Merger estimates for rotating Kerr-MOG black holes in modified gravity
In this paper, we explore the signatures of non-rotating and rotating black hole mergers in the matter-free modified gravity. First, we solve the unstable circular null orbits and the innermost stable circular timelike orbits via the geodesic motion. The characteristic quantities of these orbits are systematically analyzed by varying the black hole spin and the scalar field parameter of the gravity. Then based on it, we study the ringdown modes from the light ring/quasinormal modes correspondence. The final spins of the merged black holes are also estimated with the Buonanno-Kidder-Lehner recipe. Several black hole merging cases are investigated in detail. All these results show that the black hole mergers are closely dependent of the scalar field parameter of the gravity.
pacs:04.70.Dy, 04.25.dg, 04.50.Kd
Recently, gravitational waves have been directly observed by LIGO and Virgo Abbott (), and have opened a new window to probe the fundamental questions of our Universe and gravitational theory. As expected, these gravitational waves were produced by merging binary black holes or neutron stars. Particularly, the exact determination of the masses, spins, and other parameters for the initial and final black holes will greatly improve our test near the strong gravitational regimes. Moreover, different modified gravities are also expected to be tested or restricted by using gravitational wave observations.
Although recent observations incline to support general relativity (GR) Abbott (); AbbottBP (); Yunes (), some subtle potential deviations may be witness in the coming years when more merging events are detected and higher improved signal-to-noise is achieved. In order to discriminate different relevant gravity theories by the observations, one needs first to examine their differences. Importantly, it is necessary to identify the key signatures of the waveforms of the black hole mergers for these theories, which is also very useful to exactly determine the dynamic properties and nature of these initial and final black holes.
On the other hand, there is an interesting modified gravity (MOG), the scalar-tensor-vector gravity theory proposed by Moffat Moffat0 (). Besides the metric tensor fields, extra massive vector field and scalar fields were introduced. The scalar fields strengthen the gravitational attraction, while the vector field produces an effective repulsive gravitational force. Such theory can be treated as an alternative to GR for dealing with the galaxy rotation curves and galaxy clusters without introducing the dark matter Rahvar (); MoffatR (); Toth ().
Particularly, many recent works were devoted to explored the novel properties of the non-rotating and rotating MOG black holes Moffat3 (). Black hole shadow was briefly examined in Ref. Moffat4 (). The thermodynamics of these black holes were studied in Ref. Mureika (). The geodesics and accretion disk were investigated in Refs. Lee (); Sharif (); Hussain (); Perez (); Sheoran (). Misaligned spin merging black holes were discussed in Ref. Moffat5 (). And the quasinormal modes (QNMs) were calculated in Ref. Manfredi (). All these results imply that there exists a significant difference between MOG and GR.
In the present work, we would like to consider the Kerr-MOG black hole merger, and two issues of the merger are studied. The first one is the gravitational wave emission patterns at the last stage, i.e., the ringdown stage, of the merger. In the eikonal limit, the QNMs are associated with the unstable circular null geodesics, the light ring, of the black hole in the asymptotically flat spacetime (for recent progress, see Refs. Cardoso (); Glampedakis (); Kokkotas ()). Although there exists a small deviation between the numerical data Khanna (); Konoplya (), it is still a good approximation. According to this light ring/QNM correspondence, the real part of the QNM is related to the angular velocity of the massless particle orbiting around the light ring. And its imaginary part is related to the Lyapunov exponent of the light ring. So through studying the light ring from the null geodesics, we can obtain the ringdown modes of the black hole merger. This has a theoretical guidance to distinguish this MOG from GR by probing the merger dynamics at the ringdown stage.
Another issue of the merger we will pursue is the estimate of the final black hole spin. For this section, we adopt the Buonanno-Kidder-Lehner (BKL) receipt Buonanno (), which is based on the angular momentum and mass conservations. And such receipt has gained a great success on estimating the final black hole spin. Moreover, this approach has two advantages. First, it considers the orbital angular momenta and both initial spins of the merged black holes. Second, it can be applied to general mass ratio mergers. Then through solving the innermost stable circular orbits (ISCO) from the timelike geodesics, one can obtain the final spin of the black hole following this BKL receipt. The study has been carried out for the Kerr black hole and Einstein-Maxwell-dilaton black hole mergers. Comparing with the numerical simulations, the result shows that the BKL formula is quite accurate Buonanno (); Chatrabhuti (). More importantly, such approach can also be easily generalized to other different merger configurations, and some interesting and novel dynamics might be explored immediately. For example, the “flip” phenomenon was discussed with such receipt. Moreover, it was found that it is impossible to spin-up a black hole to extremal values through merger scenarios Buonanno (). However, considering the energy loss, a near extreme spinning black hole can be produced by equal-mass maximally spinning aligned mergers Kesden ().
This paper is organized as follows. In Sec. II, we give a brief review of the black hole metric and its geodesics. Based on the geodesics, we calculate the radius and minimum impact parameter for the light rings. Further, we study the QNMs at the ringdown stage in Sec. III. In Sec. IV, we first numerically calculate the ISCO. Then based on the result, we follow the BKL receipt and estimate the final black hole spins for several merger configurations. Finally, the conclusions are presented in Sec. V.
Ii Kerr-MOG black hole and geodesics
The action of the scalar-tensor-vector gravity is formulated as Moffat0 ()
where is a Proca-type massive vector field with mass , and are two scalar fields, and their potentials are respectively and . denotes the matter action. The tensor field , and it satisfies the following equations
The energy-momentum tensor for the vector field is
Since the effect of the mass of the vector field displays at kiloparsec scales from the source, it can be neglected for a black hole solution. Moreover, we can also treat as a constant independent of the spacetime coordinates. With this hypothesis, the action will be simplified and easy to deal with,
The corresponding field equation reads
and is corresponded to the Newton’s gravitational constant as with being a dimensionless parameter. So can be used to measure the deviation of MOG from GR.
with the metric functions given by
For simplicity, we adopt . The parameter is related to the ADM mass of the black hole as Sheoran ()
Solving , we can obtain the radius of the black hole horizon.
It is clear that such black hole can possess two horizons for , one degenerate horizon for , and no horizon related to naked singularity for . So the black hole has a maximum spin
It is worth noting that this Kerr-MOG black hole will reduce to the Kerr black hole when . On the other hand, if we set , it will be a static black hole with two horizons . Further taking , the Schwarzschild black hole solution will be recovered.
In this rotating Kerr-MOG background, the geodesics of a test particle with unit mass is Moffat3 ()
The sign functions and are independent from each other. And is the carter constant. The values of are 1 and 0 for massive particle and photon, respectively. Here, we consider the motion limited in the equatorial plane. So we have and . Then we can reexpress Eq. (18) as the following form
where the effective potential is given by
Iii Ringdown modes
The final stage of a black hole merger is known as the ringdown stage. In this stage, the final black hole settles to a stationary one, and it is characterized by linearized vibrational modes, i.e., the QNMs. In the eikonal limit, such mode is associated with the spacetime’s light ring. And this light ring/QNM correspondence is effective for an asymptotically flat spacetime. The modes of are considered to be the most powerful emitters during the gravitational waves and thus are most easily detectable by observatories, so we mainly focus on them in this paper. Fortunately, these modes are also the easiest ones to fit the eikonal approximation. More interestingly, these modes with positive or negative are related with the equatorial prograde or retrograde orbital motion.
where , , , and are the quantum overtone number, angular momentum, angular velocity, and Lyapunov exponent of the light ring, respectively. Moreover and can be calculated with the geodesics
with being the radius of the light ring. Here the effective potential for the photon reads
The characteristic parameters of the light rings are determined by the conditions
Substituting the effective potential into these conditions, one will get
Solving them, we can obtain the radius of the light rings and the minimum impact parameter . For the non-rotating black hole , we have
When , the above result reduces to the Schwarzschild black hole case, i.e.,
On the other hand, for the rotating black hole with , due to the dragging effect, the light rings will be different for prograde and retrograde cases. Although it is impossible to obtain the analytical formula, we can numerically solve these conditions (27). And the results are presented in Figs. 1 and 2 for the prograde and retrograde orbits. For the prograde orbit, we find that both and decrease with the spin and the parameter . For the retrograde orbit, increases with the spin and decreases with , while decreases with and increases with .
For the non-rotating black hole, the angular velocity and Lyapunov exponent of the unstable null geodesics are analytically obtained
For the rotating black hole, the results are shown in Figs. 3 and 4. For the prograde orbit, with the increase of spin , the angular velocity increases, while the Lyapunov exponent decreases. Moreover, increases with . The Lyapunov exponent increases with for small , and decreases with for large . For the retrograde orbit, increases with spin and decreases with . On the other hand, the Lyapunov exponent increases with . And for small fixed , slightly decreases with the spin . However, for large fixed , first increases, and then decreases with .
Iv Final spin estimation
iv.1 BKL recipe
Here we would like to briefly review the BKL recipe, which mainly depends on the conservations and is very easily to be understood. First, it considers that the mass of this merged system is conserved. Therefore, the mass of the final black hole equals to the total mass of the initial black holes, i.e.,
To the first order, this is a good approximation. For example, the total radiated energy remains very small, about , in the gravitational wave observations Abbott ().
On the other hand, for a reasonable merger, the two black holes orbit around each other during the first stage, i.e., the inspiral stage. Due to the energy loss via a gravitational wave emission, the binary orbit contracts gradually and the system evolves quasi-adiabatically. During this stage, one can assume that the individual spins of the black holes will remain constant. Once the ISCO radius is reached, the orbit becomes unstable. This will lead to a merger, and then a final black hole is formed. At this stage, the radiation of the angular momentum with respect to that of the system is small. So the angular momentum of the system can be treated as a conserved quantity. Moreover, it is justified to estimate the contribution of the orbital angular momentum to the final black hole spin by adopting the orbital angular momentum of a test particle orbiting at the ISCO of the final black hole. Finally, the dimensionless final black hole spin can be expressed as Buonanno ()
where denotes the orbital angular momentum of a test particle with reduced mass orbiting at the ISCO of the final black hole with spin . Without loss of generality, we can assume , and then the final spin can also be reexpressed in the following convenient form Buonanno ()
where is the dimensionless spins of the initial black holes and is the angular momentum of a unit mass test particle. The mass parameter and has a value in the range of . By solving the timelike geodesics, we can obtain the radius and the orbital angular momentum of the ISCO. Then with given the initial mass parameter and spins and , the final black hole spin can be calculated with Eq. (38). Importantly, since the parameter deforms the gravitational constant charactered by the spacetime background, we fix before and after the black hole merger.
As mentioned above, in order to obtain the final spin of the black hole, we first need to solve the ISCO. So here we will consider such special orbit for a test particle. From the geodesics, the conditions to determine the ISCO are
Plunging the effective potential, the conditions reduce to
For the non-rotating black hole, we can express the angular momentum and energy in terms of the radius of the ISCO as
And the radius of the ISCO is given by
where . When , one will get
for the Schwarzschild black hole. On the other hand, when , the analytical result can be obtained for the Kerr black hole with , see Ref. Bardeen (). While when , no analytical result is available. Nevertheless, we can numerically solve Eq. (39). The results are listed in Figs. 5 and 6 for prograde and retrograde ISCOs, respectively. For the prograde ISCO, both the angular momentum and radius decreases with the black hole spin and the parameter . For the retrograde ISCO, we clearly see that the angular momentum decreases and the radius increases with the spin . However, increases and decreases with .
iv.3 Equal-spin merger
Based on the study of the ISCO for the black hole, we would like to estimate the final spin parameter of the merger. First, we consider a simple case that the two black holes have the equal spin, i.e., . Therefore, the final spin (38) will be of the form
Given and , we can solve the final spin from the above equation. Here we plot the final spin as a function of the mass parameter for =0, 0.2, 0.4, and 0.65, respectively, in Fig. 7. For the case of =0, it describes a merger of two non-rotating black holes. If these two black hole have the same mass parameter , then we will obtain for the Kerr black hole with =0. Interestingly, this value will decrease when is nonvanishing. For example, 0.6368, 0.6160, 0.5990, 0.5847, 0.5724 for =0.2, 0.4, 0.6, 0.8, 1, respectively. We can also find that the final spin increases with , while decreases with . If these black holes have initial spins, the result are the similar (see Figs. 7, 7, and 7).
On the other hand, we would like to examine the behavior of the final spin when the two merged black holes approach its extremal case. For this purpose, we list the final spin for in Fig. 8. From it, one can clearly see that the final spin decreases with the mass parameter , which is quite different from that of Fig. 7. This is a novel result. And it is also consistent with the result given in Refs. Buonanno (); Chatrabhuti ().
Next, we consider another simple case that the two initial black holes have the same mass, i.e., . Then we present the final spin for different initial values of the spin in Fig. 9. Positive or negative denotes that the initial spin is either aligned or anti-aligned with respect to the initial orbital angular momentum. From the figure, it is clear that the final spin increases with from negative to positive values. For fixed , the final spin decreases with . For example, varying =0, 0.2, 0.4, 0.6, 0.8, 1, 0.4819, 0.4599, 0.4430, 0.4296, 0.4186, 0.4094 and 0.8281, 0.7935, 0.7645, 0.7391, 0.7158, 0.6936. Moreover, we can also obtain the result that no matter the initial spin is alignment or anti-alignment, the final spin of the black hole is always aligned with the initial orbital angular momentum.
iv.4 Unequal-spin merger
In this subsection, we would like to consider the unequal spin case but with equal mass. So we set , , and . Adopting these values, the final spin will be in the following form
Taking one of the black hole spins as 0.5, and regarding that it is aligned or anti-aligned with respect to the initial orbital angular momentum, we illustrate the value of the final spin parameter for equal-mass black holes in Fig. 10. For varies from -1 to 1, we can see that the final spin increases or decreases with for aligned or anti-aligned case. It is also obvious that the final spin decreases with for both cases.
iv.5 Generic spin configuration merger
Now, we consider a generic spin configuration merger, which means that the orbit plane of the ISCO can be inclined with respect to the final total angular momentum. For this case, the calculation of the orbital contribution to the total angular momentum is required to perform a numerical integration of generic geodesics in the black hole background or use the radial potential for some certain quasi-adiabatic spherical orbits Chandrasekhar (). On the other hand, one can alternatively adopt the fit formula given in Ref. Hughes (). Here we choose the latter one. Then the orbital angular momentum of the inclined orbit reads Hughes (); Buonanno ()
where the inclination angle measures the angle between the final spin and the orbital angular momentum. Let us consider a simple case that these two merged black holes has the same masses and spins. Thus the final spin can be solved as
Here we show the final spin as a function of for different values of and in Fig. 11. From it, we can see that the angle has a significant influence on the final spin . And with the increase of , the final spin increases.
In this paper, we studied the main features of the Kerr-MOG black hole merger in a modified gravity. The results show a significant dependence on the dimensionless scalar field parameter .
At first, we explored the property and geodesics for the Kerr-MOG black hole. For the non-rotating case with , the black hole has two horizons, which is different from the Schwarzschild black hole case. On the other hand, compared to the rotating Kerr black hole, the maximum spin will be reduced by , i.e., . Then the geodesics was obtained, which also shows an -dependent property.
Based on the null geodesics, we investigated the gravitational waves at the ringdown stage, which can be effectively approximated by the light rings. The real and imaginary parts are, respectively, described by the angular velocity and the Lyapunov exponent . Our result implies that, for the prograde orbit, increases with , and increases with for small while decrease with for large . For the retrograde orbit, decreases with , while increases with it.
Next, using the result of the ISCO for the massive particles, we estimated the final black hole spin via the BKL recipe. Several especially interesting cases were explored.
For the case of the equal-spin merger, the final spin decreases with the dimensionless scalar field parameter . It was also found that the final spin increases with for low initial spin. While for the initial spin approaches its maximum value, will not increase but decrease with , see Fig. 8. So we can expect that there may exist a critical value of that the final spin is independent of the mass parameter . Certainly, such critical value will be reduced by . Further, if these two merged black hole have equal masses, the final spin will also reduced by . Nevertheless, no matter the initial spins is alignment or anti-alignment, the final spin of the black hole is always aligned with the initial orbital angular momentum.
For the case of the unequal-spin merger, we found that the final spin increases or decreases with for aligned or anti-aligned case. Moreover, for both the cases, the final spin presents a monotonically decreasing behavior with the increase of the dimensionless scalar field parameter .
We also considered the generic spin configuration merger, which allows the orbit plane has an inclination angle with respect to the final total angular momentum. Then we adopted the fit formula given in Ref. Hughes () to modify the orbital angular momentum. Considered the two black holes have equal mass and spin, we studied the final spin as the initial spin for different inclination angle and . The result implies that the final spin increases with the inclination angle .
In summary, our study reveals that for this modified gravity, the black hole merger closely depends on its characteristic parameter, i.e., the dimensionless scalar field parameter . We expected that such influence of can be determined by the gravitational wave detection in the near future.
This work was supported by the National Natural Science Foundation of China (Grants No. 11675064, No. 11522541, No. 11375075 and No. 11205074).
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