Importance of torsion and invariant volumes in Palatini theories of gravity

We study the field equations of extensions of General Relativity formulated within a metric-affine formalism setting torsion to zero (Palatini approach). We find that different (second-order) dynamical equations arise depending on whether torsion is set to zero i) a priori or ii) a posteriori, i.e., before or after considering variations of the action. Considering a generic family of Ricci-squared theories, we show that in both cases the connection can be decomposed as the sum of a Levi-Civita connection and terms depending on a vector field. However, while in case i) this vector field is related to the symmetric part of the connection, in ii) it comes from the torsion part and, therefore, it vanishes once torsion is completely removed. Moreover, the vanishing of this torsion-related vector field immediately implies the vanishing of the antisymmetric part of the Ricci tensor, which therefore plays no role in the dynamics. Related to this, we find that the Levi-Civita part of the connection is due to the existence of an invariant volume associated to an auxiliary metric $h_{\mu\nu}$, which is algebraically related with the physical metric $g_{\mu\nu}$.Comment: 14 one-column pages, no figures; v2: some minor changes and typos corrections, new references adde

Brane-world and loop cosmology from a gravity-matter coupling perspective

We show that the effective brane-world and the loop quantum cosmology background expansion histories can be reproduced from a modified gravity perspective in terms of an $f(R)$ gravity action plus a $g(R)$ term non-minimally coupled with the matter Lagrangian. The reconstruction algorithm that we provide depends on a free function of the matter density that must be specified in each case and allows to obtain analytical solutions always. In the simplest cases, the function $f(R)$ is quadratic in the Ricci scalar, $R$, whereas $g(R)$ is linear. Our approach is compared with recent results in the literature. We show that working in the Palatini formalism there is no need to impose any constraint that keeps the equations second-order, which is a key requirement for the successful implementation of the reconstruction algorithm.Comment: 8 pages, revtex4-1 styl