Fermion Propagator in Cosmological Spaces with Constant Deceleration

We calculate the fermion propagator in FLRW spacetimes with constant deceleration $q=\epsilon-1$, $\epsilon=-\dot{H}/H^{2}$ for excited states. For fermions whose mass is generated by a scalar field through a Yukawa coupling $m=g_{\mathrm{\scriptscriptstyle{Y}}} \phi$, we assume $\phi \propto H$. We first solve for the mode functions by splitting the spinor into a direct product of helicity and chirality spinors. We also allow for non-vacuum states. We normalise the spinors using a consistent canonical quantisation and by requiring orthogonality of particle and anti-particle spinors. We apply our propagator to calculate the one loop effective action and renormalise using dimensional regularisation. Since the Hubble parameter is now treated dynamically, this paves the way to study the dynamical backreaction of fermions on the background spacetime.