  We prove an upper bound on the energy density of the dilute
spin-$\frac{1}{2}$ Fermi gas capturing the leading correction to the kinetic
energy $8\pi a \rho_\uparrow\rho_\downarrow$ with an error of size smaller than
$a\rho^{2}(a^3\rho)^{1/3-\varepsilon}$ for any $\varepsilon > 0$, where $a$
denotes the scattering length of the interaction. The result is valid for a
large class of interactions including interactions with a hard core. A central
ingredient in the proof is a rigorous version of a fermionic cluster expansion
adapted from the formal expansion of Gaudin, Gillespie and Ripka (Nucl. Phys.
A, 176.2 (1971), pp. 237--260).
