Compact stars have a crucial role to play in our understanding of ultra dense and isospin asymmetric matter. Born in the extremely bright explosion of a massive star at the last stage of its life, neutron stars can present masses as high as twice the mass of the Sun, in a radius of around ten kilometers, thus gathering matter in their interior in extreme conditions of density and gravity. The nuclear physics probed by observing neutron stars is complementary in terms of density and isospin asymmetry to any terrestrial experiment. Multi-messenger astronomy is used to extract information on the interior of neutron stars, their structure and their composition. Connecting the observation of macroscopic parameters to state of the art constraints on the equation of state of dense matter can be done via hydrostatics and hydrodynamics modelling in General Relativity.

After a quick introduction to neutron star’s astrophysical features and how to model them within the theory of General Relativity, we will discuss how detection and theory of both neutron star astrophysics and nuclear physics can help us probe ultra dense matter. I will present publicly available software relevant to equation of state inference in the context of next generation of gravitational wave telescopes. Finally, we will show that almost equation of state independent relations that have been used to extract astrophysical features of neutron stars are partly irrelevant for gravitional wave signal detected with Cosmic Explorer and Einstein Telescope sensitivity.