  We consider a Bardeen-Cooper-Schrieffer (BCS)-like model in the inflationary
background. We show that with an axial chemical potential, the attractive
quartic fermion self-interaction can lead to a BCS-like condensation. In the
rigid-de Sitter (dS) limit of inflation where backreaction from the inflaton
and graviton is neglected, we perform the first computation of the
non-perturbative effective potential that includes the full spacetime curvature
effects in the presence of the chemical potential, subject to the mean-field
approximation whose validity has been checked via the Ginzburg criterion. The
corresponding BCS phase transition is always first-order, when the varying
Hubble is interpreted as an effective Gibbons-Hawking temperature of dS
spacetime. In the condensed phase, the theory can be understood from UV and IR
sides as fermionic and bosonic, respectively. This leads to distinctive
signatures in the primordial non-Gaussianity of curvature perturbations.
Namely, the oscillatory cosmological collider signal is smoothly turned off at
a finite momentum ratio, since different momentum ratios effectively probe
different energy scales. In addition, such BCS phase transitions can also
source stochastic gravitational waves, which are feasible for future
experiments.
