BBN to Late-Time Acceleration in $f(T,\mathcal{L}_m)$ Gravity

Abstract: We present, to our knowledge, the first systematic study of early-late cosmic evolution and acceleration in the framework of $f(T,\mathcal{L}_m)$ gravity, an extension of teleparallel theories coupling torsion with the matter Lagrangian. By incorporating the Big-Bang Nucleosynthesis (BBN) bound on the freeze-out temperature, we obtain a tight constraint on the inverse-torsion parameter, ensuring consistency with early-time physics. Employing Markov Chain Monte Carlo analyses with progressively richer observational datasets, CC, Union3, and SN22 supernovae, we constrain a well-motivated model and reconstruct key cosmological functions. The reconstructed Hubble and distance modulus functions show excellent agreement with the observations, confirming the observational viability of the model. The model successfully reproduces the observed late-time expansion history, yielding a transition from deceleration to acceleration through the deceleration parameter. The effective equation of state is found to remain negative throughout, with present values $w_0 > -1$, indicating a quintessence-like behavior rather than a cosmological constant or phantom regime. These results highlight the ability of $f(T,\mathcal{L}_m)$ gravity to mimic the concordance scenario while allowing controlled deviations in the expansion history.

• The paper derives a BBN constraint of 0.2321 ≤ α ≤ 0.2346 for an inverse-torsion f(T,Lₘ) model, sharply limiting deviations from general relativity during radiation domination.
• The paper fits cosmic chronometers, Union3, and Pantheon+SH0ES data with MCMC methods, finding H₀ values of 69.1–72.61 km/s/Mpc and dataset-dependent α estimates.
• The paper finds quintessence-like effective dark energy with w₀ > −1 and acceleration transitions at zₜ = 0.36–0.59, while highlighting unresolved tension between the SN22 α preference and BBN.

This paper presents a combined early- and late-time cosmological analysis of $f(T,\mathcal{L}_m)$ gravity, a teleparallel extension in which the torsion scalar is nonminimally coupled to the matter Lagrangian. The authors derive the background field equations, impose Big-Bang Nucleosynthesis (BBN) constraints on the freeze-out temperature, and constrain an inverse-torsion model using MCMC fits to cosmic chronometers (CC), Union3 supernovae, and Pantheon+SH0ES (SN22) data. They report that the model reproduces the observed expansion history while predicting a quintessence-like effective equation of state with $w_0 > -1$ and an earlier deceleration-to-acceleration transition than $\Lambda$CDM.

Framework and background dynamics

The theory is built on teleparallel geometry: tetrads $e^{\mu}_{\ i}$ define the Weitzenböck connection, whose torsion tensor and superpotential combine into the torsion scalar $$T = T^{\lambda}_{\ \alpha\beta}S_{\lambda}^{\ \alpha\beta}$$. The action is

$$\mathcal{S}=\frac{1}{16\pi G}\int d^4x\, e\,[T+f(T,\mathcal{L}_m)] + \int d^4x\, e\,\mathcal{L}_m,$$

generalizing the earlier specific coupling of Harko et al. (Harko et al., 2014) to an arbitrary functional dependence on both $T$ and $\mathcal{L}_m$, following Cruz et al. (Cruz et al., 21 Sep 2025). Varying with respect to the tetrad yields modified Friedmann equations containing derivatives $f_T$, $f_{TT}$, $w_0 > -1$0, and $w_0 > -1$1. Rewriting in GR form introduces an effective dark-energy sector with density

$w_0 > -1$2

The authors adopt $w_0 > -1$3, noting that even then the energy-momentum source term persists, so the gravitational sector remains intrinsically non-conservative; choosing $w_0 > -1$4 would instead generate an explicit extra force from the coupling. This choice of matter Lagrangian is an assumption of the analysis, and the paper does not quantify how results change under the alternative convention.

BBN constraint on the inverse-torsion parameter

The early-Universe test follows the standard freeze-out argument: weak-interaction conversion rates scale as $w_0 > -1$5 with $w_0 > -1$6, while the radiation-era Hubble rate scales as $w_0 > -1$7, giving a GR freeze-out temperature $w_0 > -1$8 MeV. In $w_0 > -1$9 gravity, the extra dark-energy contribution modifies $\Lambda$0 by $\Lambda$1, shifting the freeze-out temperature by

$\Lambda$2

linearized under the assumption $\Lambda$3 during radiation domination. Requiring $\Lambda$4, derived from primordial light-element abundances, restricts the inverse-torsion parameter to the narrow interval $\Lambda$5. This is the central early-time result: BBN alone pins $\Lambda$6 to roughly a percent-level window, which any viable parameter set must respect.

Model and late-time observables

The specific model is

$\Lambda$7

an effective $\Lambda$8 structure that recovers TEGR at high torsion while activating deviations only at late times. Imposing $\Lambda$9 eliminates one free parameter, fixing $e^{\mu}_{\ i}$0 in terms of $e^{\mu}_{\ i}$1, $e^{\mu}_{\ i}$2, and $e^{\mu}_{\ i}$3, so the reduced parameter space is $e^{\mu}_{\ i}$4. The resulting Hubble function is analytic:

$e^{\mu}_{\ i}$5

Observational constraints

Bayesian inference uses the affine-invariant emcee sampler with GetDist post-processing, over three dataset combinations: 34 CC points (19 uncorrelated plus 15 correlated via their covariance matrix), Union3 (2087 SNe Ia from 24 datasets, binned distance moduli with full covariance), and Pantheon+SH0ES (1701 spectroscopically confirmed SNe Ia with Cepheid-host calibration). The $e^{\mu}_{\ i}$6 posteriors are:

Dataset	$e^{\mu}_{\ i}$7 (km/s/Mpc)	$e^{\mu}_{\ i}$8
CC	$e^{\mu}_{\ i}$9	$$T = T^{\lambda}_{\ \alpha\beta}S_{\lambda}^{\ \alpha\beta}$$0
CC+Union3	$$T = T^{\lambda}_{\ \alpha\beta}S_{\lambda}^{\ \alpha\beta}$$1	$$T = T^{\lambda}_{\ \alpha\beta}S_{\lambda}^{\ \alpha\beta}$$2
SN22	$$T = T^{\lambda}_{\ \alpha\beta}S_{\lambda}^{\ \alpha\beta}$$3	$$T = T^{\lambda}_{\ \alpha\beta}S_{\lambda}^{\ \alpha\beta}$$4

A notable tension emerges here: the SN22-preferred value $$T = T^{\lambda}_{\ \alpha\beta}S_{\lambda}^{\ \alpha\beta}$$5 lies well below the BBN-allowed window $$T = T^{\lambda}_{\ \alpha\beta}S_{\lambda}^{\ \alpha\beta}$$6, while CC and CC+Union3 values approach it. The paper states that preferred values "lie close to the BBN-allowed interval for all three dataset combinations," but the SN22 best fit differs from the BBN upper bound by more than $$T = T^{\lambda}_{\ \alpha\beta}S_{\lambda}^{\ \alpha\beta}$$7 given its quoted uncertainty. This discrepancy between the early-time bound and the supernova-only preference is not explicitly resolved or discussed quantitatively, and it constitutes the most significant internal tension in the analysis.

Reconstructed cosmological functions

Using best-fit parameters, the reconstructed $$T = T^{\lambda}_{\ \alpha\beta}S_{\lambda}^{\ \alpha\beta}$$8 matches the 34 CC points and the distance modulus matches both the 1701 SN22 points and the Union3 compilation across $$T = T^{\lambda}_{\ \alpha\beta}S_{\lambda}^{\ \alpha\beta}$$9. The reconstructed deceleration parameter shows a smooth transition from $$\mathcal{S}=\frac{1}{16\pi G}\int d^4x\, e\,[T+f(T,\mathcal{L}_m)] + \int d^4x\, e\,\mathcal{L}_m,$$0 to $$\mathcal{S}=\frac{1}{16\pi G}\int d^4x\, e\,[T+f(T,\mathcal{L}_m)] + \int d^4x\, e\,\mathcal{L}_m,$$1, with transition redshifts:

Dataset	$$\mathcal{S}=\frac{1}{16\pi G}\int d^4x\, e\,[T+f(T,\mathcal{L}_m)] + \int d^4x\, e\,\mathcal{L}_m,$$2	$$\mathcal{S}=\frac{1}{16\pi G}\int d^4x\, e\,[T+f(T,\mathcal{L}_m)] + \int d^4x\, e\,\mathcal{L}_m,$$3	$$\mathcal{S}=\frac{1}{16\pi G}\int d^4x\, e\,[T+f(T,\mathcal{L}_m)] + \int d^4x\, e\,\mathcal{L}_m,$$4
CC	$$\mathcal{S}=\frac{1}{16\pi G}\int d^4x\, e\,[T+f(T,\mathcal{L}_m)] + \int d^4x\, e\,\mathcal{L}_m,$$5	$$\mathcal{S}=\frac{1}{16\pi G}\int d^4x\, e\,[T+f(T,\mathcal{L}_m)] + \int d^4x\, e\,\mathcal{L}_m,$$6	$$\mathcal{S}=\frac{1}{16\pi G}\int d^4x\, e\,[T+f(T,\mathcal{L}_m)] + \int d^4x\, e\,\mathcal{L}_m,$$7
CC+Union3	$$\mathcal{S}=\frac{1}{16\pi G}\int d^4x\, e\,[T+f(T,\mathcal{L}_m)] + \int d^4x\, e\,\mathcal{L}_m,$$8	$$\mathcal{S}=\frac{1}{16\pi G}\int d^4x\, e\,[T+f(T,\mathcal{L}_m)] + \int d^4x\, e\,\mathcal{L}_m,$$9	$T$0
SN22	$T$1	$T$2	$T$3

The effective EoS remains negative throughout, with $T$4 in all cases, indicating quintessence-like rather than cosmological-constant or phantom behavior. The transition redshifts are systematically lower than typical $T$5CDM expectations ($T$6–$T$7), particularly for SN22 ($T$8); the authors interpret this as an earlier acceleration onset relative to $T$9CDM and suggest it may be relevant to late-time cosmological tensions, though they do not demonstrate this quantitatively against, e.g., DESI BAO or growth data.

Limitations and open questions

Several caveats bear directly on the conclusions. First, the analysis is purely background-level: no perturbation, growth-of-structure, CMB, or weak-lensing constraints are included, so claims about resolving $\mathcal{L}_m$0 or $\mathcal{L}_m$1 tensions remain untested within this work. Second, the BBN derivation assumes $\mathcal{L}_m$2 and Boltzmann rather than Fermi-Dirac statistics for the weak rates, standard but approximate treatments. Third, the mismatch between the SN22-fitted $\mathcal{L}_m$3 and the BBN window is not reconciled; whether joint early-plus-late likelihoods would shift the posterior toward the BBN interval, or whether the model fails this consistency check, is left open. Fourth, the choice $\mathcal{L}_m$4 versus $\mathcal{L}_m$5 affects the continuity equation structure, and the sensitivity of the constraints to this convention is not explored.

Conclusion

The paper provides the first systematic early-to-late cosmological treatment of $\mathcal{L}_m$6 gravity, deriving a tight BBN bound $\mathcal{L}_m$7 and demonstrating observational viability of an inverse-torsion model against CC, Union3, and Pantheon+SH0ES data. Its principal phenomenological findings—quintessence-like $\mathcal{L}_m$8 and an accelerated-phase onset earlier than in $\mathcal{L}_m$9CDM—are consistent across datasets, though the quantitative disagreement between the SN22-fitted $f_T$0 and the BBN window, and the absence of perturbative tests, remain open questions that future multi-probe analyses must address.