Universal Relations and Correlation Analysis of Proto-Neutron Star Properties in Energy-Momentum Squared Gravity

Abstract

Proto-neutron stars (PNSs) are the hot, lepton-rich remnants of the core collapse supernovae, which go through a cooling phase and become cold, stable Neutron stars (NSs). Since PNSs are also superdense objects with strong gravitational fields, we can use them to probe general relativity (GR) in the high-curvature regime, similar to NSs. In this study, we analyze the macroscopic properties like mass, radius, compactness, tidal deformability, $f$-mode oscillations and gravitational binding energy of PNSs using four different relativistic mean-field (RMF) equations of state (EOSs) with fixed entropy per baryon ($S$ =1, 2) and varying the lepton fractions ($Y_l$). The variation of $S$ and $Y_l$ has a noticeable effect on these properties. Extending our study beyond GR, we explore these effects within the framework of Energy-Momentum Squared Gravity (EMSG). This modified gravity theory adds the squared energy-momentum terms to the field equations with a free parameter $α$. In the weak-field regimes, EMSG remains indistinguishable from GR, but in the strong-field regimes, such as PNSs or NSs, it shows measurable deviations. Varying the free parameter $α$, we observe significant changes in the macroscopic properties of the PNSs. After that, we focus on the universal relations of the macroscopic properties and the correlations of the universal relations. We find that, despite significant changes in the macroscopic properties induced by the variations of $S$, $Y_l$ and $α$, the correlations remain strong and nearly unaffected.

Figures (8)

• Figure 1: Left: Variation of effective pressure ($P_\mathrm{eff}$) with effective energy density ($\mathcal{E}_\mathrm{eff}$) for NITR, IOPB-I, MODEL I and IUFSU EOSs, represented by different colours. Right: The effective squared sound speeds ($c^2_{s(\mathrm{eff})}$) are presented as functions of baryon number density ($n_B$). In both panels, the zoom plots refer to the clear visibility of the effect of $S$, $Y_l$ and $\alpha$.
• Figure 2: Left: The Mass-radius relation of PNSs, for differnet $S$ and $Y_l$ values. The astrophysical observable constraints on mass and radius from PSR J0740+6620 Miller_2021Riley_2021, and NICER data for PSR J0030+0451 Miller_2019Riley_2019 are represented by colored regions. In the colour bar, the variation of $\alpha$ has been shown. Right: Variation of the difference in $R$ for maximum and minimum values of $\alpha$ with fixed mass represented with different coloured markers.
• Figure 3: Left: The variation of $f$-mode frequency ($f_f$) with the mass ($M$) of PNSs, for differnet $S$ and $Y_l$ values. The error bars represent the observational constraints from GW170817 GW170817 and GW190814 events GW190814. In the colour bar, the variation of $\alpha$ has been shown. Right: Variation of the difference in $f_f$ for maximum and minimum values of $\alpha$ with fixed mass represented with different coloured markers.
• Figure 4: Left: The variation of gravitational binding energy per mass ($B/M$) with the mass ($M$) of PNSs, for differnet $S$ and $Y_l$ values. In the colour bar, the variation of $\alpha$ has been shown. Right: Variation of the difference in $B/M$ for maximum and minimum values of $\alpha$ with fixed mass represented with different coloured markers.
• Figure 5: $f_f M_{1.4}$-$C$ (upper) and $f_f M_{1.4}$-$\Lambda$ (lower) relations for four different EOSs with the variation of $\mathrm{S}$ and $Y_l$ with a fixed value of $\alpha=+5.01$ (left), $\alpha=0$ (middle) and $\alpha=-5.01$ (right). Each of the lower panels of the six figures contains the plot for residuals ($\Delta$).