We discuss the universal behavior linked to the Goldstone mode associated with the spontaneous breaking of time-translation symmetry in many-body systems, in which the order parameter traces out a limit cycle. We show that this universal behavior is closely tied to Kardar-Parisi-Zhang physics, which can strongly affect the scaling properties in all dimensions. Our Letter predicts and rationalizes the emergence of Kardar-Parisi-Zhang in numerous systems such as nonreciprocal phases in active matter, active magnets, driven-dissipative quantum systems, and synchronization of oscillators.

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• Nonequilibrium statistical mechanics
• Time crystals
• Kardar–Parisi–Zhang equation
• Path-integral methods