Dynamical system and statefinder analysis of cosmological models in f(T, B) gravity

Abstract: This study systematically investigates the cosmological dynamics of two well-motivated functional forms in $f(T,B)$ gravity within a flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe. Here $T$ denotes the torsion scalar and $B$ the boundary term, with the special choice $f(T,B) = - T + B$ recovering General Relativity. We focus on a multiplicative power-law model $f(T,B) = c_1 T<sup>αB<sup>β$ and an additive mixed power-law model $f(T,B) = c_2 T^α+ c_3 B<sup>β$. Using dynamical system techniques, we construct autonomous systems and identify de Sitter attractors that naturally explain late-time cosmic acceleration. Analytical stability conditions for these fixed points are derived, and numerical simulations reveal characteristic evolutionary patterns, such as spiral trajectories and damped oscillations in the additive mixed power-law model. Furthermore, statefinder diagnostics are applied to quantitatively distinguish these models from the standard $Λ$CDM paradigm and other dark energy scenarios. The results indicate that $f(T,B)$ gravity offers a theoretically consistent and observationally distinguishable geometric framework for explaining cosmic acceleration, presenting a compelling alternative to conventional dark energy models.

• The paper develops autonomous dynamical systems without the auxiliary parameter λ, showing that multiplicative and additive power-law f(T,B) models can produce stable de Sitter attractors and late-time acceleration.
• The multiplicative model remains in the Chaplygin-gas statefinder region, while the additive model can cross the ΛCDM point and exhibit damped Chaplygin-to-quintessence oscillations.
• The analysis transfers to f(Q,C) gravity at the FLRW background level, but matter-era dynamics, perturbations, and observational constraints remain necessary to assess viability.

Overview and motivation

This paper investigates the late-time cosmological dynamics of $f(T,B)$ gravity — the teleparallel extension of General Relativity (GR) in which the gravitational Lagrangian is an arbitrary function of both the torsion scalar $T$ and the boundary term $B$ — within a spatially flat FLRW universe. The framework is motivated by the identity $R = -T + B$, which makes $f(T,B)$ the complete teleparallel analog of $f(R)$ gravity: it contains $f(R)$ as the special case $f(-T+B)$, while also encompassing TEGR ($f=0$) and pure $f(T)$ gravity. The authors analyze two functional forms introduced previously by Bahamonde and Capozziello: a multiplicative power-law model $T$0, in which torsion and boundary contributions are inseparably coupled, and an additive mixed power-law model $T$1, which permits a clean separation of their individual dynamical roles.

The central methodological contribution is the construction of fully autonomous dynamical systems without invoking the commonly used auxiliary parameterization $T$2. Instead, the authors introduce dimensionless variables including $T$3, $T$4, $T$5, and $T$6, together with the Friedmann constraint $T$7. This choice closes the system exactly once the functional form of $T$8 is specified, avoiding phenomenological assumptions about the expansion history.

Fixed points and stability

For the multiplicative model, the autonomous system reduces to three variables $T$9 and admits a single fixed point $B$0, existing whenever $B$1. This point is a de Sitter solution with $B$2, $B$3, $B$4, and scale factor $B$5; it is always accelerating. Analytical stability conditions are derived from the eigenvalues of the Jacobian, yielding a detailed parameter-space classification: for example, stability holds for $B$6 when either $B$7 or $B$8. For the representative choice $B$9, the point is a stable node, so the power-law model generically explains late-time acceleration through a purely geometric dark energy sector.

For the additive mixed power-law model, the four-variable system possesses a line of non-hyperbolic equilibria $R = -T + B$0 parametrized by $R = -T + B$1 (with existence requiring $R = -T + B$2), plus the point $R = -T + B$3 at $R = -T + B$4. Both are de Sitter points with $R = -T + B$5. Because each point carries one zero eigenvalue, linear analysis alone is inconclusive; the authors report that the center manifold theorem could not be applied because nonlinear terms do not vanish near equilibrium, so stability is assessed by requiring negative real parts of the remaining eigenvalues and confirmed numerically via phase portraits. For $R = -T + B$6, trajectories spiral into the attractor; for $R = -T + B$7, the effective equations of state exhibit damped oscillations before settling at $R = -T + B$8. These oscillatory transients are a distinctive prediction absent from simpler $R = -T + B$9 or $f(T,B)$0 cosmologies.

An additional structural observation is that, since the affine connection vanishes in FLRW geometry, the modified Friedmann equations here coincide with those of $f(T,B)$1 gravity, so the entire dynamical analysis transfers directly to that non-metricity-based formulation.

Statefinder diagnostics

The statefinder pair $f(T,B)$2 and $f(T,B)$3 is used to discriminate the models against standard scenarios, where $f(T,B)$4 corresponds to $f(T,B)$5CDM, $f(T,B)$6 to Chaplygin gas, and $f(T,B)$7 to quintessence. Closed-form expressions for $f(T,B)$8 and $f(T,B)$9 are derived for both models in terms of the dynamical variables.

Three results stand out. First, both models asymptote to the $f(R)$0CDM fixed point, consistent with the de Sitter attractors found dynamically. Second, the multiplicative model's trajectory remains confined to the Chaplygin gas region throughout its evolution before reaching $f(R)$1CDM — a direct consequence of the inseparable $f(R)$2–$f(R)$3 coupling. Third, the additive model shows qualitatively richer behavior: for $f(R)$4 its trajectory crosses the $f(R)$5CDM point and undergoes damped transitions between Chaplygin gas–like and quintessence-like regimes, with a spiral structure visible in the $f(R)$6–$f(R)$7 plane. The paper argues, plausibly but only qualitatively, that this contrast stems from whether the Lagrangian couples or separates $f(R)$8 and $f(R)$9, making the structural form of $f(R)$0 itself observationally distinguishable through higher-derivative expansion data.

Limitations and open questions

Several caveats qualify the results. All identified fixed points correspond exclusively to dark-energy-dominated epochs; no matter-dominated or radiation-dominated critical points appear in the reduced systems, so the analysis does not yet demonstrate a viable full sequence of cosmic eras — the authors explicitly flag extension to matter-dominated phases as necessary future work. Stability conclusions for the additive model rest on eigenvalue sign conditions plus numerical phase portraits rather than rigorous analytic proof, owing to the failure of the center manifold method at the non-hyperbolic points. The parameter choices illustrating oscillatory behavior, notably $f(R)$1, are extreme, and no observational constraints on $f(R)$2, $f(R)$3, or $f(R)$4 are imposed; confrontation with cosmic chronometers, BAO, and growth-of-structure data remains outstanding. Finally, perturbations, structure formation, and early-universe applications (inflation, singularity avoidance) lie outside the scope of the background-level analysis presented.

Conclusion

The paper establishes that both multiplicative and additive power-law $f(R)$5 cosmologies possess stable de Sitter attractors capable of driving late-time acceleration geometrically, with analytically characterized stability regions in $f(R)$6 space. Its principal technical advance is an autonomous dynamical formulation free of the $f(R)$7 parameterization, and its principal physical finding is that the additive form supports transient Chaplygin-gas-to-quintessence transitions and damped oscillations that the multiplicative form lacks, features cleanly exposed by statefinder diagnostics. Whether these distinguishing signatures survive observational constraints and perturbative scrutiny remains the key open question raised by the work.