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

We consider a Bose gas at density ρ > 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 ρ 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 ρ² (ρ a³)1/2.