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

We consider a Bose gas at density > 0, interacting through a repulsive potential V L² (R³) with scattering length a > 0. We prove an upper bound for the free energy of the system, valid at low temperature T a. Combined with the recent lower bound obtained in HabHaiNamSeiTri-23, our estimate resolves the free energy per unit volume up to and including the Lee--Huang--Yang order a ² (a³) ^1/2.