Radial and Non-Radial Oscillations of Protoneutron Stars with Hyperonic Composition

TL;DR

This work models protoneutron stars with a density-dependent RMF EoS including hyperons, across neutrino-trapped to cold-catalyzed stages, to quantify radial and non-radial oscillations using both full GR and the Cowling approximation. It demonstrates that f-mode frequencies and damping times are sensitive to thermal and compositional evolution, and that hyperons modestly raise frequencies while softening the EoS; p1-modes are also affected by temperature but less so by hyperons. A key contribution is the introduction of a robust universal relation based on the effective compactness $ ilde{\eta}= ext{sqrt}(M^3/I)$ that links $f$-mode properties across all evolutionary stages, highlighting the limitations of cold-NS universals for hot PNSs. The study also analyzes radial profiles and crust-core coupling, and assesses gravitational-wave detectability, showing that Galactic events are accessible to current detectors while next-generation observatories extend reach significantly, with the results providing a framework to constrain dense-matter EoS through GW asteroseismology.

Abstract

This paper explores radial and non-radial oscillations of protoneutron stars (PNSs) as they evolve from hot, neutrino-rich configurations through deleptonization to cold, catalyzed states. The equation of state (EoS) is modeled using a density-dependent relativistic mean-field framework, with stellar evolution characterized by changes in entropy and lepton fraction. Both nucleonic and hyperonic compositions are considered. Non-radial $f$- and $p_1$-mode oscillations are computed using both the Cowling approximation and the full General Relativistic framework. Trapped neutrinos initially increase the error in the Cowling approximation for $f$-modes, which decreases during deleptonization and rises again in the cold phase. In contrast, $p_1$-mode errors peak during intermediate stages due to evolving pressure and density gradients. The emergence of hyperons modestly raises oscillation frequencies in both modes. Existing universal relations for $f$-mode frequency and damping time lack model independence for PNSs, motivating a more robust relation. In particular, our proposed universal relation involving the moment of inertia and $\tildeη$ shows strong agreement across all evolutionary phases, offering a temperature-sensitive, model-independent scaling for asteroseismology. Radial oscillations of a $1.4\,M_\odot$ PNS are also studied for different EoSs. Our results show that displacement ($ξ$) and pressure perturbation ($η$) profiles are highly sensitive to thermal state, composition, and compactness. Hyperonic stars show higher frequencies, altered node structures, and stronger pressure perturbations due to EoS softening. Differences in frequency separation $Δν_n$ and fundamental frequency $ν_0$ between nucleonic and hyperonic models provide clear observational diagnostics for probing the interiors of PNSs and constraining the EoS of dense matter.

Figures (11)

• Figure 1: Total gravitational mass ($M$) of a cold NS (T = 0) as well as for a PNS capturing the evolutionary stages of neutrino-trapped, $\beta$-equilibrated stellar matter, as a function of its radius ($R$). The different PNS stages are characterized by different entropy per baryon ($s_B$) and lepton fraction ($Y_l$), and are compared with a neutrino-transparent star where $s_B = 2$ and $Y_{\nu_e} = 0$. Solid (dashed) lines correspond to the nucleonic (hyperonic) EoSs. The 1$\sigma$ (68%) confidence intervals for the 2D mass-radius posterior distributions of the millisecond pulsars PSR J0030+0451 Riley:2019ydaMiller:2019cac and PSR J0740+6620 Riley:2021pdlMiller:2021qha, obtained from NICER X-ray observations, are included. Furthermore, the plot presents the most recent NICER constraints on the mass and radius of PSR J0437$-$4715 Choudhury:2024xbk.
• Figure 2: The temperature distribution in the stellar matter for nucleon-only and nucleon-hyperon admixed PNSs at different stages of the PNS evolution.
• Figure 3: Gravitational Mass ($M$) versus fundamental frequency ($f$) of non-radial oscillation modes for different stages of the PNS evolution with nucleons (solid lines) and with hyperons (dashed lines). The left panel represents results obtained using the Cowling approximation, while the right panel shows calculations based on the full General Relativity (GR) framework.
• Figure 4: Left: Damping time ($\tau$) as a function of gravitational mass ($M$) for non-radial $f$-mode oscillation. Right: Non-radial $f$-mode frequency as a function of Compactness ($M/R$) at different stages of the PNS evolution for nucleons (solid lines) and hyperons (dashed lines).
• Figure 5: Left: The $f$-mode frequency as a function of the mean stellar density. The IR1 (brown) and IR2 (gray) are the fits from the earlier study, represented by a dot-dashed line Rather:2024nry, while "Our fit" (dotted line) represents the fit from the current work. Right: Normalized damping time of the $f$-mode as a function of the stellar compactness. The solid (dashed) lines correspond to the nucleonic (hyperonic) EoSs for different stages of the PNS evolution.