  Weyl fermions of spin $\frac12$ minimally coupled to Einstein's gravity in 4
dimensions cannot be produced purely gravitationally in an expanding Universe
at tree level. Surprisingly, as we showed in a recent letter [1], this changes
at gravitational 1-loop when cosmic perturbations, like a gravitational wave
background, are present. Such a background introduces a new scale, thereby
breaking the fermions' conformal invariance. This leads to a non-vanishing
gravitational self-energy for Weyl fermions at 1-loop and induces their
production. In this paper, we present an extended study of this new mechanism,
explicitly computing this effect using the in-in formalism. We work in an
expanding Universe in the radiation-dominated era as a fixed background.
Gravitational wave-induced fermion production has rich phenomenological
consequences. Notably, if Weyl fermions eventually acquire mass, and assuming
realistic - and potentially detectable - gravitational wave backgrounds, the
mechanism can explain the abundance of dark matter in the Universe. More
generally, gravitational-wave induced freeze-in is a new purely gravitational
mechanism for generating other feebly interacting fermions, e.g. right-handed
neutrinos. We show that this loop level effect can dominate over the
conventional - tree-level - gravitational production of superheavy fermions in
a sizable part of the parameter space.
