High Probability of Detecting Lensed Supermassive Black Hole Binaries by LISA
Zucheng Gao, Xian Chen, Yi-Ming Hu, Jian-Dong Zhang, Shunjia Huang
DDraft version February 23, 2021
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High Probability of Detecting Lensed Supermassive Black Hole Binaries by LISA
Zucheng Gao, Xian Chen,
1, 2
Yi-Ming Hu, Jian-dong Zhang, and Shunjia Huang Astronomy Department, School of Physics, Peking University, Beijing 100871, P. R. China Kavli Institute for Astronomy and Astrophysics at Peking University, Beijing 100871, P. R. China TianQin Research Center for Gravitational Physics and School of Physics and Astronomy, Sun Yat-sen University (Zhuhai Campus),Zhuhai 519082, P. R. China
ABSTRACTGravitational lensing of gravitational waves (GWs) is a powerful probe of the matter distributionin the universe. Here we study the lensing effect induced by dark matter (DM) halos on the GWsignals from merging massive black holes, and we revisit the possibility of detection using the LaserInterferometer Space Antenna (LISA). In particular, we include the halos in the low-mass range of10 − M (cid:12) since they are the most numerous according to the cold DM model. In addition, weemploy the matched-filtering technique to search for weak diffraction signatures when the MBHBshave large impact parameters ( y ∼ ). We find that about (20 − − M (cid:12) and the redshift range of 4 −
10 should show detectable wave-optics effects. Theuncertainty comes mainly from the mass function of DM halos. Not detecting any signal during theLISA mission would imply that DM halos are significantly more massive than 10 M (cid:12) . Keywords: gravitational waves — gravitational lensing: strong — dark matter INTRODUCTIONGravitational lensing events are unique probes of thedistribution of matter in the universe (Schneider et al.1992). Like light, gravitational waves (GWs) couldalso be lensed by intervening matter (Lawrence 1971;Cyranski 1974; Sonnabend 1979; Markovi´c 1993). Theprospect of detecting the lensing of GWs is promis-ing given the increasing number of GW events discov-ered in the recent years by the Laser InterferometerGravitational-wave Observatory (LIGO) and the Virgodetectors (Abbott et al. 2016, 2019; Abbott et al. 2020).One major difference between GW and light is thatthe former usually has a much longer wavelength. Forexample, LIGO/Virgo are sensitive to the GWs witha wavelength of O (10 ) km. It is comparable to orlonger than the characteristic sizes of many astrophysi-cal objects, such as stars or intermediate-massive blackholes (IMBHs). If lensed by these objects, the GWsin the LIGO/Virgo band would behave like light in thewave-optics limit (Ohanian 1974; Bontz & Haugan 1981; [email protected]@pku.edu.cn Nakamura 1998; Nakamura & Deguchi 1999). In thiscase, wave diffraction would modify the amplitude andphase of the GWs, producing a characteristic “beat-ing pattern” in the frequency domain of the waveform(Takahashi & Nakamura 2003). In addition, the wave-optics effect could also smear the plane of GW polar-ization (Cusin & Lagos 2020) and produce beat pat-terns in the time-domain waveform (Yamamoto 2005;Hou et al. 2020). These effects, in principle, could allowLIGO/Virgo to detect massive stars, IMBHs, and thedense cores of globular clusters and dark-matter (DM)halos (Moylan et al. 2008; Cao et al. 2014; Takahashi2017; Christian et al. 2018; Dai et al. 2018; Diego et al.2019; Jung & Shin 2019; Liao et al. 2019; Meena &Bagla 2020; Oguri & Takahashi 2020). However, so farno strong evidence of lensing effects has been officiallyreported by LIGO/Virgo (Hannuksela et al. 2019), sug-gesting that the lensing probability is relatively low.The Laser Interferometer Space Antenna (LISA) is afuture space-based mission aiming at detecting the GWsin the milli-Hertz (mHz) band (Amaro-Seoane et al.2017). One of its major targets is the merger of twomassive black holes (MBHs), preferentially in the massrange of 10 ∼ M (cid:12) . Because of the superb sensitiv-ity, LISA could detect MBH mergers up to a redshift of a r X i v : . [ a s t r o - ph . C O ] F e b Gao et al.
20 with a signal-to-noise ratio (SNR) as high as 10 − (Amaro-Seoane et al. 2017). Such a high redshift sug-gests that gravitational lensing by the large-scale struc-ture is no longer negligible for LISA (Takahashi 2006;Yoo et al. 2007). The long wavelength and high SNRalso indicate that the diffraction effects in the wave-optics limit, which is relatively weak for LIGO/Virgosources, may become significant for LISA.For LISA, the lenses which produce the diffraction ef-fects are mainly low-mass dark-matter (DM) halos, aswell as the subhalos in massive main halos (Takahashi &Nakamura 2003; Takahashi 2004). Takahashi & Naka-mura (2003) considered the DM halos in the mass rangeof 10 ∼ M (cid:12) and estimated that the lensing proba-bility for each MBH merger in the LISA band is about10 − ∼ − . However, the cold DM (CDM) model pre-dicts that the most abundant halos are those in the massrange of 10 ∼ M (cid:12) (e.g. Cooray & Sheth 2002; Hanet al. 2016). These small halos are not included in thecalculation of Takahashi & Nakamura (2003). More-over, Takahashi & Nakamura (2003) imposed a maxi-mum impact parameter y = 3 for the sources, to ensurea strong, detectable diffraction effect. This criterion,however, may be too strict. The high SNR of LISAsources would enable us to detect weak lensing signalseven when the impact parameter is larger. For this rea-son, the lensing probability could have been significantlyunderestimated in the past.Motivated by the importance of the wave-optics ef-fects for LISA sources, we revisit the calculation of thelensing probability for the MBH binaries in the LISAband. In particular, we improve the lens model by in-cluding the halos and subhalos at the lower mass end andwe also adopt the matched-filtering technique to searchfor weak lensing signals. We remark that although thecalculation is based on LISA, the conclusion also ap-plies to space-borne GW missions with slightly differentfrequency coverage, such as TianQin (Mei et al. 2020).The paper is organized as follows. In Section 2, wedescribe our method, including the calculation of thelensing signal in the wave-optics limit, the mass func-tion of halos and subhalos, the probability of lensing, aswell as the matched-filtering technique. Then in Sec-tion 3 we quantify the difference between the lensedand unlensed signals and derive a criterion for detect-ing the diffraction effect. Using this criterion, we es-timate the lensing probability for the MBH mergers inthe LISA band in Section 4. We discuss the importanceof DM models in Section 5 and conclude in Section 6.Throughout the paper, we assume a flat ΛCDM cos-mology with the parameters Ω m = 0 .
27, Ω Λ = 0 . lenssource observer ⃑𝜉⃑𝜂 𝐷 ! 𝐷 "! 𝐷 " Figure 1.
Physical picture of GW lensing. The vectors η and ξ denote the two positions on the source plane and onthe lens plane. For illustrative purposes the two vectors arealigned but in principle they are not. The angular diameterdistances D L , D S , and D LS are measured in the rest frameof the observer. H = 72kms − Mpc − , σ = 0 .
9. Therefore, in our work h := H / (100 kms − Mpc − ) = 0 . METHOD2.1.
Lensing Model
For simplicity, we assume a singular isothermal sphere(SIS) profile for our DM halos and subhalos. In thiscase, we can follow Takahashi & Nakamura (2003) tomodel the diffraction of GWs. Using a more realisticNavarro-Frenk-White (NFW) profile usually leads to aslighter smaller magnification factor (Takahashi & Naka-mura 2003).The basic picture of lensing of GWs is illustrated inFigure 1, where D L , D S , and D LS denote the angu-lar diameter distances to the lens, to the source, andtheir difference. All these quantities are measured inthe frame of the observer. Using these distances, theEinstein radius ξ on the lens plane can be expressedwith ξ = 4 πσ v D L D LS /c D S , where σ v is the velocitydispersion of the lens.The magnification factor is defined as F ( ω, η ) := ˜ φ Lobs ( ω, η )˜ φ obs ( ω, η ) , (1)where ˜ φ Lobs ( ω, η ) and ˜ φ obs ( ω, η ) are the lensed and un-lensed gravitational wave amplitudes in the Fourierspace, ω is the observed angular frequency of the GW,and η is the position vector on the source plane (“impactparameter” hereafter). Taking the cosmological redshiftinto account, the magnification factor can be calculatedwith F ( ω, η ) = ω (1 + z L ) D S πi D L D LS (cid:90) d ξ exp[ iω (1 + z L ) t d ( ξ , η )] , (2) etecting lensed SMBHBs by LISA z L is the redshift of the lens and t d is the time de-lay caused by lensing. The time delay can be computedwith t d ( ξ , η ) = D L D S D LS (cid:12)(cid:12)(cid:12)(cid:12) ξ D L − η D S (cid:12)(cid:12)(cid:12)(cid:12) − ˆ ψ ( ξ ) + ˆ φ m ( η ) , (3)where ˆ φ m ( η ) denotes the arrival time of the unlensedGW, which is approximately D S [1 + | η /D S | / /c , andˆ ψ is the deflection potential which solves the equation ∇ ξ ˆ ψ ( ξ ) = 8 π Σ( ξ ) , (4)with ∇ ξ the two-dimensional Laplacian and Σ( ξ ) themass surface density of the lens.For the purpose of numerical calculation, we definethe dimensionless positions as x = ξ ξ ; y = D L ξ D S η . (5)The corresponding dimensionless frequency is w = 4 GM L (1 + z L ) ω/c , (6)where M L is the so-called “lens mass”, which is de-fined as the mass enclosed by a circle of the Einsteinradius in the lens plane. In the SIS model, we have M L = 4 π σ v D L D LS / ( GD S c ) (Appendix A). Using theabove nondimensional quantities, the time delay can berewritten as T ( x , y ) = D L D LS D S ξ − t d ( ξ , η )= 12 | x − y | − D L D LS D S ξ − ˆ ψ ( ξ ) + D L D LS D S ξ − ˆ φ m ( η ) . (7)It follows that the nondimensional amplification factoris F ( w, y ) = w πi (cid:90) d x exp[ iwT ( x , y )] . (8)In the SIS model the last equation can be calculatedwith F ( w, y ) = − iwe iwy / (cid:90) ∞ dx xJ ( wxy ) × exp[ iw ( 12 x − x + φ m ( y ))] , (9)where φ m ( y ) = y + 1 / J is the zeroth-order Besselfunction. The corresponding amplification factor | F | and phase-change factor θ F are | F | = √ F F ∗ , θ F = − i ln[ F/ | F | ] , (10)where F ∗ is the complex conjugate of F . −6 −5 −4 −3 −2 w | F | y=100y=250 −6 −5 −4 −3 −2 w −0.002−0.001 θ F y=100y=250 Figure 2.
The amplification factor (upper panel) and thephase-change factor (lower panel) as a function of the di-mensionless frequency according to the SIS model. Differentlines correspond to different impact parameters ( y ). Figure 2 shows the dependence of | F | and θ F on the di-mensionless frequency w and impact factor y . We noticethree results which are important for the later estima-tion of the lensing probability.First, the amplification factor | F | in general decreaseswith increasing impact parameter y . However, evenwhen y is relatively large, e.g., y = 250, the amplifi-cation factor converges to the value in the geometriclimit and is not 1, and the phase-change factor does notvanish. The implication is that even though the wave-optics effect is weak, it may still be detectable if theSNR of the event is sufficiently large. We will study thecriterion for detecting such a weak signal in the latersections. The previous works, however, normally adoptan upper limit between y = 1 and 10 to estimate thelensing probability (e.g., Takahashi & Nakamura 2003).Such a small value could cause an underestimation ofthe number of lensing events.Second, when y is fixed, both | F | and θ F could varysignificantly due to the change of w . In particular, thecritical value of y , above which the diffraction effect be-comes undetectable, depends on w , which, according to Gao et al.
Equation (6), depends on the lens mass, lens redshift,and the GW frequency. We will take such a dependenceinto account in the following sections. These results in-dicate that it is oversimplified to use an single value of y to estimate the lensing probability in the diffractionlimit, as is often the case in the previous works.Third, the peaks of the amplification factor and thephase-change angles shift to smaller w as y increases. Asa result, for large impact parameters, i.e., y = 100 − w ∼ (10 − − − ). Sucha small dimensionless frequency corresponds to a low-mass lens according to Equation (6), which is about M L (1 + z L ) (cid:39) M (cid:12) (cid:16) w − (cid:17) (cid:18) f − Hz (cid:19) − . (11)The corresponding halo mass is also small, about 10 − M (cid:12) according to Appendix A. The above relation-ships suggest that the majority of the diffraction eventsdetected by LISA should be induced by small halos, be-cause (i) the lensing probability increases with y and(ii) when y is large only small halos produce strongdiffraction effect.2.2. DM Halos and Subhalos
The lenses of our interest are those DM halos as smallas 10 − M (cid:12) . The last section has shown that theyinduce an observable diffraction effect to the mHz GWsin the LISA band. Two types of DM halos fall in thismass range.The first type reside in the low-density regions of theuniverse. They predominate the low-mass end of themass function of ordinary DM halos (e.g. Wang et al.2020). To compute the number density of these halos,we adopt the mass function density dn/dM h derived inCooray & Sheth (2002), where n denotes the numberdensity of halos in unit of Mpc − and M h is the halomass. Note that by convention dn/dM h has a unit ofM − (cid:12) Mpc − h .The second type of DM halos fall in our interestedmass range are the substructures of those massive DMhalos. These substructures are often referred to as “sub-halos”. Numerical simulations show that given the mass M h of a main halo, the masses of the subhalos follow apower-law distribution with a universal power-law index(e.g. Gao et al. 2004a,b; Diemand et al. 2004; Libeskindet al. 2005; Giocoli et al. 2008)). Following Han et al.(2016), we write the mass function of the subhalos as dN ( < R ) d ln m = A ( R ) M ( < R ) m (cid:20) mm (cid:21) − α , (12)where N ( < R ) is the number of subhalos within a radiusof R of the main halo, m is the mass of the subhalo, A ( R ) is a normalization factor, M ( < R ) is the total mass en-closed by the radius R , and m = 10 M (cid:12) and α = 0 . R vir of the main halo, regardless of their spatial dis-tribution within the main halo. Therefore, we shouldreplace R with R vir when using Equation (12). Byconstruction, the total mass within the virial radius is M ( < R vir ) = M h . This leaves A ( R vir ) the only quantitythat is undetermined. We notice that Figure 15 of Hanet al. (2016) gives the value of A ( < R ) which shows thatat R = R vir the value converges to 0 .
01 for a wide rangeof halo mass, from M h = 10 h − M (cid:12) to 10 h − M (cid:12) .For this reason, we adopt A ( < R vir ) = 0 .
01 for our latercalculations.Knowing the mass function of subhalos in one mainhalo, we can calculate the mass function density at agiven redshift for all the subhalos of the same mass with dn sub dm ( m, z L ) = (cid:90) dM (cid:48) h dn ( M (cid:48) h , z L ) dM (cid:48) h dNd ln m m . (13)Such a quantity is useful for our later calculation ofthe lensing probability. Correspondingly, the total massfunction density contributed by the halos and subhalosof a mass of M h is ξ lens ( M h , z ) = dn sub dM h ( M h , z ) + dndM h ( M h , z ) . (14)Note that to find the lens mass M L corresponding to ahalo mass M h , the relationship derived in Appendix Ais applied.Figure 3 shows the mass function density predicted byEquation (14). We can see that the mass function den-sity has little evolution from redshift z = 2 to 6 (upperpanel), and it is more sensitive to the halo mass (lowerpanel). We note that in general halos are more numer-ous than subhalos. Nevertheless, we include subhalos inthe calculation for completeness.2.3. Calculation of the Lensing Probability
To calculate the lensing probability, we have to specify(i) the number of lenses of different masses at each red-shift and (ii) the solid angle these lenses cover in whichwe can detect the diffraction of GW.For (i), we start with the halo mass function den-sity derived in the previous section, ξ lens ( M h , z ). Sincethe SIS model predicts a unique relationship betweenthe lens mass and halo mass, M h ( M L , z L , z S ) (see Ap-pendix A), we can rewrite ξ lens as a function of thelens mass, i.e., ξ lens ( M h ( M L , z L , z S ) , z L ). Using this etecting lensed SMBHBs by LISA M h [M ⊙ ] −10 −8 −6 −4 −⊙ ξ l e n s [ M − ⊙ M p c − h ] z = ⊙z = 4z = 6z = 8 z L −9 −8 −7 −6 −5 −4 −3 −2 −1 ξ l e n s [ M − ⊙ M p c − h ] M h ⊙ 10 M ⊙ M h ⊙ 10 M ⊙ M h ⊙ 10 M ⊙ Figure 3.
The mass function of the small halos which couldproduce diffraction effects as a function the halo mass (upperpanel) or redshift (lower panel). new mass function, we can calculate the number of lensin the mass range ( M L , M L + dM L ) and redshift bin( z L , z L + dz L ) per unit solid angle using the equation d N lens ( M L , z L , z S ) dM L dz L d Ω = ξ lens ( M h ( M L , z L , z S ) , z L ) × χ ( z L ) dχdz dM h dM L , (15)where χ ( z ) is the comoving distance for redshift z .Figure 4 shows the result of Equation (15) integratedover a redshift range of [0 , z Lmax ] and above a certainlens mass. The source is assumed to be at z S = 4.It is clear that the number of lenses in a solid angleincreases with redshift, and small lenses (e.g., M L ∼ M (cid:12) ) are the most numerous. Therefore, we expectthat small halos contribute most of the lensing events.We have considered the lenses as small as 10 M (cid:12) becausethey correspond to a halo mass of about 10 M (cid:12) (seeAppendix A).As for (ii), suppose y crit is the critical impact param-eter in the source plane within which the effect due tothe diffraction of GW is detectable. In the lens plane,the critical impact parameter corresponds to an angular Lmax d N / d Ω ( < z L m a x )[ s r − ] M L > 10M ⊙ M L > 100M ⊙ M L > 1000M ⊙ Figure 4.
Cumulative distribution of lenses per unit solidangle. The three curves from top to bottom are counting thelenses above three different minimum masses, i.e., M Lmin =(10 , , M (cid:12) . In this example, the source is at z S = 4. size of θ ( M S , M L , z L , z S ) = ξ y crit /D L . (16)We have written θ as a function of the source mass M S and redshift z S to highlight the dependence of y crit onthe “loudness” of the source. Therefore, the lensing ef-fect is detectable within a solid angle of πθ = πy × GM L D LS c D L D S (17)towards the lens, where we have used the relation ξ = 2 (cid:114) GM L D L D LS c D S (18)from the SIS model. Summing up all possible lenses be-tween the source and the observer, we derive the lensingprobability–the probability of detecting the diffractioneffect in a given GW source–as P = (cid:90) z S dz L (cid:90) πθ × d N lens ( M L , z L , z S ) dM L dz L d Ω dM L . (19)In principle, the integration should be performed overall possible lens masses. In practice, we restrict the in-tegration within a mass range [ M Lmin , M
Lmax ]. The up-per and lower limits are functions of lens redshift z L ,which should be determined by evaluating the promi-nence of the diffraction effect. Only those lenses pro-ducing a detectable diffraction effect should be counted.The following subsection explains how we quantify thedetectability of the diffraction effect.2.4. Signal and Matched Filtering
The magnification factor derived in Section 2.1 is afunction of GW frequency. To use it, we need to first
Gao et al. derive the unlensed GW signal in the frequency domain.This is done by a Fourier transformation,˜ h ( f ) = (cid:90) e πift h ( t ) dt, (20)of the GW strain h ( t ) in the time domain.For illustrative purposes, we show in Figure 5 the char-acteristic strain of three MBH mergers (solid curves).We assume equal-mass mergers for simplicity and thetotal masses are M S = 10 , 10 , and 10 M (cid:12) respec-tively. The source redshift is fixed at z S = 4 in theseexamples. The waveforms are generated using the “IM-RPhenomC” model in the PyCBC package (Santamar´ıaet al. 2010; Nitz et al. 2020), excluding the effect of BHspin. Because we assume circular orbits for the MBH bi-naries, the merger time is about 1 . f − / M − / (1 + z S )years (Peters & Mathews 1963), where f mHz is the GWfrequency in unit of mHz and M is the total BH massin unit of 10 M (cid:12) . It is shorter than the canonical life-time of LISA (5 years) except in the case of the smallestBHs. −4 −3 −2 −1 f [Hz] −21 −20 −19 −18 −17 C h a r a c t e r i s t i c S t r a i n M S =10 M ⊙ M S =10 ⊙ M ⊙ M S =10 M ⊙ Figure 5.
Characteristic strains of unlensed (solid lines)and lensed (dot-dashed lines) MBH binaries. The lines ofdifferent colors correspond to different total masses of thebinaries. The black dashed line is the square root of thespectral noise density of LISA, i.e., (cid:112) fS h ( f ), adopted fromRobson et al. (2019). We now integrate the characteristic strain in Figure 5to derive the SNR of each merger. The calculation takesadvantage of an inner product (Finn 1992; Cutler &Flanagan 1994) which is defined as (cid:104) h | h (cid:105) = 2 (cid:90) ∞ ˜ h ∗ ( f )˜ h ( f ) + ˜ h ( f )˜ h ∗ ( f ) S h ( f ) df, (21)where h and h are two waveforms, S h ( f ) is the one-sided power spectral density for LISA (from Robson et al. 2019), and the star symbols denote the com-plex conjugates. The SNR of a signal h is defined asSNR := (cid:112) (cid:104) h | h (cid:105) .Figure 6 shows the SNR of different MBH mergersat different redshift. We see that when the total massis higher than about 10 M (cid:12) and the source redshift islower than 10, the source in general has a SNR muchhigher than 10. These events are detectable by LISA(Amaro-Seoane et al. 2017). In the following we studythe lensing signals of these events. S S N R M S = 10 M ⊙ M S = 10 M ⊙ M S = 10 M ⊙ M S = 10 M ⊙ SNR=10 M S [M ⊙ ]0⊙5050075010001⊙5015001750⊙000 S N R z S = 4z S = 6z S = 8z S = 10SNR=10 Figure 6.
SNR as a function of the redshift (upper panel)or the total mass (lower panel) of the source.
The strain of the lensed signals are shown in Figure 5as the dot-dashed lines. In the calculation, we assumedthat the lens has a mass of M L = 10 M (cid:12) and is at aredshift of z L = 2. The impact parameter is set to y = 1to maximize the effect in these examples. In this case,we can discern by eye that the lensed signals differ fromthe unlensed ones.In more general cases, the impact parameters aremuch larger than 1 so that the diffraction effects aremuch more difficult to discern by eye (e.g., see Fig-ure 2). To quantify a small difference between two wave-forms, we define δh := h − h . We deem the two wave-forms to be discernible when the SNR of the difference etecting lensed SMBHBs by LISA (cid:104) δh | δh (cid:105) > DIFFERENCE BETWEEN LENSED ANDUNLENSED SIGNALSThe significance of the diffraction effect on the lensingsignal depends on five parameters. Two of them arerelated to the source, i.e., the total mass of the MBHbinary M S and the source redshift z S . Two are relatedto the lens, i.e., the lens mass M L and redshift z L . Thefinal one is the impact parameter y . In this section, weinvestigate the dependence of the inner product (cid:104) δh | δh (cid:105) on these parameters.Figure 7 shows the dependence of (cid:104) δh | δh (cid:105) on thesource mass M S . Comparing it with the lower panelof Figure 6, we find that the inner product behaves sim-ilarly as the SNR of the unlensed signal. The reasonis that higher SNR normally makes it easier to iden-tify the diffraction effect. Moreover, we find that when M S (cid:46) M (cid:12) , the inner product becomes smaller thanunity. For this reason, we will restrict our later analysisto M S ≥ M (cid:12) . Note that we are already consider-ing a relatively conservative example in the sense thatthe lens mass is small, M L = 16 M (cid:12) [so that the term ωM L in Eq.6 becomes f (100 M (cid:12) )], and the impact pa-rameter is relatively large ( y = 100). Increasing the lensmass and decreasing the impact parameter could bothenhance the inner product to (cid:104) δh | δh (cid:105) > M S < M (cid:12) .The dependence of (cid:104) δh | δh (cid:105) on the lens mass M L isshown in Figure 8. We see a wave-like behavior ofthe inner product. It is correlated with the behaviorof the amplification factor, as has been shown in Fig-ure 2. As M L increases to 10 M (cid:12) , (cid:104) δh | δh (cid:105) convergesto a constant value which depends on both the sourceand the lens redshifts. Such a convergence is due tothe limit of the geometric optics. For a variety of com-binations of z S and z L , we find that (cid:104) δh | δh (cid:105) becomessmaller than 1 when M L is smaller than about 10 M (cid:12) .This result indicates that such small lenses should notproduce a detectable diffraction signature. Therefore, in M S [M ⊙ ]01020304050 ⟨ δ h | δ h ⟩ z S ⟩4⊙z L ⟩2⊙y⟩100z S ⟩6⊙z L ⟩3⊙y⟩100z S ⟩⟨⊙z L ⟩4⊙y⟩100z S ⟩10⊙z L ⟩5⊙y⟩100⟨δh|δh⟩⟩1 Figure 7.
The inner product (cid:104) δh | δh (cid:105) as a function of thetotal mass of the source. The solid lines of difference colorscorrespond to different combinations of source redshift, lensredshift, and impact parameter. The lens mass is fixed to M L = 16 M (cid:12) in this example. The black dashed line marksthe place where (cid:104) δh | δh (cid:105) = 1. M L [M ⊙ ]012345678 ⟨ δ h | δ h ⟩ z ⟩ ⟨4⊙z L ⟨2⊙y⟨100z ⟩ ⟨6⊙z L ⟨3⊙y⟨100z ⟩ ⟨8⊙z L ⟨4⊙y⟨100z ⟩ ⟨10⊙z L ⟨5⊙y⟨100⟨δh|δh⟩⟨1 Figure 8.
The same as Fig. 7 but varying the lens mass M L while fixing the source mass at M S = 10 M (cid:12) . the following we do not consider lens masses significantlysmaller than 10 M (cid:12) . Moreover, when M L is significantlygreater than 10 M (cid:12) , the system enters the geometric-optics limit since w starts to exceed 10 − (1+ z L ) accord-ing to Equation (11). In this limit, the amplificationfactor converges to (cid:112) /y for all frequencies (Taka-hashi & Nakamura 2003). As a result, the diffractioneffect is suppressed. Therefore, we do not consider thelens masses significantly greater than 10 M (cid:12) either.Figure 9 shows the inner product as a function of thelens redshift. We find that (cid:104) δh | δh (cid:105) increases as the lensredshift approaches the source redshift. This behavioris caused by the fact that the lensing effect is in generalstronger when the lens and the source are closer. Wealso find that the sources at higher redshift in generalproduce a smaller (cid:104) δh | δh (cid:105) . This result stems from thedecrease of the SNR as the source redshift increases.Finally, we show the dependence of (cid:104) δh | δh (cid:105) on theimpact parameter y in Figure 10. In general, the innerproduct decreases with the impact parameter. Interest- Gao et al. L ⟨ δ h | δ h ⟩ z S ⟩ 10⟨ y ⟩ 100z S ⟩ 8⟨ y ⟩ 100z S ⟩ 6⟨ y ⟩ 100z S ⟩ 4⟨ y ⟩ 100⟨δh|δh⟩ ⟩ 1 Figure 9.
Dependence of the inner produce (cid:104) δh | δh (cid:105) on thelens redshift z L . In this example, the source mass is chosento be M S = 10 M (cid:12) and the lens mass is M L = 16 M (cid:12) . Theimpact parameter is fixed at y = 100 for easier comparison. ingly, for a variety of lensing systems, we find that theinner product remains higher than 1 as long as y (cid:46) ⟨ δ h | δ h ⟩ z S = 4⟨ z L = 2z S = 6⟨ z L = 3z S = 8⟨ z L = 4z S = 10⟨ z L = ⟩⟨δh|δh⟩ = 1 Figure 10.
The inner-product (cid:104) δh | δh (cid:105) as a function of thedimensionless impact parameter y . In these examples, wehave chosen M S = 10 M (cid:12) and M L = 16 M (cid:12) .4. LENSING PROBABILITYHaving understood the dependence of the inner prod-uct (cid:104) δh | δh (cid:105) on the parameters M S , z S , M L , z L , and y ,we can now include the halo mass function and calculatethe probability that a MBH binary in the LISA band has (cid:104) δh | δh (cid:105) > P ( (cid:104) δh | δh (cid:105) > M S , z S , M L , z L ), we first calculate the crit-ical impact parameter y crit which produces exactly (cid:104) δh | δh (cid:105) = 1. Figure 11 shows the critical impact pa-rameter as a function of the lens redshift. We can see that in a large redshift range, y crit is higher than 10 .Moreover, sources at lower redshifts have higher y crit ,since the SNR is higher. L y c r i t z S = 4, M L = 16M ⊙ z S = 6, M L = 16M ⊙ z S = 8, M L = 16M ⊙ z S = 10, M L = 16M ⊙ Figure 11.
The critical impact parameter which gives (cid:104) δh | δh (cid:105) = 1 as a function of the lens redshift. Differentlines correspond to different source redshift. In the calcula-tion, we fix the lens mass to 16 M (cid:12) and the source mass to10 M (cid:12) . Then we replace θ in the integrand of Equation (19)with θ ( y crit ) and integrate to give the probability P ( y
1, which suggests that y crit ( M L > M (cid:12) ) isgreater than y crit ( M L = 16 M (cid:12) ). Correspondingly, thelensing probability derived with y crit ( M L = 16 M (cid:12) ) is alower limit as well. Note that although we fix M L whencalculating y crit ( M L ), we do not fix the value of M L inthe halo mass function of Equation (19).These lower limits of the lensing probability are givenin Table 1. We restrict M L to the mass range [16 , . × ] M (cid:12) because, as has been mentioned above, neitherthe smaller lenses nor the larger ones produce signifi-cant diffraction effect. The corresponding halo mass isbetween 10 and 10 M (cid:12) . In the calculation, we assumethat the total source mass is M S = 10 M (cid:12) and we findthat the total probability is 0 .
28, 0 .
34, 0 .
37, and 0 . z S = 4, 6, 8, 10. The lensing probabil-ity would be even higher if we adopt a source mass of etecting lensed SMBHBs by LISA Table 1.
Lensing Probability Contributed by DifferentLenses z S P (M L = 1 . × ∼ . × M (cid:12) ) 0.04 0.05 0.05 0.06 P (M L = 1 . × ∼ . × M (cid:12) ) 0.08 0.10 0.11 0.11 P (M L = 1 . × ∼ . × M (cid:12) ) 0.16 0.19 0.21 0.23 P (M L = 1 . × ∼ . × M (cid:12) ) 0.28 0.34 0.37 0.40 M S = 10 M (cid:12) because the inner product (cid:104) δh | δh (cid:105) wouldbe greater, as is shown in Figure 7. However, for asource mass of 10 M (cid:12) or higher, the lensing probabil-ity would be lower than those given in Table 1, becauseFigure 7 shows that the corresponding inner product issmaller than 1. The above results indicate that for thoseMBHs in the mass range between 10 M (cid:12) and 10 M (cid:12) , itis probable that LISA could detect the diffraction effect.Table 1 also shows that the biggest contribution tothe lensing probability comes from the lenses within thehighest mass range, between 1 . × and 1 . × .These lenses are the least abundant according to Fig-ure 3. Nevertheless, their contribution is the largest be-cause their strong gravity leads to a large lensing crosssection. IMPACT OF DM MODELSWe notice that Takahashi & Nakamura (2003) deriveda lensing probability of 10 − ∼ − for the MBH bi-naries in the LISA band. It is two to three orders ofmagnitude smaller than our estimation. The discrep-ancy stems from the different mass range of the lensesthat are considered in the two works.Takahashi & Nakamura (2003) considered the lensesin the mass range of M L = 10 ∼ M (cid:12) , which corre-sponds to a halo mass of 10 ∼ M (cid:12) . Note thatsuch lenses are already in the geometric-optics limit.Moreover, they assumed a critical impact parameter of y crit = 3. In our model, we considered the lenses in themass range of 16 ∼ . × M (cid:12) . The correspondinghalo mass is 10 ∼ M (cid:12) . These lenses produce wave-optics effect in the lensing signal, and we have shownthat the effect is detectable even for a large impact pa-rameter of y ∼
200 (Figure 11).The difference of the lens masses produces two conse-quences in the estimation of the lensing probability. (i)In our model, the solid angle within in which the lens-ing signal is detectable is about 20 times smaller thanthe choice of Takahashi & Nakamura (2003), since it isproportional to y M L according to Equation (17). (ii)Our lenses are 2000 ∼ Lmax d N / d Ω ( < z L m a x )[ s r − ] z S = 4z S = 6z S = 8z S = 10 Lmax C u m u l a t i v e N u m b e r R a t i o z S = 4z S = 6z S = 8z S = 10 Figure 12.
Upper panel: Cumulative distribution of thenumber of lenses as a function of the upper limit of lensredshift z Lmax . The solid lines represent the lenses in themass range of M L = 16 ∼ . × M (cid:12) , while the dashedone represent the lenses of 10 ∼ M (cid:12) . The four colorsrefer to four different source redshifts. Lower panel: Theratio between the two kinds of cumulative lens numbers. seen in Figure 12. Given these two consequences, wecan understand the result that our lensing probability isabout 100 times higher than that derived in Takahashi& Nakamura (2003).The above comparison suggests that the lensing prob-ability is sensitive to the abundance of small halos.Since different DM models predict very different num-ber density for small halos, we now investigate the de-pendence of the lensing probability on the lower bound-ary of the halo mass function. To simulate the effectof different DM models, we cut off the integration ofEquation (19) at different lower boundaries M hmin andcount only those halo with M h > M hmin . The resultis shown in Figure 13. We see a sharp cut off aroundM hmin = 10 M (cid:12) . This result indicates that LISA is un-likely to detect the diffraction effect of a MBH binary ifthe smallest halos are all above 10 M (cid:12) , as the warm-DMmodels would predict (e.g. Lovell et al. 2014). SUMMARY CONCLUSION0
Gao et al. M hmin [M ⊙ ]10 −3 −2 −1 L e n s i n g ⊙ P r o b a b ili t y z S = 4z S = 6z S = 8z S = 10 Figure 13.
The total lensing probability as a function ofthe minimum halo mass. Different colors refer to differencesource redshift.
In this work, we studied the lensing signals of theMBH binaries in the LISA band. We focus on thewave-optics effect and found that it is produced mainlyby the DM halos and subhalos in the mass range of10 ∼ M (cid:12) . Using the matched-filtering technique,we showed that the effect could be discernible by LISAeven when the source has an impact parameter as largeas y = 200. Such a large impact parameter substantiallyenhances the probability of detecting the diffraction sig-natures. According to our preliminary estimation, ifCDM predominates the matter content of the universe,we could already detect the diffraction effect after 3 − ∼ M (cid:12) arediscovered by LISA. If, on the other hand, warm DMpredominates, the chance of detecting the diffraction ef-fect would be diminished by at least one order of mag-nitude. Therefore, looking for the wave-optics effects inLISA events could help us constrain the DM models. As a final remark, we note that our model of the lens-ing signal and the criterion of discerning the diffractioneffect are based on ideal assumptions. For example, wedo not consider multiple lenses along the line of sighteven though we have found a relatively high lensingprobability. Moreover, we have assumed that the devi-ation of the detected signal from the model template issolely due to gravitational lensing, while for real LISAobservation, other factors, such as confusion betweenmultiple events or inaccuracy of the waveform template,could also contribute to the deviation. We will addressthese caveats in a future work.ACKNOWLEDGMENTSThis work is supported by the National ScienceFoundation of China grants No 11873022, 11991053,and 11805286. X.C. is partly supported by theStrategic Priority Research Program of the ChineseAcademy of Sciences, Grant No. XDB23040100 andNo. XDB23010200. Y.H. is partly supported by theNational Key Research and Development Program ofChina (No. 2020YFC2201400), and Guangdong Ma-jor Project of Basic and Applied Basic Research (GrantNo. 2019B030302001). We especially thank Liang Daiat University of California, Berkeley for many insight-ful discussions on the theory of wave-opitcal lensingmodel and precious comments on an early version ofthis manuscript. We also extremely thank Xiao Guo atNational Astronomical Observatories of China (NAOC)for providing us with valuable suggestions on variousproblems encountered in this work.APPENDIX A. RELATION BETWEEN HALO MASS AND LENS MASS IN THE SIS MODELWe assume that DM halo follows an SIS density profile, ρ ( r ) = σ v πGr , (A1)where σ v is the velocity dispersion. According to this profile, the total mass of the halo, M h , is related to the virialradius r vir ( M h , z h ), which is a function of the mass M h and redshift z h , as M h = 2 σ v G r vir ( M h , z h ) . (A2)The lens mass is defined as the mass enclosed by a circle on the lensing plane with a radius of ξ = 4 πσ v D L D LS /c D S ,which is known as the Einstein radius. To calculate the lens mass M L , we use the the surface density of an SIS projectedon the lensing plane, Σ( ξ ) = σ v / (2 Gξ ) (Takahashi & Nakamura 2003), and derive M L = 4 π σ v D LS D L GD S c . (A3) etecting lensed SMBHBs by LISA σ v and we find that M h = r vir ( M h , z h ) (cid:115) M L D S c π GD LS D L . (A4)Figure 14 shows the relationship between these two masses and the dependence on the redshift of the halo. L M h [ M ⊙ ] z S ⊙ 4z S ⊙ 6z S ⊙ 8z S ⊙ 10 M L [M ⊙ ] M h [ M ⊙ ] z S = 4⊙ z L = 2.0z S = 6⊙ z L = 3.0z S = 8⊙ z L = 4.0z S = 10⊙ z L = 5.0 Figure 14.
The left panel shows the halo mass as a function of the redshift when the lens mass is fixed to 100 M (cid:12) . Differentlines correspond to different redshift of the source. The right panel shows the halo mass as a function of lens mass M L . Differentlines correspond to different combinations of the source and lens redshifts. Gao et al.
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