  We derive the first positivity bounds for low-energy Effective Field Theories
(EFTs) that are not invariant under Lorentz boosts. "Positivity bounds" are the
low-energy manifestation of certain fundamental properties in the UV -- to date
they have been used to constrain a wide variety of EFTs, however since all of
the existing bounds require Lorentz invariance they are not directly applicable
when this symmetry is broken, such as for most cosmological and condensed
matter systems. From the UV axioms of unitarity, causality and locality, we
derive an infinite family of bounds which (derivatives of) the $2\to2$ EFT
scattering amplitude must satisfy even when Lorentz boosts are broken (either
spontaneously or explicitly). We apply these bounds to the leading-order EFT of
both a superfluid and the scalar fluctuations produced during inflation,
comparing in the latter case with the current observational constraints on
primordial non-Gaussianity.
