A second order upper bound to the free energy of the two dimensional Bose gas

We consider a two-dimensional Bose gas in the dilute regime where $\rho a^2$ is small. For temperatures below the Berezinskii-Kosterlitz-Thouless critical temperature, we derive an explicit upper bound for the free energy density using Bogoliubov theory. Our result captures the contribution of quasiparticle modes with dispersion relation $\sqrt{p^4+8\pi \rho \delta p^2}$ and where $\delta =2\mathrm{/}(\mathrm{\mid }\log (\rho a^2)\mathrm{\mid }+\log \mathrm{\mid }\log (\rho a^2)\mathrm{\mid })$.