Upper Bound for the Free Energy of Dilute Bose Gases at Low Temperature

We consider a Bose gas at density $\rho >0$, interacting through a repulsive potential $V\in L^2({\mathbb{R}}^3)$ with scattering length $\mathfrak{a}>0$. We prove an upper bound for the free energy of the system, valid at low temperature $T\lesssim \rho \mathfrak{a}$. Combined with the recent lower bound obtained in \cite{HabHaiNamSeiTri-23}, our estimate resolves the free energy per unit volume up to and including the Lee–Huang–Yang order $\mathfrak{a}{\rho }^2(\rho {\mathfrak{a}}^3{)}^{1\mathrm{/}2}$.