Properties of hypermassive neutron stars formed in mergers of spinning binaries

We present numerical simulations of binary neutron star mergers, comparing irrotational binaries to binaries of NSs rotating aligned to the orbital angular momentum. For the first time, we study spinning BNSs employing nuclear physics equations of state, namely the ones of Lattimer and Swesty as well as Shen, Horowitz, and Teige. We study mainly equal mass systems leading to a hypermassive neutron star (HMNS), and analyze in detail its structure and dynamics. In order to exclude gauge artifacts, we introduce a novel coordinate system used for post-processing. The results for our equal mass models show that the strong radial oscillations of the HMNS modulate the instantaneous frequency of the gravitational wave (GW) signal to an extend that leads to separate peaks in the corresponding Fourier spectrum. In particular, the high frequency peaks which are often attributed to combination frequencies can also be caused by the modulation of the $m=2$ mode frequency in the merger phase. As a consequence for GW data analysis, the offset of the high frequency peak does not necessarily carry information about the radial oscillation frequency. Further, the low frequency peak in our simulations is dominated by the contribution of the plunge and the first 1–2 bounces. The amplitude of the radial oscillations depends on the initial NS spin, which therefore has a complicated influence on the spectrum. Another important result is that HMNSs can consist of a slowly rotating core with an extended, massive envelope rotating close to Keplerian velocity, contrary to the common notion that a rapidly rotating core is necessary to prevent a prompt collapse. Finally, our estimates on the amount of unbound matter show a dependency on the initial NS spin, explained by the influence of the latter on the amplitude of radial oscillations, which in turn cause shock waves.

Figure 1

The polar coordinate system constructed for analysis at a time 0.9 ms after the merger, for model SHT-M2.0-I. The black radial lines are the coordinate lines of constant $\varphi$, and the black circles mark coordinate lines of constant $r_c$. Both are spaced equidistantly. For comparison, we show the density profile as color plot, and a coordinate circle with respect to simulation coordinates (dashed red curve).

Figure 2

Comparison of density moments $P_2^{\rho }$ computed either using the new coordinate system (thick green line) or the simulation coordinates (thin red line). Shown are the results for models LS220-M1.5-I (top panel) and SHT-M2.0-I (bottom panel).

Figure 3

Proper separation of maximum density locations versus their orbital phase. The phase has been aligned at a fixed separation of $d=20M_{\infty }$. The symbols are marking the start of the simulations.

Figure 4

Dimensionless BH spin versus initial NS spinup rate $\mathrm{\Delta }{F}_{\mathrm{R}}$ (see main text). Our results are shown in comparison to earlier results [22] for the case of an ideal gas EOS. The vertical line marks the rotation rate of the double pulsar PSR J0737-3039A.

Figure 5

Evolution of the HMNS average circumferential radius ${\overline{R}}_c$. The time coordinate is relative to the merger time $t_m$.

Figure 6

Time evolution of the HMNS rotation for models SHT-M2.0-I (top panel), and LS220-M1.5-I (lower panel). The black solid line shows the average angular velocity ${\overline{\mathrm{\Omega }}}_{\mathrm{R}}$. The shaded area is bounded by the extrema of the angular velocity ${\omega }_{\mathrm{R}}$ in the orbital plane inside the HMNS (ignoring mass densities below ${10}^{-2}$ of the maximum one). The difference $\mathrm{\Delta }\mathrm{\Omega }$ between maximum and minimum is given by the red curve. The dotted line shows the pattern angular velocity $\frac{1}{2}{\omega }_2$ of the $l=m=2$ mode, computed from the phase velocity of $P_2^{\rho }$.

Figure 7

Average rotation profile and Keplerian angular velocity profile for models SHT-M2.0-I and -S (top panel), and LS220-M1.5-I and -S (bottom panel). The solid curves mark the angular velocity ${\omega }_{\mathrm{R}}$ in the orbital plane versus circumferential radius $r_c$. The dotted lines show the contribution $-{\overline{\beta }}^{\varphi }$ of the frame dragging. The dash-dotted curves are an estimate of the angular velocity of test particles in corotating circular orbits. The vertical dash-dotted lines mark the radius where the mass density falls below 5% of the central one. For comparison, the circles mark radius and equatorial orbital velocity of cold, uniformly rotating stars with the same central rotation rate and central rest mass density. The diamonds mark models at zero-temperature with the same central rest mass density, uniformly rotating at Keplerian rate. The red squares mark the maximum mass (Keplerian) model for uniformly rotating cold NSs.

Figure 8

Density profile in the equatorial plane of the time and $\varphi$-averaged HMNS for models SHT-M2.0-I (top panel), and LS220-M1.5-I (bottom panel). For comparison, we show the density profiles of uniformly rotating NS stars with same central density and central rotation rate as the HMNS, and same central density but maximum rotation rate. For the LS220 case, we also show a model with same mass as the HMNS, but a $j$-const rotation law (see main text). The density is scaled by $r_c^2$ to better visualize the contribution of each spherical shell to the total mass. Note however that since the models are not spherically symmetric, simply taking the integrals of the curves would overestimate the total mass inside a given radius.

Figure 9

Time evolution of the density moment amplitudes $|P_m^{\rho }|$ for models SHT-M2.0-I (top panel) and LS220-M1.5-I (bottom panel). The amplitudes are normalized to the maximum of the $m=2$ moment.

Figure 10

Gravitational wave strain at distance 100 Mpc for models SHT-M2.0-I (left panel), SHT-M2.0-S (middle panel), and SHT-M2.2-I (right panel). The waveforms were extracted at radius $r=916\mathrm{km}$.

Figure 11

Gravitational wave strain at distance 100 Mpc for models LS220-M1.5-I (left panel), LS220-M1.5-S (middle panel), and LS220-M1.8-I (right panel). The waveforms were extracted at radius $r=756\mathrm{km}$.

Figure 12

(Top panel) Fourier spectrum of the toy signal, with values $f_s=3.3\mathrm{kHz}$, $\mathrm{\Delta }f=1.3\mathrm{kHz}$, ${\tau }_s=8\mathrm{ms}$, ${\tau }_m=3\mathrm{ms}$, $f_m=0.6\mathrm{kHz}$. (Bottom panel) Corresponding instantaneous frequency. The vertical dashed lines mark the maxima of the Fourier spectrum, the vertical dotted lines the first five extrema of the instantaneous frequency.

Figure 13

(Top panel) GW effective strain spectrum for model SHT-M2.0-I. The thick blue curve shows the spectrum of the full signal at nominal distance of 100 Mpc. The red solid line shows the spectrum of the late signal, starting 2 ms (plus time of flight) after the time of the first density maximum during the merger, $t_{\mathrm{m}}$. The green dash-dotted line depicts $f^2{\tilde{P}}_2^{\rho }(f)$, where ${\tilde{P}}_2^{\rho }$ is the Fourier spectrum of the fluid moment $P_2^{\rho }$, rescaled to match the maximum amplitude. (Bottom panel) Evolution of the instantaneous frequencies of the GW signal ${\mathrm{\Psi }}_4$ (thick blue line), and the moment $P_2^{\rho }$ (green dash-dotted line). The GW signal has been shifted to compensate for the time of flight.

Figure 14

Like Fig. 13, but showing model LS220-M1.5-I.

Figure 15

Like Fig. 13, but for unequal-mass model LS220-M1.4-U.

Figure 16

(Top panel) GW effective strain spectrum at nominal distance of 100 Mpc for irrotational model SHT-M2.0-I compared to the same spectrum for spinning model SHT-M2.0-S. The GW signal from the longer simulation was cut such that the signal duration after the merger matches the shorter simulation. (Bottom panel) Evolution of the corresponding instantaneous frequencies.

Figure 17

Like Fig. 16, but for models LS220-M1.5-I and -S.

Figure 18

Specific entropy in the orbital plane, 2.5 ms after the merger, for model SHT-M2.0-I. The regions where matter is unbound according to the geodesic criterion are bounded by the black lines, the cyan lines enclose regions unbound according to the Bernoulli criterion. Regions covered by the artificial atmosphere are colored blue.

Figure 19

Thermal evolution of model LS220-M1.5-I from merger until BH formation. The color corresponds to the $\varphi$-averaged temperature in the orbital plane as a function of circumferential radius and coordinate time. The green line marks the density averaged circumferential radius ${\overline{R}}_c$.

Figure 20

Time evolution of the estimate for the mass of unbound matter, according to the geodesic criterion. The decreasing parts (dotted lines) are unphysical artifacts caused by the artificial atmosphere, the maximum serves as a lower bound for the real value.