Equation of State of Neutron Rich Matter

How does subatomic matter organize itself? Neutron stars are cosmic laboratories uniquely poised to answer this fundamental question that lies at the heart of nuclear science. Newly commissioned rare isotope facilities, telescopes operating across the entire electromagnetic spectrum, and ever more sensitive gravitational-wave detectors will probe the properties of neutron-rich matter with unprecedented precision over an enormous range of densities. Yet, a coordinated effort between observation, experiment, and theoretical research is of paramount importance for realizing the full potential of these investments. Theoretical nuclear physics provides valuable insights into the properties of neutron-rich matter in regimes that are not presently accessible to experiment or observation. In particular, nuclear density functional theory is likely the only tractable framework that can bridge the entire nuclear landscape by connecting finite nuclei to neutron stars. Although density functional theory is very successful in predicting the isoscalar properties of nuclei, the isovector sector is poorly constrained. Thus, this dissertation is focused on how to best constrain the isovector sector of the density functional. The electric dipole polarizability is an isovector observable that probes the dynamic response of the system. In order to incorporate it into the fitting protocol of the functional, we need to compute the $m_{-1}$ moment of the excitation strength through ground-state properties with an isovector dipole field added to the original mean-field Lagrangian. To solve this problem, we developed a fully relativistic approach based on the Dirac oscillator basis that can be applied to self-consistent calculations that include deformed potentials. While the electric dipole polarizability is an observable from the nuclear collective excitations, with remarkable advancements in experimental physics, it is now possible to gain valuable insights into the neutron distribution using exclusively electroweak probes. With calculations based on relativistic density functional theory, we found that both parity-violating elastic electron scattering and coherent elastic neutrino-nucleus scattering will play a vital role in constraining the weak form factor of nuclei which is closely related to the neutron density of nuclei. This will ultimately lead to powerful constraints on the isovector sector of the density functional. Finally, we also show that the difference in the charge radii of mirror nuclei has a strong correlation with the neutron skin of neutron-rich nuclei, which is another strong isovector indicator. This means elastic electron scattering on unstable nuclei can also be used to constrain the isovector component of the density functional.