Tidal deformability and gravitational-wave phase evolution of magnetized compact-star binaries

The evolution of the gravitational-wave phase in the signal produced by inspiralling binaries of compact stars is modified by the nonzero deformability of the two stars. Hence, the measurement of these corrections has the potential of providing important information on the equation of state of nuclear matter. Extensive work has been carried out over the last decade to quantify these corrections, but it has so far been restricted to stars with zero intrinsic magnetic fields. While the corrections introduced by the magnetic tension and magnetic pressure are expected to be subdominant, it is nevertheless useful to determine the precise conditions under which these corrections become important. To address this question, we have carried out a second-order perturbative analysis of the tidal deformability of magnetized compact stars under a variety of magnetic-field strengths and equations of state describing either neutron stars or quark stars. Overall, we find that magnetically induced corrections to the tidal deformability will produce changes in the gravitational-wave phase evolution that are unlikely to be detected for a realistic magnetic field i.e., $B\sim {10}^{10}–{10}^{12}\mathrm{G}$. At the same time, if the magnetic field is unrealistically large, i.e., $B\sim {10}^{16}\mathrm{G}$, these corrections would produce a sizeable contribution to the phase evolution, especially for quark stars. In the latter case, and if the neglected higher-order terms will remain negligible also for very high magnetic fields, the induced phase differences would represent a unique tool to measure the properties of the magnetic fields, providing information that is otherwise hard to quantify.

Figure 1

Dimensionless magnetically modified tidal deformability $\delta \mathrm{\Lambda }$ shown as a function of magnetic field strength and compactness $M/R$ of the star. The left panel refers to the neutron-star EOS APR, while the right one to the EOS MIT2cfl, and is therefore representative of quark stars. Two different colors are used account for the different signs of $\delta \mathrm{\Lambda }$ and indicate that while a magnetic field increases the tidal deformation for weak gravitational fields, it opposes it in stronger gravity.

Figure 2

Left panel: dimensionless magnetic tidal deformability $\delta \mathrm{\Lambda }$ shown as a function of the stellar compactness and for a fixed magnetic field of $B={10}^{15}\mathrm{G}$. The top part refers to quark-star EOSs, while the bottom one to representative neutron-star EOSs. Marked with dots are the positions of stars with $M=1.4M_{\odot }$. Right panel: relative weight of the dimensionless magnetic tidal deformability $\delta \mathrm{\Lambda }$ when compared with the tidal deformability ${\mathrm{\Lambda }}^{\mathrm{T}}$. Different lines refer to different EOSs.

Figure 3

Left panel: evolution of the GW phase differences $\mathrm{\Delta }\varphi$ relative to binaries with zero magnetic field or with $B={10}^{15}\mathrm{G}$. Different lines refer to different EOSs, but are all relative to a binary with masses $m_1=m_2=1.4M_{\odot }$, that enters the detector at an initial frequency of 60 Hz. The inset shows a magnification near the merger. Right panel: final phase difference at the merger $\mathrm{\Delta }\varphi {|}_{\mathrm{merg}}$ shown as a function of the tidal deformability of a star with $M=1.4M_{\odot }$. The top part refers neutron stars, while the bottom part to quark stars; the color code used is the same as in the left panel and hints to a linear behavior for neutron stars.

Figure 4

Evolution of the GW phase differences $\mathrm{\Delta }\varphi$ shown as a function of different magnetic-field strengths. Also in this case, the data refers to a representative binary with masses $m_1=m_2=1.4M_{\odot }$, that enters the detector at an initial frequency of 60 Hz. The left panel refers to the neutron-star EOS APR, while the right one to the EOS MIT2cfl, and is therefore representative of quark stars.

Figure 5

Evolution of the GW phase differences $\mathrm{\Delta }\varphi$ during the early inspiral (i.e., for frequencies up to 1 kHz). Shown as a comparison are the other the high-order non-magnetic corrections, i.e., the odd-parity tidal correction $j_2$, the even-parity spin-tidal coupling, and the magnetic-moment correction. The blue-shaded region refers to magnetic-field strengths between ${10}^{15}\mathrm{G}$ and ${10}^{16}\mathrm{G}$. All the data refers to a representative binary with masses $m_1=m_2=1.4M_{\odot }$, with the left panel relative the APR EOS and the right one to the MIT2cfl EOS.

Figure 6

Broken universal relation between the magnetic tidal deformability and the dimensionless moment of inertia $\overline{I}≔I/M^3$ for different EOSs and for a magnetic field of $B={10}^{15}\mathrm{G}$ (different magnetic-field strengths will only rescale the vertical axis but do not change the functional behavior). The loss of universality between $\delta \mathrm{\Lambda }$ and $\overline{I}$ does not impact significantly the overall universality between $\mathrm{\Lambda }$ and $\overline{I}$ (see inset).