  We present solutions of the BCS gap equation in the channels ${}^1S_0$ and
${}^3P_2-{}^3F_2$ in neutron matter based on nuclear interactions derived
within chiral effective field theory (EFT). Our studies are based on a
representative set of nonlocal nucleon-nucleon (NN) plus three-nucleon (3N)
interactions up to next-to-next-to-next-to-leading order (N$^3$LO) as well as
local and semilocal chiral NN interactions up to N$^2$LO and N$^4$LO,
respectively. In particular, we investigate for the first time the impact of
subleading 3N forces at N$^3$LO on pairing gaps and also derive uncertainty
estimates by taking into account results for pairing gaps at different orders
in the chiral expansion. Finally, we discuss different methods for obtaining
self-consistent solutions of the gap equation. Besides the widely-used
quasi-linear method by Khodel et al. we demonstrate that the modified Broyden
method is well applicable and exhibits a robust convergence behavior. In
contrast to Khodel's method it is based on a direct iteration of the gap
equation without imposing an auxiliary potential and is straightforward to
implement.

Nuclear Theory;High Energy Astrophysical Phenomena