We generalize unitarity bounds on operator dimensions in conformal field
theory to field theories with spacetime dependent couplings. Below the energy
scale of spacetime variation of the couplings, their evolution can strongly
affect the physics, effectively shifting the infrared operator scaling and
unitarity bounds determined from correlation functions in the theory. We
analyze this explicitly for large-N double-trace flows, and connect these to
UV complete field theories. One motivating class of examples comes from our
previous work on FRW holography, where this effect explains the range of
flavors allowed in the dual, time dependent, field theory.Comment: 38 page
