  Recently, Holmes and Perkins identified conditions which ensure that for a
class of critical lattice models the scaling limit of the range is the range of
super-Brownian motion. One of their conditions is an estimate on a spatial
moment of order higher than four, which they verified for the sixth moment for
spread-out lattice trees in dimensions $d>8$. Chen and Sakai have proved the
required moment estimate for spread-out critical oriented percolation in
dimensions $d+1>4+1$. We prove estimates on all moments for the spread-out
critical contact process in dimensions $d>4$, which in particular fulfills the
spatial moment condition of Holmes and Perkins. Our method of proof is
relatively simple, and, as we show, it applies also to oriented percolation and
lattice trees. Via the convergence results of Holmes and Perkins, the upper
bounds on the spatial moments can in fact be promoted to asymptotic formulas
with explicit constants.
