  Background: Neutron-skin thickness is an excellent indicator of isovector
properties of atomic nuclei. As such, it correlates strongly with observables
in finite nuclei that depend on neutron-to-proton imbalance and the nuclear
symmetry energy that characterizes the equation of state of neutron-rich
matter. A rich worldwide experimental program involving studies with rare
isotopes, parity violating electron scattering, and astronomical observations
is devoted to pinning down the isovector sector of nuclear models. Purpose: We
assess the theoretical systematic and statistical uncertainties of neutron-skin
thickness and relate them to the equation of state of nuclear matter, and in
particular to nuclear symmetry energy parameters. Methods: We use the nuclear
superfluid Density Functional Theory with several Skyrme energy density
functionals and density dependent pairing. To evaluate statistical errors and
their budget, we employ the statistical covariance technique. Results: We find
that the errors on neutron skin increase with neutron excess. Statistical
errors due to uncertain coupling constants of the density functional are found
to be larger than systematic errors, the latter not exceeding 0.06 fm in most
neutron-rich nuclei across the nuclear landscape. The single major source of
uncertainty is the poorly determined slope L of the symmetry energy that
parametrizes its density dependence. Conclusions: To provide essential
constraints on the symmetry energy of the nuclear energy density functional,
next-generation measurements of neutron skins are required to deliver precision
better than 0.06 fm.
