Lensing rates of gravitational wave signals displaying beat patterns detectable by DECIGO and B-DECIGO
Shaoqi Hou, Pengbo Li, Hai Yu, Marek Biesiada, Xi-Long Fan, Seiji Kawamura, Zong-Hong Zhu
LLensing rates of gravitational wave signals displaying beat patterns detectable by(B-)DECIGO
Shaoqi Hou
School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072, China
Hai Yu
Department of Astronomy, School of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai, 200240, China
Marek Biesiada
Department of Astronomy, Beijing Normal University, Beijing 100875, China andNational Centre for Nuclear Research, Pasteura 7, 02-093 Warsaw, Poland
Xi-Long Fan
School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072, China
Seiji Kawamura
Department of Physics, Nagoya University, Nagoya, Aichi 464-8602, Japan
Zong-Hong Zhu ∗ School of Physics and Technology, Wuhan University, Wuhan, Hubei 430072, China andDepartment of Astronomy, Beijing Normal University, Beijing 100875, China (Dated: September 18, 2020)The coherent nature of gravitational wave emanating from a compact binary system makes itpossible to detect some interference patterns in two (or more) signals registered simultaneously bythe detector. Gravitational lensing effect can be used to bend trajectories of gravitational waves,which might reach the detector at the same time. Once this happens, a beat pattern may form,and can be used to obtain the luminosity distance of the source, the lens mass, and cosmologicalparameters such as the Hubble constant. Crucial question is how many such kind of events couldbe detected. In this work, we study this issue for the future space-borne detectors: DECIGO andits downscale version, B-DECIGO. It is found out that there can be a few tens to a few hundredsof lensed gravitational wave events with the beat pattern observed by DECIGO and B-DECIGOper year , depending on the evolution scenario leading to the formation of double compact objects.In particular, black hole-black hole binaries are dominating population of lensed sources in whichbeat patterns may reveal. However, DECIGO could also register a considerable amount of lensedsignals from binary neutron stars, which might be accompanied by electromagnetic counterparts.Our results suggest that, in the future, lensed gravitational wave signal with the beat pattern couldplay an important role in cosmology and astrophysics.
I. INTRODUCTION
Einstein’s prediction of gravitational waves (GWs)[1] has been verified by the detection of GWs byLIGO/Virgo Collaborations [2]. These observationsmarked the new era of GW astronomy and multimessen-ger astrophysics [2, 3]. Together with its electromagneticcounterpart, the GW could shed new light on cosmology,since its source can be used as the standard siren to ac-curately measure the luminosity distance [4]. It is alsointeresting to know that it may not be necessary to relyon the electromagnetic counterpart to determine the red-shift of the source as discussed in Refs. [5–8]. The GWcan also serve as a probe into fundamental physics [9],such as the nature of gravity [10] and spacetime [11]. ∗ [email protected] The GW travels at the speed of light in vacuum c (i.e. along null geodesic) as predicted by general rel-ativity [12]. If there is a massive enough object nearits path, the trajectory of the GW is bent due to thecurvature of spacetime produced by this object. Thisis the gravitational lensing effect [13, 14]. Although thelight can also be gravitationally lensed, one should un-derstand that there are several differences between thelensing of the light and that of the GW. First, GWs usu-ally have much longer wavelengths than the light. Sec-ond, the GW produced by a compact binary system isnearly monochromatic, so it is coherent; in contrast, thelight emitted by a star is incoherent. Because of the longwavelengths of GWs, the lens should be very massive inorder for the geometric optics to be applicable. For exam-ple, the mass of the lens should be greater than 10 M (cid:12) for the ground-based detectors, which are operating inthe frequency range 1 ∼ Hz; for LISA (frequency a r X i v : . [ g r- q c ] S e p range 10 − ∼ . M (cid:12) , atleast [15, 16]. Although we will refer to the wave na-ture of GWs, we focus on the geometric optics regime.In this regime, the lensed GW has magnified amplitudeand its polarization plane gets rotated [17]. However,the deflection angle is very small, so the rotation of thepolarization plane can be ignored.As an effect of the gravitational lensing, there can bemultiple paths along which GWs reach the detector. TheGWs along different paths experience distinct gravtita-tional time dilation, and the lengths of the paths are notthe same, so there exist time delays ∆ t between them[13]. If it happens that, during some time window, theinterferometer simultaneously detects the GWs comingfrom the same source along different trajectories, inter-ference patterns may occur [18, 19]. During the inspiralphase, frequency of the GW evolves very slowly. Con-sequently, there might be a small frequency difference∆ f between the GWs coming from the lensed images ofthe source. Therefore, if ∆ f is small enough the inter-ference results in a beat pattern, which can be used toextract useful information (e.g., lensing time delay ∆ t and the magnifications) and further to measure the trueluminosity distance of the source, the lens mass, and cos-mological parameters as discussed in Ref. [19]. In typicalcases of galaxy lensing, time delay ∆ t might be of orderof a few days to a few months. Therefore, it is highlyimpossible for a ground-based detector operating at highfrequencies to simultaneously observe the lensed GWstraveling along different paths. This is because the GWfrom the final merger phase detectable in the frequencyrange of the ground-based detector lasts for a few hun-dred seconds at most. However, there is no difficulty forthe space-borne detector sensitive to low GW frequenciesto observe the beat pattern. Therefore, it is very inter-esting to study the prospects of future space-borne GWdetectors to register the beat patterns from lensed GWsignals.As a starting point, one should estimate how manylensed GW events exhibiting the beat pattern to be de-tected by the space-borne detector per year. If the detec-tion rate is large enough, it would definitely be importantto study such phenomenon further. In the past, Sereno et al. calculated the number of lensed GW events thatmight be observed by LISA [20]. They found out thatthere can be at most 4 lensed GW events in a 5-year mis-sion. Since they did not specialize the particular eventswith the beat pattern, one expects that those with thebeat pattern should be fewer than 4. Nevertheless, it isworth to note that even with such a low detection rate,some interesting constraints on cosmological parameterscan still be derived according to Ref. [21]. One expectsthat with a higher detection rate, the constraints can beimproved. The low detection rate is related to the factthat the number of the massive black hole binaries, themain targets of LISA, is only on the order of 10 [22, 23].On the contrary, there are many more less massive com-pact binaries, whose merger rate is larger by a few or- ders of magnitude [2, 24, 25]. These mergers might beeasier detected by a second space-borne interferometer,DECi-hertz Interferometer Gravitational wave Observa-tory (DECIGO) [5, 26, 27].DECIGO is a planed Japanese space-borne interferom-eter, which is sensitive to GWs at mHz to 100 Hz. Sinceits sensitive band is higher than that of LISA, DECIGOis capable of observing GWs from much less massive bi-naries. Moreover, some of these binaries would also betargets of ground-based detectors, such as LIGO/Virgoand Einstein Telescope (ET) [28]. Therefore, DECIGOand ground based detectors can observe some merging bi-nary systems jointly (but at different times, of course) tomake the multiband GW astronomy possible [29]. Likethe LISA Pathfinder, a downscale version of DECIGO, B-DECIGO, will also be operating in the similar frequencyband, but will be less sensitive [30, 31]. In this work,we discuss the detection rates of the lensed GW eventsexhibiting the beat pattern observable by (B-)DECIGO.The predictions of the GW lensing rate have been for-mulated by several authors for different interferometers.As mentioned above, Sereno et al. estimated how manylensed GW events can be detected by LISA [20]. Forground-based detectors, Ref. [32] predicted that therewould be only 1 lensed GW event per year for aLIGOat its design sensitivity, but ET can detect about 80events. Refs. [33–37] all discussed the lensing rate forET, and concluded that there are 50 ∼
100 events peryear. Recently, Ref. [38] predicted the lensing rates for(B-)DECIGO, but the possibility of the beat pattern wasnot considered. This work will fill in the gap.Gravitational lensing has many applications other thanthose discussed in Ref. [19]. For instance, one can detectdark matter [39–43], constrain the speed of light [44, 45],determine the cosmological constant [20, 21, 46], examinethe wave nature of GWs [47–49], and localize the hostgalaxies of strong lensed GWs [50, 51] using gravitationallensing. Although no gravitational lensing signals havebeen detected in the observed GW events, the advent ofmore sensitive GW detectors might make it possible soon[52].This work is organized in the following way. We willstart with a brief review of the formation of the beatpattern due to the lensing effect in Sec. II. Then, (B-)DECIGO will be introduced in Sec. III. Section IV dis-cusses the lens model and how to calculate the opticaldepth, and the lensing rates are computed in Sec. V. Inthe end, there is a short conclusion in Sec. VI. We choosea units such that c = 1. II. THE BEAT PATTERN
In this section, we shall briefly review the formation ofthe beat pattern due to the gravitational lensing effect ofGWs. For more detail, please refer to Ref. [19].We will assume that the lens is described by a sin-gular isothermal sphere (SIS), which models the earlytype galaxies, because the early type galaxies contributeto the strong lensing probability dominantly [53]. Theline-of-sight velocity dispersion σ v of stars in the galaxycharacterizes the lensing effect. As shown in Fig. 1, theGW produced by the source S can travel in two paths,labeled by 1 and 2, to arrive at the observer O due to thepresence of a lens L. β is the misalignment angle betweenthe optical axis OL and the would-be viewing directionOS if there were no lens. Deflected rays form two angles, θ ± , with OL at the observer, which are given by [13] θ ± = β ± θ E , (1)where θ E = 4 πσ v D LS /D S is the angular Einstein radius,and D LS and D S are the angular diameter distances indi-cated in the Fig. 1. In further considerations concerningmerger rates we will assume flat ΛCDM model, in whichthe angular diameter distance D A ( z ) between the Earthand a celestial object at the redshift z is [54] D A ( z ) = 1 H (1 + z ) (cid:90) z dz (cid:48) h ( z (cid:48) ) (2)where H is the Hubble constant, h ( z ) = [Ω m (1 + z ) +Ω Λ ] / is the dimensionless expansion rate. In orderto comply with population synthesis model used fur-ther in this paper, we assume Ω m = 0 . H =70 km s − Mpc − . Therefore, in our shorthand nota-tion: D L = D A ( z L ), and D S = D A ( z S ) with z L and z S the redshifts of the lens and the source, respectively. D LS = H (1+ z S ) (cid:82) z S z L dz (cid:48) h ( z (cid:48) ) is the angular diameter distancebetween the lens and the source. β SO L D L D LS D S θ + θ - FIG. 1. The geometry of a lens. Two GW rays, 1 and 2, orig-inating from the source S, travel along two trajectories andchange their directions near the lens L. Eventually, they arriveat the detector at O. Vertical lines represent the observer, lensand source planes, from the left to the right. Thick dashedline is the optical axis, and the thin dashed line would be theviewing direction if there were no lens. The angle betweenthe two dashed lines is called the misalignment angle β . TheGW rays form the angles θ + and θ − with the optical axisat the observer. The distances D S , D L , and D LS are angulardiameter distances. Lensed GW signals are magnified, and the magnifica- tion factors of the GW amplitudes are given by µ ± = (cid:115)(cid:12)(cid:12)(cid:12)(cid:12) θ ± /θ E | θ ± /θ E | − (cid:12)(cid:12)(cid:12)(cid:12) . (3)Finally, the GW rays travel along paths of differentlengths, and they experience different time dilation dueto the gravitational potential of the lens, so they arriveat the observer at different times. The time delay is∆ t = 32 π σ v (1 + z L ) D L D LS D S βθ E . (4)One can see that σ v , appearing in Eqs. (1), (3), and (4),indeed characterizes the lensing effect of a SIS model.The time delay ∆ t typically ranges from a few days toa few months. For example, one can assume that the GWsource is at z S = 2, and the lens at z L = 1. Such value of z S is suggested by the fact that the redshift distributionof detectable neutron star-neutron star mergers (NS-NS)is maximal near z = 2, the redshift of black hole-blackhole mergers (BH-BH) peaks around z = 4 [37] and thelensing probability is maximal roughly for a lens half-way between the source and observer. Taking σ v as acharacteristic velocity dispersion σ ∗ = 161 ± .
03 weeks < ∆ t < . <β < .
25 arcsecond. Note that β < θ E ≈ .
27 arcsecondin order that the interferometer can “hear” two GWs.One can reasonably expect that ∆ t is much longer thanthe duration of GW signal observed by ground-based in-terferometers. Of course, one may imagine a case where β is very small, of the order of 10 − − − arcsecond,such that ∆ t ∼ is of order of a few seconds, and the beatpattern forms, as displayed in Fig. 2 in Ref. [19]. But theprobability for such case is extremely low. Therefore, itis very unlikely to use ground-based interferometers toobserve the interference patterns in GW events lensed bya SIS. However, the signals observed by LISA and (B-)DECIGO usually last for several months or even years.So, there is no difficulty for them to simultaneously de-tect two lensed GW signals in some time window. Thesesignals would interfere with each other and form an in-terference pattern in the time domain. If the strains forthe lensed GWs are h ( t ) and h ( t ), the total strain issimply h ( t ) = h ( t ) + h ( t ) . (5)Suppose the frequencies of h and h are f and f , re-spectively. Without the loss of generality, let f > f ,i.e., we assume h arrives earlier than h . Their differ-ence ∆ f = f − f is much smaller than both f and f in the inspiral phase, due to slow evolution of the GWfrequency during the lensing time delay. Therefore, thebeat pattern could show up during the inspiral phase,as discussed in Ref. [19]. As the binary system evolves,the GW frequency increases, so the beat pattern has asmaller and smaller period. Eventually, the beat patterndisappears, and a generic interference pattern is left.Taking into account the orbital motion of the space-borne interferometer, the beat pattern would have morecomplicated behavior than that described above. So onemay want to consider the cases with small enough ∆ t such that the impact on the beat pattern due to thechanging orientation of the constellation plane is smallenough, and the analysis is easier. Of course, ∆ t shouldnot be too small; otherwise, the probability for suchevents would be negligible again. So in this work, wewould like to mainly consider the lensing events with∆ t = 1 month.The Fourier transformation of the strain is used tocalculate the signal-to-noise ratio (SNR). Let h ( t ) beFourier transformed to˜ h ( f ) = µ + (cid:90) ∞−∞ h u ( t ) e i πft d f = µ + ˜ h u ( f ) , (6)where h u ( t ) would be the strain if there were no lens,and ˜ h u ( f ) is its Fourier transform. Then the frequencydomain waveform ˜ h ( f ) for h ( t ) = µ − µ + h ( t − ∆ t ) is˜ h ( f ) = e i πf ∆ t µ − ˜ h u ( f ) . (7)So the amplitude of the total waveform is | ˜ h ( f ) | = (cid:113) µ + µ − + 2 µ + µ − cos(2 πf ∆ t ) | ˜ h u ( f ) | . (8)This suggests that in the frequency domain, the ampli-tude of the total waveform is also oscillating with a “pe-riod” 1 / ∆ t . III. DECIGO AND B-DECIGO
In this work, we estimate the lensing rate of lensed GWevents with beat patterns detectable by (B-)DECIGO, sothis section briefly reviews the detectors characteristics.DECIGO is supposed to have a configuration of four clus-ters of spacecrafts. Each cluster would consists of threedrag-free satellites, separated from each other by 1000km and forming an equilateral triangle. All four clus-ters would orbit around the Sun with a period of 1 yr.DECIGO was originally proposed in Ref. [5]. Over thefollowing years, it evolved somehow, and now, its currentobjectives and design can be found in Ref. [27, 31].According to Yagi & Seto [56], a triangular cluster isequivalent to two L-shaped interferometers rotated by45 ◦ with uncorrelated noise. The noise spectrum for asingle effective L-shape DECIGO is S h ( f ) =10 − × (cid:34) . (cid:18) f f p (cid:19) + 4 . × − × f − f /f p ) + 5 . × − f − (cid:35) Hz − , (9) where f p = 7 .
36 Hz.B-DECIGO is the “DECIGO Pathfinder”, and hasonly one cluster of spacecrafts. The distance betweenspacecrafts is also smaller, which is 100 km. Its sen-sitivity is of course lower and can be described by thefollowing effective noise spectrum [57], S h ( f ) =10 − × [4 .
040 + 6 . × − f − + 6 . × − f ] Hz − . (10)Fig. 2 shows the characteristic strains √ f S h for these twodetectors. For comparison, the signals of GW150914 andGW170817 are also plotted, using PyCBC [58]. Although -3 -2 -1 -24 -23 -22 -21 -20 -19 -18 DECIGOB-DECIGOGW150914GW170817
FIG. 2. Characteristic strains of for DECIGO (solid bluecurve) and B-DECIGO (dashed red curve). The dot-dashed and the dotted curves are signals of GW150914 andGW170817, respectively. not shown in this figure, both of the two signals will laterend with a merger well beyond the sensitivity bands of(B-)DECIGO, but accessible to LIGO/Virgo and defi-nitely also to next generation of ground-based detectors.As one can see, both GW150914 and GW170817 wouldbe detected by (B-)DECIGO in their inspiral phase. So ifGW150914, for instance, were gravitationally lensed witha one-month time delay, then one may observe the beatpattern shown in Fig. 3. This figure displays the beatpattern (the black curve) formed due to the interferencebetween two GW rays (the blue and the red curves) trav-eling in different paths because of a suitable SIS lens.Here, for the purpose of demonstration, we only con-sider the quadruple contribution to the waveform, andassume the GWs incident the detector nearly perpendic-ularly and the inclination angle is zero.Once one knows the Fourier transform ˜ h ( f ) of a signal h ( t ), one can calculate the SNR ρ of it [59], ρ = 4 (cid:90) ∞ | ˜ h ( f ) | S h ( f ) d f. (11)If ρ > ρ th , a threshold SNR, one may claim a detectionof the GW. This condition may not necessarily mean one
200 400 600 800 1000 1200 1400 1600-1.5-1-0.500.51 10 -23 h h h FIG. 3. A schematic diagram showing the time domain wave-form of the beat pattern. The binary system is assumed to beGW150914, and the GW is gravitationally lensed by a suit-able SIS lens. The time delay is assumed to be 1 month. Theblue and the red curves are for the strains of the GW rayspropagating in two different trajectories, and the black curveis for the interfered wave. can easily extract some useful information from the beatpattern. For that purpose, one expects that the higherthe SNR is, the easier the extraction can be done. How-ever, we will not determine the least SNR for an accurateextraction in this work, in spite of its importance.
IV. THE LENS MODEL AND THE OPTICALDEPTH
The lens model is chosen to be the SIS. The elementarycross section for lensing is [21] s cr = 16 π σ (cid:18) D L D LS D S (cid:19) ( y − y ) . (12)Here, y = β/θ E , and y max is its maximal value, deter-mined by requirement that lensed GW signals could bedetected as displaying the beat pattern. Concerning theminimal value, arising when the geometric optics approx-imation breaks down, we assume y min = 0 as suggestedby Ref. [21]. To determine y max , one first notes that y max ≤ ρ should be greater than a threshold usually assumedas ρ th = 8. This might be too low to extract useful infor-mation from the beat pattern. However, for our purposeit would be sufficient to assume this standard threshold.It can be easily adjusted in the following calculations.Third, the time delay ∆ t = y ∆ t z , with [20]∆ t z ≡ π σ D L D LS D S (1 + z L ) . (13) should be small enough, say less than ∆ t m = 1 month.Now, according to Eqs. (8) and (11), one knows that ρ = 4 (cid:90) ∞ µ + µ − + 2 µ + µ − cos(2 πf ∆ t ) S h ( f ) | ˜ h u ( f ) | d f. (14)The cosine term in the integrand is highly oscillatingcompared to | ˜ h u ( f ) | in the frequency domain, so one mayignore it for the purpose of estimating the lensing rate,and the total GW SNR is ρ ≈ (cid:113) µ + µ − ρ int = (cid:112) /yρ int with ρ int the intrinsic SNR, ρ int = 4 (cid:90) ∞ | ˜ h u ( f ) | S h ( f ) d f. (15)Therefore, one has y max = min (cid:26) y , ∆ t m ∆ t z (cid:27) , (16)with y = min { , ρ /ρ } .Since (B-)DECIGO will operate for a limited amountof time, the actual cross section used to calculate theoptical depth is [21] s ∗ cr = 16 π (cid:16) σc (cid:17) (cid:18) D L D LS D S (cid:19) (cid:18) y − t z T s y (cid:19) , (17)where T s is the survey time, and set to 4 years [38]. Thedifferential optical depth is given by [20] ∂ τ∂z L ∂σ = d n d σ s ∗ cr d t d z L , (18)where τ is the optical depth, z L is the redshift of the lens, n is the lens number density, t is the cosmological timeand d n/ d σ is modeled as a modified Schechter function[55] d n d σ = n ∗ σ ∗ β Γ( α/β ) (cid:18) σσ ∗ (cid:19) α − exp (cid:34) − (cid:18) σσ ∗ (cid:19) β (cid:35) (19)where Γ( x ) is the gamma function, and n ∗ = 8 . × − h Mpc − , σ ∗ = 161 ± , (20) α = 2 . ± . , β = 2 . ± . . (21)Further, by the definition of the redshift [54],1 + z = a a , (22)one can determine the final factor in Eq. (18), which isd t d z L = − z L ) H L , H L = H ( z L ) . (23)So now, one can calculate the differential optical depthd τ / d z L using Eq. (18),d τ d z L = (cid:90) ∞ ∂ τ∂z L ∂σ d σ. (24)The complexity of Eqs. (16) and (18) makes the aboveintegration very difficult. But since ∆ t z is an increasingfunction of σ according to (13), there exists a value σ such that if σ < σ , y max = y , while if σ ≥ σ , y max = ∆ t m / ∆ t z . This σ is given by σ = (cid:20) ∆ t m π y D S D L D LS
11 + z L (cid:21) / , (25)obtained from the condition y = ∆ t m / ∆ t z . Then, onecan carry out the integration (24) by dividing the inte-gration range into two parts, separated by σ . This givesd τ d z L = 16 π y n ∗ Γ( α/β ) (cid:16) σ ∗ c (cid:17) c (1 + z L ) H L (cid:18) D L D LS D S (cid:19) (cid:20) Γ (cid:18) α + 4 β (cid:19) P (cid:18) u , α + 4 β (cid:19) − t ∗ y T s × Γ (cid:18) α + 8 β (cid:19) P (cid:18) u , α + 8 β (cid:19)(cid:21) + ( c ∆ t m ) n ∗ π Γ( α/β ) (cid:18) cσ ∗ (cid:19) cH L (cid:18) − t m T s (cid:19) Q (cid:18) u , α − β (cid:19) , (26)where ∆ t ∗ is given by Eq. (13) with σ replaced by σ ∗ , u = ( σ /σ ∗ ) β , and P ( x, a ) = 1Γ( a ) (cid:90) x ξ a − e − ξ d ξ, (27) Q ( x, a ) = (cid:90) ∞ x ξ a − e − ξ d ξ, (28)are the incomplete gamma functions. The optical depthis thus τ ( z S ) = (cid:90) z S d τ d z L d z L . (29)This integration can be performed numerically. V. LENSING RATES
In this section, we will estimate the yearly detec-tion rate of lensed GW events displaying the beat pat-tern. We consider GWs emitted by the double com-pact objects (DCOs), which include NS-NS, black hole-neutron star binaries (BH-NS) and BH-BH. FollowingRef. [35], we use the intrinsic coalescence rate ˙ n ( z n ( z ) was calculated based on well-motivated assump-tions about star formation rate, galaxy mass distribution,stellar populations, their metallicities and galaxy metal-licity evolution with redshift (“low-end” and “high-end”cases). The binary system evolves from zero-age mainsequence to the compact binary formation after super-nova (SN) explosions. Since the formation of the com-pact object is related to the physics of common envelope(CE) phase of evolution and on the SN explosion mech-anism, and both of them are uncertain to some extent,Ref. [60] considered four scenarios: standard one – based on conservative assumptions, and three of its modifica-tions — Optimistic Common Envelope (OCE), delayedSN explosion and high BH kicks scenario. For more de-tails, see [60] and references therein. The chirp massesare assumed: 1 . M (cid:12) for NS-NS, 3 . M (cid:12) for BH-NS, and6 . M (cid:12) for BH-BH. These values are the average chirpmasses for these different binary systems given by thepopulation synthesis [61]. The median chirp masses usedin previous papers according to [61] were: 1 . M (cid:12) for NS-NS, 3 . M (cid:12) for BH-NS, and 6 . M (cid:12) for BH-BH. However,these were values obtained under the assumption of solarmetallicity of initial binary systems. Such scenario wasabsolutely right guess before the first detections of GWs.Now, the data gathered by the LIGO/Virgo detectorshave significantly modified these guesses demonstratingthat observed chirp masses (prticular of BH-BH systems)are much higher. Therefore, guided by the real data gath-ered so far we will adopt different values. According to[2] we will assume the median value of BH-BH systemschirp masses reported in their Table III. Since the dataon BH-NS systems is more scarse, we will take the valueof [2]. In summary, we take the following vlaues as rep-resentative of typical chirp masses of DCO inspirallingsytems: 1 . M (cid:12) for NS-NS, 6 . M (cid:12) for BH-NS, and24 . M (cid:12) for BH-BH.Matched filtering is used to identify GW events. TheSNR for a single detector can be approximately deter-mined with [62] ρ = 8Θ r d L ( z S ) (cid:18) M z . M (cid:12) (cid:19) / (cid:112) ζ ( f max ) , (30)where Θ is the orientation factor, M z = (1 + z S ) M is the chirp mass registered by the detector, M is thechirp mass at the source frame, d L ( z S ) is the luminositydistance, and finally, r is the detector’s characteristicdistance. The function ζ ( f max ) is ζ ( f max ) = 1 x / (cid:90) f max ( πM (cid:12) ) ( πM (cid:12) f ) / S h ( f ) d f, (31)where x / is nothing but the above integration with theupper limit being infinity. Since DCO inspiralling sys-tems studied in this work pass the sensitivity bands of(B-)DECIGO, one assumes that ζ ( f max ) = 1 [38]. Thecharacteristic distance parameter r is determined by r = 5192 π / c (cid:18) GM (cid:12) (cid:19) / (cid:90) ∞ d ff / S h ( f ) . (32)From this, one finds out that r = 6709 Mpc for DE-CIGO, and r = 535 Mpc for B-DECIGO.The orientation factor Θ is defined asΘ = 2 (cid:113) F (1 + cos ι ) + 4 F × cos ι, (33)where F + and F × are the antenna pattern functions forthe + and × polarizations, respectively, given by [63] F + = 12 (1 + cos θ ) cos 2 φ cos 2 ψ − cos θ sin 2 φ sin 2 ψ, (34) F × = 12 (1 + cos θ ) cos 2 φ sin 2 ψ + cos θ sin 2 φ cos 2 ψ. (35)In these expressions, θ and φ are the polar and azimuthalangles of the spherical coordinate system, which centersat the detector and whose z -axis is perpendicular to thedetector plane. ι is the inclination angle between the GWpropagation direction and the angular momentum of thebinary system, and finally, ψ is called the polarizationangle. These angles ( θ, φ, ι, ψ ) are uncorrelated and uni-formly distributed, so the probability distribution P (Θ)of Θ is approximated by [64] P (Θ) = 5Θ(4 − Θ) , (36)for 0 < Θ <
4, and P (Θ) = 0, otherwise.The differential beat rate is given by [35] ∂ ˙ N∂z S ∂ρ = 4 π ˙ n ( z S )(1 + z S ) d ( z S ) H ( z S ) τ ( z S ) P Θ ( x ( z S , ρ )) x ( z S , ρ ) ρ , (37)where x ( z, ρ ) = ρρ th (1 + z ) − / d L ( z ) r (cid:18) . M (cid:12) M (cid:19) / . (38)The yearly detection rate is thus˙ N = (cid:90) z max d z S (cid:90) ∞ d ρ ∂ ˙ N∂z S ∂ρ , (39)and the differential yearly detection rates are defined tobe ∂ ˙ N∂ρ = (cid:90) z max d z S ∂ ˙ N∂z S ∂ρ , (40) ∂ ˙ N∂z S = (cid:90) ∞ d ρ ∂ ˙ N∂z S ∂ρ . (41) Figure 4 shows the relative differential detection rates N ∂ ˙ N∂ρ v.s. ρ for (B-)DECIGO. The evolutionary modelfor DCOs is the standard one with the “low-end” metal-licity scenario. As one can see low SNR events of threedifferent types of DCOs dominate for B-DECIGO. ForDECIGO, although the relative differential rate for NS-NS type DCOs peaks at a small SNR ( ρ < ρ th ), thecurves for the remaining types of DOCs are more flatand reach maximum at higher SNRs. ; _ N @ _ N @ ; A ; = 8 NS-NS with DECIGONS-NS with B-DECIGOBH-NS with DECIGOBH-NS with B-DECIGOBH-BH with DECIGOBH-BH with B-DECIGO
FIG. 4. The relative differential detection rates N ∂ ˙ N∂ρ v.s. ρ for (B-)DECIGO. The evolutionary model for DCOs is thestandard one with the “low-end” metallicity scenario. Thepurple dot-dashed line is at ρ = 8. Figure 5 displays the relative differential detectionrates N ∂ ˙ N∂z S v.s. z S for (B-)DECIGO. The evolutionarymodel for DCOs is also the standard one with the “low-end” metallicity scenario. From this figure, one can seethat lensed GW events observable in DECIGO are dom-inated by NS-NS and BH-NS binaries at z S = 2 ∼ z S = 4 ∼
5. On the other hand, differ-ential lensing rates for B-DECIGO peak at the slightlylower redshifts, respectively. The earlier launch of B-DECIGO would provide valuable information.Table I displays the yearly detection rate ˙ N for lensedGW events with the beat pattern from the inspiralingDCOs of different classes. As shown in the table, we con-sider all four scenarios with both the low-end and high-end metallicity evolution assumed. From this table, onefinds out that lensed GWs generated by BH-BH binarysystems dominate in most cases, except for the High BHkicks scenario, for which lensed GWs from NS-NS bina-ries contribute the most. One interesting result is that inall cases, there are at least 10 lensed GW events with thebeat pattern from the NS-NS binaries per year. This cre-ates possibility that at least for some of them electromag-netic counterparts could be detected allowing to identifythe host galaxy and measure the redshift. Hence, thecosmological parameters could be measured from themaccording to Ref. [19]. Of course, one should also try to z s _ N @ _ N @ z s NS-NS with DECIGONS-NS with B-DECIGOBH-NS with DECIGOBH-NS with B-DECIGOBH-BH with DECIGOBH-BH with B-DECIGO
FIG. 5. The relative differential detection rates N ∂ ˙ N∂z S v.s. z S for (B-)DECIGO. The evolutionary model for DCOs is thestandard one with the “low-end” metallicity scenario.˙ N Stand. Opt. CE Del. SN BH kicksNS-NSlow-end metallicity 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . take advantage of the dominating BH-BH binary systemsusing statistical methods as discussed in Ref. [21].Table II shows the yearly detection rate for lensed GWevents with the beat pattern for B-DECIGO. Comparedwith Table I, it has similar features but the rates aresmaller. This is due to the lower designed sensitivity.However, there is still a considerable amount of lensedevents dominated by signals from BH-BH systems. Theonly exception is the High BH kicks scenario, leadingto suppression of BH-BH formation rate. In such a casethe perspectives for the B-DECIGO to detect lensed GWwith a beat pattern are poor.One may concern that in order to detect the beat pat-terns with enough accuracy to determine the luminos-ity distance, the lens mass and cosmological parameters, ˙ N Stand. Opt. CE Del. SN BH kicksNS-NSlow-end metallicity 0 .
05 0 .
63 0 .
06 0 . .
09 0 .
59 0 .
10 0 . .
61 2 .
98 0 .
83 0 . .
19 2 .
67 0 .
62 0 . . . . . . . .
14 4 . .
46 220 . .
89 5 . .
88 201 . .
86 4 . SNR threshold ρ th should be bigger than usually assumedvalue ρ th = 8 . It turns out, however, that by increasing ρ th by a factor of 10 or even 100, the detection rates forthe standard scenario case remain the same. This canbe understood by realizing the fact that Eq. (39) can berewritten in the following form,˙ N = (cid:90) z max d z S (cid:90) ∞ d R ∂ ˙ N∂z S ∂R , (42)where R = ρ/ρ th , and the new integrand is given byreplacing ρ by R in the denominator of the last factorin Eq. (37). According to Eq. (38), x ( z, ρ ) is actuallya function of R . Moreover, the optical depth τ ( z S ) alsodepends on R ; see the definition of y below Eq. (16),and also Eq. (26). Therefore, the new integrand ∂ ˙ N∂z S ∂R isindeed a function of R , and ˙ N is insensitive to the choiceof ρ th for any DCO evolutionary scenario. VI. CONCLUSION
In this work, we analyzed how many lensed GW eventswith the beat pattern can be detected by (B-)DECIGOevery year. It turns out that there are many more lensedevents from DCOs than those observable by LISA [20].Among different binary types of DCOs, BH-BH binariescontribution is dominating in most evolutionary modelsof DCOs. Nevertheless, there is still a considerable num-ber of lensed GWs expected from NS-NS and BH-NS bi-naries, which can be used together with their electromag-netic counterparts to study the cosmology accurately. Infact, the lensed GWs from BH-BH binaries are also valu-able with the statistics method [21], even though thereare no electromagnetic counterparts.Another point worth mentioning is that it is very ad-vantageous to study cosmology with the beat pattern,because high redshift binaries ( z S = 3 ∼
6) contribute alot to the total detection rates. In Ref. [19], the authorsdiscussed how to use the beat pattern to measure the lu-minosity distance of the GW source, the mass of the lens,and some cosmological parameters (e.g., H ). In princi-ple, these measurements can be very accurate. However,these studies were based on the simple lens models: thepoint-mass model and SIS. One may expect that simi-lar opportunity will emerge in more complicated and re-alistic lens mass profiles. This deserves further study.Moreover some complications arising in realistic situa-tion were also omitted, such as the small SNRs for someGW events, the intrinsic scatter in the lens profile andthe cosmic shear, etc.. One has to properly address theseissues in the real measurements in order to guarantee ac-curacy of the method. In this work, we only estimatethe lensing rate, which at the order of magnitude levelwould not be affected much by these factors. Our predic- tions raise hopes of detecting beat patterns in forthcom-ing (B-)DECIGO missions and motivates to undertakemore realistic studies of this phenomenon. ACKNOWLEDGMENTS
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