I will describe how causality implies certain non-perturbative analyticity and exponential boundedness conditions on correlators of relativistic QFTs, in a mixed (t,k) representation. I will then discuss their implications for correlators in Lorentz-breaking backgrounds, including finite-density states and cosmological spacetimes, and show how they can be used to derive a positivity condition on inflationary theories. Along the way, I will compare with the case of S-matrix positivity in flat space Lorentz-invariant theories.