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

For a spinless Bose gasmoving 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/kT⁽⁰⁾_c = η, where n is the density, V₀ the space integral of the interaction potential and T ⁽⁰⁾_c 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 = T⁽⁰⁾_c the upper bound on f is proportional to η^1/3, while for η^1/3 ≪ 1 - T⁽⁰⁾ _c /T ≪ 1 it is proportional to η/(1 - (T⁽⁰⁾ _c /T)^3/2)².