Modified Teleparallel Gravity

• Modified Teleparallel Gravity is a framework that generalizes the teleparallel equivalent of general relativity by using torsion scalars and boundary terms instead of curvature.
• It employs the tetrad formalism with a curvature-free connection, enabling extensions like f(T) and f(T,B) that yield second- or higher-order field equations.
• These models offer novel cosmological dynamics, including late-time cosmic acceleration, bouncing scenarios, and modified gravitational structures for compact objects.

Modified teleparallel gravity encompasses a broad class of gravitational theories that generalize the teleparallel equivalent of general relativity (TEGR) by constructing the action from torsion scalars and their higher-order or boundary term modifications, rather than from the spacetime curvature of the Levi–Civita connection as in GR. These theories use the tetrad (vierbein) formalism and a curvature-free (but torsionful) affine connection. Modifications are often introduced either via an arbitrary function of the torsion scalar (f(T)), via couplings to boundary terms (B), via higher-derivative torsional invariants, or by incorporating couplings to matter, Gauss–Bonnet terms, or scalar fields. These constructions yield a rich phenomenology with distinctive implications for cosmology, astrophysics, local Lorentz invariance, and the structure of gravitational degrees of freedom.

1. Teleparallel Gravity Framework and the Basis for Modification

• Geometric Structure: Teleparallel gravity replaces the role of curvature in mediating gravitational effects with torsion, using an independent affine connection constrained to have vanishing curvature (the Weitzenböck connection) but nonzero torsion.
• Dynamical Variables: The fundamental fields are the tetrad (e^a_μ) and, in covariant formulations, an independent flat spin connection ω^a_{bμ}. The metric is reconstructed as $g_{\mu\nu} = \eta_{ab} e^a_\mu e^b_\nu$.
• Torsion Scalars: The teleparallel equivalent of GR (TEGR) employs the torsion scalar $T$, defined by contractions of the torsion tensor. It relates to the Ricci scalar R via the boundary term B as $R = -T + B$.
• Generalization: Modified teleparallel theories generalize the gravitational Lagrangian beyond the TEGR limit. A generic action takes the form

$$S = \frac{1}{2\kappa} \int d^4x \, e \, f(\text{torsional invariants}, B, \mathcal{T}, \ldots) + S_m \,,$$

where $f$ can be a function of T, its irreducible quadratic invariants, the boundary term B, additional invariants (such as teleparallel Gauss–Bonnet T_G), and possible couplings to the trace of the matter energy-momentum tensor $\mathcal{T}$ (Hohmann, 2022, Bahamonde et al., 2017, Bahamonde et al., 2016, Vilhena et al., 2023, Kadam, 19 Jan 2025).

2. Major Classes of Modified Teleparallel Gravity Theories

• f(T) Gravity: The simplest extension, f(T) gravity, replaces the TEGR torsion scalar with an arbitrary function f(T). The resulting field equations remain second order, unlike the fourth order of metric f(R) gravity. However, f(T) construction generically breaks local Lorentz invariance due to the frame-dependent nature of T (Bahamonde et al., 2015, Krssak, 2017, Golovnev, 2018). These theories are attractive for cosmological applications, as they can provide late-time acceleration and avoid the Dolgov–Kawasaki instability, which constrains f(R) models (Behboodi et al., 2012, Kadam, 19 Jan 2025).
• f(T,B) and f(T,T_G,B_G) Theories: To interpolate between f(T) and f(R) gravity (and to explore higher-derivative extensions), models include both T and the boundary term B: f(T,B) (Bahamonde et al., 2015, Bahamonde et al., 2016, Bahamonde et al., 2019, Bahamonde et al., 2020). The unique Lorentz-invariant subclass is f(-T+B) = f(R), while generic f(T,B) models have higher-order equations and richer cosmological phenomenology. Analogously, teleparallel Gauss–Bonnet terms (T_G) and their boundary terms (B_G) allow constructions such as f(T,T_G) or the most generic f(T,B,T_G,B_G) models (Bahamonde et al., 2016, Kadam, 19 Jan 2025).
• General Quadratic and Irreducible-Torsion Models: A program of constructing the most general quadratic-in-torsion theories yields actions built from independent invariants: vector, axial, and tensor parts of the torsion tensor (Bahamonde et al., 2017, Bahamonde et al., 2020, Vilhena et al., 2023). The action may thus be a function f(T_ax, T_ten, T_vec), with the TEGR and New General Relativity (NGR) limits recovered for specific choices. Extensions may include parity-violating and higher-order invariants.
• Scalar–Teleparallel Theories and Matter Couplings: Scalar-torsion extensions (f(T,φ), teleparallel Horndeski analogs) introduce scalar fields coupled via T, B, or both (Kadam, 19 Jan 2025). Further generalizations include dependencies on the trace of the energy-momentum tensor ($f(T, B, \mathcal{T})$), which can arise from semi-classical quantum corrections (Chen et al., 2021, Bahamonde et al., 2016).

3. Symmetry Realizations and Covariant Formulation

• Tetrad and Spin Connection Choice: Modified teleparallel theories are sensitive to the choice of tetrad (and, in the non-covariant formulation, the choice of vanishing spin connection), with only certain “good tetrads” admitting solutions in f(T)/f(T,B) models (Krssak, 2017, Hohmann et al., 2019). The covariant approach, with an independent flat spin connection, allows local Lorentz invariance to be restored at the level of the action, but only for specific classes (e.g., f(-T+B)=f(R)), while the condition

$$Q_{ab}(h,\omega) = \partial_\nu T \cdot \partial_\mu [h h_{[a}^\mu h_{b]}^\nu] = 0$$

governs the consistent selection of tetrads and connections (Krssak, 2017).

• Role of Symmetry: Imposing spacetime symmetries via Cartan geometry clarifies which tetrads / connections are compatible, singling out “good tetrads” in cosmological scenarios that automatically solve the antisymmetric field equations. This is particularly crucial in f(T) cosmology (Hohmann et al., 2019).
• Bigravity Interpretation: Covariant modified teleparallel theories can be interpreted as effective bigravity models, where the dynamical and reference tetrads play distinct gravitational and inertial roles (Krssak, 2017).

4. Cosmological Dynamics and Applications

• Cosmic Acceleration: Many f(T), f(T,B), and scalar-torsion models naturally produce a late-time accelerating universe. For instance, f(T) = T + ε φ(T) models with small ε are observationally viable and free of matter instabilities (Behboodi et al., 2012, Karimzadeh et al., 2019). The dynamical systems approach applied to f(T,φ), f(T,B), and higher-derivative extensions provides a systematic classification of critical points representing radiation, matter, and de Sitter phases, yielding full cosmological histories closely matching ΛCDM expectations or enabling phantom-like behaviors without invoking exotic fields (Kadam, 19 Jan 2025, Otalora et al., 2016).
• Higher-Derivative and Gauss–Bonnet Corrections: Models with $F(T, (\nabla T)^2, \Box T)$ reproduce all cosmic epochs and can yield both de Sitter and scaling attractors (Otalora et al., 2016). Teleparallel Gauss–Bonnet terms T_G, appearing in f(T,T_G) or f(T,B,T_G,B_G), enrich cosmological dynamics and allow for connections with higher-curvature gravity and new phenomenological signatures (Bahamonde et al., 2016).
• Bounce Cosmologies: Certain f(T) forms (including Born–Infeld–inspired models) yield modified Friedmann equations equivalent to loop quantum cosmology effective dynamics, enabling nonsingular bounces solely from torsion corrections and avoiding the need for exotic matter (Casalino et al., 2020).
• Phantom Behavior and Observational Viability: f(T) and f(T,B) models can be calibrated to reproduce observed effective equations of state (e.g., w_0 ≈ -1.03 from Planck2018), producing phantom-like expansion with stable matter perturbations in subspaces of the parameter space (Karimzadeh et al., 2019).
• Neutron Stars and Compact Objects: Appropriate parameterizations in quadratic f(T) or NGR-type theories can modify the effective gravitational constant in the presence of matter, shifting the mass-radius relationship and maximum mass of neutron stars, with possible observational constraints from high-mass compact objects (Vilhena et al., 2023).

5. Formal Properties, Stability, and Degrees of Freedom

• Field Equations: f(T) gravity is unique among f(T,B) models in retaining second-order field equations but generically breaks local Lorentz invariance (Bahamonde et al., 2015). Other models (f(T,B), f(T,T_G), etc.) are fourth order and may violate Lorentz invariance unless reduced to specific forms (e.g., f(R) gravity via f(-T+B)).
• Stability and Matter Instabilities: Unlike f(R) gravity, modified teleparallel theories such as f(T) are free of the Dolgov–Kawasaki instability. The linear perturbation equation for T is first order in time and always exhibits damping, regardless of the form of f(T), provided the modification parameter is small (Behboodi et al., 2012).
• Strong Coupling and Additional Modes: f(T) gravity is known to possess extra degrees of freedom that appear non-dynamical at quadratic order and only “switch on” at quartic order in perturbation theory; this is a manifestation of the strong coupling problem. Modified effective field theory (EFT) constructions can lower the order at which these degrees become dynamical (e.g., to cubic in the action), providing better perturbative control and a classification of models by the perturbative order of activation (Hu et al., 4 Oct 2024).
• Disformal and Conformal Transformations: Disformal transformations of the tetrad, which generalize conformal transformations by including scalar derivatives, do not in general eliminate local Lorentz–breaking terms in f(T) gravity, though they offer insight and a pathway toward scalar–torsion theories (Golovnev et al., 2019). Conformal analysis reveals that only TEGR or f(-T+B)=f(R) models admit an Einstein frame transformation; generic f(T_ax, T_ten, T_vec) models do not (Bahamonde et al., 2017).

6. Extensions, Nonminimal Couplings, and Prospects

• Quantum Fluctuations and Semi-Classical Corrections: Effective quantum corrections in the tetrad give rise to nonminimal torsion–matter couplings, resulting in theories of the form f(T,B,𝓣). These promote new effective energy exchange terms between the gravitational and matter sectors and enable transitions between expanding and accelerating cosmological phases (Chen et al., 2021).
• Novel Parity-Violating and Birefringent Theories: Couplings to topological terms such as the Nieh–Yan invariant and axion-like fields introduce parity violation in gravitational waves, potentially leading to distinctive birefringence and correlated scalar–tensor spectra, especially in cosmological inflation (Li et al., 2023).
• Exact Solutions, Spherical Symmetry, and Noether Symmetries: Explicit spherically symmetric solutions—black holes, compact stars—can be obtained in f(T,B) and general torsion-invariant theories using Noether symmetry methods, providing solution branches distinct from GR and offering observational discriminants (Bahamonde et al., 2019, Bahamonde et al., 2020).
• Phenomenological Constraints and Observational Signatures: Mass-radius diagrams for neutron stars, corrections to planetary perihelion shifts, light bending, and Shapiro delay all provide distinct, parameter-dependent signatures for confronting modified teleparallel models with astrophysical data (Bahamonde et al., 2020, Vilhena et al., 2023). Cosmological perturbations show modified growth rates and effective gravitational constants, with possible impact on large-scale structure and lensing (Bahamonde et al., 2020).

7. Summary and Outlook

Modified teleparallel gravities provide a broad and versatile framework for extending GR through torsional degrees of freedom, with f(T), f(T,B), and more general f(T_ax, T_ten, T_vec, B) models offering a spectrum of field equations, symmetry properties, and phenomenological consequences. They are distinguished by unique properties regarding local Lorentz invariance, stability of matter and cosmological solutions, degree-of-freedom structure, and strong coupling behavior. Theoretical advances—including covariant formulation, symmetry-based tetrad classification, EFT analysis of strong coupling, and the use of Noether symmetries—have clarified the viable model space and observational implications, paving the way for rigorous constraints and further model-building using precision cosmology, gravitational-wave, and compact object data (Behboodi et al., 2012, Bahamonde et al., 2015, Krssak, 2017, Bahamonde et al., 2017, Otalora et al., 2016, Bahamonde et al., 2016, Hohmann et al., 2019, Bahamonde et al., 2019, Bahamonde et al., 2020, Bahamonde et al., 2020, Casalino et al., 2020, Chen et al., 2021, Li et al., 2023, Vilhena et al., 2023, Hu et al., 4 Oct 2024, Kadam, 19 Jan 2025).