A second order upper bound for the ground state energy of a hard-sphere gas in the Gross-Pitaevskii regime

We prove an upper bound for the ground state energy of a Bose gas consisting of N hard spheres with radius a/N, moving in the three-dimensional unit torus Λ. Our estimate captures the correct asymptotics of the ground state energy, up to errors that vanish in the limit N → ∞. The proof is based on the construction of an appropriate trial state, given by the product of a Jastrow factor (describing two-particle correlations on short scales) and of a wave function constructed through a (generalized) Bogoliubov transformation, generating orthogonal excitations of the Bose-Einstein condensate and describing correlations on large scales.