Probing nuclear bubble structure via neutron star asteroseismology

We consider torsional oscillations that are trapped in a layer of spherical-hole (bubble) nuclear structure, which is expected to occur in the deepest region of the inner crust of a neutron star. Because this layer intervenes between the phase of slab nuclei and the outer core of uniform nuclear matter, torsional oscillations in the bubble phase can be excited separately from usual crustal torsional oscillations. We find from eigenmode analyses for various models of the equation of state of uniform nuclear matter that the fundamental frequencies of such oscillations are almost independent of the incompressibility of symmetric nuclear matter, but strongly depend on the slope parameter of the nuclear symmetry energy $L$. Although the frequencies are also sensitive to the entrainment effect, i.e., what portion of nucleons outside bubbles contribute to the oscillations, by having such a portion fixed, we can successfully fit the calculated fundamental frequencies of torsional oscillations in the bubble phase inside a star of specific mass and radius as a function of $L$. By comparing the resultant fitting formula to the frequencies of quasi-periodic oscillations (QPOs) observed from the soft-gamma repeaters, we find that each of the observed low-frequency QPOs can be identified either as a torsional oscillation in the bubble phase or as a usual crustal oscillation, given generally accepted values of $L$ for all the stellar models considered here.