  We study a gas of point particles with hard-core repulsion in one dimension
where the particles move freely in-between elastic collisions. We prepare the
system with a uniform density on the infinite line. The velocities $\{v_i; i
\in \mathbb{Z} \}$ of the particles are chosen independently from a thermal
distribution. Using a mapping to the non-interacting gas, we analytically
compute the equilibrium spatio-temporal correlations $\langle v_i^m(t)
v_j^n(0)\rangle$ for arbitrary integers $m,n$. The analytical results are
verified with microscopic simulations of the Hamiltonian dynamics. The
correlation functions have ballistic scaling, as expected in an integrable
model.
