Stability of symmetric teleparallel scalar-tensor cosmologies with alternative connections

In symmetric teleparallel geometry the curvature and torsion tensors are assumed to vanish identically, while the dynamics of gravity is encoded by nonmetricity. Here the spatially homogeneous and isotropic connections that can accompany the flat Friedmann-Lemaitre-Robertson-Walker metric come in three sets. As the trivial set has received much attention, we focus on the two alternative sets, which introduce an extra degree of freedom into the equations. Working in the context of symmetric teleparallel scalar-tensor gravity with generic nonminimal coupling and potential, we show that the extra free function in the connection cannot play the role of dark matter nor dark energy, but it drastically alters the scalar field behavior. We determine the restrictions on the model functions which permit the standard cosmological scenario of successive radiation, dust matter, and scalar potential domination eras to be stable. However, the alternative connections also introduce a rather general possibility of the system meeting a singularity in finite time.