Almost Optimal Upper Bound for the Ground State Energy of a Dilute Fermi Gas via Cluster Expansion

We prove an upper bound on the energy density of the dilute spin- $$ Fermi gas capturing the leading correction to the kinetic energy $$ with an error of size smaller than $$ for any $$ , 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 et al. (Nucl Phys A 176(2):237-260, 1971. 10.1016/0375-9474(71)90267-3).