Numerical relativity simulations of binary neutron stars

We present a new numerical relativity code designed for simulations of compact binaries involving matter. The code is an upgrade of the BAM code to include general relativistic hydrodynamics and implements state-of-the-art high-resolution-shock-capturing schemes on a hierarchy of mesh refined Cartesian grids with moving boxes. We test and validate the code in a series of standard experiments involving single neutron star spacetimes. We present test evolutions of quasiequilibrium equal-mass irrotational binary neutron star configurations in quasicircular orbits which describe the late inspiral to merger phases. Neutron star matter is modeled as a zero-temperature fluid; thermal effects can be included by means of a simple ideal gas prescription. We analyze the impact that the use of different values of damping parameter in the Gamma-driver shift condition has on the dynamics of the system. The use of different reconstruction schemes and their impact in the post-merger dynamics is investigated. We compute and characterize the gravitational radiation emitted by the system. Self-convergence of the waves is tested, and we consistently estimate error bars on the numerically generated waveforms in the inspiral phase.

Figure 1

Evolution of the central rest-mass density of model A0. The picture shows the normalized value ${\rho }_c(t)/{\rho }_c(0)$ for different reconstruction schemes.

Figure 2

Convergence in L1 norm of evolutions of model A0. The first three panels from the top left to the bottom right show the evolution of the L1 distance $\Vert \rho (t)-\rho (0){\Vert }_1$. Different lines refer to different resolutions. Different panels refer to the three different reconstructions considered: MC2, PPM, and CENO. The last panel (bottom right) shows in a log-log plot the L1 distances as a function of $1/h$ at $t=8\mathrm{ms}$ for the three reconstructions MC2 (solid-red line), PPM (dashed-blue line) and CENO (solid-green line).

Figure 3

Same as in Fig. 2 but results are computed in the Cowling approximation.

Figure 4

Hamiltonian violation during the evolution of model A0. The first three panels from the top left to the bottom right show the time evolution of the 2-norm of the Hamiltonian constraint computed on the finest grid level which fully contain the star. Different lines refer to different resolutions. Different panels refer to the three different reconstructions considered: MC2, PPM, and CENO. The last panel shows the profile in the $x$ direction of the Hamiltonian constraint at time $t\sim 10\mathrm{ms}$ for the maximum resolution and three reconstructions MC2 (solid-red line), PPM (dashed-blue line) and CENO (solid-green line). Note the different scales of the plots.

Figure 5

Evolution of the central rest-mass density in the migration test. The picture shows the normalized value ${\rho }_c(t)/{\rho }_c(0)$ in the case of an ideal gas EoS (red solid line) and polytropic EoS (blue dashed line).

Figure 6

Shock formation in the migration test. The picture shows the profiles of the specific internal energy $ϵ$ and of the component $v^y$ of the velocity at time $t\sim 1.1\mathrm{ms}$, i.e. soon after the shock formation. Note that only $1/10$ of the numerical grid is shown.

Figure 7

Dynamics of the collapse to a black hole with puncture gauge. The picture shows the evolution of the central rest-mass density (top panel) and of the central lapse (bottom panel). Both are normalized by the central value of the initial data.

Figure 8

Evolution of different mass quantities in the collapse test. The picture shows the irreducible mass (solid line) of the final black hole and the rest mass of the matter (dashed line) normalized to their initial values at levels 5 and 7.

Figure 9

Profile of rest-mass density of the evolved boosted model A0. The picture shows the rest-mass density profile normalized to the initial value at time $t=1.72\mathrm{ms}$ for evolutions corresponding to different values of the $\eta$ parameter in the Gamma-driver shift (dashed colored lines). The profiles are shifted back to the initial position of the star, the initial data is also plotted (black solid line).

Figure 10

Self-convergence test of the evolution of a boosted star. The pictures shows the differences between the profiles of the rest-mass density at $t=3.95\mathrm{ms}$ evolved with different resolutions. The labels of the solid lines “low,” “med,” and “high” in the legend correspond to resolutions $h_4=0.4$, 0.201, 0.278. The difference between medium and high resolution is scaled for second order (dashed line) and first order (dotted line). The gamma-driver condition employs $\eta =0$.

Figure 11

Dynamics of the G2P14 evolution. The picture shows contour plots in the $x-y$ plane of the rest-mass density $\rho$ and the velocity $v^i$ at different times. Data refer to run H3.

Figure 12

Orbital motion of G2P14. The figure shows the evolution of the coordinate (top left) and proper (bottom left) separation of the binary and the star track (right) of the two stars for different grid settings.

Figure 13

Evolution of the central rest-mass density in BNS simulations. Runs G2P14 are reported for the three highest resolutions while run G2P14hot for H1.

Figure 14

Hamiltonian violation in G2P14 evolution. The figure shows the evolution of the L2 norm of the Hamiltonian constraint for different resolutions.

Figure 15

Conservation of rest mass in BNS simulations. The figure shows the evolution of the rest mass normalized to the initial value for G2P14 and G2P14hot evolutions. Runs G2P14 are reported for the three highest resolutions, G2P14hot for resolution H1.

Figure 16

Shock formation in G2P14hot evolution. The figure shows the quantity $K=p/{\rho }^{\Gamma }$ normalized by its initial value on the orbital plane at time $t=1275$ (6.256 ms) in a ${log}_{10}$ scale. Data refer to run H1.

Figure 17

Dependence of the dynamics on the use of different reconstruction schemes. The figure shows the maximum of the rest-mass density obtained from G2P14 evolutions with different reconstruction methods. Reference vertical line corresponding to $\max \rho$ of run H3.

Figure 18

Effect on orbital dynamics of different choices of the parameter $\eta$ in the Gamma-driver shift condition. The top panel shows the star track of one star. The central panel shows the evolution of the coordinate separation while the bottom panel shows the evolution of the proper separation of the binary. Each line refers to a different G2P140 evolution with a different $\eta$. Data refer to H1 runs.

Figure 19

Effect on the final BH mass and coordinate radius of the parameter $\eta$ in the Gamma-driver shift condition. The top panel shows the irreducible mass of the final BH normalized by the ADM mass of the system as computed from the apparent horizon finder. The bottom panel shows the coordinate radius of the apparent horizon. Each line refers to a different G2P140 evolution with a different $\eta$. Data refer to H1 runs.

Figure 20

${\psi }_4$ waveforms from G2P14 evolution. The figure shows the real part (top panel) and the imaginary part (bottom panel) of the $r{\psi }_{22}^4$ waveform extracted at $r=200$ from the H3 run. The amplitude is also shown in both panels.

Figure 21

Metric waveforms from G2P14 evolution. The figure shows the real part (top panel) and the imaginary part (bottom panel) of the $r{h}_{22}$ waveform extracted at $r=200$ from H3 run. The amplitude is also shown in both panels. The waveforms are plotted versus the retarded time $t_{\mathrm{ret}}\equiv t-r_{*}$ where $r_{*}$ is the tortoise radius corresponding to $r$. The metric waveform is computed with the FFI and cutoff frequency ${\nu }_0=0.002$.

Figure 22

Gravitational wave amplitude $|r{h}_{22}|$ from G2P14 evolution. The figure shows the amplitude computed with the FFI ${\nu }_0=0.002$ (red solid line) and ${\nu }_0=0.00035$ (green dashed-dotted line) and with the CTI where a cubic (blue dashed line) and linear polynomial (black dotted line) correction is used.

Figure 23

Comparison between waveforms from G2P14 and G2P14hot evolutions. The top panel shows the real part of $h_{22}$ extracted at $r=200$ for the two evolutions. The bottom panel shows the amplitudes. Vertical lines mark the time of the first contact between the stars and the merger time for G2P14 evolution. The waveforms are plotted against the retarded time $t_{\mathrm{ret}}\equiv t-r_{*}$. Data refer to run H1.

Figure 24

Gravitational wave frequency from G2P14 and G2P14hot evolutions. The figure shows $M{\omega }_{22}$ computed from $h_{22}$ waveforms. In the top panel the blue solid line refers to G2P14 and the red dashed line to G2P14hot. The vertical black solid lines mark, respectively, the merger time and the apparent horizon formation for G2P14. The vertical black dashed lines mark, respectively, the merger time and the apparent horizon formation for G2P14hot. The bottom panel gives a detail of the top panel showing the inspiral part. The horizontal green solid line marks the value of $M{\omega }_{22}$ at the merger for G2P14. Data refer to run H1.

Figure 25

${\psi }_4$ waveforms from G2P14 evolution at different resolutions. The figure shows the real part of the $r{\psi }_{22}^4$ waveform extracted at $r=200$ from runs H1-3.

Figure 26

Convergence of phase and amplitude in $r{\psi }_{22}^4$ waveform from G2P14 evolution. The figure shows the difference between runs H1 and H2 (blue solid line), between runs H2 and H3 (green solid-dotted line) as well as the difference between H2 and H3 (red dashed line) scaled for second-order convergence. Top panel displays the logarithm of amplitude differences, bottom panel displays the logarithm of phase differences. The vertical line indicates the merger.