Axial and polar stability of neutron stars in scalar-tensor theories with disformal coupling

In the present work, we study the radial and non-radial perturbative stability of neutron stars in which the matter is disformally coupled to the metric. First, we derive the gravitational and the fluid equations of the neutron star in a static and spherically symmetric background. Then, we calculate the second-order expansion of the action that describes the dynamics of both axial and polar modes. From the resulting expressions, we derive the conditions to avoid gradient and ghost instabilities at the center of the star and at spatial infinity. In addition, a numerical analysis is performed to investigate the stability of a particular model with a constant disformal function denoted as $\mathrm{\Lambda }$ in the whole space time. We found that the chosen model is stable against the gradient instability in a small range of the constant $\mathrm{\Lambda }$.