  We consider minimal dark matter scenarios featuring momentum-dependent
couplings of the dark sector to the Standard Model. We derive constraints from
existing LHC searches in the monojet channel, estimate the future LHC
sensitivity for an integrated luminosity of 300 fb$^{-1}$, and compare with
models exhibiting conventional momentum-independent interactions with the dark
sector. In addition to being well motivated by (composite) pseudo-Goldstone
dark matter scenarios, momentum-dependent couplings are interesting as they
weaken direct detection constraints. For a specific dark matter mass, the LHC
turns out to be sensitive to smaller signal cross-sections in the
momentum-dependent case, by virtue of the harder jet transverse-momentum
distribution.
