Ghost instabilities and strong coupling in quadratic non-metricity theories

We revisit the framework of Newer General Relativity, defined by all independent quadratic invariants of the non-metricity tensor, including the unique quadratic parity-violating term. We analyze linear perturbations around a flat FLRW background and find that the theory generically exhibits ghost instabilities and/ or propagates more degrees of freedom than in the Minkowski limit, signalling strong coupling. There are two notable exceptions: the Symmetric Teleparallel Equivalent of General Relativity (STEGR) and the transverse-diffeomorphism-invariant gravity subclass, both of which are supplemented by the parity-violating operator. However, since the parity-violating term explicitly breaks (transverse) diffeomorphism invariance, we show, using both the Dirac-Bergmann procedure and the Cartan-Kuranishi algorithm, that the parity-violating extension of STEGR propagates eight degrees of freedom at the fully non-linear level.