESI for Model-I.

Figure 2: Posterior probability distributions of $5\sigma$7 for Model I from all major dataset combinations, showing intermediate values and reduced uncertainty with DESI and DES SN5YR.

Key numerical results:

• $5\sigma$8 (CC+PN+SDSS); $5\sigma$9 (CC+PN+DESI)
• $\Lambda$0 (CC+PN(SH0ES)+SDSS); $\Lambda$1 (CC+PN(SH0ES)+DESI)
• DES SN5YR combinations further tighten $\Lambda$2 uncertainties to sub-percent levels.

Model II: Nonlinear Saturation

Functional form:

$\Lambda$3

with normalization and evolution equations ensuring gradual transition and stability at all epochs.

Contour plots and $\Lambda$4 posteriors show Model II can accommodate $\Lambda$5 for Pantheon-based datasets, but—upon inclusion of DES SN5YR and DESI—converges toward Planck $\Lambda$6 values.

Figure 3: Contour plot for the combined dataset for CC, Pantheon, Pantheon+SHOES, BAO(SDSS), DESI for Model-II.

Figure 4: Posterior probability distributions of $\Lambda$7 for Model II across all dataset combinations.

Numerically:

• $\Lambda$8 (CC+PN+SDSS); $\Lambda$9 (CC+PN+DESI)
• $H_0 > 73$0 (CC+DES+SDSS); $H_0 > 73$1 (CC+DES+DESI), closely matching Planck CMB values.

Comparative Assessment

Radar plots of $H_0 > 73$2 and information criterion deltas (AIC/BIC) show:

• Model I (logarithmic) delivers intermediate $H_0 > 73$3 values, statistically competitive with $H_0 > 73$4CDM, with lowest $H_0 > 73$5AIC/$H_0 > 73$6BIC for most dataset combinations.
• Model II (saturation) yields tighter clustering near Planck $H_0 > 73$7, with variability depending on dataset, occasionally outstripping Model I in $H_0 > 73$8AIC for DES-based combinations.

Figure 5: Inferred $H_0 > 73$9 values for Model I, Model II, and $^{-1}$0CDM across six datasets, illustrating model/data sensitivity.

Figure 6: Variation of $^{-1}$1AIC and $^{-1}$2BIC for Model I and Model II versus $^{-1}$3CDM across dataset combinations.

Tables of best-fit parameters emphasize that Model I maintains negative $^{-1}$4 (deviating from $^{-1}$5CDM) and balances $^{-1}$6 tension with statistical support, while Model II's $^{-1}$7 approaches zero when high-precision data are included.

$^{-1}$8CDM Reference

Analysis of $^{-1}$9CDM itself across identical datasets confirms $^{-1}$0's sensitivity to adopted probes:

• $^{-1}$1--$^{-1}$2 km s$^{-1}$3 Mpc$^{-1}$4 for Pantheon/BAO;
• $^{-1}$5 km s$^{-1}$6 Mpc$^{-1}$7 for Pantheon+SH0ES/SDSS;
• DESI datasets provide narrower confidence intervals.

Figure 7: Contour plot for the combined dataset for CC, Pantheon, Pantheon+SH0ES, BAO(SDSS), DESI for $^{-1}$8CDM.

Figure 8: Posterior probability distributions of $^{-1}$9 for $H_0 \approx 68.5$0CDM from all dataset combinations.

Implications and Future Directions

The findings establish modified gravity via nonmetricity, particularly the logarithmic $H_0 \approx 68.5$1 model, as a viable framework for alleviating $H_0 \approx 68.5$2 tension. This approach maintains statistical parity with $H_0 \approx 68.5$3CDM under AIC/BIC while predicting $H_0 \approx 68.5$4 values favoring reconciliation between local and early-Universe determinations, provided dataset combinations are carefully chosen.

Practically, the logarithmic $H_0 \approx 68.5$5 correction offers a geometric mechanism for mimicking dynamical dark energy behavior, without explicit dark energy components or distorting early universe physics. The nonlinear saturation model is effective at restoring $H_0 \approx 68.5$6CDM-like predictions as dataset precision increases, implying a robustness to observational constraints but less flexibility for tension mitigation.

Upcoming high-precision cosmological surveys (BAO, SN Ia, CC) will be critical for distinguishing between these model classes, and for testing the relevance of geometric corrections. Further theoretical work could address the microphysical interpretation of the $H_0 \approx 68.5$7 degrees of freedom and their implications for structure formation and gravitational wave propagation.

Conclusion

The nonmetricity-based $H_0 \approx 68.5$8 gravity models provide compelling alternatives to $H_0 \approx 68.5$9CDM for explaining cosmic acceleration and mitigating $^{-1}$0 tension. The logarithmic model, in particular, delivers intermediate and robust $^{-1}$1 values across diverse dataset combinations and minimal information criterion penalties, marking it as a preferred candidate for partial tension resolution. Model selection under more extensive and precise datasets will determine whether purely geometric corrections suffice for cosmological concordance or necessitate new physics beyond GR and the standard cosmological model.