$f$ and $p$ mode oscillation of proto-neutron stars with systematic variation of the nucleon effective mass
Atanu Guha, Debashree Sen, Hana Gil, Hajime Togashi, Chang Ho Hyun
TL;DR
This paper addresses how finite-temperature effects and uncertainties in the nucleon isoscalar effective mass $μ_S^*$ shape the EoS and non-radial oscillations of proto-neutron stars (PNSs). Employing the KIDS density functional, it constructs isentropic EoS across entropy values $S=1,2,3,4$ with four $μ_S^*$ values and fixed $μ_V^*=1$, then computes GR and Cowling $f$- and $p_1$-mode frequencies, along with universal relations involving $ω_f M$, compactness $C$, and tidal deformability $Λ$. The results show thermal effects reduce oscillation frequencies relative to cold neutron stars, with larger shifts at higher $S$ and smaller $μ_S^*$, while the $ω_f M$–$C$ relation shifts upward with increasing $S$ and the $ω_f M$–$Λ$ relation remains comparatively stable; GR systematically lowers $f_f$ compared to Cowling by ~15–20%. These findings imply that GW observations of PNS oscillations could reveal the evolutionary stage and constrain the finite-$T$ nuclear EoS, though a fully self-consistent finite-$T$ EoS and dynamic simulations are needed for more realistic modeling.
Abstract
We develop equation of state (EoS) of proto-neutron stars (PNSs) at various stages of evolution by varying entropy per baryon $S$, using the Korea-IBS-Daegu-SKKU density functional model. With finite values for both temperature and density, we systematically investigate the influence of nucleon effective mass on EoS of PNSs, for different values of isoscalar effective mass $μ_S^*$. For high entropy values, we aim to replicate conditions of failed core-collapse supernovae forming black holes. At each stage of evolution, structural and non-radial oscillation (fundamental $f$-mode and first pressure $p_1$-mode) properties are computed under isentropic conditions by varying $μ_S^*$. We focus on the effects of $S$ and $μ_S^*$ on oscillation frequencies $f_f$ and $f_{p_1}$ adopting complete general relativistic formalism and Cowling approximation. Thermal effects reduce the values of $f_f$ and $f_{p_1}$ of PNSs compared to those of cold NSs, consequently detection of the former gets facilitated. For high-mass PNSs, this reduction is more pronounced for $f_{p_1}$ than $f_f$. Moreover, lower values of $μ_S^*$ reduce $f_f$ and $f_{p_1}$ further. Universality of mass-scaled angular frequency ($ω_fM$) with compactness ($C$) and tidal deformability ($Λ$) are obtained as non-linear fits that shift upwards (downwards) in $ω_fM-C$ ($ω_fM-Λ$) plane for increasing values of $S$. For fixed $S$, the universality is also retained for variation of $μ_S^*$. $S$ shows stronger correlation than $μ_S^*$ with structural and oscillation properties of (P)NSs. Strength of correlation of $S$ is more prominent with $f_{p_1}$ than $f_f$ while the trend is opposite for $μ_S^*$. These findings suggest that detection of oscillation frequencies by upcoming GW detectors, could potentially indicate the evolutionary stage of a star during its transition from supernova to cold NS.