Modified Teleparallel Gravity Models

• Modified Teleparallel Gravity Models are gravitational theories that encode gravitation via torsion instead of curvature, generalizing TEGR.
• They extend the gravitational action to include functions of torsion, boundary terms, and higher-order invariants, introducing new dynamics and degrees of freedom.
• These models provide frameworks for addressing cosmic acceleration, dark energy phenomenology, and modified gravitational wave propagation beyond standard GR.

Modified teleparallel gravity models encompass a diverse class of gravitational theories formulated in the teleparallel framework, where gravitation is encoded in torsion rather than curvature. These models generalize the teleparallel equivalent of General Relativity (TEGR) by allowing the gravitational Lagrangian to depend not just on the linear torsion invariant, but also on more general functions of torsion, its boundary contributions, and higher-order invariants or matter couplings. The inclusion of these generalizations enables the exploration of phenomenology beyond GR, facilitates the data-driven testing of extensions that may explain cosmic acceleration, and connects with various theoretical proposals in both high-energy and cosmological contexts.

1. Foundations of Teleparallel Gravity and Its Modifications

Teleparallel gravity is constructed on a manifold equipped with a tetrad field $e^a_\mu$ and a connection with vanishing curvature (typically the Weitzenböck connection), yielding a nonzero torsion tensor

$$T^\alpha_{\;\mu\nu} = e_a^\alpha (\partial_\mu e^a_\nu - \partial_\nu e^a_\mu) .$$

The torsion scalar $T$ provides the Lagrangian density for TEGR: $$T = \frac{1}{4} T^\rho_{\;\mu\nu} T_\rho^{\;\mu\nu} + \frac{1}{2} T^\rho_{\;\mu\nu} T^{\nu\mu}_{\;\;\;\rho} - T_{\rho\mu}^{\;\;\;\rho} T^{\nu\mu}_{\;\;\;\nu} .$$ The Ricci scalar $R$ of GR is related to $T$ by a boundary term $B$: $$R = -T + B , \qquad B = \frac{2}{e} \partial_\mu (e T^\mu) .$$ Modified teleparallel gravity models introduce generalizations by considering gravitational actions of the form $f(T)$, $f(T,B)$, $f(T, T_G)$, or more generally $f(T, B, T_G, B_G, \mathcal{T})$, where $T_G$ is the teleparallel analogue of the Gauss–Bonnet invariant, $B_G$ is its associated boundary term, and $\mathcal{T}$ is the trace of the energy–momentum tensor (Kofinas et al., 2014, Bahamonde et al., 2015, Bahamonde et al., 2016). These extensions admit new dynamical features, degrees of freedom, and symmetry properties.

2. Classes of Modified Teleparallel Gravity Theories

The principal classes and their characteristic structures are as follows:

Theory	Lagrangian Form	Key Properties
$f(T)$	$f(T)$	Second order, Lorentz breaking
$f(R)$ (teleparallel equiv.)	$f(-T+B)$	Fourth order, Lorentz invariant
$f(T,B)$	$f(T,B)$	Unifies $f(T)$ and $f(R)$
$f(T, T_G)$	$f(T, T_G)$	Involves quartic torsion, new d.o.f.
$f(T, B, T_G, B_G, \mathcal{T})$	General	Encodes trace and higher invariants

• $f(T)$ Gravity: Depends solely on the torsion scalar; field equations remain second order due to the first-derivative structure of $T$, but the theory breaks local Lorentz invariance as neither $T$ nor $f(T)$ are invariant under local Lorentz transformations (Bahamonde et al., 2015, Bahamonde et al., 2017).
• $f(R)$ in Teleparallel Form: Achieved via $f(T,B)$ with $f(-T+B) = f(R)$; the only Lorentz-invariant possibility within this class, resulting in fourth-order field equations due to the second-derivative dependence of $B$ (Bahamonde et al., 2015).
• $f(T, T_G)$ Gravity: Incorporates a teleparallel equivalent of the Gauss–Bonnet invariant, $T_G$, which is highly nontrivial (quartic in torsion). In four dimensions, $T_G$ reduces to a topological invariant (does not affect bulk equations of motion), but in dimensions $D>4$ or for general $F(T,T_G)$, genuinely new dynamics emerge (Kofinas et al., 2014).
• Generalized Forms / Trace Extensions: The action may include boundary terms and couplings to the trace of the energy–momentum tensor, providing a unified framework encompassing modified gravity models with matter-geometry couplings or higher-order invariants (Bahamonde et al., 2016).

3. Mathematical Structure and Symmetries

A central feature is the relationship between the torsion-based and the curvature-based formulations. The combination $-T+B$ restores diffeomorphism and local Lorentz invariance (mirroring $R$). For Gauss–Bonnet analogues: $G = -T_G + B_G ,$ where $G$ is the usual Gauss–Bonnet invariant and $T_G$ is its torsion-based counterpart, constructed as a quartic function of the contorsion 1-forms $K$ and the vielbein $e^a$: $$T_G = -\frac{1}{(D-4)!}\, \epsilon_{a_1\dots a_D}[K K K K + \cdots] \wedge e^{a_5} \wedge \cdots \wedge e^{a_D} .$$ In four dimensions, $T_G$ is a total derivative: $T_G^{(D=4)} = dB ,$ implying topological invariance in this case (Kofinas et al., 2014, Bahamonde et al., 2016).

In $f(T,B)$, only $f(-T+B)$ yields local Lorentz invariance—deviation from this specific form always results in explicit Lorentz breaking. $f(T)$ gravity is the unique modified teleparallel model with strictly second-order field equations; all other generalizations involving $B$, $T_G$, or trace couplings generically increase the order of the equations (Bahamonde et al., 2015, Bahamonde et al., 2017).

4. Cosmological Dynamics and Phenomenology

Modified teleparallel gravity models, especially those including boundary or higher-order terms, offer mechanisms to address both early- and late-time cosmology:

• Cosmological Evolution: Autonomous dynamical system techniques applied to cosmological equations reveal rich phase spaces with critical points corresponding to radiation, matter, and dark energy epochs. Late-time attractors often correspond to de Sitter-like acceleration (with deceleration parameter $q<0$) in agreement with current observations. Model parameters can be tuned to produce a viable thermal history with transitions among cosmic epochs (Kadam, 19 Jan 2025, Bahamonde et al., 2015, Wright, 2016).
• Dark Energy Phenomenology: Effective torsional fluid interpretations allow for "phantom crossing" scenarios where the equation-of-state parameter for the effective torsion component crosses $w=-1$ without introducing pathological phantom fields (Karimzadeh et al., 2019).
• Matter Coupling and Trace Extensions: Nonminimal couplings between torsion and the matter trace $\mathcal{T}$ (as arise from semiclassical quantum corrections) generate nontrivial energy transfers between gravity and matter, yielding decelerating-to-accelerating transitions after matter–dominance (Chen et al., 2021).
• Unified Models: The Chaplygin gas can be mimicked by an appropriate $f(T)$, achieving a unification of dark matter and dark energy at the effective, geometric level (Sahlu et al., 2019).

5. Gravitational Waves, Local Tests, and Perturbative Structure

• Gravitational Wave Propagation: In $f(T,B)$, the gravitational wave equation is modified only by friction (Planck mass run rate), while the speed remains luminal, consistent with multimessenger constraints: $$\ddot{h}_{ij} + (3+\nu)H\,\dot{h}_{ij} + \frac{k^2}{a^2} h_{ij} = 0 , \quad \nu = \frac{1}{f_T} \frac{d f_T}{d\ln a} .$$ Stability requires $f_T<0$ (Bahamonde et al., 2020).
• Solar System Bounds: Spherically symmetric, weak‐field solutions in various models produce corrections to the Schwarzschild metric impacting photon sphere size, perihelion shift, and light deflection. Constraints on the functional parameters ($\alpha, \beta, \gamma$, etc.) are imposed by high‐precision data (VLBI, Cassini, radar echo, perihelion precession) (Bahamonde et al., 2020, Bahamonde et al., 2020).
• Cosmological Perturbations: In $f(T)$ and scalar–torsion models, linear perturbations reveal that no extra propagating degrees of freedom arise in homogeneous backgrounds; the antisymmetric field equations act as extra constraints, algebraically fixing "Lorentz sector" modes and leading to a rescaling of the effective gravitational constant in structure formation: $$\hat\delta'' + H\,\hat\delta' = \frac{4\pi G}{f_T} a^2 \rho\,\hat\delta$$ for dust evolution (Golovnev et al., 2018, Bahamonde et al., 2020).
• Degeneracies and Uniqueness: The uniqueness of the "Einstein frame" is lost except for TEGR and the teleparallel equivalent of $f(R)$. In all other cases, residual nonminimal couplings or phantom behavior persists after conformal transformation (Wright, 2016, Bahamonde et al., 2017).

6. Mathematical and Theoretical Extensions

• Irreducible Torsion Invariants and NGR: By constructing Lagrangians as functions $f(T_{\rm ax}, T_{\rm ten}, T_{\rm vec})$ of the squares of the irreducible axial, tensor, and vector components of the torsion tensor, one obtains the most general viable second-order teleparallel theories. Special cases recover New General Relativity (NGR) and provide the general framework for analyzing degrees of freedom (Bahamonde et al., 2017).
• Boundary and Gauss–Bonnet Terms: The inclusion of boundary terms $B$ and higher-order invariants such as $T_G$, along with their associated boundary terms $B_G$, allows a precise teleparallel generalization of curvature-based modifications such as $f(R, G)$, unifying various metric and teleparallel models (Bahamonde et al., 2016, Kofinas et al., 2014).
• Branching Behavior and Degrees of Freedom: The classification of degrees of freedom depends on the structure of the Lagrangian and its derivatives. For some special classes (e.g., certain relations among $f_T, f_B$), branching reduces the number to that of standard $f(R)$ gravity, while for generic $f(T,B)$ or $f(T, T_G)$ more degrees of freedom may propagate (Bahamonde et al., 2020).

7. Theoretical Outlook and Open Issues

Modified teleparallel gravity models provide a versatile and mathematically rich framework. Their suitability for explaining cosmic acceleration, the dark sector, and early universe singularity resolution (e.g., in bounce models) has motivated substantial phenomenological investigations. However, several theoretical issues require ongoing scrutiny:

• Explicit local Lorentz violation (in most classes) and its observable consequences.
• Strong coupling and the precise degree of freedom count, particularly in nonlinear $f(T)$ and higher-derivative models.
• The uniqueness of "good" tetrads for given cosmological or astrophysical situations, especially regarding background symmetry and perturbations.
• Ghost freedom in the broader space of New GR models and trace-coupled theories.

The inclusion of boundary and higher-order torsion terms (as in $f(T, B)$ or $f(T, T_G)$), as well as nonminimal couplings motivated by semiclassical quantum corrections, continues to expand the landscape of modified teleparallel gravity, with ongoing research focused on both observational signatures and foundational consistency (Kadam, 19 Jan 2025, Chen et al., 2021, Otalora et al., 2016, Casalino et al., 2020).