Gravitational wave constraints on the shape of neutron stars

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 could not have been made just using the upper bounds on the $l=m=2$ deformation from electromagnetic observations.