An upper bound on the condensate fraction in a Bose gas with weak repulsion

For a spinless Bose gas moving in three-dimensional free space with an interaction V(r) which is everywhere positive, I show that the condensate fraction f is bounded above by an expression parametrized by the quantity nV0/kTc(0)≡η, where n is the density, V0 the space integral of the interaction potential and Tc(0) the critical temperature of the corresponding noninteracting gas. The only assumption needed is that fluctuations in the interacting system have the standard extensivity properties. In particular, for T=Tc(0) the upper bound on f is proportional to η1/3, while for η1/3<<1-Tc(0)/T<<1 it is proportional to η/(1-(Tc(0)/T)3/2)2. (author)