Scalar–Torsion Gravity Models

• Scalar–torsion theories are gravitational models that couple torsion and scalar degrees of freedom, unifying elements of scalar–tensor and teleparallel frameworks.
• They provide rich phenomenology by mimicking radiation, matter, and dark energy behaviors through nonminimal couplings and derivative interactions.
• The models employ advanced invariance principles, stability analyses, and disformal transformations to address early universe inflation and compact object solutions.

Scalar–torsion theories are gravitational models in which the spacetime torsion, either as an independent geometric feature or as encoded through gauge structure, is coupled to one or more scalar degrees of freedom. These frameworks arise naturally in the generalized Poincaré Gauge Theory (PGT) and in covariant teleparallel gravity, and provide modified gravity models with phenomenology encompassing cosmology, compact objects, and gravitational wave signatures. Scalar–torsion models support a vast landscape of theoretical structures, ranging from the minimal quadratic PGT spin–0⁺ sector to the general scalar–torsion $L(T,X,Y,\phi)$ class. They unify and generalize elements found in scalar–tensor, teleparallel, and gauge-theoretic gravities.

1. Scalar–Torsion Structures in Poincaré Gauge Theory

Scalar–torsion theory in its original realization is exemplified in the quadratic Poincaré Gauge Theory as the spin–0⁺ sector. The fundamental variables are the orthonormal coframe $e^i$ and Lorentz connection $\Gamma_i{}^j$, acting as gauge potentials on a Riemann–Cartan manifold $U_4$. The quadratic action supporting the propagating "scalar–torsion" mode is

$$\mathcal{L}_G = \tfrac{a_0}{2}\,R\,\eta + \tfrac{b}{24}\,R^2\,\eta + \tfrac{a_1}{8}\,{}^{(1)}T^i \wedge \star\,{}^{(1)}T_i,$$

with $R$ the affine curvature scalar, $T^i$ the torsion 2-form decomposed into irreducibles, and $\star$ the Hodge dual on forms. The $a_1$ and $b$ coefficients control the propagation and positivity of the 0⁺ (scalar–torsion) mode. Dynamical equations for the FLRW background reduce to

$e^i$0

with $e^i$1 the Hubble parameter, $e^i$2 the trace torsion, and $e^i$3. These equations yield effective fluid quantities $e^i$4 for the geometric torsion sector, leading to an emergent fluid equation of state

$e^i$5

Analytic solutions in the constant-curvature case interpolate $e^i$6 across cosmic epochs, while the full positive-definite kinetic branch presents $e^i$7 evolution exhibiting radiation-like behavior at high redshift, dust-mimicking at intermediate epochs, and transition through the $e^i$8 phantom divide at late times, as confirmed by explicit numerical integration (Tseng et al., 2012, Tseng, 2018).

2. Scalar–Torsion Theories in Covariant Teleparallel Gravity

The scalar–torsion sector in teleparallel gravity is constructed on a tetrad $e^i$9, a flat spin connection $\Gamma_i{}^j$0, and scalar(s) $\Gamma_i{}^j$1, with the torsion tensor $\Gamma_i{}^j$2 and the Weitzenböck connection defining the geometric structure. The general class of Lorentz-invariant theories is built from the scalars

$\Gamma_i{}^j$3

The most general action is $\Gamma_i{}^j$4. Field equations follow from independent variations in the tetrad, spin connection, and $\Gamma_i{}^j$5. The antisymmetric (local Lorentz) condition for the spin connection is equivalent to the antisymmetric sector of the tetrad equations, ensuring ghost-freedom and covariance (Hohmann et al., 2018, Hohmann, 2018).

Scalar–torsion models in this paradigm admit closure under conformal and disformal transformations, giving rise to transformation rules for the coupling functions and maintaining the form of the field equations. For single- and multi-scalar extensions, the scalar–torsion action and equations display full covariance with respect to tetrad and scalar field redefinitions, with additional derivative-coupling structures like $\Gamma_i{}^j$6 encoding novel kinetic mixings not present in curvature-based scalar-tensor theories (Hohmann et al., 2018, Hohmann, 2018, Hohmann, 2019).

3. Phenomenology: Cosmology, Inflation, and Cosmic Evolution

Scalar–torsion models are characterized by rich early- and late-time cosmological behavior. In the quadratic PGT spin–0⁺ sector, the effective torsion fluid naturally mimics radiation $\Gamma_i{}^j$7, pressureless matter $\Gamma_i{}^j$8, and dark energy $\Gamma_i{}^j$9 across cosmological epochs, with late-time cosmic acceleration achievable without introducing fundamental scalar matter (Tseng et al., 2012, Tseng, 2018).

Scalar–torsion $U_4$0 and $U_4$1 theories support non-minimal ($U_4$2) coupling, allowing the realization of unified dark sector scenarios. Dynamical system analysis confirms the existence of a sequence of critical points corresponding to matter-like ($U_4$3), scaling solutions, and attractors with $U_4$4 (de Sitter), confirming a series of phases: radiation, dust-matter, and accelerated expansion (Leon et al., 2022, Skugoreva et al., 2014). In nonminimal coupling cases, the system generically exhibits a crossing of the phantom divide ($U_4$5) and presents a late-time de Sitter attractor. For inflation, slow roll reconstruction procedures in $U_4$6 models determine viable non-minimal coupling functions and potentials consistent with observed values of $U_4$7 and $U_4$8 (Gonzalez-Espinoza et al., 2021).

Stability analyses across the scalar–torsion model space, incorporating the Sorkin–Schutz action for matter, show the absence of ghost, gradient, and tachyonic instabilities for relevant parameter ranges in $U_4$9 cosmologies, with sound speed $$\mathcal{L}_G = \tfrac{a_0}{2}\,R\,\eta + \tfrac{b}{24}\,R^2\,\eta + \tfrac{a_1}{8}\,{}^{(1)}T^i \wedge \star\,{}^{(1)}T_i,$$0 and the possibility of stable scaling matter regimes in early cosmology (Gonzalez-Espinoza et al., 2021, Leon et al., 2022).

4. Compact Objects and High-Energy Applications

Exact static spherically symmetric and black hole solutions have been constructed in both PGT and teleparallel scalar–torsion frameworks. Non-minimal derivative couplings $$\mathcal{L}_G = \tfrac{a_0}{2}\,R\,\eta + \tfrac{b}{24}\,R^2\,\eta + \tfrac{a_1}{8}\,{}^{(1)}T^i \wedge \star\,{}^{(1)}T_i,$$1 yield new classes of (A)dS, hyperscaling-violating Lifshitz, and wormhole-like solutions, evading standard no-hair theorems due to nontrivial torsion-scalar backgrounds (Kofinas, 2015, Kofinas et al., 2015). In compact star models, isotropic self-gravitating solutions in higher-order $$\mathcal{L}_G = \tfrac{a_0}{2}\,R\,\eta + \tfrac{b}{24}\,R^2\,\eta + \tfrac{a_1}{8}\,{}^{(1)}T^i \wedge \star\,{}^{(1)}T_i,$$2 gravities demonstrate sensitivity to tetrad choices, with only specific interior anisotropic or isotropic profiles admitting equilibrium and stability (Nashed, 2016). In higher dimensions, static spacetimes in scalar–torsion theories with non-minimal derivative coupling generally present naked singularities and constant scalar curvature vacua, with absence of event horizons as a generic outcome (Gunara et al., 2020).

5. Symmetry, Conformal Invariance, and Teleparallel Analogy to Scalar–Tensor Gravity

Scalar–torsion models possess rich invariance properties. In the Riemann–Cartan reformulation of GR, the scalar–torsion sector is organized through Cartan (conformal plus gauge) transformations acting as $$\mathcal{L}_G = \tfrac{a_0}{2}\,R\,\eta + \tfrac{b}{24}\,R^2\,\eta + \tfrac{a_1}{8}\,{}^{(1)}T^i \wedge \star\,{}^{(1)}T_i,$$3, $$\mathcal{L}_G = \tfrac{a_0}{2}\,R\,\eta + \tfrac{b}{24}\,R^2\,\eta + \tfrac{a_1}{8}\,{}^{(1)}T^i \wedge \star\,{}^{(1)}T_i,$$4, with full Cartan invariance of the action and observables. The theory admits reinterpretation as a scalar–tensor theory with Brans–Dicke parameter $$\mathcal{L}_G = \tfrac{a_0}{2}\,R\,\eta + \tfrac{b}{24}\,R^2\,\eta + \tfrac{a_1}{8}\,{}^{(1)}T^i \wedge \star\,{}^{(1)}T_i,$$5 and intrinsic invariance under conformal reparametrizations (Fonseca-Neto et al., 2012).

Teleparallel scalar–torsion theories exhibit a deep analogy with scalar–tensor gravity: the torsion scalar $$\mathcal{L}_G = \tfrac{a_0}{2}\,R\,\eta + \tfrac{b}{24}\,R^2\,\eta + \tfrac{a_1}{8}\,{}^{(1)}T^i \wedge \star\,{}^{(1)}T_i,$$6 replaces the curvature scalar $$\mathcal{L}_G = \tfrac{a_0}{2}\,R\,\eta + \tfrac{b}{24}\,R^2\,\eta + \tfrac{a_1}{8}\,{}^{(1)}T^i \wedge \star\,{}^{(1)}T_i,$$7, and the entire conformal frame and invariant machinery transfers, with a one-to-one correspondence established for model classification, action functionals, and observable invariants (Hohmann, 2018, Emtsova et al., 2019). Post-Newtonian analysis confirms the indistinguishability of minimally coupled scalar–torsion models from GR at the parameterized post-Newtonian level, while deviations for non-minimal couplings respect experimental bounds for appropriate coupling regimes (Flathmann et al., 2019, Emtsova et al., 2019).

6. Cosmological Perturbations, Linear Stability, and Disformal Structure

Perturbative analysis in scalar–torsion gravity requires Lorentz-covariant formulations (including a non-trivial spin connection) to maintain the correct number of degrees of freedom and avoid spurious Lorentz-violating effects. The system of perturbed equations for both $$\mathcal{L}_G = \tfrac{a_0}{2}\,R\,\eta + \tfrac{b}{24}\,R^2\,\eta + \tfrac{a_1}{8}\,{}^{(1)}T^i \wedge \star\,{}^{(1)}T_i,$$8 and $$\mathcal{L}_G = \tfrac{a_0}{2}\,R\,\eta + \tfrac{b}{24}\,R^2\,\eta + \tfrac{a_1}{8}\,{}^{(1)}T^i \wedge \star\,{}^{(1)}T_i,$$9 (pure nonminimal coupling) provides closed, well-posed evolution for both background and first-order cosmological perturbations, with algebraic gravitational slip controlled by the structure of the torsion-scalar coupling (Toporensky et al., 2021).

The quadratic scalar-torsion action, classified under disformal symmetry, admits 23 independent invariants at quadratic order in derivatives, structurally analogous to the healthy beyond-Horndeski (DHOST) actions in curvature-based gravity. Closure under disformal maps enables movement within model space while controlling higher-derivative pathologies via degeneracy conditions, offering further model-building flexibility (Hohmann, 2019).

7. Theoretical Implications and Observational Prospects

Scalar–torsion theories provide a geometric alternative to the introduction of fundamental matter fields for explaining early- and late-time cosmological phenomenology, unifying radiation, matter, and dark energy behavior within the gravitational sector. The theories possess sufficient richness to reproduce and generalize scalar–tensor, $R$0, and gauge gravity phenomenology, whilst exhibiting distinctive signatures such as nontrivial effective EoS evolution, possible phantom crossing, and deviations in the growth of structure.

High-precision observational probes (PPN, lensing, GW) tightly constrain the space of viable scalar–torsion models, but unique structures—such as derivative couplings or disformal invariance—continue to provide fertile ground for exploring deviations from general relativity in both weak and strong gravity regimes (Tseng, 2018, Kofinas, 2015). The mathematical structure of scalar–torsion models, their cosmological and astrophysical solutions, and their symmetry properties establish them as a theoretically robust and phenomenologically rich sector of contemporary gravitational research.