The Horndeski Lagrangian brings together all possible interactions between gravity and a scalar field that yield second-order field equations in four-dimensional spacetime. As originally proposed, it only addresses phenomenology without torsion, which is a non-Riemannian feature of geometry. Since torsion can potentially affect interesting phenomena such as gravitational waves and early Universe inflation, in this paper we allow torsion to exist and propagate within the Horndeski framework. To achieve this goal, we cast the Horndeski Lagrangian in Cartan's first-order formalism, and introduce wave operators designed to act covariantly on p-form fields that carry Lorentz indices. We find that nonminimal couplings and second-order derivatives of the scalar field in the Lagrangian are indeed generic sources of torsion. Metric perturbations couple to the background torsion and new torsional modes appear. These may be detected via gravitational waves but not through Yang-Mills gauge bosons.

In this work we present a generalized Brans-Dicke lagrangian including a nonminimally coupled Gauss-Bonnet term without imposing the vanishing torsion condition. In the resulting field equations, the torsion is closely related to the dynamics of the scalar field, i.e., if non-minimally coupled terms are present in the theory, then the torsion must be present. For the studied lagrangian we analyze the cosmological consequences of an effective torsional fluid and we show that this fluid can be responsible for the current acceleration of the universe. Finally, we perform a detailed dynamical system analysis to describe the qualitative features of the model, we find that accelerated stages are a generic feature of this scenario.

Scalar-tensor gravity theories with a nonminimal Gauss-Bonnet coupling typically lead to an anomalous propagation speed for gravitational waves, and have therefore been tightly constrained by multimessenger observations such as GW170817/GRB170817A. In this paper we show that this is not a general feature of scalar-tensor theories, but rather a consequence of assuming that spacetime torsion vanishes identically. At least for the case of a nonminimal Gauss-Bonnet coupling, removing the torsionless condition restores the canonical dispersion relation and therefore the correct propagation speed for gravitational waves. To achieve this result we develop a new approach, based on the first-order formulation of gravity, to deal with perturbations on these Riemann-Cartan geometries.topological invariants, e.g., the Pontryagin or Gauss-Bonnet (GB) terms, motivated by effective field theories, string theory, and particle physics [15]. From a phenomenological viewpoint, the scalar-Pontryagin modification to GR-also known as Chern-Simons modified gravity-is an interesting extension that might explain the flat galaxy rotation curves dispensing with dark matter [16], while leaving the propagation speed of GWs unaffected [17]. This interaction generates nontrivial effects when rotation is included [18][19][20][21][22][23], providing a smoking gun in future GW detectors [24][25][26][27]. The couplings between scalar fields and the GB term, on the other hand, have been studied in different setups and several solutions that exhibit spontaneous scalarization have been reported [28][29][30][31][32][33][34][35][36][37][38][39][40][41]. Their stability, however, depends on the choice of the coupling between the scalar field and the GB term [42][43][44]. In spite of this, the theory is experimentally disadvantaged from an astrophysical viewpoint, since it develops an anomalous propagation speed for GWs [45].Scalar-tensor theories have been formulated in geometries that depart from the pseudo-Riemannian framework several times in the past. In particular, the gravitational role of Riemann-Cartan (RC) geometries, characterized by curvature and torsion, was first discussed by Cartan and Einstein themselves [46], and later on in the framework of gauge theories of gravitation [47][48][49][50]. Within its simplest formulation-the Einstein-Cartan-Sciama-Kibble (ECSK) theory-torsion is a nonpropagating field sourced only by the spin density of matter. The nonminimal coupling of scalar fields to geometry dramati-arXiv:1910.00148v3 [gr-qc]

In this paper, we formulate the Mimetic theory of gravity in first-order formalism for differential forms, i.e., the mimetic version of Einstein-Cartan-Sciama-Kibble (ECSK) gravity. We consider different possibilities on how torsion is affected by Weyl transformations and discuss how this translates into the interpolation between two different Weyl transformations of the spin connection, parameterized with a zero-form parameter λ. We prove that regardless of the type of transformation one chooses, in this setting torsion remains as a non-propagating field. We also discuss the conservation of the mimetic stress-energy tensor and show that the trace of the total stress-energy tensor is not null but depends on both, the value of λ and spacetime torsion.