  Neutron matter is an ideal laboratory for nuclear interactions derived from
chiral effective field theory since all contributions are predicted up to
next-to-next-to-next-to-leading order (N$^3$LO) in the chiral expansion. By
making use of recent advances in the partial-wave decomposition of three-
nucleon (3N) forces, we include for the first time N$^3$LO 3N interactions in
many-body perturbation theory (MBPT) up to third order and in self-consistent
Green's function theory (SCGF). Using these two complementary many-body
frameworks we provide improved predictions for the equation of state of neutron
matter at zero temperature and also analyze systematically the many-body
convergence for different chiral EFT interactions. Furthermore, we present an
extension of the normal-ordering framework to finite temperatures. These
developments open the way to improved calculations of neutron-rich matter
including estimates of theoretical uncertainties for astrophysical
applications.
