并合双星系统的引力波理论模型

The gravitational wave models for binary compact objects

引力波直接探测已经被LIGO成功实现.在GW150914的数据处理中引力波理论模型起到了关键作用.理论模型不仅从原始数据的噪声中挖掘出引力波信号,使得探测结果的置信度达到5.1σ,而且还证认出其波源是并合的双黑洞.除了LIGO这样的地面引力波探测器,还有脉冲星计时计划和空间引力波探测计划等.在所用的这些引力波探测计划中,理论模型都扮演着信号提取和波源反演的重要角色.随着引力波天文学的逐渐开展,理论模型研究的重要性和紧迫性将变得越来越明显.双星系统是上述所有引力波探测计划最为重要和最为现实的波源.本文对双星系统的引力波理论模型研究现状作一个概括性介绍,并指出存在的问题和以后可能的研究方向.

The gravitational wave detection GW150914 has been realized by LIGO. The theoretical model played an important role in the data analysis. The model not only extracts the signal enveloped in the detector noise, but also recognizes that the source is a binary-black-hole merger. Besides the ground-based detectors like LIGO, pulsar timing arrays and space-based detectors detect gravitational wave in different frequency band. Soon the FAST telescope will partly serve for pulsar timing arrays in the coming year. In the near future, SKA project will strongly enhance the gravitational wave detection with pulsar timing array. Regarding to the space-based detectors, e LISA project is going on in Europe. And the LISA pathfinder works quite well which implies that most instrument techniques for e LISA are ready. Besides e LISA, there are two more plans for space-based detectors in China including Taiji and Tianqin. Among all of these gravitational wave detection projects, theoretical models are very important for signal extraction and parameters inversion. Along the development of the gravitational wave astronomy, the theoretical model research becomes more and more important and urgent. Binary compact objects are among the most important and the most realistic gravitational wave sources for all the above mentioned gravitational wave detection projects. We briefly describe the research status of theoretical models for binary compact objects in this paper. Post Newtonian method, perturbation theory of black hole and numerical relativity are all introduced. Typically these three methods are applied to the inspiral, ringdown and merger stage respectively. And more we also introduced effective one body method which combines the results of all these three methods and constructs a full model for the whole inspiral-merger-ringdown（IMR） process. This model is called effective one body numerical relativity（EOBNR） model which is essential in the data analysis of GW150914. Currently, post Newtonian method has achieved 4 PN order for the conservative dynamics and 3.5 PN order for gravitational radiation part. Unfortunately the PN order is less for spinning black holes. With higher PN order, ones expect the description is more accurate, and it is valid for nearer separated binary compact objects. Perturbation theory for black holes includes two formalisms. One is metric perturbation which corresponds to the Regge-Wheeler-Zerilli equation. One is curvature perturbation which corresponds to the Teukolsky equation. For Schwarzschild black hole these two formalisms are equivalent. The analytical solution to the Teukolsky equation is not clear yet. In most applications, ones use numerical methods to treat the Teukolsky equation. After the merger of two black holes, the space-time can always be looked as a perturbation of a Kerr black hole. This stage corresponds to the ringdown. So the ringdown stage can always be treated by perturbation theory. In addition, if the masses of the two black hole are extremely different, the effect of the small black hole can be treated as a perturbation to the big black hole. Different to the ringdown description, ones have to care about the motion of the small black hole which is affected by the back reaction of the gravitational wave. In addition, ones need to consider the Teukolsky equation with source. Differently in ringdown case, ones need only care about the initial state of the Teukolsky equation which is determined by the binary merger. Different to both the PN method and the perturbation method, numerical relativity（NR） solves the full Einstein equation without any analytical approximation. In this sense, NR is a robust method to treat any gravitational wave sources even beyond binary compact objects. But ones have to care about two issues when using NR method. One is the massive computational cost. The another is the solution accuracy. Currently spectral method is more efficient and more accurate than finite difference method. But spectral method is less flexible to treat such as precession binary black hole, eccentric binary black hole, large mass ratio binary black hole and other cases. And more, spectral method meets essential difficulties when matter couples to the Einstein equations. So it is desirable to develop a new numerical method which can combine both the advantages of spectral method and finite difference method.