On the cosmological constant as a quantum operator

Abstract

We regard the cosmological fluid within an exponentially expanding FLRW spacetime as the probability fluid of a nonrelativistic Schroedinger field. The scalar Schroedinger particle so described has a mass equal to the total (baryonic plus dark) matter content of the Universe. This procedure allows a description of the cosmological fluid by means of the operator formalism of nonrelativistic quantum theory. Under the assumption of radial symmetry, a quantum operator proportional to $1/r^2$ represents the cosmological constant $\Lambda$. The experimentally measured value of $\Lambda$ is one of the eigenvalues of $1/r^2$. Next we solve the Poisson equation $\nabla^2U=\Lambda$ for the gravitational potential $U(r)$, with the cosmological constant $\Lambda(r)=1/r^2$ playing the role of a source term. It turns out that $U(r)$ includes, besides the standard Newtonian potential $1/r$, a correction term proportional to $\ln r$ identical to that appearing in theories of modified Newtonian dynamics.