Strong acoustic turbulence and the speed of Bose-Einstein condensation

The final stage of Bose-Einstein condensation in a large volume occurs through coarsening--growth of individual correlated patches. We present analytical arguments and numerical evidence that in the momentum space this growth corresponds to strong turbulence of the particle number, with a turbulent cascade towards the infrared and the power law $n(k) \propto k^{-(d+1)}$ in $d=2,3$ spatial dimensions. As a corollary, we find that the correlation length grows linearly in time, with the speed of order of the speed of Bogoliubov's quasiparticles (phonons). We use these results to estimate the speed of BEC in atomic vapors and in galactic dark matter.