Investigating the I-Love-Q and $w$-mode universal relations using piecewise polytropes

Neutron stars are expected to have a tight relation between their moment of inertia ($I$), tidal deformability ($\lambda$, which is related to the Love number), and rotational mass quadrupole moment ($Q$) that is nearly independent of the unknown equation of state (EOS) of cold dense matter. These and similar relations are often called “universal”, and they have been used for various applications including analysis of gravitational wave data. We extend these studies using piecewise polytropic representations of dense matter, including for so-called twin stars that have a second branch of stability at high central densities. The second-branch relations are less tight, by a factor of $\sim 3$, than the relations found in the first stable branch. We find that the relations on both branches become tighter when we increase the lower limit to the maximum mass for the EOS under consideration. We also propose new empirical relations between $I$, $\lambda$, $Q$, and the complex frequency $\omega ={\omega }_R+i{\omega }_I$ of the fundamental axial $w$-mode, and find that they are comparably tight to the I-Love-Q correlations.

Figure 1

Example mass-radius relations for the equations of state we consider. The solid blue parts of the curves show the first branch of stability, and the solid magenta part of one of the curves shows the second branch of stability. The dotted portions of the curves show regions that are unstable. The central density of the star increases along each curve from the bottom right to the top left.

Figure 2

The ${\log }_{10}\overline{Q}-{\log }_{10}\overline{I}$ relation and its residuals. Top left: our full first-branch set of neutron stars. Top right: only the first-branch stars constructed using EOS with maximum gravitational mass $\ge 2M_{\odot }$. In both of the top figures, the bottom panel shows the fractional difference from the best fourth-order fit [see Eq. (1)], as well as the overall root-mean-square (rms) deviation. Bottom left: only the stars constructed from a second high-density stable branch. Bottom right: only the stars constructed from a second high-density stable branch using an EOS with a maximum gravitational mass $\ge 2M_{\odot }$. For the bottom figures, the middle panel shows the fractional differences and rms relative to our fourth-order fit to the second-branch data, and the bottom panel shows the fractional differences and rms relative to our fourth-order fit to the first-branch data. We see that increasing the minimum value of the maximum mass tightens the relations for the second-branch stars, as well as for the first-branch stars. We also see that the second-branch stars follow a slightly different relation than the first-branch stars.

Figure 3

The ${\log }_{10}\overline{Q}-{\log }_{10}\overline{\lambda }$ relation and its residuals. The panels correspond to the panels in Fig. 2. We find, in agreement with [10], that the $\overline{Q}-\overline{\lambda }$ relation is not as tight as the $\overline{Q}-\overline{I}$ relation. This statement is also true when we focus exclusively on neutron stars in the second stable branch of central density.

Figure 4

The relation between the real component ${\omega }_R$ of the frequency of the $w$-mode and ${\log }_{10}\overline{Q}$. The panels correspond to those of Fig. 2. The correlations are tightened when we impose a lower limit on the maximum mass, and the second-branch fit to the second-branch data is tighter than the first-branch fit to the second-branch data. However, the tightening is not as pronounced as it is for the $\overline{Q}-\overline{\lambda }$ and $\overline{Q}-\overline{I}$ relations.

Figure 5

The relation between the imaginary component ${\omega }_I$ of the frequency of the $w$-mode and ${\log }_{10}\overline{Q}$. The panels correspond to those of Fig. 2. The residuals show significant systematics; for example, the spread at low $\overline{Q}$ is much greater than at higher $\overline{Q}$. The maximum in ${\omega }_I$ is a common feature observed in sequences of other families of quasinormal modes, such as $f$-modes [75]. We find that the maximum corresponds to $M/R\approx 0.15$ for all equations of state.