Reconstruction of the neutron star equation of state from $w$-quasinormal modes spectra with a piecewise polytropic meshing and refinement method

In this paper we present a new approach to the inverse problem for relativistic stars using quasinormal modes and the piecewise polytropic parametrization of the equation of state. The algorithm is a piecewise polytropic meshing and refinement method that reconstructs the neutron star equation of state from experimental data of the mass and the $wI$-quasinormal modes. We present an algorithm able to numerically calculate axial quasinormal modes of neutron stars in an efficient way. We use an initial mesh of 27440 equations of state in a 4-volume of piecewise polytropic parameters that contains most of the candidate equations of state used today. The refinement process drives us to the reconstruction of the equation of state with a certain precision. Using the reconstructed equation of state, we calculate predictions for tidal deformability and slow rotation parameters (moment of inertia and quadrupole moment, for example). In order to check the method with an explicit example, we use as input data a few (five) configurations of a given equation of state. We reconstruct the equation of state in a quite good approximation, and then we compare the curves of physical parameters from the original equation of state and the reconstructed one. We obtain a relative difference for all of the parameters smaller than 2.5% except for the tidal deformability, for which we obtain a relative difference smaller than 6.5%. We also study constraints from GW170817 event for the reconstructed equation of state.

Figure 1

Real and imaginary part of the phase function $g$ vs $r$ inside and outside the star for a fundamental $wI$ mode with $\nu =8.003\mathrm{kHz}$ and $\tau =29.7892\mu \mathrm{s}$, obtained for SLy EOS with central energy density ${\epsilon }_0={10}^{18}\mathrm{kg}/{\mathrm{m}}^3$ ($M=1.4169M_{\odot }$).

Figure 2

Frequency of the fundamental $wI$ mode vs mass for the different EOSs considered.

Figure 3

Predictions for tidal deformability given by the different realistic EOSs considered, under the assumption that both components are neutron stars (for the low-spin scenario). Contours enclosing 90% and 50% of the probability density are shown as dashed lines (both curves taken from Ref. [3]).

Figure 4

Reconstruction of SLy EOS from its polytropic parameters. Pink vertical lines represent fixed density values where $\mathrm{\Gamma }$ changes in the curst region. Green vertical lines represent fixed density values where $\mathrm{\Gamma }$ changes in the core region. Blue vertical line represents the core-crust density transition.

Figure 5

Illustration of the inverse problem for neutron stars. The EOS is reconstructed from measurements of the frequency of the fundamental wI mode and the mass of 5 different neutron stars.

Figure 6

Graphical illustration of the local refinement of the initial mesh from the three EOSs with smallest values of $e_i$. The image represents a simplified model with 2 polytropic parameters $X=[1,8{]}_8$ and $Y=[1,9{]}_9$.

Figure 7

Scheme of the piecewise polytropic meshing and refinement method for the inverse problem.

Figure 8

Moment of inertia vs quadrupole moment for the 27440 EOSs generated (colors randomly chosen).

Figure 9

I-Q relation for the 27440 EOSs generated (colors randomly chosen).

Figure 10

I-Love relation for the 27440 EOSs generated (colors randomly chosen).

Figure 11

Frequency of the fundamental $wI$ mode vs its damping time for the 27440 EOSs generated (colors randomly chosen).

Figure 12

$\Im \overline{\omega }$ vs $\Re \overline{\omega }$ for the 27440 EOSs generated (colors randomly chosen). The scaling with the square root of the central pressure is quite independent of the EOS.

Figure 13

$\Im {\omega }^{*}$ vs $\Re {\omega }^{*}$ for the 27440 EOSs generated (colors randomly chosen). The scaling with the mass is quite independent of the EOS.

Figure 14

$\Im \overline{\omega }$ vs ${\overline{\lambda }}^{\mathrm{tid}}$ for the 27440 EOSs generated (colors randomly chosen).

Figure 15

$\Im {\omega }^{*}$ vs ${\overline{\lambda }}^{\mathrm{tid}}$ for the 27440 EOSs generated (colors randomly chosen).

Figure 16

Frequency of the fundamental $wI$ mode vs mass for the 27440 EOSs generated (colors randomly chosen).

Figure 17

Top panel: frequency of the fundamental $wI$ mode vs mass for the input data (blue diamonds) and the reconstructed EOS (black circles). Bottom panel: relative difference between the input data and the reconstructed EOS.

Figure 18

Predictions for tidal deformability given by the reconstructed GNH3 EOS, under the assumption that both components are neutron stars. Contours enclosing 90% and 50% of the probability density are shown as dashed lines (both curves taken from Ref. [3]).