We continue the recent study of the possibility of constructing a consistent infrared modification of gravity by treating the vierbein and connection as independent dynamical fields.We present the generalized Fierz-Pauli equation that governs the propagation of a massive spin-2 mode in a model of this sort in the backgrounds of arbitrary torsionless Einstein manifolds. We show explicitly that the number of propagating degrees of freedom in these backgrounds remains the same as in flat space-time. This generalizes the recent result that the Boulware-Deser phenomenon does not occur in de Sitter and anti-de Sitter backgrounds.We find that, at least for weakly curved backgrounds, there are no ghosts in the model. We also briefly discuss the interaction of sources in flat background.

We consider the model of modified gravity with dynamical torsion. This model was previously found to have promising stability properties about various backgrounds. Here we study the stability of linear perturbations about the self-accelerating solution. We apply the (3 + 1)-decomposition and consider the scalar sector of perturbations. We find that the number of degrees of freedom is equal to 2, which is the same as in Minkowski background. However, there is at least one instability in the scalar sector, if the value of background torsion is large enough. This does not rule out the possibility of stable self-acceleration with torsion of the order of the effective cosmological constant.

We consider a model belonging to the class of gravities with dynamical torsion. The model is free of ghosts and gradient instabilities about Minkowski and torsionless Einstein backgrounds. We find that at zero cosmological constant, the model admits a self-accelerating solution with non-Riemannian connection. Small value of the effective cosmological constant is obtained at the expense of the hierarchy between the dimensionless couplings.

We study spherically symmetric solutions in a four-parameter Einstein-Cartan-type class of theories. These theories include torsion, as well as the metric, as dynamical fields, and contain only two physical excitations (around flat spacetime): a massless spin-2 excitation and a massive spin-2 one (of mass m2 ≡ κ). They offer a geometric framework (which we propose to call "torsion bigravity") for a modification of Einstein's theory that has the same spectrum as bimetric gravity models. We find that the spherically symmetric solutions of torsion bigravity theories exhibit several remarkable features: (i) they have the same number of degrees of freedom as their analogs in ghost-free bimetric gravity theories (i.e. one less than in ghost-full bimetric gravity theories); (ii) in the limit of small mass for the spin-2 field (κ → 0), no inverse powers of κ arise at the first two orders of perturbation theory (contrary to what happens in bimetric gravity where 1/κ 2 factors arise at linear order, and 1/κ 4 ones at quadratic order). We numerically construct a high-compactness (asymptotically flat) star model in torsion bigravity and show that its geometrical and physical properties are significantly different from those of a general relativistic star having the same observable Keplerian mass.