Comparison of Equations of State for Neutron Stars with First-Order Phase Transitions: A Qualitative Study

The equation of state is fundamental in describing matter under the extreme conditions characteristic of neutron stars and is central to advancing our understanding of dense matter physics. A critical challenge, however, lies in accurately modelling first-order phase transitions while ensuring thermodynamic consistency and aligning with astrophysical observations. This study explores two frameworks for constructing EoSs with first-order phase transitions: the polytropic interpolation method and the randomized speed-of-sound interpolation approach. It is found that the mass-radius relation and pressure vs. energy density relation are blind towards the thermodynamic consistency check. The polytropic interpolation method can exhibit discontinuities in the chemical potential for first-order phase transition, raising concerns regarding potential causality violations and thermodynamic inconsistencies. In contrast, the speed of sound interpolation approach ensures continuity in the chemical potential, offering a more thermodynamically consistent and reliable framework. Moreover, the sound speed method effectively captures the softer segment of the mass-radius spectrum, a capability not achieved by the consistent piecewise-polytropic approach due to its monotonic stiffness constraints. The speed of sound definition involving number density and chemical potential reveals the thermodynamic inconsistency, making it a more consistent and robust definition. These findings underscore the importance of thermodynamic consistency in EoS construction and highlight the advantages of the randomized speed-of-sound method for modelling phase transitions in dense matter.