$m=1$ instability and gravitational wave signal in binary neutron star mergers

We examine the development and detectability of the $m=1$ instability in the remnant of binary neutron star mergers. The detection of the gravitational mode associated with the $m=1$ degree of freedom could potentially reveal details of the equation of state. We analyze the postmerger epoch of simulations of both equal- and nonequal-mass neutron star mergers using three realistic, microphysical equations of state and neutrino cooling. Our studies show such an instability develops generically and within a short dynamical time to strengths that are comparable to or stronger than the $m=2$ mode, which is the strongest during the early postmerger stage. We estimate the signal to noise ratio that might be obtained for the $m=1$ mode and discuss the prospects for observing this signal with available Earth-based detectors. Because the $m=1$ occurs at roughly half the frequency of the more powerful $m=2$ signal and because it can potentially be long lived, targeted searches could be devised to observe it. We estimate that with constant amplitude direct detection of the mode could occur up to a distance of roughly 14 Mpc, whereas a search triggered by the inspiral signal could extend this distance to roughly 100 Mpc.

Figure 1

The norm of a given mode $(l=2,m)$ of the gravitational radiation described by ${\mathrm{\Psi }}_4^0$ as a function of time for different mass ratios with the DD2 EoS. Overall, the $(l=2,m=1)$ mode achieves saturation earlier for unequal-mass than equal-mass binaries. Notice that the $m=1$ modes decay more slowly with time than the corresponding $(l=2,m=2)$ modes. Here, the merger time is chosen when the stars first come into contact.

Figure 2

Same as Fig. 1 for $\mathrm{q}=0.85$ with three different EoS (SFHo, DD2, and NL3). Although the qualitative behavior is similar for all the cases, the strength of the signal is stronger for the softest EoS. Additionally, the decay rate of the $m=1$ mode has a mild dependence on the EoS though this observation might be affected by other physics effects—such as the Tayler instability. Notice that the remnant with the SFHo EoS collapses to a black hole roughly 8 ms after merger.

Figure 3

The panels show three profiles of the average mass density in the equatorial plane, extracted from annuli of width 290 m centered on the origin at radii of 4.4 (red curve), 7.4 (black curve), and 10.3 km (blue curve) from the $q=0.85$ simulation with DD2 EoS. The density profiles $\approx 2\mathrm{ms}$ after merger (top panel) indicate a strong, $m=2$ bar mode modulation; at $\approx 16\mathrm{ms}$ (bottom panel) on the other hand, the density variations a are dominated by a single $m=1$ mode. The eigenmode can be seen to have a trailing spiral character as the peak in the profile shifts to a slightly smaller azimuth as the radius increases.

Figure 4

Fourier decomposition of rest mass density extracted from an annulus at 3.3 km and 370 m wide about the instantaneous center of mass from simulations of a symmetric binary (top panel), a binary with mass ratio of $q=0.85$ (middle panel), and $q=0.76$ (bottom panel). These three simulations were evolved with the DD2 EoS. The amplitude of modes with $m$ from 1 to 4 are shown normalized by the largest value of the zeroth component of the Fourier decomposition in the annulus.

Figure 5

Fourier transform of the ${\mathrm{\Psi }}_4^0$ modes $(l=2,m)$ shown in Fig. 1 for the DD2 EoS. For this EoS, the dominant radiative mode after merger is the $(l=2,m=2)$ mode, peaking at a frequency $f_{\mathrm{peak}}=2.3-2.6\mathrm{kHz}$. Notice that, in contrast, the $(l=2,m=1)$ mode peaks around $f_{\mathrm{m}1}\approx f_{\mathrm{peak}}/2\approx 1.3\mathrm{kHz}$. This peak becomes more prominent for small mass ratios.