A First Post-Friedmann Extension of the Schrödinger Approach to Cosmic Structure Formation

Abstract

We extend the Schrödinger approach to large-scale structure formation beyond the Newtonian regime by working at first post-Friedmann (1PF) order. The standard Schrödinger--Poisson system gives a useful reformulation of the dynamics of a self-gravitating pressureless fluid, but it corresponds to the leading post-Friedmann, or Newtonian, limit. It therefore misses the relativistic corrections that enter at next-to-leading order and become relevant on horizon scales and for high-precision cosmological surveys. Starting from the 1PF continuity and Euler equations in a flat $Λ$CDM background, we identify the conserved density variable associated with covariant mass conservation. In terms of this variable, the continuity equation takes a Newtonian-like conservative form. However, even for vanishing covariant vorticity, the spatial velocity field in the cosmological frame contains a transverse 1PF component. Thus the full 1PF mass flux cannot be represented solely by the gradient of a scalar phase. We show that the Schrödinger-like formulation at 1PF order requires an effective vector potential fixed by this transverse velocity component. This vector potential contains the post-Friedmann metric vector perturbation, related to relativistic frame-dragging effects, together with nonlinear scalar terms required by the zero-vorticity condition. Equivalently, when the equation is written in scalar form, these corrections appear as an imaginary contribution to the effective potential. At leading order our system reduces to the usual Schrödinger--Poisson formulation, while at 1PF order it provides a relativistic extension of the Schrödinger description of cold matter dynamics.