  We derive exact relations between the Renyi entanglement entropies and the
particle number fluctuations of spatial connected regions in systems of N
noninteracting fermions in arbitrary dimension. We prove that the asymptotic
large-N behavior of the entanglement entropies is proportional to the variance
of the particle number. We also consider 1D Fermi gases with a localized
impurity, where all particle cumulants contribute to the asymptotic large-N
behavior of the entanglement entropies. The particle cumulant expansion turns
out to be convergent for all integer-order Renyi entropies (except for the von
Neumann entropy) and the first few cumulants provide already a good
approximation. Since the particle cumulants are accessible to experiments,
these relations may provide a measure of entanglement in these systems.
