Inspiraling black-hole binary spacetimes: Challenges in transitioning from analytical to numerical techniques

We explore how a recently developed analytical black-hole binary spacetime can be extended using numerical simulations to go beyond the slow-inspiral phase. The analytic spacetime solves the Einstein field equations approximately, with the approximation error becoming progressively smaller the more separated the binary. To continue the spacetime beyond the slow-inspiral phase, we need to transition. Such a transition was previously explored at smaller separations. Here, we perform this transition at a separation of $D=20M$ (large enough that the analytical metric is expected to be accurate), and evolve for six orbits. We find that small constraint violations can have large dynamical effects, but these can be removed by using a constraint-damping system like the conformal covariant formulation of the Z4 system. We find agreement between the subsequent numerical spacetime and the predictions of post-Newtonian theory for the waveform and inspiral rate that is within the post-Newtonian truncation error.

Figure 1

Schematic diagram of the zones. BH1 and BH2 are denoted by solid black dots, where the orbital separation is $r_{12}$. The BZs are denoted with gray shells, the outer one representing the FZ/NZ BZ, and the two inner ones representing the NZ/IZ BZs (see also Table 1). The IZ, NZ and FZ are also shown in the figure.

Figure 2

The metric function $W$ at $t=0$ plotted versus $x$, showing the effects of stuffing the BH interior and the same function at $t=165M$ (but plotted versus $y$). The more regular shape of $W$ near the center of the BH at $t=165M$ is typical of moving puncture simulations (note that the puncture is offset from the $y$ axis by $0.01M$). For reference, the function $W$ for a Bowen-York puncture simulation (solid curve) when the puncture crosses the $x$ axis for the second time is included (the plot of the Bowen-York data has been shifted). Note how the stuffed $W$ appears to evolve to be very similar to the standard trumpet $W$ typical of puncture initial data.

Figure 3

A set of timelike geodesics initially equally spaced in $x$ and normal to the $t=0$ hypersurface (note that one of the BHs is centered at $x=10M$ at $t=0$). Here the proper time of each geodesic is plotted as a function of the geodesic’s spatial position. The solid (black) curves correspond to geodesics evolved on the numerical spacetime using BSSN, the dot-dashed (cyan) curves correspond to the geodesics evolved on the numerical spacetime using CCZ4, and the dotted (red) curves are for geodesics evolved with the second-order analytical metric. The plot to the right zooms in on a typical geodesic near the start of the simulation. Note how the numerical spacetime geodesics smoothly deviate from their analytical spacetime counterparts and there is no noticeable difference between the test particle trajectories for the BSSN and CCZ4 spacetimes at these early times. The rapid change in gauge near the start of the simulation is apparent in the smooth change in the geodesic seen at a proper time of about $\tau =3M$ in the inset.

Figure 4

The unphysical mass of the binary versus binary separation $D$ for the second-order analytical data (i.e., initial data) as measured using the integrals $m_{\text{unphys}}$ and $m_{\mathrm{abs}}$ The horizontal axis is in units of $D/M$, where $D$ is the separation of the binary. The larger mass is $m_{\mathrm{abs}}$ Note that while a significant amount of unphysical matter is present, it is spread out such that only a fraction of it is absorbed by the BHs (see Fig. 5). Note also that there is a near equal amount of positive and negative mass (which is why $m_{\text{unphys}}$ is about a factor of 20 smaller than $m_{\mathrm{abs}}$.). Rather than plotting $m_{\text{unphys}}$, we plot $-m_{\text{unphys}}$, since the net unphysical mass is actually negative.

Figure 5

The Hamiltonian constraint violation in the vicinity of the two BHs in the binary for binary separations of $D=20M$ (top) and $D=100M$ (bottom). First-order metrics are shown to the left, and second-order metrics to the right. The very small white circles at the centers of the BHs are the horizons.

Figure 6

The individual apparent horizon masses for a BHB with initial separation of 20M for both CCZ4 (black) and BSSN (red) evolutions of the second-order analytical data. The BSSN curves show very large oscillations, while the CCZ4 curves show a much smaller linear growth. The large mass oscillations in the BSSN run are due to absorption of constraint violations. Note how the effects changing the evolution system from BSSN to CCZ4 are much larger than truncation error effects.

Figure 7

The Hamiltonian constraint averaged over the horizon as a function of time. This linear log plot shows the absolute value of the constraint violation for the BSSN and CCZ4 evolutions of the second-order analytic metric, as well as the evolution of Bowen-York data (BY) using CCZ4. The BSSN, CCZ4, and BY data points are colored according to the sign of the constraint violation. The lower plot shows the early-time behavior.

Figure 8

The Hamiltonian constraint averaged over the horizon, the flux of the momentum constraint violations through the horizon, and the time derivative of the horizon mass for the BSSN simulation. The Hamiltonian and change of rate of the horizon mass have been multiplied by constant positive factors. Note the near-perfect correlation of horizon mass change and constraint violation on the horizon. In the figure, solid curves correspond to a resolution of $h_0/1.2$, while dotted curves correspond to a resolution of $h_0/{1.2}^2$.

Figure 9

The Hamiltonian constraint averaged over the horizon and the time derivative of the horizon mass for the CCZ4 simulation. Note how, unlike in Fig. 8, these two are not correlated. Here, both the constraint violation and $\dot{m}$ appear to be converging to a very small value but from opposite directions.

Figure 10

Time evolution of the $L^2$ norms of the constraint violations for BSSN and CCZ4 evolutions. Note how the BSSN Hamiltonian constraint remains relatively high while the CCZ4 constraints quickly fall to about ${10}^{-7}$. Also shown are the $L^2$ norms of the constraint violations for an evolution of Bowen-York data with CCZ4. The Bowen-York data leads to constraint violations that are on average a factor of 2 below constraint violations for the new data.

Figure 11

The $L^2$ norm of Hamiltonian constraint violations and the $L^2$ norm of the Euclidean magnitude of the momentum constraint violations. The momentum constraints have been rescaled by a factor of $(h_0/h{)}^4$, where $h$ is the base resolution of a particular run and $h_0$ is the base resolution of the coarsest simulation. Note that the Hamiltonian violations have not been rescaled. The spikes occurring roughly every 350M are due to the high-frequency initial gauge wave reflections off the mesh refinement boundaries. Here too, the $L^2$ norms have been restricted to the interior of a sphere of radius $30M$ about the origin and outside the two horizons.

Figure 12

The Hamiltonian constraint on the $xy$ plane near the two BHs at $t=200M$ for the BSSN (left) and CCZ4 (right) evolutions of the second-order analytical data starting at a separation of $D=20M$. The scale is logarithmic and goes from ${10}^{-7}$ to ${10}^{-3}$. The absolute value of the constraint violations for the BSSN simulation match closely the violations on the initial slice, while the CCZ4 violations are three orders of magnitude smaller. In each plot the interiors of the BHs have been masked out. The constraint violations at $t=0$ are given in the top right plot in Fig. 5.

Figure 13

The mass of the individual horizons versus time for the CCZ4 simulations using the standard fifth-order dissipation, seventh-order dissipation, and fifth-order dissipation of Bowen-York data. The linear trend in the mass, while converging to a small value, is substantially larger than that for Bowen-York.

Figure 14

The rate of horizon mass increase versus time. Here D5 indicates that fifth-order dissipation was used, while D7 indicates that seventh-order was used.

Figure 15

The SPD versus time as calculated using a CCZ4 evolution of the new data both before and after the eccentricity reduction procedure. For reference, a BSSN evolution of the low-eccentricity data is also given. Constraint violation drives the eccentricity for this latter case.

Figure 16

The ($\ell =2$, $m=2$) mode of the waveform ($r{\psi }_4$) extracted at $r=800M$ and $r=1600M$, a linear extrapolation of these to $r=\infty$, and an extrapolation to $r=\infty$ using perturbative techniques. Both the standard extrapolation in $r$ and the perturbative approach give very similar waveforms.

Figure 17

The ($\ell =2$, $m=2$) mode of the waveform ($r{\psi }_4$) extracted at $r=800M$ at two resolutions. Both the raw waveform and the extrapolation to infinity are shown. The dominant error is the extrapolation error, which manifests itself predominately as a phase error.

Figure 18

The SPD and PN separation versus time as calculated using a CCZ4 evolution of the new data and an older BSSN evolution of Bowen-York data. The SPD is always larger than the coordinate (and hence PN) separations. We shift the SPDs downward by $3.25M$ so that they agree with initial PN separation at $t=0$.

Figure 19

The frequency and magnitude of the ($\ell =2$, $m=2$) mode of $r{\psi }_4$ as measured at $r=1600M$ in the full numerical simulation and various PN predictions. Here we use either the 1.5 PN or 3.5 PN expressions for $\dot{r}$ and either the full 3.5 PN expression for $h_{22}$ (as a function of $r$ and $\omega$) of Faye et al. [62], or we truncate to 1 PN order. We use the 3 PN expression for $\omega$ in all cases.

Figure 20

The numerical waveform from the new data, a numerical waveform from equivalent Bowen-York data, and the PN waveform. Here the numerical waveforms are shifted by $r^{*}$ (the tortoise coordinate at the extraction radius), and the PN waveform is unshifted. The phase agreement is good after $t=0$ but breaks down prior to the initial data pulse (for the new data), despite the fact that the NR frequency is in close agreement with PN for the whole waveform. The jump in phase between the early and late part of the waveform is likely due to the use of the flat-space retarded time in constructing the early waveform.

Figure 21

A plot of the real part of the ($\ell =2$, $m=2$) mode of $r{\psi }_4$ (shifted in time by $-r$) for Bowen-York and the hybrid data here (denoted by PN). The waveforms extracted at $r=400M$ show a high-frequency pulse near $t=-100M$ due to an unresolved gauge wave. This high-frequency pulse is dissipated away and not visible in the waveform extracted at $r=1600M$.