Scalarization of neutron stars with realistic equations of state

We consider the effect of scalarization on static and slowly rotating neutron stars for a wide variety of realistic equations of state, including pure nuclear matter, nuclear matter with hyperons, hybrid nuclear and quark matter, and pure quark matter. We analyze the onset of scalarization, presenting a universal relation for the critical coupling parameter versus compactness. We find that the onset and the magnitude of the scalarization are strongly correlated with the value of the gravitational potential (the metric component $g_{tt}$) at the center of the star. We also consider the moment-of-inertia–compactness relations and confirm universality for the nuclear matter, hyperon and hybrid equations of state.

Figure 1

Total mass $M$ (in solar masses $M_{\odot }$) versus the physical radius $R_s$ (in km) of the neutron star models for all EOSs considered: The first two rows show the five hyperon EOSs (H4, BHZBM, GNH3, WCS1, WCS2) and the polytropic EOS, the last two rows contain the two nuclear EOSs (SLy, APR4), the two quark EOSs (WSPHS1, WSPHS2), and the four hybrid EOSs (WSPHS3, ALF2, ALF4, BS4). The solid black lines represent the GR configurations. The dashed red and orange lines represent the scalarized solutions for $A_1=e^{\frac{1}{2}\beta {\phi }^2}$ with ${\beta }_1=-4.8$ and ${\beta }_2=-4.5$, respectively. The solid blue and purple lines represent the scalarized solutions for $A_2=1/\cosh (\sqrt{-\beta }\phi )$ with the same values of ${\beta }_1=-4.8$ and ${\beta }_2=-4.5$.

Figure 2

Scalar field charge ${\omega }_A$ versus the total mass $M$ (in solar masses $M_{\odot }$) of the neutron star models for all EOSs considered: The first two rows show the five hyperon EOSs (H4, BHZBM, GNH3, WCS1, WCS2) and the polytropic EOS, the last two rows contain the two nuclear EOSs (SLy, APR4), the two quark EOSs (WSPHS1, WSPHS2), and the four hybrid EOSs (WSPHS3, ALF2, ALF4, BS4). The solid black lines represent the GR configurations. The dashed red and orange lines represent the scalarized solutions for $A_1=e^{\frac{1}{2}\beta {\phi }^2}$ with ${\beta }_1=-4.8$ and ${\beta }_2=-4.5$, respectively. The solid blue and purple lines represent the scalarized solutions for $A_2=1/\cosh (\sqrt{-\beta }\phi )$ with the same values of ${\beta }_1=-4.8$ and ${\beta }_2=-4.5$. Note that the scalar charge can be positive and negative.

Figure 3

The critical value of the coupling parameter $\beta$ versus the compactness $\mathcal{C}=M/R_s$ for all EOSs considered (except BS4). The grey curve is a fit to the function $f=c_0+c_1(M/R{)}^{-1}+c_4(M/R{)}^{-4}$.

Figure 4

Scalar field charge ${\omega }_A$ versus the compactness $\mathcal{C}=M/R_s$ (in units of $c=G_{*}=1$) of the neutron star models for all EOSs considered: The five hyperon EOSs (H4, BHZBM, GNH3, WCS1, WCS2), the polytropic EOS, the two nuclear EOSs (SLy, APR4), the two quark EOSs (WSPHS1, WSPHS2), and the four hybrid EOSs (WSPHS3, ALF2, ALF4, BS4). The coupling function is $A_1=e^{\frac{1}{2}\beta {\phi }^2}$ with ${\beta }_2=-4.5$ in (a) and ${\beta }_1=-4.8$ in (b). Note that the scalar charge can be positive and negative.

Figure 5

Scalar field charge ${\omega }_A$ (a) and scalar field at the center ${\phi }_0$ (b) versus the metric function $g_{tt}(0)$ at the center of the neutron star models for all EOSs considered: The five hyperon EOSs (H4, BHZBM, GNH3, WCS1, WCS2), the polytropic EOS, the two nuclear EOSs (SLy, APR4), the two quark EOSs (WSPHS1, WSPHS2), and the four hybrid EOSs (WSPHS3, ALF2, ALF4, BS4). The coupling function is $A_1=e^{\frac{1}{2}\beta {\phi }^2}$ with ${\beta }_1=-4.8$ and ${\beta }_2=-4.5$, respectively. The critical values ${\beta }_{\mathrm{cr}}$ are also indicated. Note that the scalar charge and the scalar field can be positive and negative.

Figure 6

Moment of inertia $I$ (in $10^{45}\mathrm{g}{\mathrm{cm}}^2$) versus the total mass $M$ (in solar masses $M_{\odot }$) of the neutron star models for all EOSs considered: The first two rows show the five hyperon EOSs (H4, BHZBM, GNH3, WCS1, WCS2) and the polytropic EOS, the last two rows contain the two nuclear EOSs (SLy, APR4), the two quark EOSs (WSPHS1, WSPHS2), and the four hybrid EOSs (WSPHS3, ALF2, ALF4, BS4). The solid black lines represent the GR configurations. The dashed red and orange lines represent the scalarized solutions for $A_1=e^{\frac{1}{2}\beta {\phi }^2}$ with ${\beta }_1=-4.8$ and ${\beta }_2=-4.5$, respectively. The solid blue and purple lines represent the scalarized solutions for $A_2=1/\cosh (\sqrt{-\beta }\phi )$ with the same values of ${\beta }_1=-4.8$ and ${\beta }_2=-4.5$.

Figure 7

Moment of inertia $I$ versus the compactness $\mathcal{C}=M/R_s$ in the slow rotation approximation for two different normalizations, $I/M{R}_s^2$ (a) and $I/M^3$ (b), for all 14 EOSs, for both coupling functions $A_i$, as well as for GR. The upper panels show the scaled values of $I$ (symbols) together with the fitted curves (lines) of the universal relations (excluding the two quark EOSs). The lower panels exhibit the deviations from the fitted values, $|1-I/I_{\mathrm{fit}}|$.

Figure 8

The value of $\beta$ at the bifurcation versus the central pressure ${\tilde{p}}_0$ for two polytropic EOSs. The respective critical values ${\beta }_{\mathrm{cr}}$ are indicated by horizontal lines.

Figure 9

The critical value ${\beta }_{\mathrm{cr}}$ versus the trace of the energy-momentum tensor $\widetilde{ϵ}-3{\tilde{p}}_0$, for all EOSs considered (except BS4). The grey curve is a fit to the function $f=d_0+d_1(\widetilde{ϵ}-3{\tilde{p}}_0)$, with $d_0=-4.353$, $d_1=0.024$, and reduced ${\chi }^2$ of $1.13\times {10}^{-6}$.