Topology of the Superconducting Heart of Neutron Stars: Effects of Microphysics and Gravitational-Wave Signatures

Abstract

We present a general-relativistic study of the distribution of proton superconductivity in strongly magnetized neutron stars (NSs), using the XNS code to solve the coupled Einstein-Maxwell equations. We investigate equilibrium configurations with both toroidal and poloidal magnetic field geometries and incorporate complex many-body effects through microscopically derived proton pairing gaps. The models employ equations of state (EoS) obtained from microscopic many-body theory - including realistic two- and three-body nuclear interactions - as well as from relativistic mean-field approaches. We compare superconducting topologies across our collection of EoS and explore the influences of magnetic field geometry in stellar models parameterized by central density. Our models confirm the absence of $S$-wave superconductivity in the inner core and, importantly, reveal that non-superconducting regions exhibit complex three-dimensional geometries: doughnut-shaped for toroidal fields and prolate-shaped for poloidal fields -- spatial structures that are inherently absent in one-dimensional analyses. We also compute magnetic deformations and ellipticities for several millisecond pulsars (MSPs), estimating their continuous gravitational wave strain. While these MSPs remain undetectable by current detectors, next-generation instruments such as the Einstein Telescope and Cosmic Explorer may detect their signals, opening an observational window into internal superconductivity and internal magnetic field of NSs, as well as the fundamental microphysics of dense matter.