Klein–Gordon flow on Friedmann–Lemaïtre–Robertson–Walker spacetimes

We study a new approach to generally covariant quantum mechanics applied in the case of a Friedmann–Lemaïtre–Robertson–Walker 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.