Klein-Gordon flow on FLRW spacetimes

We study a new approach to generally covariant quantum mechanics applied in the case of an FLRW cosmological background. For positive spatial curvature we find a discrete series of solutions of the Klein-Gordon equation that can reasonably be called gravitationally bound `cosmological atom' states. For all cases of curvature, these modes, as well as more conventional atomic spatial modes bound by an external potential, extend to solutions of the Klein-Gordon equations viewed as stationary modes of Klein-Gordon quantum mechanics where wavefunctions are over spacetime and evolution is with respect to an external `geodesic time' parameter $s$. For general nonstationary states with fixed spatial eigenvector, the theory reduces to a novel 1-dimensional quantum system on the time $t$ axis with potential $1/a(t)^2$, where $a(t)$ is the Friedmann expansion factor. Its behaviour, and hence the evolution of spatial states, changes critically when the Hubble constant exceeds $2/3$ of the particle mass, as typically occurs during inflation. We also find washout of the evolution of spatial observables at late times and a backward-traveling reflected mode generated when the value of $H$ transitions to a larger value.