Gravitational waveform model based on photon motion for spinning black holes

The waveforms from binary black hole mergers include inspiral, merger, and ringdown parts. Usually, the inspiral waveform can be obtained by calibrating from post-Newtonian approximation; The merger and ringdown ones can be gotten from the quasinormal modes with black hole perturbation theory. However, for more general black holes, the calculation of the quasinormal modes is not trivial. In this paper we use the photon sphere to get the quasinormal modes of spinning black holes. Then we connect the ringdown wave with the inspiral part to get full waveforms and compare with the ones from numerical relativity. We find that they match with each other very well. In principle this method can be extended to some general compact objects. As an example, the ringdown waveforms from parametrized axisymmetric black holes are obtained. We also use this method to get the ringdown signals of the accelerating final black hole; this is due to gravitational recoil during the merger. Even for the extreme cases, the acceleration due to the recoil cannot produce detectable effects.

Figure 1

Comparison of ${\omega }_R$ calculated from the photon sphere (PS) and black hole perturbation method.

Figure 2

Comparison of ${\omega }_I$ calculated from the PS and black hole perturbation method.

Figure 3

The minimum allowable value of ${\mathrm{\Omega }}_0$ in the BOB model with different spins under Kerr metric.

Figure 4

Comparison of PSI ($\mathrm{\Psi }$) and SEOBNRv4 waveforms with different spins. In the inspiral part, the two kind waveforms are totally overlapped due to the PSI model including the EOB inspiral waves.

Figure 5

Comparison of PSI waveforms with SEOBNRv4 and SXS for binary black holes systems with different spins. ${\chi }_1$ and ${\chi }_2$ represents the dimensionless spin of the first and second black hole, respectively.

Figure 6

The photon circular orbital radius and ${\omega }_R$ with different values of ${\delta }_1({\delta }_2)$ in the equatorial plane under the KRZ metric. The left figure shows the effect of ${\delta }_1$ or ${\delta }_2$ on the photon circular orbital radius with two different spin cases: $a=0.00$ and $a=0.50$. The right figure shows the effect of ${\delta }_1$ or ${\delta }_2$ on the ${\omega }_R$ under spin $a=0.50$. It should be noted that when we study the influence of one of the $\delta$, the other $\delta$ we take as 0.

Figure 7

The ringdown waveform of perturbed KRZ black holes with various values of ${\delta }_1$ and ${\delta }_2$. The top panels show the ringdown waveform with various ${\delta }_1$ (fixing ${\delta }_2=0.00$ and $a=0.50$); The bottom panels show the ringdown waves with various ${\delta }_2$ (fixing ${\delta }_1=0.00$ and $a=0.50$).

Figure 8

The mismatch (1-FF) of ringdown waves between the Kerr and KRZ black holes with varied parameters ${\delta }_1$ and ${\delta }_2$.

Figure 9

The radius of the photon circular orbits for varied acceleration parameter $A$.

Figure 10

The value of ${\omega }_R$ under accelerating black hole ($a=0.01$) versus acceleration $A$.

Figure 11

The value of ${\omega }_I$ under accelerating black hole ($a=0.01$) versus acceleration $A$.

Figure 12

The ringdown signals of accelerating and no-acceleration Kerr BHs ($a=0.01$). The left panel is the ringdown with $A=5\times {10}^{-4}$, and the right panel shows the results for $A=4\times {10}^{-3}$.