  A lower bound is derived on the free energy (per unit volume) of a
homogeneous Bose gas at density $\rho$ and temperature $T$. In the dilute
regime, i.e., when $a^3\rho \ll 1$, where $a$ denotes the scattering length of
the pair-interaction potential, our bound differs to leading order from the
expression for non-interacting particles by the term $4\pi a (2\rho^2 -
[\rho-\rho_c]_+^2)$. Here, $\rho_c(T)$ denotes the critical density for
Bose-Einstein condensation (for the non-interacting gas), and $[ ]_+$ denotes
the positive part. Our bound is uniform in the temperature up to temperatures
of the order of the critical temperature, i.e., $T \sim \rho^{2/3}$ or smaller.
One of the key ingredients in the proof is the use of coherent states to extend
the method introduced in [arXiv:math-ph/0601051] for estimating correlations to
temperatures below the critical one.
