Gravitational wave asteroseismology of dark matter hadronic stars

The influence of the dark matter mass ($M_{\chi }$) and the Fermi momentum ($k_F^{\mathrm{DM}}$) on the $f_0$-mode oscillation frequency, damping time parameter, and tidal deformability of hadronic stars are studied by employing a numerical integration of hydrostatic equilibrium, nonradial oscillation, and tidal deformability equations. The matter inside the hadronic stars follows the NL3* equation of state. We obtain that the influence of $M_{\chi }$ and $k_F^{\mathrm{DM}}$ is observed in the $f_0$-mode, damping tome parameter, and tidal deformability. Finally, the correlation between the tidal deformability of the GW170817 event with $M_{\chi }$ and $k_F^{\mathrm{DM}}$ are also investigated.

Figure 1

Mass, normalized in Sun’s masses, versus radius for some values of $M_{\chi }$ and $k_F^{\mathrm{DM}}$. The curve in magenta represents the case without dark matter. The curves of the same color represent the results for a fixed value of $k_F^{\mathrm{DM}}$ and the curves of the same type depict results for a fixed value of $M_{\chi }$. The solid curves correspond to $M_{\chi }=50\mathrm{MeV}$, dashed curves correspond to $M_{\chi }=200\mathrm{MeV}$, and solid curves with circles correspond to the case $M_{\chi }=500\mathrm{MeV}$. Red and blue regions represent the masses and radii intervals (95%) from PSR $\mathrm{J}0030+0451$ and PSR $\mathrm{J}0740+6620$ measured by NICER [70, 71, 72, 73]. The magenta horizontal line includes all observed neutron stars over 2 solar masses, which include the pulsars PSR $J1614-2230$, PSR $\mathrm{J}0348+0432$, and PSR $\mathrm{J}0740+6620$ [74, 75, 76].

Figure 2

Tidal deformability versus mass different values of $M_{\chi }$ and $k_F^{\mathrm{DM}}$. The curve in magenta stands for the case without dark matter. The curves with the same type of color represent results obtained with the same value of $k_F^{\mathrm{DM}}$ and the curves of the same type depict the results found with the same value $M_{\chi }$. The solid curves correspond to $M_{\chi }=50\mathrm{MeV}$, dashed curves correspond to $M_{\chi }=200\mathrm{MeV}$, and solid curves with circles correspond to the case $M_{\chi }=500\mathrm{MeV}$. The vertical solid line depicts ${\mathrm{\Lambda }}_{1.4}=190_{-120}^{+390}$ published in Ref. [23].

Figure 3

Fundamental mode frequency $f_0$ and damping time of fundamental mode $\tau$ against the total mass $M/M_{\odot }$ are shown on the top panel and versus the tidal deformability $\mathrm{\Lambda }$ on the bottom panel. In all cases, different values of $M_{\chi }$ and $k_F^{\mathrm{DM}}$ are employed. The curves in magenta depict the case without dark matter. The curves of the same color represent the results for a fixed value of $k_F^{\mathrm{DM}}$ and the curves of the same type stand results for a fixed value of $M_{\chi }$. The solid curves correspond to $M_{\chi }=50\mathrm{MeV}$, dashed curves correspond to $M_{\chi }=200\mathrm{MeV}$, and solid curves with circles correspond to the case $M_{\chi }=500\mathrm{MeV}$. In the bottom panels, the dashed vertical lines represent the tidal deformability obtained from the event GW170817, i.e., ${\mathrm{\Lambda }}_{1.4}=190_{-120}^{+390}$, reported Ref. [23].

Figure 4

The relative difference between the frequency of the fundamental mode in the full linearized equations and the relativistic Cowling approximation, $\mathrm{diff}=100\times (f_0-f_0^{\mathrm{cow}})/f_0$, versus the total mass for all $M_{\chi }$ and $k_F^{\mathrm{DM}}$ considered in Fig. 4.

Figure 5

The $f_0$-mode oscillation frequency, damping time parameter, and tidal deformability against the Fermi momentum of dark matter are shown respectively on the left, middle, and right panels. The results connected on straight dashed lines on red, blue, and black correspond to $M_{\chi }=50\mathrm{MeV}$, 200 MeV, and 500 MeV. All values presented concern the equilibrium configurations with total masses $M=1.4M_{\odot }$. The horizontal straight line represent ${\mathrm{\Lambda }}_{1.4}=190_{-120}^{+390}$ obtained by LVC in Ref. [23].

Figure 6

In the Left, middle, and bottom panels are respectively plotted ${\mathrm{\Lambda }}_2\times {\mathrm{\Lambda }}_1$, $f_{0(2)}\times f_{0(1)}$, and ${\tau }_2\times {\tau }_1$ for the components of the event GW170817 for different values of $k_F^{\mathrm{DM}}$ and $M_{\chi }$. The wine lines on the left panel depict the LIGO-Virgo confidence curves of 50% and 90% levels [22] and the solid diagonal appearing on the left, middle, and right panels correspond respectively to ${\mathrm{\Lambda }}_1={\mathrm{\Lambda }}_2$, $f_{0(1)}=f_{0(2)}$, and ${\tau }_1={\tau }_2$.