We consider a simple model-independent extension of the minimal KSVZ axion model, augmenting the Peccei - Quinn effective potential with a dimension-6 operator invariant under the Peccei - Quinn symmetry. We show that this operator can trigger a first-order phase transition and study its cosmological consequences. We map out the parameter space, consistent with present observational constraints, in which the phase transition produces a gravitational wave signal observable at current and future interferometers, and we investigate the resulting dark matter phenomenology. Finally, we present specific ultraviolet completions that generate the dimension-6 operator, each leading to distinct phenomenological signatures, which motivates our model-independent approach.

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