We show that there is a direct relation between upper limits on (or potential
future measurements of) the m = 2 quadrupole moments of slowly rotating neutron
stars and the l = m = 2 deformation of the star's surface, in full general
relativity, to first order in the perturbation. This relation only depends on
the star's structure through its mass and radius. All one has to assume about
the star's constituents is that the stress-energy tensor at its surface is that
of a perfect fluid, which will be true with good accuracy in almost all the
situations of interest, and that the magnetic field configuration there is
force-free, which is likely to be a good approximation. We then apply this
relation to the stars which have direct LIGO/Virgo bounds on their m = 2
quadrupole moment, below the spin-down limit, and compare with the expected
surface deformations due to rotation. In particular, we find that LIGO
observations have constrained the Crab pulsar's l = m = 2 surface deformation
to be smaller than its l = 2, m = 0 deformation due to rotation, for all the
causal equations of state we consider, a statement that would not have been
able to be made just using the upper bounds on the l = m = 2 deformation from
electromagnetic observations.Comment: 13 pages, 3 figures, version accepted by PRD, including expanded
discussion of the calculation and a few correction