Horndeski Teleparallel Gravity Overview

• Horndeski Teleparallel Gravity is a framework that blends scalar-tensor Horndeski theory with teleparallel gravity by replacing curvature with torsion.
• It interpolates between TEGR, f(T) gravity, and nonminimal scalar–torsion couplings, offering viable models for late-time acceleration and bouncing cosmologies.
• The theory facilitates well-tempered self-tuning to screen vacuum energy and predicts distinctive gravitational wave and perturbation signatures.

Horndeski teleparallel gravity denotes a class of gravitational theories generalizing both Horndeski scalar–tensor gravity and the teleparallel approach to gravity, offering a unified framework for exploring modified gravitational dynamics based on torsion rather than curvature. This family of theories includes and interpolates between the teleparallel equivalent of General Relativity (TEGR), f(T) gravity, and nonminimal scalar–torsion couplings analogous to those appearing in the general Horndeski Lagrangian. By leveraging the additional structural freedom of torsion and boundary terms, these models can evade some of the critical theoretical and observational limitations faced by their curvature-based counterparts, such as those resulting from gravitational wave constraints. Horndeski teleparallel models have been developed systematically for both cosmological and astrophysical scenarios, permitting late-time acceleration, crossing of the phantom barrier, stable bouncing cosmologies, well-tempered screening of vacuum energy, and a generally enriched phenomenology.

1. Foundations: Teleparallel Geometry and the Horndeski Structure

Teleparallel gravity reformulates gravitation using torsion and tetrad fields as the primary variables, replacing the central role of curvature in metric-based General Relativity. The torsion scalar $T$ is constructed from the antisymmetric part of the Weitzenböck (flat) connection:

$T = T^{\alpha}_{\;\;\mu\nu} S_\alpha^{\;\;\mu\nu}$

where $S_\alpha^{\;\;\mu\nu}$ is the superpotential, built from torsion and contortion, and the metric is reconstructed via %%%%2%%%%. TEGR and GR differ only by a boundary term: $R = -T + B$ with $B = -2 \bar{\nabla}^\mu T^\beta_{\;\;\mu\beta}$.

The standard Horndeski action in curvature-based gravity is the most general scalar–tensor theory guaranteeing second-order equations of motion, and comprises contributions: $\mathcal{L} = \sum_{i=2}^5 \mathcal{L}_i$ with explicit nonminimal couplings and higher-derivative terms for a single scalar field $\phi$.

In Horndeski teleparallel gravity, the key construction is to reproduce these operator structures using torsion invariants and their irreducible pieces, replacing the Ricci scalar $R$ by $-T+B$, and to allow arbitrary scalar–torsion and derivative couplings without introducing higher-order pathologies.

2. Generalized Teleparallel Theory and Interpolation

A core formulation is the "generalized teleparallel theory" (GTT), in which the gravitational sector is described by a scalar quantity,

$$\mathcal{T} = -T - 2 a_1 \bar{\nabla}^\mu T^\beta_{\;\;\mu\beta}$$

where $a_1$ is a real parameter. For $a_1=1$, $\mathcal{T}$ coincides with the Ricci scalar up to sign; for $a_1=0$, $\mathcal{T}=-T$, yielding standard $f(T)$ gravity. This interpolation renders the GTT action,

$$S_{\text{GTT}} = \int d^4x\, \frac{e}{2\kappa^2} f(\mathcal{T}) + L_{\mathrm{matter}}$$

a bridge between locally Lorentz invariant $f(R)$ models and $f(T)$ theories that typically lack full local Lorentz invariance. The GTT structure thus embodies the flexibility found in Horndeski theory, managing both the order of field equations and local symmetry properties (Junior et al., 2015).

3. Structure of Teleparallel Horndeski Lagrangians

The teleparallel analogue of Horndeski gravity is constructed by

• using the Weitzenböck connection and tetrad variables,
• encoding the scalar field and its derivative couplings using torsion invariants, and
• including new scalar contractions beyond those available in $f(T)$ or standard scalar–torsion theories.

The generalized action takes the schematic form

$$\mathcal{S}_{\mathrm{BDLS}} = \int d^4x\, e\left[\mathcal{L}_{\mathrm{Tele}} + \sum_{i=2}^{5} \mathcal{L}_i + \mathcal{L}_{\mathrm{m}}\right]$$

where

• $$\mathcal{L}_{\mathrm{Tele}}=G_{\mathrm{Tele}}(\phi,X,T,T_{\mathrm{ax}},T_{\mathrm{vec}},I_2,J_1,J_3,\ldots)$$,
• the $\mathcal{L}_i$ mimic the Horndeski operators (now recast in the teleparallel setting), and
• the additional invariants such as $I_2$ are constructed from contractions of torsion pieces with derivatives of $\phi$ (Bahamonde et al., 2020, Bahamonde et al., 2019).

A rich set of new scalar invariants involving the irreducible components of torsion allows for an expanded Lagrangian sector $L_{\mathrm{Tele}}$ not present in the standard metric-based Horndeski action (Bahamonde et al., 2019).

4. Dynamics, Stability, and Cosmological Evolution

Parameter Space and Constraints

Horndeski teleparallel gravity preserves second-order field equations through judicious arrangement of couplings. The full parameter space is significantly larger than standard Horndeski, due to the additional torsion invariants. Crucially, after the observation of GW170817 and the constraint $|c_T/c-1| \lesssim 10^{-15}$ on the speed of gravitational waves, only a narrow class of curvature-based Horndeski models survived. In the teleparallel variant, extra torsion-dependent operators appear in the tensor perturbation action: $c_T^2 = \frac{\mathfrak{F}_T}{\mathcal{G}_T}$ where $\mathcal{G}_T$ and $\mathfrak{F}_T$ are explicit functions of Horndeski and teleparallel Lagrangian derivatives. The additional degrees of freedom in $G_{\mathrm{Tele}}$ can be tuned such that $c_T=1$ even in the presence of $G_4$ and $G_5$ couplings that would otherwise be excluded (Bahamonde et al., 2019, Capozziello et al., 2023, Ahmedov et al., 2023).

Well-Tempering and Screening Mechanisms

Horndeski teleparallel gravity enables the construction of well-tempered self-tuning models. These solutions utilize a degeneracy of the field equations to dynamicaly screen an arbitrarily large vacuum energy, leading to a stable late-time de Sitter or Minkowski vacuum (Bernardo et al., 2021, Bernardo et al., 2021). The mechanism exploits an on-shell degeneracy condition $\mathcal{Y}\mathcal{D}-\mathcal{C}\mathcal{Z}=0$ between the highest derivative coefficients in the scalar and Hubble evolution equations, ensuring that the contribution from vacuum energy is cancelled by the adjustment of the scalar field and torsion couplings.

Cosmological Dynamics and Phase Space

Cosmological phase space analyses show that teleparallel Horndeski models admit attractor solutions representing all standard cosmic epochs (radiation, matter, and dark energy domination) and can smoothly interpolate between decelerating and accelerating phases. The presence of non-minimal scalar–torsion and boundary term couplings allows for late-time de Sitter attractors even in the absence of a cosmological constant, as well as a greater capacity to cross the phantom divide or describe bouncing and cyclic scenarios (Kadam, 19 Jan 2025, Duchaniya, 2 Oct 2025, Ahmedov et al., 2023).

Perturbation Theory and Observational Viability

Second-order cosmological perturbation theory, including a full scalar–vector–tensor decomposition in gauge-invariant formalism, has been developed for teleparallel Horndeski gravity (in particular the BDLS theory) (Ahmedov et al., 2 Dec 2024). The quadratic actions for each sector expose novel torsion-induced kinetic couplings and gradient terms (with coefficients $D_1$, $D_2$, and analogous matrices in the scalar sector). Stability against ghosts and Laplacian instabilities imposes positivity of these kinetic and gradient coefficients. Scalar perturbations can alter structure formation and effective gravitational coupling, while tensor perturbations (gravitational waves) receive torsion contributions that can, in principle, be tested observationally.

The so-called $\alpha$-parameter formalism ($\alpha_K$, $\alpha_B$, $\alpha_M$, $\alpha_T$), standard in effective field theory approaches, is generalized in the teleparallel context, with the teleparallel sector introducing additional contributions to the expressions for these parameters, further enlarging the viable model space (Ahmedov et al., 2023).

5. Machine Learning, Noether Symmetries, and Systematic Classification

Machine learning and Gaussian process regression have begun to be used to reconstruct Lagrangian functions $f(T, B, \phi, X)$ nonparametrically from cosmological data, offering complementary avenues for constraining viable teleparallel Horndeski models (Bahamonde et al., 2021).

A comprehensive taxonomy of Horndeski teleparallel cosmologies using Noether symmetry analysis enables systematic selection and identification of integrable models. The imposition of symmetry invariance conditions yields strong restrictions on the free functions in the Lagrangian, thereby identifying subclasses with analytic solutions, exact scaling behaviors, or conserved quantities (Dialektopoulos et al., 2021).

6. Special Theories and Extensions: Boundary Terms, Nonmetricity, and Effective Field Theory

Boundary terms, such as the divergence $B$ appearing in $R = -T + B$, as well as further extensions involving the teleparallel equivalent of the Gauss–Bonnet term $T_G$ and its boundary $B_G$, are instrumental for cosmological and dynamical richness (Kadam, 19 Jan 2025). The most general actions involve arbitrary functions $f(T,B,T_G,B_G, \phi, X)$, providing additional degrees of freedom to control cosmic evolution and late-time acceleration (Duchaniya, 2 Oct 2025).

Symmetric teleparallel Horndeski gravity further generalizes the framework by considering gravity fully decoupled from both curvature and torsion, with nonmetricity $Q_{\lambda\mu\nu}$ as the carrier of gravitational dynamics, and constructing the most general k-essence and kinetic gravity braiding actions that remain second order (Bahamonde et al., 2022).

Effective field theory expansions for teleparallel gravity have revealed a substantially richer landscape of higher-order corrections than for GR, owing to non-symmetric tetrad variables and the distinct structure of torsion. This leads to new physical operators in the high-energy theory, albeit suppressed by the heavy scale $\Lambda$, and paves the way for distinguishing GR and TEGR (and their scalar–tensor extensions) observationally at sufficiently high energy (Mylova et al., 2022).

7. Outlook: Observational Implications and Theoretical Significance

Horndeski teleparallel gravity accommodates cosmological models describing accelerated expansion, dark energy evolution with phantom crossing, successful screening of vacuum energy via well-tempered solutions, and non-singular bounces that bypass the no-go theorems present in standard Horndeski frameworks (Ahmedov et al., 2023). The enlarged space of Lagrangian operators, made possible by torsion and boundary terms, enables restoration of phenomenologically attractive sectors of Horndeski gravity previously excluded by observational constraints, particularly in the gravitational wave sector. Cosmological perturbation theory in this framework is sufficiently developed for direct comparison with structure formation and multi-messenger astrophysical data, and the theory is poised for future confrontation with large-scale surveys, gravitational wave experiments, and beyond-standard-cosmology observations.

The teleparallel-Horndeski merger thus represents a robust, technically rich, and phenomenologically fertile ground for modified gravity, advancing both theoretical understanding and the quest for observable deviations from the standard cosmological model.