# Motion deviation of test body induced by spin and cosmological constant in extreme mass ratio inspiral binary system

## Abstract

The future space-borne detectors will provide the possibility to detect gravitational waves emitted from extreme mass ratio inspirals of stellar-mass compact objects into supermassive black holes. It is natural to expect that the spin of the compact object and cosmological constant will affect the orbit of the inspiral process and hence lead to the considerable phase shift of the corresponding gravitational waves. In this paper, we investigate the motion of a spinning test particle in the spinning black hole background with a cosmological constant and give the order of motion deviation induced by the particle’s spin and the cosmological constant by considering the corresponding innermost stable circular orbit. By taking the neutron star or kerr black hole as the small body, the deviations of the innermost stable circular orbit parameters induced by the particle’s spin and cosmological constant are given. Our results show that the deviation induced by particle’s spin is much larger than that induced by cosmological constant when the test particle locates not very far away from the black hole, the accumulation of phase shift during the inspiral from the cosmological constant can be ignored when compared to the one induced by the particle’s spin. However when the test particle locates very far away from the black hole, the impact from the cosmological constant will increase dramatically. Therefore the accumulation of phase shift for the whole process of inspiral induced by the cosmological constant and the particle’s spin should be handled with caution.

## 1 Introduction

Gravitational waves have been directly detected by the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo from merging black holes and inspiraling neutron star binaries[1, 2, 3, 4, 5]. These systems have been detected with a mass ratio of the order of 1 (as predicted by [6], who also predicted that these binaries should predominately have low spin values and essentially be circular). Since intermediate-mass ratio inspirals are already detectable by ground-based detectors, see [7], we will have to wait for space-borne detectors such as the Laser Interferometer Space Antenna (LISA) [8, 9], DECIGO[10, 11], Taiji [12, 13], and Tianqin [14] to detect extreme-mass ratio inspirals, i.e. the progressive inspiral of a stellar-mass compact object on to a supermassive black hole, see [15].

For an extreme-mass ratio inspiral system (EMRI), a test particle with the motion along geodesic is the simplest description for the small body. While a small body always possesses spin angular momentum in such system, in order to describe the EMRI system more accurately, the motion of the small body can be envisaged as a spinning test body inspiraling into a supermassive black hole. A test particle without spin can be treated as a point-like particle and its motion in a curved spacetime is described by geodesics. However, when the reaction of the test particle is considered, the motion does not comply with a geodesic [16, 17, 18]. Likewise, the motion of a spinning test particle does not follow a geodesic because of the additional spin-curvature force [19, 20]. As the descriptions for the spinning test particle, the spin of the test particle indeed make contributions to its motion and the corresponding spin should be considered.

Like the spin of a test particle, a non-zero cosmological constant also affects the motion of the test particle. The observations [22, 23] have shown that the cosmological constant is positive and non-zero with a confidence of \(P(\varLambda >0)=99\%\). Therefore contributions to the motion of the particle in the black hole background from the non-zero cosmological constant should be considered. The cosmological constant was firstly proposed by Einstein in order to obtain a static universe. A black hole with a negative cosmological constant will have a thermodynamic behavior, and there exists a phase transition between the stable large black hole and the thermal gas phase [24]. The AdS/CFT correspondence and Holography has been addressed by the works of [25, 26], and the small-large black hole phase transition in the charged or rotating AdS black hole backgrounds was also investigated in Refs. [27, 28, 29].

We should note that the magnitudes of the particle’s spin and current observed cosmological constant are very tiny and the corresponding motion deviations caused by them can be ignored. That’s why most of the works about the EMRI are always described by a test particle with the motion along the geodesic trajectories in a black hole background. While the key thing should be noted here is that the deviation of the orbital angular frequency induced by the particle’s spin and cosmological constant will accumulate and lead to considerable phase shift during the long time inspiraling process of the small body in the EMRI system, this phenomena was also reported in Ref. [30]. Therefore, if we want to investigate the motion of a test particle more accurately in the EMRI system, the relationship between particle motion and both particle’s spin and cosmological constant should be clarified.

Circular orbits below the innermost stable circular orbit (ISCO) are unstable, and it is always regarded as the beginning of the merger of the binary. The motion behavior of particles in ISCO is closely related to the nature of black holes and we will use it to investigate the motion deviation induced by the spin of test particle and the non-zero cosmological constant. The ISCO of a spinning test particle in the Schwarzschild, Kerr, and Kerr–Newman (KN) black hole backgrounds have been investigated in Refs. [31, 32, 33]. Equatorial circular orbits and ISCOs in different black hole backgrounds have been investigated systematically in the related literature (see Refs. [16, 21, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64]). The motion of the spinning test particle in the Horndeski theory was also investigated in Ref. [65]. Some works have addressed the motion of a spinning test particle with non-zero cosmological constant in Schwarzschild and Kerr-de Sitter spacetimes [66, 67, 68, 69], for which the equilibrium conditions and nonequatorial circular orbits were investigated.

In this paper, we will investigate the motion deviation induced by the spin of a test particle and the non-zero cosmological constant by considering the motion of a spinning test particle in a rotating black hole with non-zero cosmological constant. Firstly, we review the equations of motion for a spinning test particle in curved spacetime and derive the corresponding four-momentum and tangent vector along the trajectory in a Kerr-dS/AdS black hole background in Sect. 2. By setting the particle’s spin and cosmological constant to zero, we can get the original no-deviation geodesic motion of the test particle in Kerr black hole background, the motion deviation can be obtained by comparing the results that for the original case and non-zero particle’s spin and cosmological constant case. In Sect. 3, we derive the ISCO for the spinning test particle in Kerr-dS/AdS black hole, with these results in hand, the motion of small body with non-zero spin and cosmological constant can be obtained. Thanks to the real magnitudes of the particle’s spin and cosmological constant are very tiny and we can simplify our results in the linear order approach of particle’s spin and cosmological constant. Then we have the analytic angular frequency with the linear order approach, which is used to naively estimate the corresponding magnitude of the phase shift induced by them. Finally, a brief summary and conclusion are given in Sect. 4.

## 2 Motion of a small body in black hole background

In this section, we solve the equations of motion of a spinning test particle in a Kerr-dS/AdS black hole background, where a unified treatment between the cases of geodesics in a Kerr-dS/AdS black hole and spinning test particle in a Kerr-dS/AdS black hole is presented. We will use the “pole-dipole” approximation to describe the motion of the spinning test particle. The corresponding equations of motion for the spinning test particle, also known as the Mathisson–Papapetrou–Dixon (MPD) equations, can be found in Refs. [70, 71, 72, 73, 74, 75, 76, 77, 78]. The four-velocity \(u^\mu \) and the four-momentum \(P^\mu \) in this scenario are not parallel [74, 79, 80]. The four-momentum keeps timelike along the trajectory and satisfies \(P^\mu P_\mu =-m^2\) with *m* the mass of the test particle, while the four-velocity might be superluminal [74, 79, 80] if the spin of the test particle is too large. When the multi-pole effects are considered, the superluminal problem can be avoided [81, 82, 83, 84, 85]. The gravitational radiation of a spinning test particle has been calculated by using the perturbation method, see Refs. [30, 57, 86, 87]. The collisional Penrose process with spinning particles were also investigated in Refs. [88, 89].

*M*, and \(a=\frac{J}{M}\) are the cosmological constant, mass, and spin of the black hole, respectively. We set the gravitational constant \(G=1\) and the speed of light \(c=1\).

*s*is the spin angular momentum of the test particle and the spin perpendicular to the equatorial plane. With the non-vanishing spin tensor, the non-zero spatial component of the spin angular momentum is [74]

*z*stands for the direction of the spin for the test particle.

*e*and

*j*are the energy and total angular momentum of the spinning test particle. One can verify the relations \(S^{\mu \nu }{\xi }_{\mu ;\nu }=S^{\mu \nu }\xi ^\beta \partial _\nu g_{\beta \mu }\) and \(S^{\mu \nu }{\eta }_{\mu ;\nu }=S^{\mu \nu }\eta ^\beta \partial _\nu g_{\beta \mu }\) for the two Killing vectors.

## 3 ISCO of a spinning particle in Kerr-dS/AdS black hole background and estimation of phase shift

We know that the motion of a test particle in a central field can be solved in terms of the radial coordinate in the Newtonian dynamics [34, 35]. And the motion of a test particle in the black hole background can also be solved by using the effective potential method in general relativity.

The ISCO parameters of the spinning test particle in the black hole background with \(a=0\) and \(\bar{\varLambda }=0\)

\(\bar{s}\) | \(\frac{r_{\text {ISCO}}}{M}\) | \(\bar{j}_{1\text {ISCO}}\) | \(\bar{e}_{\text {ISCO}}\) |
---|---|---|---|

\(1\times 10^{-9}\) | 6.0000000019 | 3.4641016146 | 0.9428090416 |

\(1\times 10^{-8}\) | 6.0000000168 | 3.4641016104 | 0.9428090417 |

\(1\times 10^{-7}\) | 6.0000001636 | 3.4641015679 | 0.9428090432 |

\(1\times 10^{-6}\) | 6.0000016334 | 3.4641011437 | 0.9428090576 |

\(1\times 10^{-5}\) | 6.0000163303 | 3.4640969010 | 0.9428092020 |

\(1\times 10^{-4}\) | 6.0001633001 | 3.4640544744 | 0.9428106453 |

The ISCO parameters of the spinning test particle in the black hole background with \(a=0\) and \(\bar{s}=0\)

\(\bar{\varLambda }\) | \(\frac{r_{\text {ISCO}}}{M}\) | \(\bar{j}_{1\text {ISCO}}\) | \(\bar{e}_{\text {ISCO}}\) |
---|---|---|---|

\(1\times 10^{-9}\) | 6.0000006484 | 3.4641014903 | 0.9428090246 |

\(1\times 10^{-8}\) | 6.0000064802 | 3.4641003679 | 0.9428088719 |

\(1\times 10^{-7}\) | 6.0000648048 | 3.4640891442 | 0.9428073445 |

\(1\times 10^{-6}\) | 6.0006484677 | 3.4639769053 | 0.9427920710 |

\(1\times 10^{-5}\) | 6.0065271010 | 3.4628543148 | 0.9426393359 |

\(1\times 10^{-4}\) | 6.0692859436 | 3.4514357844 | 0.9411003230 |

The ISCO (counter-rotating orbit) parameters of the spinning test particle in the black hole background with \(a=0.25\) and \(\bar{\varLambda }=0\)

\(\bar{s}\) | \(\frac{r_{\text {ISCO}}}{M}\) | \(\bar{j}_{1\text {ISCO}}\) | \(\bar{e}_{\text {ISCO}}\) |
---|---|---|---|

\(1\times 10^{-9}\) | 6.7948537709 | \(-\) 3.6855902359 | 0.9496770967 |

\(1\times 10^{-8}\) | 6.7948537859 | \(-\) 3.6855902313 | 0.9496770968 |

\(1\times 10^{-7}\) | 6.7948539372 | \(-\) 3.6855901854 | 0.9496770980 |

\(1\times 10^{-6}\) | 6.7948554486 | \(-\) 3.6855897260 | 0.9496771092 |

\(1\times 10^{-5}\) | 6.7948705614 | \(-\) 3.6855851315 | 0.9496772214 |

\(1\times 10^{-4}\) | 6.7950216919 | \(-\) 3.6855391874 | 0.9496783434 |

The ISCO (counter-rotating orbit) parameters of the spinning test particle in the black hole background with \(a=0.25\) and \(\bar{s}=0\)

\(\bar{\varLambda }\) | \(\frac{r_{\text {ISCO}}}{M}\) | \(\bar{j}_{1\text {ISCO}}\) | \(\bar{e}_{\text {ISCO}}\) |
---|---|---|---|

\(1\times 10^{-9}\) | 6.7948554128 | \(-\) 3.6855900062 | 0.9496770728 |

\(1\times 10^{-8}\) | 6.7948651402 | \(-\) 3.6855882896 | 0.9496768779 |

\(1\times 10^{-7}\) | 6.7949626058 | \(-\) 3.6855713142 | 0.9496749402 |

\(1\times 10^{-6}\) | 6.7959416478 | \(-\) 3.6854025138 | 0.9496556180 |

\(1\times 10^{-5}\) | 6.8058299155 | \(-\) 3.6837127936 | 0.9494623240 |

\(1\times 10^{-4}\) | 6.9147050109 | \(-\) 3.6663300559 | 0.9475046091 |

The ISCO (co-rotating orbit) parameters of the spinning test particle in the black hole background with \(a=0.25\) and \(\bar{\varLambda }=0\)

\(\bar{s}\) | \(\frac{r_{\text {ISCO}}}{M}\) | \(\bar{j}_{1\text {ISCO}}\) | \(\bar{e}_{\text {ISCO}}\) |
---|---|---|---|

\(1\times 10^{-9}\) | 5.1559328132 | 3.2099521130 | 0.9331177083 |

\(1\times 10^{-8}\) | 5.1559328269 | 3.2099521091 | 0.9331177084 |

\(1\times 10^{-7}\) | 5.1559329658 | 3.2099520706 | 0.9331177103 |

\(1\times 10^{-6}\) | 5.1559343514 | 3.2099516854 | 0.9331177292 |

\(1\times 10^{-5}\) | 5.1559482101 | 3.2099478341 | 0.9331179181 |

\(1\times 10^{-4}\) | 5.1560867956 | 3.2099093195 | 0.9331198064 |

The ISCO (co-rotating orbit) parameters of the spinning test particle in the black hole background with \(a=0.25\) and \(\bar{s}=0\)

\(\bar{\varLambda }\) | \(\frac{r_{\text {ISCO}}}{M}\) | \(\bar{j}_{1\text {ISCO}}\) | \(\bar{e}_{\text {ISCO}}\) |
---|---|---|---|

\(1\times 10^{-9}\) | 5.1559331501 | 3.2099520350 | 0.9331176953 |

\(1\times 10^{-8}\) | 5.1559361936 | 3.2099513292 | 0.9331175786 |

\(1\times 10^{-7}\) | 5.1559666327 | 3.2099442709 | 0.9331164123 |

\(1\times 10^{-6}\) | 5.1562711583 | 3.2098736874 | 0.9331047488 |

\(1\times 10^{-5}\) | 5.1593301318 | 3.2091677670 | 0.9329881144 |

\(1\times 10^{-4}\) | 5.1913776247 | 3.2020998180 | 0.9318217663 |

*s*and

*m*are the angular momentum and mass of the compact object. The mass of the central supermassive black hole in this system is assumed to be \(10^6 M_\odot \), where the parameter \(M_\odot \) is the solar mass. The spin angular momentum of the test body satisfies \(s=bm^2\), and the dimensionless spin parameter reads

*c*, and \(\varOmega _\varLambda \) are the Hubble constant, speed of light, and dark energy density parameter today. The current observational values of these parameters from Planck collaboration are [101] (see Table A.1)

| \(\varOmega _0\) | \(\phi _\varLambda \) | \(\phi _s\) |
---|---|---|---|

− 0.2 | \(\frac{0.059144}{M}\) | \(-\frac{1.35394}{M}\) | \(-\frac{0.01480}{M}\) |

0 | \(\frac{1}{6\sqrt{6}M}\) | \(-\sqrt{\frac{3}{2}}\frac{1}{M}\) | \(-\frac{5}{288M}\) |

0.2 | \(\frac{0.079974}{M}\) | \(-\frac{1.08631}{M}\) | \(-\frac{0.02077}{M}\) |

*r*, while the impact that from the cosmological constant is opposite.

*M*is the mass of the supermassive black hole.

Although we have shown that the contribution induced by cosmological constant will increase with radius, however when the test particle locates at the range of Eq. (64), the contribution from the particle’s spin is still far greater than the one from the cosmological constant.

## 4 Summary and conclusion

In this paper, we investigated the motion deviation of the small body induced by the particle’s spin and the non-zero cosmological constant. To derive the corrections induced by the particle’s spin and the cosmological constants with a unified treatment, we solved the equation of motion for a spinning test particle by using the MPD equation with Tulczyjew spin-supplementary condition in the equatorial plane of the Kerr-dS/AdS black hole background, and gave the four-momentum and four-velocity of the spinning test particle. Since the radial component of the tangent vector \(\dot{r}\) and the radial component of the four-momentum \(P^r\) are parallel, we derived the effective potential by decomposing the radial component of the four-momentum and used it to obtain the ISCO of the spinning test particle.

By using the ISCO of the test particle, we estimated how they are changed by the non-vanishing particle’s spin and cosmological constant. We numerically investigated the corrections for the ISCO that from the particle’s spin and cosmological constant with small order values, we found that the same order particle’s spin and cosmological constant can make different order of contributions to the motion of the test particle. By considering a small Kerr black hole or a rotating neutron star as the small test spinning body and considering the current observations of the cosmological constant, we got two different order of magnitudes of parameters \(\bar{s}={s}/{(mM)}\) and \(\bar{\varLambda }=\varLambda ~M^2\). The linear order analytic corrections to the angular frequency induced by the particle’s spin and cosmological constant in the background of Kerr black hole with arbitrary black hole spin *a* were given, and it will work well due to the so tiny true values of the particle’s spin and cosmological constant. With our linear order correction (50), we estimated the magnitudes of phase shift induced by the particle’s spin and cosmological constant over a period at the ISCO. We also showed that ratio of the impacts that from the particle’s spin and cosmological constant does not hold everywhere and depends on the location of the orbit. The analytic ratio of the contribution induced by between the cosmological constant and the particle’s spin was still given. It will be useful for the more accurate phase shift of the whole inspiraling process for the dynamical EMRI system with gravitational emission.

## Notes

### Acknowledgements

We thank the anonymous referee’s useful suggestions for this paper, which played a great role to the improvement of this paper. This work was supported in part by the National Natural Science Foundation of China (Grants nos. 11875151, 11675064, and 11522541), the Strategic Priority Research Program on Space Science, the Chinese Academy of Sciences (Grant no. XDA15020701), and the Fundamental Research Funds for the Central Universities (Grant nos. lzujbky-2019-ct06, lzujbky-2018-k11, and lzujbky-2017-it68). Y.P. Zhang was supported by the scholarship granted by the Chinese Scholarship Council (CSC). PAS acknowledges support from the Ramóny Cajal Programme of the Ministry of Economy, Industry and Competitiveness of Spain, as well as the COST Action GWverse CA16104. This work was supported by the National Key R&D Program of China (2016YFA0400702) and the National Science Foundation of China (11721303).

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