The Gross-Pitaevskii equations of a static and spherically symmetric condensate of gravitons.
In this paper we consider the Dvali and G\'omez assumption that the end state of a gravitational collapse is a Bose-Einstein condensate of gravitons. We then construct the two Gross-Pitaevskii equations of a static and spherically symmetric configuration of the condensate.
These two equations correspond to the constrained minimisation of the gravitational Hamiltonian with respect to the redshift and the Newtonian potential, per given number of gravitons. We find that the effective geometry of the condensate is the one of a gravastar (a DeSitter star) with a sub-Planckian cosmological constant. Thus, the condensate is always quantum and weakly coupled, no matter its size.
Finally, applying our findings to the current observable Universe, we find that the emergent cosmological constant of the condensate, inversely proportional to the square of the visible mass, matches unexpectedly well the observational value.
Publisher URL: http://arxiv.org/abs/1711.01282