The most general EFT describing a metric (in unitary gauge) with time dependent spatial-diffeomorphisms (hence FRW backgrounds) with at most two derivatives (sufficient condition for no Ostrogradski ghost) is coupled to a Sorkin-Schutz matter action (which can describe a perfect fluid). By expanding this action to quadratic order, one can derive the conditions necessary for the absence of ghosts, gradient instabilities and tachyons.

Stability conditions are a useful and efficient way of restricting the viability space of field theories. Even before any comparison is made with data, mathematical consistency is enough to contrain the free EFT functions in a fairly general representation of modified gravity.

Through a number of simple examples, it is demonstrated that singularities in the Einstein frame may appear as regular evolution in the Jordan frame, and vice versa. Explicitly, in the Jordan frame, a single scalar field is conformally coupled to the curvature. Different potentials are then considered, such as a cosmological constant (of either sign) and the hyperbolic potential believed to support a bounce.

The authors identify two existing approaches in the literature to crossing singularities (the 'two-scalar field' approach 1004.0752, and the 'variable gravity' approach 1308.1019) and claim that this method of transitioning between Einstein and Jordan frames represents a third approach. They point out one limitation: in these examples (and in the other two approaches) one requires Weyl symmetry. Furthermore, they use a scalar field redefinition which becomes non-invertible precisely at the singularities they are hoping to cross - and so really the singularity is still there, just 'hidden' inside the field redefinition. This is also related to the question of whether Einstein and Jordan frames are equivalent in the presence of matter (see e.g. 1609.01824 for a recent discussion).