  Using a solitonic model of nuclear matter, the BPS Skyrme model, we compare
neutron stars obtained in the full field theory, where gravitational back
reaction is completely taken into account, with calculations in a mean-field
approximation using the Tolman-Oppenheimer-Volkoff approach. In the latter
case, a mean-field-theory equation of state is derived from the original BPS
field theory. We show that in the full field theory, where the energy density
is non-constant even at equilibrium, there is no universal and coordinate
independent equation of state of nuclear matter, in contrast to the mean-field
approximation. We also study how neutron star properties are modified by going
beyond mean field theory, and find that the differences between mean field
theory and exact results can be considerable. Further, we compare both exact
and mean-field results with some theoretical and phenomenological constraints
on neutron star properties, demonstrating thus the relevance of our model even
in its most simple version.
