We present a novel theory of gravity by considering an extension of symmetric teleparallel gravity. This is done by introducing, in the framework of the metric-affine formalism, a new class of theories where the nonmetricity $Q$ is nonminimally coupled to the matter Lagrangian. More specifically, we consider a Lagrangian of the form $L\sim f_1(Q)+f_2(Q)L_M$, where $f_1$ and $f_2$ are generic functions of $Q$, and $L_M$ is the matter Lagrangian. This nonminimal coupling entails the nonconservation of the energy-momentum tensor, and consequently the appearance of an extra force. The formulation of the gravity sector in terms of the $Q$ instead of the curvature may result in subtle improvements of the theory. In the context of nonminimal matter couplings, we are therefore motivated to explore whether the new geometrical formulation in terms of the $Q$, when implemented also in the matter sector, would allow more universally consistent and viable realizations of the nonminimal coupling. Furthermore, we consider several cosmological applications by presenting the evolution equations and imposing specific functional forms of the functions $f_1(Q)$ and $f_2(Q)$, such as power-law and exponential dependencies of the nonminimal couplings. Cosmological solutions are considered in two general classes of models, and found to feature accelerating expansion at late times.

We compute the complete post-Newtonian limit of both the metric and Palatini formulations of $f(R)$ gravities using a scalar-tensor representation. By comparing the predictions of these theories with laboratory and solar system experiments, we find a set of inequalities that any lagrangian $f(R)$ must satisfy. The constraints imposed by those inequalities allow us to find explicit bounds to the possible nonlinear terms of the lagrangian. We conclude that in both formalisms the lagrangian $f(R)$ must be almost linear in $R$ and that corrections that grow at low curvatures are incompatible with observations. This result shows that modifications of gravity at very low cosmic densities cannot be responsible for the observed cosmic speed-up.

In this work we show that the gravity Lagrangian $f(R)$ at relatively low curvatures in both metric and Palatini formalisms is a bounded function that can only depart from the linearity within the limits defined by well-known functions. We obtain those functions by analyzing a set of inequalities that any $f(R)$ theory must satisfy in order to be compatible with laboratory and solar system observational constraints. This result implies that the recently suggested $f(R)$ gravity theories with nonlinear terms that dominate at low curvatures are incompatible with observations and, therefore, cannot represent a valid mechanism to justify the cosmic speedup.

We discuss two aspects of $f(R)$ theories of gravity in metric formalism. We first study the reasons to introduce a scalar-tensor representation for these theories and the behavior of this representation in the limit to General Relativity, $f(R)\to R$. We find that the scalar-tensor representation is well behaved even in this limit. Then we work out the exact equations for spherically symmetric sources using the original $f(R)$ representation, solve the linearized equations, and compare our results with recent calculations of the literature. We observe that the linearized solutions are strongly affected by the cosmic evolution, which makes very unlikely that the cosmic speedup be due to $f(R)$ models with correcting terms relevant at low curvatures.