Modified Teleparallel $f(T)$ Gravity, DESI BAO and the $H_0$ Tension

Abstract: We investigate whether late-time modifications of gravity in the teleparallel framework can impact the current tension in the Hubble constant $H_0$, focusing on $f(T)$ cosmology as a minimal and well-controlled extension of General Relativity. We consider three representative $f(T)$ parametrisations that recover the teleparallel equivalent of General Relativity at early times and deviate from it only at late epochs. The models are confronted with unanchored Pantheon+ Type~Ia supernovae, DESI DR2 baryon acoustic oscillations, compressed Planck cosmic microwave background distance priors, and redshift-space distortion data, allowing us to jointly probe the background expansion and the growth of cosmic structures. Two of the three models partially shift the inferred value of $H_0$ towards local measurements, while the third worsens the discrepancy. This behaviour is directly linked to the effective torsional dynamics, with phantom-like regimes favouring higher $H_0$ and quintessence-like regimes producing the opposite effect. A global statistical comparison shows that the minimal $f(T)$ extensions considered here are not favoured over $Λ$CDM by the combined data. Nevertheless, our results demonstrate that late-time torsional modifications can non-trivially redistribute current cosmological tensions among the background and growth sectors.

• The paper evaluates three f(T) gravity models to see if late-time cosmological modifications can reduce the Hubble constant tension, finding that while some, enhance underlying data consistency, none statistically outperform ΛCDM.
• Phantom-like models (f1(T) and f3(T)) tend to push the inferred H₀ upward, while the quintessence-like model (f2(T)) pushes it downward, hard-coding a dichotomy rooted in their different gravitational coupling regimes.
• across all considered datasets, the Akaike Information Criterion (AIC) showed decisive evidence against the analyzed new models relative to ΛCDM, They are, stand-inadequate models for the resolution of the $H0$ tension.

Overview and motivation

This paper examines whether late-time modifications of gravity in the teleparallel framework can alleviate the Hubble constant tension, using $f(T)$ cosmology as a minimal, one-parameter extension of General Relativity. The authors analyze three parametrisations that recover the teleparallel equivalent of General Relativity (TEGR) at early times and deviate from it only at late epochs. The models are constrained with unanchored Pantheon+ Type Ia supernovae (calibrated via a local $H_0$ prior), DESI DR2 baryon acoustic oscillations (BAO), compressed Planck CMB distance priors, and a compilation of 22 redshift-space distortion (RSD) measurements of $f\sigma_8(z)$ (2601.22225). The central finding is a dichotomy: two of the three models shift the inferred $H_0$ upward toward local distance-ladder values, while the third shifts it downward — but none is statistically favored over $\Lambda$CDM by the combined data.

Theoretical framework

The analysis is set in standard (non-covariant) $f(T)$ gravity, where the gravitational action replaces the torsion scalar $T$ of TEGR with an arbitrary function $f(T)$. For a spatially flat FLRW background with $T = -6H^2$, the modified Friedmann equation admits an effective-fluid interpretation in which the torsional sector contributes an energy density $\rho_T$, pressure $H_0$0, and an evolving equation of state $H_0$1.

At the perturbative level, working in Newtonian gauge on subhorizon scales within the quasi-static approximation, the effective gravitational coupling is $H_0$2 and the gravitational slip vanishes ($H_0$3), assuming zero scalar anisotropic stress at linear order. Matter growth then obeys a modified growth equation in which deviations from GR enter exclusively through $H_0$4. The authors note that since the relevant background and linear perturbation equations coincide between the standard and covariant formulations, they adopt the simpler non-covariant version; this sidesteps but does not resolve the known local Lorentz violation issue of the standard formulation.

The three models

All three models are constructed so that $H_0$5 is fixed algebraically at $H_0$6 through the Friedmann equation, leaving no additional free parameter beyond those of $H_0$7CDM:

• Model 1: $H_0$8, previously studied in the literature.
• Model 2: $H_0$9, also previously proposed.
• Model 3: $f\sigma_8(z)$0, a novel parametrisation inspired by an analogous $f\sigma_8(z)$1 model.

A key theoretical diagnostic separates the models into two classes. Models 1 and 3 exhibit phantom-like behavior ($f\sigma_8(z)$2) over the relevant late-time range together with $f\sigma_8(z)$3, enhancing both the expansion rate and structure growth. Model 2 exhibits quintessence-like behavior ($f\sigma_8(z)$4) with $f\sigma_8(z)$5. This dichotomy anticipates the observational results: phantom-like regimes favor higher inferred $f\sigma_8(z)$6, quintessence-like regimes the opposite.

Data and methodology

The Bayesian analysis uses MCMC sampling via Cobaya with Gelman–Rubin convergence checks. Sampled parameters are $f\sigma_8(z)$7, $f\sigma_8(z)$8, $f\sigma_8(z)$9 (with a Gaussian BBN-motivated prior), and $H_0$0 when RSD data are included. Pantheon+ is treated as unanchored, with analytic marginalization over the absolute magnitude $H_0$1 and calibration imposed through the Riess et al. prior $H_0$2. The sound horizon $H_0$3 is computed with the same fitting formula used in the DESI analyses, ensuring internal consistency. Model comparison uses the corrected Akaike Information Criterion; because all models share the same number of free parameters and data, $H_0$4 reduces to a difference in maximum likelihoods, and the BIC yields identical conclusions.

Results

The dataset-by-dataset constraints reveal a consistent pattern summarized below (values in $H_0$5 for $H_0$6):

Dataset	$H_0$7CDM	$H_0$8	$H_0$9	$\Lambda$0
SN	$\Lambda$1	$\Lambda$2	$\Lambda$3	$\Lambda$4
BAO	$\Lambda$5	$\Lambda$6	$\Lambda$7	$\Lambda$8
BAO+CMB	$\Lambda$9	$f(T)$0	$f(T)$1	$f(T)$2
SN+BAO+CMB+RSD	$f(T)$3	$f(T)$4	$f(T)$5	$f(T)$6

The corresponding $f(T)$7 for the full combination are $f(T)$8, $f(T)$9, and $T$0 for $T$1, $T$2, and $T$3 respectively — decisive evidence against all three relative to $T$4CDM. On BAO+CMB alone, the penalties are already strong ($T$5, $T$6, $T$7). Only for RSD data alone are all models statistically indistinguishable from $T$8CDM ($T$9).

Several implications follow directly. First, the phantom-like models $f(T)$0 and $f(T)$1 raise the BAO- and CMB-inferred $f(T)$2 by roughly $f(T)$3--$f(T)$4, partially closing the gap to the local distance ladder — but this improvement is not free: the residual inconsistency is transferred to the matter density, which becomes discrepant between early- and late-time probes. Second, the growth sector shows a complementary trade-off: $f(T)$5 and $f(T)$6, with $f(T)$7, yield lower RSD-inferred $f(T)$8 ($f(T)$9 and $T = -6H^2$0 versus $T = -6H^2$1 in $T = -6H^2$2CDM for the full combination), while implying larger CMB-inferred $T = -6H^2$3 and hence a worsened early–late discrepancy. Conversely, $T = -6H^2$4 potentially improves consistency in the $T = -6H^2$5 sector while aggravating the $T = -6H^2$6 tension. No single minimal model addresses both tensions simultaneously. Third, the statistical reconstruction of $T = -6H^2$7 from the full combined chains confirms these classifications: the entire $T = -6H^2$8 band for Model 2 remains above $T = -6H^2$9, while Models 1 and 3 remain below it, with narrow confidence bands indicating that the data permit only small deviations from the best-fit behaviors.

It should be noted that the paper acknowledges recent KiDS-Legacy weak-lensing results reporting no significant $\rho_T$0 tension with $\rho_T$1CDM; the relevance of the growth-sector discussion is therefore that modified gravity can introduce such discrepancies even when none exists in the standard scenario.

Limitations and open questions

The paper concedes several limitations explicitly. The use of compressed CMB distance priors rather than full Planck temperature and polarization spectra is justified only because the models reduce to standard cosmology at early times; any extension with early-time deviations would invalidate this treatment. The perturbation analysis relies on the quasi-static, subhorizon approximation and assumes vanishing scalar anisotropic stress at linear order, so the growth conclusions do not extend to horizon scales or to higher-order perturbative effects. The RSD likelihood assumes uncorrelated errors across the 22 $\rho_T$2 measurements, an idealization given that several datasets share survey systematics. Additionally, the standard (non-covariant) $\rho_T$3 formulation carries the known issue of local Lorentz violation, deferred here on the grounds that the relevant equations coincide at the level used. Open questions left by the paper include whether more general teleparallel Lagrangians, non-minimal couplings, or additional degrees of freedom can break the observed complementarity between the $\rho_T$4 and $\rho_T$5 sectors, and how refined treatments of observational systematics would alter the decisive AIC penalties found here.

Conclusion

This work provides a controlled test of three minimal, early-TEGR-recovering $\rho_T$6 parametrisations against the most current late- and early-time probes, including DESI DR2 BAO. Its principal contribution is a clear mechanistic account of how torsional dynamics control the direction of the $\rho_T$7 shift — phantom-like $\rho_T$8 raising it, quintessence-like $\rho_T$9 lowering it — and of the associated redistribution of tensions between the background and growth sectors. Quantitatively, however, the verdict is negative: with $H_0$00 between $H_0$01 and $H_0$02 for the full data combination, none of the minimal extensions considered improves on $H_0$03CDM. The paper thus establishes both the diagnostic value and the empirical insufficiency of one-parameter late-time $H_0$04 gravity as a resolution to current cosmological tensions.