High-accuracy comparison of numerical relativity simulations with post-Newtonian expansions

Abstract

Numerical simulations of 15 orbits of an equal-mass binary black hole system are presented. Gravitational waveforms from these simulations, covering more than 30 cycles and ending about 1.5 cycles before merger, are compared with those from quasi-circular zero-spin post-Newtonian (PN) formulae. The cumulative phase uncertainty of these comparisons is about 0.05 radians, dominated by effects arising from the small residual spins of the black holes and the small residual orbital eccentricity in the simulations. Matching numerical results to PN waveforms early in the run yields excellent agreement (within 0.05 radians) over the first $\sim 15$ cycles, thus validating the numerical simulation and establishing a regime where PN theory is accurate. In the last 15 cycles to merger, however, {\em generic} time-domain Taylor approximants build up phase differences of several radians. But, apparently by coincidence, one specific post-Newtonian approximant, TaylorT4 at 3.5PN order, agrees much better with the numerical simulations, with accumulated phase differences of less than 0.05 radians over the 30-cycle waveform. Gravitational-wave amplitude comparisons are also done between numerical simulations and post-Newtonian, and the agreement depends on the post-Newtonian order of the amplitude expansion: the amplitude difference is about 6--7% for zeroth order and becomes smaller for increasing order. A newly derived 3.0PN amplitude correction improves agreement significantly ($<1%$ amplitude difference throughout most of the run, increasing to 4% near merger) over the previously known 2.5PN amplitude terms.

Figures (24)

• Figure 1: Proper separation (top panel) and its time derivative (lower panel) versus time for short evolutions of the $d=30$ initial data sets 30a, 30b, and 30c (see Table \ref{['tab:ID']}). These three data sets represent zero through two iterations of our eccentricity-reduction procedure. The orbital eccentricity is reduced significantly by each iteration.
• Figure 2: Spacetime diagram showing the spacetime volume simulated by the numerical evolutions listed in Tab. \ref{['tab:Evolutions']}. The magnified view in the right panel shows how the gravitational waves are escorted to our extraction radii (see Sec. \ref{['sec:Escorting']}) after the simulation in the center has already crashed at $t\sim 3930m$, and after the estimated time of the black hole merger, which is indicated by the circle. The thin diagonal lines are lines of constant $t-r^*$; each corresponds to a retarded time at which the gravitational wave frequency $\omega$ at infinity assumes a particular value.
• Figure 3: Coordinate trajectories of the centers of the black holes. The small circles/ellipsoids show the apparent horizons at the initial time and at the time when the simulation ends and wave escorting begins. The inset shows an enlargement of the dashed box.
• Figure 4: Deviation of total irreducible mass $m(t)=2M_{\text{irr}}(t)$ from its value in the initial data. Plotted are the six different resolutions of run 30a-1.
• Figure 5: Gravitational waveform extracted at $r=240m$. From top panel to bottom: The real part of the $(2,2)$ component of $r\Psi_4$; the gravitational wave strain, obtained by two time integrals of $\mathrm{Re}(r\Psi_4)$; the frequency of the gravitational wave, Eq. (\ref{['eq:omega-definition']}); the gravitational wavelength, $\lambda=2\pi/\omega$. The vertical brown line at $t\approx 3930m$ indicates the time when "wave escorting" starts.