Free Energies of Dilute Bose gases: upper bound

We derive a upper bound on the free energy of a Bose gas system at density $\rho$ and temperature $T$. In combination with the lower bound derived previously by Seiringer \cite{RS1}, our result proves that in the low density limit, i.e., when $a^3\rho \ll 1$, where $a$ denotes the scattering length of the pair-interaction potential, the leading term of $\mathrm{\Delta }f$ the free energy difference per volume between interacting and ideal Bose gases is equal to $4\pi a(2{\rho }^2-[\rho -\text{rhoc}{]}_{+}^2)$. Here, $\text{rhoc}(T)$ denotes the critical density for Bose-Einstein condensation (for the ideal gas), and $[\cdot {]}_{+}$ $=$ $\max \{\cdot ,0\}$ denotes the positive part.