个人简介
曹周键，应用数学所副研究员。
2006年博士毕业于北京师范大学物理系，师从胡岗教授和梁灿彬教授。
2006年6月到中国科学院数学与系统科学研究院工作至今。
2012年被评为中国科学院数学与系统科学研究院“陈景润未来之星”。
2016年获国家自然科学基金委优秀青年基金资助。
研究方向
学术论文

Waveform model for an eccentric binary black hole based on the effectiveonebodynumerical relativity formalism, Physical Review D, 96, 044028 (2017).
Binary black hole systems are among the most important sources for gravitational wave detection. They are also good objects for theoretical research for general relativity. A gravitational waveform template is important to data analysis. An effectiveonebodynumericalrelativity (EOBNR) model has played an essential role in the LIGO data analysis. For future spacebased gravitational wave detection, many binary systems will admit a somewhat orbit eccentricity. At the same time, the eccentric binary is also an interesting topic for theoretical study in general relativity. In this paper, we construct the first eccentric binary waveform model based on an effectiveonebodynumericalrelativity framework. Our basic assumption in the model construction is that the involved eccentricity is small. We have compared our eccentric EOBNR model to the circular one used in the LIGO data analysis. We have also tested our eccentric EOBNR model against another recently proposed eccentric binary waveform model; against numerical relativity simulation results; and against perturbation approximation results for extreme mass ratio binary systems. Compared to numerical relativity simulations with an eccentricity as large as about 0.2, the overlap factor for our eccentric EOBNR model is better than 0.98 for all tested cases, including spinless binary and spinning binary, equal mass binary, and unequal mass binary. Hopefully, our eccentric model can be the starting point to develop a faithful template for future spacebased gravitational wave detectors.

Inspiralmergerringdown (2, 0) mode waveforms for alignedspin blackhole binaries, Classical and Quantum Gravity, 33, 155011 (2016).
Based on spin weighted spherical harmonic decomposition, the (2,+/ 2) modes dominate the gravitational waveforms generated by binary black holes. Several recent works found that other modes including (l,0) ones are also important to gravitational wave data analysis. For alignedspin binaries, these (l,0) modes are related to the memory effect of gravitational wave. Based on the postNewtonian analysis, quasinormal modes analysis and the results of numerical relativity simulations, we present a full inspiralmergerringdown gravitational waveform model for the (2,0) mode generated by binary black holes. Our model includes the quasinormal ringing part and includes the effect of a black hole’s spin. It is complementary to the previous results.

Parameter estimation of eccentric inspiraling compact binaries using enhanced postcircular model for groundbased detectors, Physical Review D, 92, 044034 (2015).
Inspiraling compact binaries have been identified as one of the most promising sources for gravitationalwave detection. These binaries are always expected to have been circularized by the gravitational radiation when they enter the detector's frequency band. However, recent studies indicate that some binaries may still possess a significant eccentricity. In light of the enhanced postcircular waveform model for eccentric binaries in the frequency domain, we do a systematic study of the possible signaltonoise ratio loss if one uses quasicircular waveform templates to analyze the eccentric signal, and revisit the problem of parameter estimation of gravitationalwave chirp signals from eccentric compact binaries. We confirm previous results from other researchers that the resulting signaltonoise ratio loss becomes larger than 5% for eccentricity bigger than 0.1 and the resulting parameter estimation bias is more than 0.1%. We study the parameter estimation accuracy for such a waveform with different initial eccentricities from 0.1 to 0.4 by using the Fisher matrix method. As expected, the eccentricity improves the parameter estimation accuracy significantly by breaking degeneracies between different parameters. Particularly, we find that the eccentricity errors improve by 2 orders of magnitude from 102 to 104 when eccentricity grows from 0.1 to 0.4, and the estimated errors of the chirp mass are about 103 for a binary black hole using the Advanced LIGO detector. For the Einstein Telescope detector, the estimated accuracy of parameters will be 2 orders of magnitude higher.

Improved fastrotating black hole evolution simulations with modified BaumgarteShapiroShibataNakamura formulation, Physical Review D, 92, 024034 (2015).
Different formulations of Einstein's equations used in numerical relativity can affect not only the stability but also the accuracy of numerical simulations. In the original BaumgarteShapiroShibataNakamura (BSSN) formulation, the loss of the angular momentum, J , is nonnegligible in highly spinning single black hole evolutions. This loss also appears, usually right after the merger, in highly spinning binary black hole simulations, The loss of J may be attributed to some unclear numerical dissipation. Reducing unphysical dissipation is expected to result in more stable and accurate evolutions. In the previous work [H.J. Yo et al., Phys. Rev. D 86, 064027 (2012).] we proposed several modifications which are able to prevent black hole evolutions from the unphysical dissipation, and the resulting simulations are more stable than in the traditional BSSN formulation. Specifically, these three modifications (M1, M2, and M3) enhance the effects of stability, hyperbolicity, and dissipation of the formulation. We experiment further in this work with these modifications, and demonstrate that these modifications improve the accuracy and also effectively suppress the loss of J , particularly in the black hole simulations with an initially large ratio of J and a square of the ADM mass.

Binary black hole simulation with an adaptive finite element method: Solving the Einstein constraint equations, Physical Review D, 91, 044033 (2015).
http://xueshu.baidu.com/s?wd=paperuri%3A%281776bdd2352bcb3d1e488f007a9c4e62%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2015PhRvD..91d4033C&ie=utf8&sc_us=1991543834140552797

Gravitational lensing effects on parameter estimation in gravitational wave detection with advanced detectors, Physical Review D, 90, 062003 (2014).
The gravitational lensing effect is important to the detection of electromagnetic signals in astrophysics. The gravitational wave lensing effect has also been found significant to gravitational wave detection in the past decade. Recent analysis shows that the lensing events for advanced detectors could be quite plausible. The black holes in our Milky Way Galaxy may play the role of lens objects. These facts motivate us to study the lensing effects on gravitational wave signals for advanced detectors. Taking advanced LIGO and Einstein Telescope for examples, we investigate the lensing effects on the parameter extraction of gravitational wave signals. Using the Markov chain Monte Carlo simulation together with matched filtering methods, we find that the lensing effect for a lens object with small mass is negligible. But when the mass of the lens object increases to larger than 1000M⊙ the lensing effect becomes important. Using the template without lensing corrections would result in loss of signal detections. In contrast if we consider templates with lensing effects, the lensed signal may provide much information about the lens black hole. These facts may give us a new way to determine the parameters of the lensing object. For example, this kind of signal may also help us estimate the mass and the distance of the supermassive black hole hosted at the center of our Galaxy.

Generalized BondiSachs equations for characteristic formalism of numerical relativity, Physical Review D, 88, 104002 (2013).
The Cauchy formalism of numerical relativity has been successfully applied to simulate various dynamical spacetimes without any symmetry assumption. But discovering how to set a mathematically consistent and physically realistic boundary condition is still an open problem for Cauchy formalism. In addition, the numerical truncation error and finite region ambiguity affect the accuracy of gravitational wave form calculation. As to the finite region ambiguity issue, the characteristic extraction method helps much. But it does not solve all of the above issues. Besides the above problems for Cauchy formalism, the computational efficiency is another problem. Although characteristic formalism of numerical relativity suffers the difficulty from caustics in the inner near zone, it has advantages in relation to all of the issues listed above. Cauchycharacteristic matching (CCM) is a possible way to take advantage of characteristic formalism regarding these issues and treat the inner caustics at the same time. CCM has difficulty treating the gauge difference between the Cauchy part and the characteristic part. We propose generalized BondiSachs equations for characteristic formalism for the Cauchycharacteristic matching end. Our proposal gives out a possible same numerical evolution scheme for both the Cauchy part and the characteristic part. And our generalized BondiSachs equations have one adjustable gauge freedom which can be used to relate the gauge used in the Cauchy part. Then these equations can make the Cauchy part and the characteristic part share a consistent gauge condition. So our proposal gives a possible new starting point for Cauchycharacteristic matching.

梁灿彬,曹周键,从零学相对论,高等教育出版社,北京(2013).
本书用浅显而直观的几何语言，较少的数学公式，对狭义相对论与广义相对论的物理基础及前沿领域作了通俗、清晰而富有趣味的介绍，对广大理工科学生和教师理解相对论大有帮助。除了相对论基础知识，本书还详细讲述了光速为什么是极限速率、高速物体的视觉形象、相对论在全球定位系统（GPS）的重要作用、虫洞和时间机器（以及弑母悖论）等饶有兴味的专题。

Binary black hole mergers in f(R) theory, Physical Review D, 87, 104029 (2013). 10.HweiJang Yo, ChunYu Lin and Zhoujian Cao, Modifications for numerical stability of black hole evolution, Physical Review D 86, 064027 (2012).
In the near future, gravitational wave detection is set to become an important observational tool for astrophysics. It will provide us with an excellent means to distinguish different gravitational theories. In the effective form, many gravitational theories can be cast into an f(R) theory. In this article, we study the dynamics and gravitational waveform of an equalmass binary black hole system in f(R) theory. We reduce the equations of motion in f(R) theory to the EinsteinKleinGordon coupled equations. In this form, it is straightforward to modify our existing numerical relativistic codes to simulate binary black hole mergers in f(R) theory. We consider a scalar field with the shape of a spherical shell containing binary black holes scalar field. We solve the initial data numerically using the Olliptic code. The evolution part is calculated using the extended AMSSNCKU code. Both codes were updated and tested to solve the problem of binary black holes in f(R) theory. Our results show that the binary black hole dynamics in f(R) theory is more complex than in general relativity. In particular, the trajectory and merger time are strongly affected. Via the gravitational wave, it is possible to constrain the quadratic part parameter of f(R) theory in the range a2<1011m2. In principle, a gravitational wave detector can distinguish between a merger of a binary black hole in f(R) theory and the respective merger in general relativity. Moreover, it is possible to use gravitational wave detection to distinguish between f(R) theory and a nonselfinteracting scalar field model in general relativity.

Modifications for numerical stability of black hole evolution, Physical Review D 86, 064027 (2012).
We experiment with several new modifications to the BaumgarteShapiroShibataNakamura formulation of Einstein’s field equation, and demonstrate how these modifications affect the stability of numerical black hole evolution. With these modifications, we obtain accurate and stable simulations of both single excised KerrSchild black holes and punctured binary black holes.

Numerical stability of the Z4c formulation of general relativity, Physical Review D 85, 124032 (2012).
We study numerical stability of different approaches to the discretization of a conformal decomposition of the Z4 formulation of general relativity. We demonstrate that in the linear, constant coefficient regime a novel discretization for tensors is formally numerically stable with a method of lines time integrator. We then perform a full set of “apples with apples” tests on the nonlinear system, and thus present numerical evidence that both the new and standard discretizations are, in some sense, numerically stable in the nonlinear regime. The results of the Z4c numerical tests are compared with those of BaumgarteShapiroShibataNakamuraOoharaKojima (BSSNOK) evolutions. We typically do not employ the Z4c constraint damping scheme and find that in the robust stability and gauge wave tests the Z4c evolutions result in lower constraint violation at the same resolution as the BSSNOK evolutions. In the gauge wave tests, we find that the Z4c evolutions maintain the desired convergence factor over many more lightcrossing times than the BSSNOK tests. The difference in the remaining tests is marginal.

Constructing effective onebody dynamics with numerical energy flux for intermediatemassratio inspirals, Physical Review D 84, 044014 (2011).
A new scheme for computing dynamical evolutions and gravitational radiations for intermediatemassratio inspirals (IMRIs) based on an effective onebody (EOB) dynamics plus Teukolsky perturbation theory is built in this paper. In the EOB framework, the dynamic essentially affects the resulted gravitational waveform for a binary compact star system. This dynamic includes two parts. One is the conservative part, which comes from effective onebody reduction. The other part is the gravitational backreaction, which contributes to the shrinking process of the inspiral of a binary compact star system. Previous works used an analytical waveform to construct this backreaction term. Since the analytical form is based on postNewtonian expansion, the consistency of this term is always checked by numerical energy flux. Here, we directly use numerical energy flux by solving the Teukolsky equation via the frequencydomain method to construct this backreaction term. The conservative correction to the leading order terms in massratio is included in the deformedKerr metric and the EOB Hamiltonian. We try to use this method to simulate not only quasicircular adiabatic inspiral, but also the nonadiabatic plunge phase. For several different spinning black holes, we demonstrate and compare the resulted dynamical evolutions and gravitational waveforms.

Numerical evolution of multiple black holes with accurate initial data, Physical Review D 82, 024005(2010).
We present numerical evolutions of three equalmass black holes using the moving puncture approach. We calculate puncture initial data for three black holes solving the constraint equations by means of a highorder multigrid elliptic solver. Using these initial data, we show the results for three black hole evolutions with sixthorder waveform convergence. We compare results obtained with the BAM and AMSSNCKU codes with previous results. The approximate analytic solution to the Hamiltonian constraint used in previous simulations of three black holes leads to different dynamics and waveforms. We present some numerical experiments showing the evolution of four black holes and the resulting gravitational waveform.

Reinvestigation of moving punctured black holes with a new code, Physical Review D 78, 124011 (2008).
We report on our code, in which the moving puncture method is applied and an adaptive/fixed mesh refinement is implemented, and on its preliminary performance on black hole simulations. Based on the BaumgarteSharpiroShibataNakamura (BSSN) formulation, uptodate gauge conditions and the modifications of the formulation are also implemented and tested. In this work, we present our primary results about the simulation of a single static black hole, of a moving single black hole, and of the headon collision of a binary black hole system. For the static punctured black hole simulations, different modifications of the BSSN formulation are applied. It is demonstrated that both the currently used sets of modifications lead to a stable evolution. For cases of a moving punctured black hole with or without spin, we search for viable gauge conditions and study the effect of spin on the black hole evolution. Our results confirm previous results obtained by other research groups. In addition, we find a new gauge condition, which has not yet been adopted by any other researchers, which can also give stable and accurate black hole evolution calculations. We examine the performance of the code for the headon collision of a binary black hole system, and the agreement of the gravitational waveform it produces with that obtained in other works. In order to understand qualitatively the influence of matter on the binary black hole collisions, we also investigate the same headon collision scenarios but perturbed by a scalar field. The numerical simulations performed with this code not only give stable and accurate results that are consistent with the works by other numerical relativity groups, but also lead to the discovery of a new viable gauge condition, as well as clarify some ambiguities in the modification of the BSSN formulation. These results demonstrate that this code is reliable and ready to be used in the study of more realistic astrophysical scenarios and of numerical relativity.

Light cone structure near null infinity of the Kerr metric, Physical Review D 75, 044003 (2007).
Motivated by our attempt to understand the question of angular momentum of a relativistic rotating source carried away by gravitational waves, in the asymptotic regime near future null infinity of the Kerr metric, a family of null hypersurfaces intersecting null infinity in shearfree (good) cuts are constructed by means of asymptotic expansion of the eikonal equation. The geometry of the null hypersurfaces as well as the asymptotic structure of the Kerr metric near null infinity are studied. To the lowest order in angular momentum, the BondiSachs form of the Kerr metric is worked out. The NewmanUnti formalism is then further developed, with which the NewmanPenrose constants of the Kerr metric are computed and shown to be zero. Possible physical implications of the vanishing of the NewmanPenrose constants of the Kerr metric are also briefly discussed.
联系方式
北京中关村东路55号数学院思源楼 612
zjcao@amt.ac.cn