Gravitational Radiation of Binaries Coalescence into Intermediate Mass Black Holes

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1 Commun Theor Phys 57 (22) 56 6 Vol 57 No January 5 22 Gravitational Radiation of Binaries Coalescence into Intermediate Mass Black Holes LI Jin (Ó) HONG Yuan-Hong ( ) 2 and PAN Yu ( ) 3 College of Physics Chongqing University Chongqing 433 China 2 College of Engineering and Communication Chongqing University Chongqing 43 China 3 College of Mathmatic and Physics Chongqing University of Posts and Telecommunications Chongqing 465 China (Received April 25 2; revised manuscript received July 2) Abstract This paper discusses the gravitation waveforms of binaries coalescence into intermediate mass black holes (about 3 times of the solar mass) We focus on the non-spinning intermediate mass black hole located less than Mpc from earth By comparing two simulation waveforms (effective one body numerical relativity waveform (EOBNR) phenomenological waveform) we discuss the relationship between the effective distance and frequency; and through analyzing large amounts of data in event we find that the phenomenological waveform is much smoother than EOBNR waveform and has higher accuracy at the same effective distance PACS numbers: 43Tv 48Nn 42Cv Key words: binary coalescence intermediate mass black hole effective one body numerical relativity waveform (EOBNR) phenomenological waveform Introduction A system composed of either two neutron stars two black holes or one of each bound together by gravity forms a binary system According to general relativity the objects will lose energy through the emission of gravitational radiation [] As a result their orbits shrink and the two stars spiral in towards one another eventually combining to form a single star most likely a black hole This process is called binary coalescence The coalescence can be divided into three phases according to how well we can model the waveform at different times The inspiral phase is defined as that time while the two stars are distinct objects orbiting around one another and the gravitational waveform emitted can be well approximated by the post-newtonian model (ie the velocities are low) The post-newtonian approximation breaks down as the stars begin their final few orbits and plunge in towards one another We refer to this as the merger phase Although numerical simulations are telling us more about the waveform produced at this stage it is still not represented by an analytic waveform We refer to this waveform as an unmodeled burst After the plunge the resulting star tries to return to a stable configuration by emitting gravitational waves in a series of quasi-normal modes These are also well modeled and this phase is known as the ringdown phase In this paper we discuss the gravitational waveforms in the whole process (ie inspiral-merger-ringdow (IMR)) [2 8] The waveform produced during the inspiral phase is colloquially known as a chirp waveform because the frequency and amplitude of the signal increases rapidly with time However the waveform produced in merger and ringdow phase can not be described as an analytic result Therefore the IMR waveform should be discussed by numerical methods In the paper we use EOBNR and phenomenological waveforms to represent IMR waveform and comparing them to each other through injection in LIGO data 2 EOBNR Waveform The model of effective one body describes the movement of a test particle in the two Schwarzschild spacetimes We use Hamilton equation to find the actual evolution of the binary system From the generalized Schwarzschild metric combined with the Hamiltonian function in effective one body model the coefficients in the post-newtonian metric can be approximately expressed as: [9] ds 2 = A(r)dt 2 + (D(t)/A(r))dr 2 + r 2 (dθ 2 + sin 2 θdϕ 2 ) () In the fourth-order post-newtonian approximation the coefficients A(r) and D(r) can be respectively expressed as: A(r) = 2GM/r + 2ηGM/r 3 + [(94/3 4π 2 /32)η z ](/r 4 ) + λη 2 /r 5 D(r) = 6η/r 2 + [7z + z 2 + 2η(3η 26)](/r 3 ) + d 4 η/r 4 (2) Supported by the Fundamental Research Funds for the Central Universities under Grant No CDJRC33 cqstarv@hotmailcom c 2 Chinese Physical Society and IOP Publishing Ltd

2 No Communications in Theoretical Physics 57 Derived from the metric the non-zero components of gravitational radiation intensity are: [] = 8 π δm 3 5 M η(mω)e iϕ π = 8 5 η(mω)2/3 e i2ϕ 2 = 2 π δm 3 7 M η(mω)e iϕ 2 = 8 π 3 7 η( 3η)(Mω)4/3 e i2ϕ 6π δm 22 = 3 7 M η(mω)e i3ϕ 3 = 63 8 πη( 3η)(Mω) 4/3 e i2ϕ 33 = 64 π 9 7 η( 3η)(Mω)4/3 e i4ϕ (3) Then it is important to match it to Ringdown phase The ringdown is the final phase of a binary black hole coalescence following the inspiral and merger A central result of general relativity is that gravitational waves are emitted from an accelerating mass It has been established using black hole perturbation theory that the waveform emitted by a perturbed black hole can be modeled as a superposition of quasi-normal modes with quasi referring to the fact that the oscillation is damped It is expected that at late times the oscillation will be dominated by a single mode Throughout this analysis we will refer to a gravitational wave emitted from a perturbed black hole as a ringdown waveform or just ringdown The ringdown waveform (far from the source) can be approximated by: h(t) = A d eff exp ( πft ) cos(2πft) (4) Q where A is the amplitude According to the stress-energy tensor equation [] this is usually expressed as: 5 ( GM ) A = 2 c 2 Q /2( + 7 ) /2 (5) 24Q 2 According to angular quantum number l and magnetic quantum number m the waveform in inspiral and merger phase can be divided to spherical harmonics functions Each one should match to the Quasinormal Mode with the same main quantum number n In the matching points the better waveform must be with greater n which contains more spherical harmonics Generally when n = N the waveform should be N order continuous differentiable at the matching points Figure shows the whole IMR waveform of h 22 (ie l = 2 m = 2 n = ) and the mass of black holes without spin are both 3M sun which are located about Mpc from us Fig (Color online) Inspiral Merge waveform (blue) is matched with Ringdown waveform (red) It can be seen that in inspiral-merger phase (blue curve) the gravitational wave frequency and amplitude increases When entering Ringdown phase (red curve) the frequency becomes stabilized and the amplitude decrease exponentially The speed of amplitude decrease depends on the mass and spin The signal process is matched filter the template parameters (effective distance frequency mass) are necessary for our research The parameters of injection signal can be recovered from coincidence test The deviation of effective distance is an important parameter for test our detection efficiency which means the efficiency of detection distance: δd eff d eff = 2[d eff(det) d eff (inj)] [d eff (det) + d eff (inj)] (6) where det means the recovered input signal corresponding to the template parameter inj represents ideal input signal Figure 2 shows the histogram of H H2 L (H: LIGO observatory with 4 km arm length in Hanford; H2:LIGO observatory with 2 km arm length in Hanford LIGO observatory with 4 km arm length in livingston) It is obvious to see a big deviation between the templates and the injections and the effective distance of the template is greater than the injections Figure 3 shows the deviation between frequency and effective distance That shows the condition of L and we can see there is a large deviation between low and high frequency It is also Singular around Hz to see the difference of effective distance is dispersed which indicates there is a big randomness between the effective distance of the template and the one to be detected

3 Communications in Theoretical Physics 58 Vol 57 3 Phenomenological Waveform The purpose of building phenomenological waveform is to use analysis method to binary construct the whole waveform in the three stages of binary coalescence[2] Despite the EOBNR waveform can theoretically give a whole wave in this process but its biggest flaw is that too much consumption of computer resources Thus this study uses the phenomenological waveform as a certain degree of approximation of the waveform In general the frequency domain components of a given frequency gravitational waves can be expressed as: h (f ) = A(f ) e iψ(f ) Fig 2 HI H2 L effective distance in EOBNR Accordingly in the process of the binary coalescence into black hole phenomenological waveform in the frequency domain is as follows: u(f ) = Aeff (α; f ) e iψeff (φ;f ) Fig 3 The relationship between frequency and effective distance difference After coincidence test for EOBNR injection the effective distance accuracy ε (ie number of found injection divided by total injection number) can be determined Figure 4 indicates: With the increase of effective distance the accuracy rate decreases gradually In the range of ( ) Mpc the accuracy can achieve ε = (7) (8) where Aeff (α; f ) is the amplitude of the GWs with such frequency The relative parameters are:[3] (f /fmerg ) 7/6 if f < fmerg Aeff (α; f ) = C (f /fmerg ) 2/3 if fmerg f < fring (9) ωl(f fring σ) if fring f < fcut fcut is the cutoff frequency of template; fmerg is the frequency at which the power-law changes from f 7/6 to f 2/3 [4] α = {fmerg fring σ fcut } C is a numerical constant dependent on the detector location; L(f fring σ) is a Lorentz function determined by center frequency fring and bandwidth σ: σ () L(f fring σ) = 2π (f fring )2 + σ 2 /4 The value of ω meets the requirement of Aeff (f ) being continuous on fring : πσ fring 2/3 () ω= 2 fmerg In addition the effective phase is Ψeff (ϕ; f ) = 2πf t + φ + 7 X ϕk f (k 5)/3 (2) k= Fig 4 Effeciency vs distance in EOBNR where t is the arrival time ϕ is initial phase φ = {φ φ2 φ3 φ4 φ5 φ6 φ7 } is the phase parameters of GWs that is the set of phenomenological parameters describing the phase of the waveform In the post-newtonian approximation the numerical constants are obtained through the Fourier transform for the gravitational waves on the direction with the highest sensitivity are: 7/6 M 5/6 fmerg 5η /2 C= (3) 24 dπ 2/3 where η is a parameter ranging from 6 to 25 In order to find the fitting factor of our phenomenological bank to a hybrid waveform as well as the best-matched

4 Communications in Theoretical Physics No parameters (αmax ϕmax ) we need to perform a maximization of the overlap M (α ϕ) which can be broken into a product of two terms M (α ϕ) = MA (α)mp (α ϕ) (4) with a MP (α ϕ) = b MA (α) = df (5) cos[ Ψ(f )] df (6) where Ψ(f ) = Ψ(f ) Ψeff (ϕ; f ) (7) In the above expressions the normalization constants a and b are defined by 2 2 A (f ) Aeff (α; f ) df df (8) a2 S (f ) h b df (9) The next step is set the maximum of αmax {fmerg fring σ fcut }max which are From Eq (2) we can see the phase is a linear function in ϕ minimizing MP becomes a least-square fit with a weighting function (2) More specifically writing Ψeff (ϕ; f ) as the following: X Ψeff (ϕ; f ) = ϕj f (5 j)/3 (22) j Using Eq (2) (7) (22) we have Mp = [ϕaϕt 2BϕT + D] (23) 2 where we have defined a matrix A a vector B and a scalar constant D such that ( i j)/3 Aij = f µ(f )df b (5 j)/3 Bj = f Ψ(f )µ(f )df b 2 D= Ψ (f )µ(f )df (24) b The maximum value of Mp is equal to Mp = [D BA B] (25) 2 Therefore ϕmax = BA (26) = 3η 2 + 5η + 6η 2 + 9η + 2 fring = 5η 2 + 8η + 2 σ= 8η 2 + η + 3 fcut = (27) Now all the parameters of waveform are fixed We need use match filter to do data analysis The results are as the following Being similar to EOBNR Fig 5 shows a big deviation between the templates and the injections and the effective distance of the template is greater than the injections Figure 6 shows us that the ideal frequency of phenomenological waveform model is around 2 Hz fmerg = If the phase difference Ψ(f ) is small we can approximate cos Ψ Ψ2 /2 then Mp can be rewritten as Ψ(f ) 2 Mp df (2) 2b µ(f ) = 59 Fig 5 The histogram of effective distance for H H2 L in Phenomenological waveform Fig 6 The relationship between frequency and effective distance difference

5 6 Communications in Theoretical Physics Vol 57 Figure 7 gives the accuracy of phenomenological waveform Comparing witnr the curve of phenomenological waveform is smoother and there is higher accuracy at the same effective distance In other words for achieving the same accuracy phenomenological waveform will be able to observe further gravitational wave source Fig 7 The detection accuracy changes with effective distance 4 Conclusions As two representative waveforms the EOBNR and phenomenological waveform have much different parameters for recovered injections After considering the distribution of noise in such frequency band we can determine the possible location of the binares The comparison between the time frequency effective distance shows that most of the points are distributed around and extend to the local limited range According to phenomenological and EOBNR waveform comparison the found injections of EOBNR are always in further area than phenomenological ones because when the EOBNR waveform is made its amplitude is lower than the phenomenological at the same injection time Therefore for the same matched template EOBNR is the signal from further source How to make EOBNR and phenomenological waveform simulation for the relic gravity waves produced during the transition from a radiation-dominated inflationary phase to a dust-dominated Friedman Robertson Walkertype expansion may be a much more challenge work in future [5] References [] M Mao Gravitational Radiation from Encounters with Compact Binaries in Globular Clusters Doctor Thesis Massachusetts Institute of Technology USA (28) 2 [2] J Abadie et al Phys Rev D 82 (2) 2 [3] B Abbott et al Phys Rev D 69 (24) 22 [4] B Abbott et al Phys Rev D 72 (25) 82 [5] B Abbott et al Phys Rev D 72 (25) 822 [6] B Abbott et al Phys Rev D 73 (25) 62 [7] B Abbott et al arxiv:7225 [8] B Abbott et al arxiv: [9] Y Pan et al Phys Rev D 8 (2) 844 [] T Damour Phys Rev D 64 (2) 243 [] JB Hartle Gravity: an Introduction to Einstein s General Relativity Addison Welsey (23) [2] R Sturani et al J Phys Conf Ser 243 (2) 27 [3] arxiv:743764; arxiv:72335 [4] A Buonanno GB Cook and F Pretorius Phys Rev D 75 (27) 248 arxiv:gr-qc/622 [5] J Li et al Commun Theor Phys 53 (2) 496