# How is it possible to produce a Bose-Einstein condensate of astrophysical dimension?

There is simple mechanism behind such a production and it suggests that a cloud made of bosons could collapse gravitationally into an equilibrium polytrope in the condensate phase. Assuming that it takes the WMAP DM cosmic mean density (times 10 since super-clusters occupy about 10% of known space), it would have a radius ~ 10^8 lys.--the size scale of a galactic super-cluster. It would be composed of bosons of neutrino mass (1 ev.)
Would it not then be reasonable, as per SUSY, to assume that dark matter consists of sterile sneutrinos at about the same mass? Of course, SUSY would have to be weakly broken for the case of this fourth "flavorless" lepton flavor

Although a Bose-Einstein condensate of such a dimension seems unlikely, an argument based on the virial theorem suggests that the formation of such condensates could be the result of a gravitationally induced phase transition-one that occurs within a boson cloud that begins collapsing in the normal phase. Assuming that the virial theorem controls the temperature of a cloud of radius R, for a fixed number of bosons in a gas of uniform density, the virial temperature takes the form $T_{vir} = \frac{C_1}{R}$. At the same time, the BE condensation temperature, being proportional to the density to the 2/3, takes the form $T_{BE}= \frac{C_2}{R^2}$. Evidently, for a shrinking cloud, there will always be a critical radius, $R_{cr}$, at which $T_{BE}$ exceeds $T_{vir}$. Instead of a transition induced by a cooling process, this mechanism would drive the transition by raising the transition temperature faster than the virial temperature. To then get the critical radius, set the virial temperature equal to $T_{BE}$, to get $$R_{cr} = \frac {\hbar}{(G^{1/2})(\rho^{1/6})(m^{4/3})}$$ Putting the WMAP density of DM (times 10) into ${\rho}$ and, say, $(3)(10^{-34})$ gm. into the boson mass m returns $R_{cr}$ ~ $10^{8}$ ly.