  We consider effective field theories (EFTs) of scalar fields with broken
Lorentz boosts, which arise by taking the decoupling and flat-space limits of
the EFT of inflation, and derive constraints that must be satisfied by the
corresponding scattering amplitudes if there is an underlying non-linearly
realised symmetry. We primarily concentrate on extended shift symmetries which
depend on the space-time coordinates, and find that combinations of scattering
amplitudes obey enhanced Adler zeros. That is, such combinations vanish as one
external momentum is taken soft, with the rate at which they vanish dictated by
the corresponding symmetry. In our soft theorem derivation, we pay particular
care to the energy and momentum-conserving delta functions that arise due to
space-time translations, and show that when acted upon by derivatives with
respect to spatial momenta, they yield a tower of soft theorems which are
ultimately required for closure of the underlying symmetry algebra. All of our
soft theorems correspond to constraints that must be satisfied by on-shell
amplitudes and, even for symmetries that depend on the time coordinate, our
soft theorems only require derivatives to be taken with respect to spatial
momenta. We perform a soft bootstrap procedure to find solutions to our soft
theorems, and compare these solutions to what we find from an off-shell
analysis using the coset construction.
