  We calculate the differential scattering rate for thermal neutrinos in a hot
and dilute gas of interacting neutrons using linear response theory. The
dynamical structure factors for density and spin fluctuations of the strongly
interacting neutron matter, expected in the neutrino decoupling regions of
supernovae and neutron star mergers, are calculated in the virial expansion for
the first time. Correlations due to nucleon-nucleon interactions are taken into
account using a pseudo-potential that reproduces measured nucleon-nucleon phase
shifts, and we find that attractive s-wave interactions enhance the density
response and suppress the spin response of neutron matter. The net effect of
neutron correlations is to strongly suppress backscattering. Moreover, we find
nearly exact scaling laws for the response functions, valid for the range $T =
5 - 10$ MeV and q < 30 MeV, allowing us to obtain analytic results for the
dynamic structure factors at second-order in the fugacity of the neutron gas.
We find that the modification of scattering rates depends on the energy and
momentum exchanged, implying that dynamical structure factors are essential to
describe neutrino decoupling in supernovae and neutron star mergers.
