Astrophysical implications of double-layer torsional oscillations in a neutron star crust as a lasagna sandwich

In the crust of a neutron star, global torsional oscillations could occur in two elastic layers. The outer and inner layers are composed of spherical and cylindrical nuclei and of cylindrical holes (tubes) and spherical holes (bubbles), respectively, while between these two layers, a phase of slab-like (lasagna) nuclei with vanishingly small elasticity is sandwiched. In this work, we update systematic calculations of the eigenfrequencies of the fundamental oscillations in the inner layer by newly allowing for the presence of tubes. We find that the frequencies still depend strongly on the slope parameter of the nuclear symmetry energy, $L$, while being almost independent of the incompressibility of symmetric nuclear matter. We also find that the fundamental frequencies in the inner layer can become smaller than those in the outer layer because the tube phase has a relatively small shear modulus and at the same time dominates the inner layer in thickness. As a result, we can successfully explain not only the quasi-periodic oscillations originally discovered in the observed X-ray afterglow of the giant flare of SGR 1806--20 but also many others recently found by a Bayesian procedure.