Gravitational Lensing of Gravitational Waves: Effect of Microlens Population in Lensing Galaxies
Anuj Mishra, Ashish Kumar Meena, Anupreeta More, Sukanta Bose, J S Bagla
MMNRAS , 1–16 (2021) Preprint 9 February 2021 Compiled using MNRAS L A TEX style file v3.0
Gravitational Lensing of Gravitational Waves: Effect of MicrolensPopulation in Lensing Galaxies
Anuj Mishra, ★ Ashish K. Meena, Anupreeta More, † Sukanta Bose and J. S. Bagla The Inter-University Centre for Astronomy and Astrophysics (IUCAA), Post Bag 4, Ganeshkhind, Pune 411007, India Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 81, Sahibzada Ajit Singh Nagar, Punjab 140306, India
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We analyse the effects of microlensing in the LIGO/Virgo frequency band due to a populationof stellar-mass microlenses and study their implications for strongly lensed gravitational wave(GW) signals. We consider a wide range of strong lensing magnifications and the correspondingsurface densities of the microlens population found in lensing galaxies, and use them to generaterealisations of the amplification factor. The methodologies for simulating amplification curvesfor both type-I (minima) and type-II (saddle) images are also discussed. We find that, onaverage, the presence of microlens population introduces a net amplification (de − amplification)in minima (saddle points) type of images in the LIGO frequency range. With increasingmicrolens density, the overall scatter and distortions increase and become significant fromrelatively lower frequencies. Comparison between IMFs suggests that although the differencesare not significant in typical cases, the bottom-heavy IMF tends to show a steeper rise in thescatter due to microlensing at higher frequencies compared to a bottom-light IMF. However,with the increase in the strong lensing magnification, the effects of microlensing becomeincreasingly significant regardless of other parameters, such as the microlens density, type ofimages or the IMF of the population. Hence, for microlensing features to be notable in GWsignal, the strong lensing magnification needs to be substantial. In some extreme cases ofstrong lensing magnification ( ∼ ∼ Key words: gravitational lensing: strong – gravitational lensing: micro – gravitational waves
The detection of gravitational waves (GWs, e.g., Abbott et al.2019, 2020) by Laser Interferometer Gravitational-Wave Observa-tory (LIGO) and Virgo opened up a new window to observe the Uni-verse. Those observatories have so far announced the detection of 50GW events in their first three observing runs, and this number willcontinue to rise with upcoming observations and detector facilitieslike the Kamioka Gravitational Wave Detector (KAGRA, Somiya2012). These observed GW events are the results of merging blackholes (BH-BH), neutron stars (NS-NS), and black hole-neutron star(BH-NS) binaries in galaxies at cosmological distances.Since these are cosmologically distant sources, the possibilityof gravitational lensing of their GW signals is quite promising. Onthe theoretical side, gravitational lensing of GWs by the galaxy and ★ E-mail: [email protected] † E-mail: [email protected] cluster scale lenses and its applications have been investigated inseveral works (e.g., Liao et al. 2017; Takahashi 2017; Li et al. 2018;Smith et al. 2018; Broadhurst et al. 2018, 2019, 2020). Searches havealso been carried out for signatures of strong lensing in the LIGOand Virgo data (e.g., Hannuksela et al. 2019; Smith et al. 2019; Daiet al. 2020; Liu et al. 2020). As the wavelength of GWs in the LIGOband is much smaller compared to the Schwarzschild radius of agalaxy or a galaxy cluster − scale lens, one can use the usual geo-metrical optics approach to calculate the strong lensing effects (e.g.,Dai & Venumadhav 2017; Dai et al. 2020; María Ezquiaga et al.2020). The strong lensing amplifies the GW signal (independentof frequency) by a factor √ 𝜇 , where 𝜇 is the strong lensing mag-nification, and hence, increases the signal-to-noise ratio (SNR) bythe same factor. The amplitude of the GW signal is inversely pro-portional to the luminosity distance to the source and proportionalto the chirp mass of the source binary. Hence, strong lensing intro-duces degeneracies in the measurement of these two parameters and © a r X i v : . [ a s t r o - ph . C O ] F e b Anuj Mishra et al. can lead to misleading results (e.g., Broadhurst et al. 2018, 2019,2020).However, this is not the only effect for lensing of GWs in theLIGO frequency range as the GW wavelength is of the order ofthe Schwarzschild radius of stellar-mass objects with mass range ∼
10 M (cid:12) − M (cid:12) (e.g., Figure 1 in Meena & Bagla 2020).As a result, the diffraction effects become essential and introducefrequency-dependent features in the GW signal (e.g., Baraldo et al.1999, Nakamura & Deguchi 1999, Takahashi & Nakamura 2003).Due to the frequency dependence, the effect of microlensing is notonly limited to the luminosity distance or chirp mass, instead, itcould affect most of the GW parameters. Additionally, Christianet al. (2018) showed that the effect of microlenses having mass (cid:38)
30 M (cid:12) can be detected with the current ground-based detectorsprovided that SNR (cid:38)
30. Meena & Bagla (2020) discussed the effectof point mass microlenses combined with strong lensing on the GWsignal and pointed out the possibility that microlensing can affecttwo counterparts of the strongly lensed signal in entirely differentways leading to misidentification of the strongly lensed GW signals.In a realistic gravitational lensing scenario, each of the stronglylensed GW signals will be affected by a population of microlensesinstead of a single microlens. In such cases, a macroimage will splitinto several microimages which will interfere and form complicatedinterference patterns (see Pagano et al. 2020, for a python basedpipeline to find microimages). The first study in this regard was doneby Diego et al. 2019, henceforth referred to as D19, in which theauthors discuss the possibility of microlensing due to a population ofmicrolenses present in a galaxy halo or the intra − cluster medium,on the strongly lensed GW signals. In D19 and their subsequentwork (Diego 2020), the microlensing effect is considered for surfacestellar density ≈
12 M (cid:12) pc − and for very high strong lensingmagnifications only, where caustics overlap and microlensing effectsbecome inevitable. However, these specific conditions may not holdfor a majority of strongly lensed signals as the local surface densitiesand strong lensing magnifications for the lensed counterparts canvary significantly.Here we look at the effect of the microlens population ondifferent counterparts of the strongly lensed GW signals. Follow-ing Vernardos (2019), we consider typical strong lensing magnifi-cations in combination with typical stellar densities in galaxy − scalelenses to determine the microlensing effects. To estimate the ampli-fication due to a microlens population, we have developed a code inMathematica following the method described in Ulmer & Goodman(1995, henceforth, referred to as UG95). We find that the microlens-ing effects become significant for higher values of strong lensingmagnification. As we will show, the contribution of microlensingis enhanced by strong lensing magnification independent of im-age type, surface microlens densities, and the stellar Initial MassFunctions (IMFs) studied in this work. As a result, the mismatchbetween lensed and unlensed signals is primarily noticeable for highstrong-lensing magnifications only.A few terms will be used throughout the paper. We refer tothe galaxy lenses as macrolenses. The embedded stars and stellarremnants in these macrolenses that perturb them on the small scaleare referred to as microlenses, and their surface density is denotedby Σ • . The images produced by the macrolens are referred to asmacroimages while their magnification is referred to as macro-magnification ( 𝜇 ). Since we measure signal amplitude in the caseof GWs rather than flux, it gets amplified by a factor √ 𝜇 in thosecases, which we refer to as macro-amplification value. Around thepositions of the microlenses, several images of a macroimage canform, which we refer to as microimages. For our analysis, we assume the redshifts of the lens and the source to be 𝑧 d = . 𝑧 s = . − scale strong lens. In §5, we present the results and discussthe importance of various factors such as macro − magnification, mi-crolens densities, and stellar IMF. The conclusions are summarizedin §6. In this section, we briefly discuss the basics of gravitational lensingin geometric (Schneider et al. 1992) and wave optics limits (Taka-hashi & Nakamura 2003). In the geometric optics limit, the gravita-tional lensing of a source at an angular diameter distance 𝐷 s due tothe presence of an intervening lens/deflector at an angular diameterdistance 𝐷 d can be described by the so-called gravitational lensequation (assuming small-angle and thin-lens approximation), 𝑦𝑦𝑦 = 𝑥𝑥𝑥 − 𝛼𝛼𝛼 ( 𝑥𝑥𝑥 ) = 𝑥𝑥𝑥 − ∇ 𝑥𝑥𝑥 𝜓 ( 𝑥𝑥𝑥 ) , (1)where 𝑦𝑦𝑦 = 𝜂𝜂𝜂𝐷 d / 𝜉 𝐷 s ≡ 𝛽𝛽𝛽 / 𝜃 and 𝑥𝑥𝑥 = 𝜉𝜉𝜉 / 𝜉 ≡ 𝜃𝜃𝜃 / 𝜃 representthe projected unlensed source position and the lensed image posi-tion on the lens/image plane (measured with respect to the opticalaxis), respectively. Here 𝜂𝜂𝜂 = 𝛽𝛽𝛽𝐷 s and 𝜉𝜉𝜉 = 𝜃𝜃𝜃𝐷 d represent physi-cal distances in the source and image planes, respectively, while 𝛽𝛽𝛽 and 𝜃𝜃𝜃 are their corresponding angular positions on the sky. Inorder to make the lens equation dimensionless, we have chosen anarbitrary length (angular) scale 𝜉 ( 𝜃 ) such that 𝜉 = 𝜃 𝐷 d . Thelens equation simply describes the vector addition and is derivedpurely from geometry where physics is contained in the deflectionterm 𝛼𝛼𝛼 ( 𝑥𝑥𝑥 ) , i.e., in the projected 2D lensing potential 𝜓 ( 𝑥𝑥𝑥 ) whichdetermines the deflection as a function of the impact parameter 𝑥𝑥𝑥 on the lens plane. The nonlinearity brought by the deflection termis what leads to the formation of multiple images of a given source.In the context of GWs, this would lead to multiple detection of thesame source which may be separated in the time domain by an orderof a few minutes to several years (e.g., see Fig. 13 in Oguri 2018).The magnification factor corresponding to these different lensedmacroimages (or, equivalently, events) is given as 𝜇 ≡ | det A | − = (cid:2) ( − 𝜅 ) − 𝛾 (cid:3) − , (2)where A 𝑖 𝑗 ≡ [ 𝜕𝑦 𝑖 / 𝜕𝑥 𝑗 ] is the Jacobian (matrix) corresponding tothe lens equation Eq. (1) and 𝜅 and 𝛾 represent the convergence andshear at the image position. Both 𝜅 and 𝛾 are functions of the lensplane coordinate 𝑥𝑥𝑥 ≡ ( 𝑥 , 𝑥 ) . Without loss of generality, one canalways choose the principle direction of the shear to be horizontallyaligned, in which case 𝛾 = | 𝛾 | = √︃ 𝛾 + 𝛾 = | 𝛾 | .For a given lensed signal, the corresponding time delay withrespect to its unlensed counterpart is given by 𝑡 d ( 𝑥𝑥𝑥, 𝑦𝑦𝑦 ) = 𝑇 s (cid:20) | 𝑥𝑥𝑥 − 𝑦𝑦𝑦 | − 𝜓 ( 𝑥𝑥𝑥 ) + 𝜙 m ( 𝑦 ) (cid:21) ≡ 𝑇 s 𝜏 d ( 𝑥𝑥𝑥, 𝑦𝑦𝑦 ) , (3)where 𝑇 s ( + 𝑧 d ) = 𝜉 𝐷 s 𝑐𝐷 d 𝐷 ds ≡ 𝑅 s0 𝑐 = 𝐺 𝑀 𝑐 , (4) MNRAS , 1–16 (2021)
L of GWs: Effect of Microlens Population 𝑧 d is the lens redshift, 𝑐 is the speed of light, 𝐷 ds is the angular diam-eter distance between the source and the lens, 𝜙 𝑚 ( 𝑦 ) is a constantindependent of lens properties and 𝑅 s0 is the Schwarzschild radiuscorresponding to the mass 𝑀 . This mass has an Einstein radius 𝜉 if placed on the lens plane. The factor 𝑇 s is the characteristic timedelay as it sets, roughly, the order of the time delay for a given lenssystem.The formalism of geometrical/ray optics described above isvalid as long as the time delay between any two images is sufficientlylarge compared to the wavelength 𝜆 of light, i.e., 𝑓 𝑡 d (cid:29)
1, where 𝑓 isthe frequency of the signal. This relation holds in a typical scenarioof strong gravitational lensing where different macroimages areformed. Gravitational lensing of gravitational waves due to galaxiesor galaxy cluster scale lenses can also be described using the aboveformalism. However, if the time delay of a lensed signal is lessthan, or of the order of, its time period, i.e., when 𝑓 𝑡 d (cid:46) 𝑅 s0 (cid:46) 𝜆 ), then wave effects are non-negligible andone has to take diffraction into account. Furthermore, when thesource and the deflector are far from the observer, one can useHuygens-Fresnel principle for the analysis of the lensing of theincoming plane wave flux, in which case every point on the lensplane act as a secondary source (Huygens point sources) and theamplitude of the signal at each point on the observer plane is thesuperposition of the signal from these various sources, leading tothe production of interference pattern. This phenomenon is knownas ‘microlensing’. We remind the reader that, although the term‘microlensing’ has some historical significance, it is generally usedfor any gravitational lensing effect by a compact object producingunresolvable (micro) images of a source (Mollerach & Roulet 2002).For an isolated point mass lens of mass 𝑀 L , the above con-dition ( 𝑓 𝑡 d (cid:46)
1) translates to, roughly, 𝑀 L (cid:46) 𝑀 (cid:12) ( 𝑓 / Hz ) − .Therefore, for gravitational waves with frequency in the LIGO range(10 − Hz), the corresponding mass range, where wave effectsbecome significant, is ∼ − M (cid:12) . This mass range is pre-dominantly responsible for the microlensing in the strongly lensedimages of a source. Whereas, for electromagnetic (EM) signals with 𝑓 ∼ − Hz, the diffraction effects become significant formass range ∼ − − − 𝑀 (cid:12) . This is a major difference in themicrolensing of EM waves and that of GWs.In a typical scenario of microlensing, one would need to con-sider the corrections arising from the wave optics (e.g., Nakamura& Deguchi 1999; Takahashi & Nakamura 2003). If we denote theratio of the observed lensed and the unlensed GW amplitude as 𝐹 ( 𝑓 , 𝑦𝑦𝑦 ) , then the amplification of the lensed signal is given by thediffraction integral (Takahashi & Nakamura 2003; Goodman 2005) 𝐹 ( 𝜈, 𝑦𝑦𝑦 ) = 𝜈𝑖 ∫ 𝑑 𝑥𝑥𝑥 exp [ 𝜋𝑖𝜈𝜏 d ( 𝑥𝑥𝑥, 𝑦𝑦𝑦 )] , (5)where 𝜏 d ( 𝑥𝑥𝑥, 𝑦𝑦𝑦 ) = 𝑡 d ( 𝑥𝑥𝑥, 𝑦𝑦𝑦 ) 𝑇 s , 𝜈 ≡ 𝜉 𝐷 s 𝐷 d 𝐷 ds 𝑓𝑐 ( + 𝑧 d ) = 𝑇 s 𝑓 . (6)Note that, the definition of dimensionless frequency, 𝜈 , and dimen-sionless time, 𝜏 d , is such that 𝜈𝜏 d = 𝑓 𝑡 d . Since 𝐹 ( 𝑓 ) is a complexvalued function, the total amplification, | 𝐹 | , and phase shift, 𝜃 𝐹 , canbe obtained through the relation 𝐹 ( 𝑓 ) = | 𝐹 | 𝑒 𝑖𝜃 𝐹 . As one can see,in wave optics, the amplification is frequency-dependent, unlike ingeometric optics, where the average magnification over a frequencyrange is independent of the frequency. In the geometric optics limit, 𝑓 (cid:29) 𝑡 − , the integral in Eq. (5) becomes highly oscillatory andonly the stationary points of the time delay surface contribute to the amplification. In that case, wave optics reduces to ray optics and asa result, the diffraction integral reduces to 𝐹 ( 𝑓 ) (cid:12)(cid:12) geo = ∑︁ 𝑖 √︃ | 𝜇 𝑗 | exp (cid:0) 𝑖 𝜋 𝑓 𝑡 𝑑, 𝑗 − 𝑖𝜋𝑛 𝑗 (cid:1) , (7)where 𝜇 𝑗 and 𝑡 𝑑, 𝑗 are, respectively, the magnification factor and thetime delay for the 𝑗 -th image. Also, 𝑛 𝑗 is the Morse index with values0, 1/2, 1 for stationary points corresponding to minima, saddles andmaxima of the time delay surface, respectively. As one can see fromabove equation, even in the geometric optics limit, gravitationallensing introduces an extra phase, the so-called Morse phase, of 𝑒 − 𝑖 𝜋 / and 𝑒 − 𝑖 𝜋 in the saddles and maxima with respect to theminima, respectively (Dai & Venumadhav 2017, María Ezquiagaet al. 2020). This phase difference can be used to search for thestrongly lensed and multiply imaged gravitational wave signals, andto constrain viable lenses (Dai et al. 2020).The diffraction integral, Eq. (5), can be solved analytically onlyfor some trivial lens models. For example, the solution for a pointmass lens of mass 𝑀 L is given by 𝐹 ( 𝜔, 𝑦 ) = exp (cid:20) 𝜋𝜔 + 𝑖𝜔 (cid:110) ln (cid:16) 𝜔 (cid:17) − 𝜙 m ( 𝑦 ) (cid:111)(cid:21) × Γ (cid:18) − 𝑖𝜔 (cid:19) 𝐹 (cid:18) 𝑖𝜔 , 𝑖𝜔𝑦 (cid:19) , (8)where 𝜔 = 𝜋𝐺 ( + 𝑧 d ) 𝑀 L 𝑓 / 𝑐 ≡ 𝜋𝜈 , 𝜙 m ( 𝑦 ) = ( 𝑥 m − 𝑦 ) / − ln ( 𝑥 m ) and 𝑥 m = (cid:16) 𝑦 + √︁ 𝑦 + (cid:17) /
2. The scale radius, 𝜉 , has beenchosen equivalent to the Einstein radius of the lens. In the geometricoptics limit, 𝑓 (cid:29) 𝑡 − , the above equation can be written as 𝐹 ( 𝑓 ) (cid:12)(cid:12) geo = √︁ | 𝜇 + | − 𝑖 √︁ | 𝜇 − | exp ( 𝜋𝑖 𝑓 Δ 𝑡 d ) , (9)where 𝜇 + and 𝜇 − are the amplification factor for primary and sec-ondary images formed due to a point mass lens and Δ 𝑡 d is the timedelay between these two images.However, in real strong lenses, the possibility of a GW signalencountering a massive isolated point lens is very less relative to ithitting a stellar (microlens) population. In that case, microlensing ofstrongly lensed images happens due to the population of point masslenses instead of a single point mass lens. As a result, the time delayfactor, 𝑡 d , in Eq. (5), is modified and includes a contribution fromthe macromodel, in terms of the convergence and shear at the imageposition, and the population of microlenses near the macroimage.The resultant time delay can be written as 𝑡 d ( 𝑥𝑥𝑥, 𝑦𝑦𝑦 ) = 𝑇 s (cid:34) ( 𝑥𝑥𝑥 − 𝑦𝑦𝑦 ) − 𝜓 total ( 𝑥𝑥𝑥 ) + 𝜙 m ( 𝑦 ) (cid:35) , (10)where 𝜓 total = 𝜓 SL + 𝜓 ML is the total lens potential with 𝜓 SL be-ing the macromodel potential and 𝜓 ML is the lens potential due tothe microlensing population embedded in the macromodel. Thesecontributions are given by 𝜓 ML ( 𝑥𝑥𝑥 ) = ∑︁ 𝑖 𝑚 𝑖 𝑀 ln | 𝑥𝑥𝑥 − 𝑥𝑥𝑥 𝑖 | ,𝜓 SL ( 𝑥𝑥𝑥 ) = 𝜅 (cid:16) 𝑥 + 𝑥 (cid:17) + 𝛾 (cid:16) 𝑥 − 𝑥 (cid:17) + 𝛾 𝑥 𝑥 , (11)where the lens plane coordinates are represented as ( 𝑥 , 𝑥 ) ; 𝑚 𝑖 and 𝑥 𝑖 denote, respectively, the mass and position of the 𝑖 -th pointmass lens in the population; 𝑀 is an arbitrary mass value as definedin Eq. (4); 𝜅 and ( 𝛾 , 𝛾 ) represent, respectively, the convergenceand the components of shear introduced due to the presence of themacrolens. The diffraction integral, Eq. (5), with the above lens MNRAS000
2. The scale radius, 𝜉 , has beenchosen equivalent to the Einstein radius of the lens. In the geometricoptics limit, 𝑓 (cid:29) 𝑡 − , the above equation can be written as 𝐹 ( 𝑓 ) (cid:12)(cid:12) geo = √︁ | 𝜇 + | − 𝑖 √︁ | 𝜇 − | exp ( 𝜋𝑖 𝑓 Δ 𝑡 d ) , (9)where 𝜇 + and 𝜇 − are the amplification factor for primary and sec-ondary images formed due to a point mass lens and Δ 𝑡 d is the timedelay between these two images.However, in real strong lenses, the possibility of a GW signalencountering a massive isolated point lens is very less relative to ithitting a stellar (microlens) population. In that case, microlensing ofstrongly lensed images happens due to the population of point masslenses instead of a single point mass lens. As a result, the time delayfactor, 𝑡 d , in Eq. (5), is modified and includes a contribution fromthe macromodel, in terms of the convergence and shear at the imageposition, and the population of microlenses near the macroimage.The resultant time delay can be written as 𝑡 d ( 𝑥𝑥𝑥, 𝑦𝑦𝑦 ) = 𝑇 s (cid:34) ( 𝑥𝑥𝑥 − 𝑦𝑦𝑦 ) − 𝜓 total ( 𝑥𝑥𝑥 ) + 𝜙 m ( 𝑦 ) (cid:35) , (10)where 𝜓 total = 𝜓 SL + 𝜓 ML is the total lens potential with 𝜓 SL be-ing the macromodel potential and 𝜓 ML is the lens potential due tothe microlensing population embedded in the macromodel. Thesecontributions are given by 𝜓 ML ( 𝑥𝑥𝑥 ) = ∑︁ 𝑖 𝑚 𝑖 𝑀 ln | 𝑥𝑥𝑥 − 𝑥𝑥𝑥 𝑖 | ,𝜓 SL ( 𝑥𝑥𝑥 ) = 𝜅 (cid:16) 𝑥 + 𝑥 (cid:17) + 𝛾 (cid:16) 𝑥 − 𝑥 (cid:17) + 𝛾 𝑥 𝑥 , (11)where the lens plane coordinates are represented as ( 𝑥 , 𝑥 ) ; 𝑚 𝑖 and 𝑥 𝑖 denote, respectively, the mass and position of the 𝑖 -th pointmass lens in the population; 𝑀 is an arbitrary mass value as definedin Eq. (4); 𝜅 and ( 𝛾 , 𝛾 ) represent, respectively, the convergenceand the components of shear introduced due to the presence of themacrolens. The diffraction integral, Eq. (5), with the above lens MNRAS000 , 1–16 (2021)
Anuj Mishra et al. potential, containing population of microlenses, cannot be solvedanalytically, in which case one has to use numerical methods, suchas in UG95 and D19, to obtain an approximate solution.
Except for the most trivial lens models, like that of an isolated pointmass, the potential 𝜓 ( 𝑥𝑥𝑥 ) takes a complicated form, in which caseno analytical form can be derived straightforwardly. Furthermore, itis highly inefficient to numerically integrate the diffraction integral,Eq. (5), because of the oscillatory nature of the integrand, andthe fact that the direct calculation of 𝐹 ( 𝑓 ) is a three-dimensionalproblem in 𝑥 , 𝑥 and 𝑓 . Hence, one needs to use a numericalmethod that is more efficient and resolves the problems mentionedabove to some extent. Such numerical methods have been describedin UG95 and D19. In the current work, we follow the methodof UG95 to calculate the magnification factor which is describedbelow in §3.1. Subsequently, we also demonstrate the validity of ourcode for both type-I (minima) macroimages and type-II (saddle)macroimages. In §3.2 and 3.3, we consider microlensing for twoelementary cases, namely, an isolated point mass lens and a pointlens situated near a type-I macroimage in the presence of an externalshear. Generally, simulating amplification curves for type-II imagesis nontrivial, and we discuss the issue separately in §3.4. We alsodescribe our methodology to deal with saddle-like macroimagesand perform numerical tests to verify our results for simple lensingconfigurations. By using the methods of contour integration and Fourier transfor-mation ( F ), UG95 splits the problem of calculating 𝐹 ( 𝑓 ) into twoparts and reduces it into two dimensions as described below. Firstly,we define Υ ( 𝜈 ) ≡ 𝑖𝐹 ( 𝜈 )/ 𝜈 . Then, we have F [ Υ ( 𝜈 )] ≡ (cid:101) 𝐹 ( 𝜏 (cid:48) ) = ∫ d 𝑥𝑥𝑥 ∫ 𝑑𝜈 exp (cid:0) 𝑖 𝜋𝜈 [ 𝜏 d ( 𝑥𝑥𝑥, 𝑦𝑦𝑦 ) − 𝜏 (cid:48) ] (cid:1) ⇒ (cid:101) 𝐹 ( 𝜏 (cid:48) ) = ∫ d 𝑥𝑥𝑥𝛿 [ 𝜏 d ( 𝑥𝑥𝑥, 𝑦𝑦𝑦 ) − 𝜏 (cid:48) ] , (12)where 𝜏 (cid:48) ≡ 𝑡 / 𝑇 s . Now, using 𝜈 = 𝑇 s 𝑓 and the fact that F − [ (cid:101) 𝐹 ( 𝜏 (cid:48) )] = Υ ( 𝜈 ) , we get 𝐹 ( 𝑓 ) = 𝑓𝑖 ∫ d 𝑡 exp ( 𝑖 𝜋 𝑓 𝑡 ) (cid:101) 𝐹 ( 𝑡 ) , (13)where ‘t’ represents the time delay value relative to an arbitraryreference time. For a type I (minima) macroimage, it is usuallymeasured relative to the global minima of the time delay surfacewhich marks the arrival of the first microimage. Whereas for type II(saddle) macroimages, we measure ‘t’ relative to the arrival of thedominant saddle image (discussed in §3.4).Eq. (13) can then be evaluated as a contour integral. The areabetween the curves defined by 𝜏 d ( 𝑥𝑥𝑥, 𝑦𝑦𝑦 ) = 𝜏 (cid:48) and 𝜏 d ( 𝑥𝑥𝑥, 𝑦𝑦𝑦 ) = 𝜏 (cid:48) + 𝑑𝜏 (cid:48) is 𝐴 = (cid:101) 𝐹 ( 𝜏 (cid:48) ) 𝑑𝜏 (cid:48) upto first order. This area can also be evaluated asan integral 𝐴 = ∮ 𝑑𝑠𝑑𝑙 , where 𝑑𝑠 is the infinitesimal length alongthe contour and 𝑑𝑙 = 𝑑𝜏 (cid:48) /|∇ 𝑥𝑥𝑥 𝜏 d | is the orthogonal distance betweenthe two contours at the point of evaluation. Moreover, there can ingeneral be more than one such contour. Thus, by comparison, wefinally get (cid:101) 𝐹 ( 𝜏 (cid:48) ) = ∑︁ 𝑘 ∮ 𝛾 𝑘 𝑑𝑠 |∇ 𝑥𝑥𝑥 𝜏 d | . (14) The summation is over all contours where 𝜏 d ( 𝑥𝑥𝑥, 𝑦𝑦𝑦 ) = 𝜏 (cid:48) . Thus, fora given time delay function (or lensing potential), we first compute (cid:101) 𝐹 ( 𝑡 ) using Eq. (14) and then inverse Fourier transform it back toget the required 𝐹 ( 𝑓 ) , as in Eq. (13). Also, from Eq. (14), one cansee that (cid:101) 𝐹 ( 𝑡 ) is a smooth function except at critical time 𝑡 𝑖 wherethe images form, i.e., where |∇ 𝑥𝑥𝑥 𝜏 d | =
0. The reader is referred tothe appendix of UG95 for the method to handle these singularities.
Since we have the analytic form of 𝐹 ( 𝑓 ) for an isolated point masslens, Eq. (8), it can be used as an initial testing ground for our numer-ical code based on the above formalism. Hence, in this subsection,we compare 𝐹 ( 𝑓 ) generated via two independent methods: analyt-ical and numerical. We consider a 100 M (cid:12) point mass lens placedat a lens redshift 𝑧 d = .
5, and a gravitational wave source placedat 𝑧 s =
2. The analysis has been done for four different non-zerosource positions: 𝑦 = 𝛽 / 𝜃 ∈ { . , . , . , . } . In the caseof a point mass lens, two images of opposite parities are alwaysformed, where positive and negative parities correspond to the min-imum and saddle point of the time delay surface, respectively. Themagnification of each image is 𝜇 ± = / ± ( 𝑦 + )/( 𝑦 √︁ 𝑦 + ) ,whereas the (dimensionless) time delay between them is Δ 𝜏 d = 𝑦 √︁ 𝑦 + / + ln (( √︁ 𝑦 + + 𝑦 )/( √︁ 𝑦 + − 𝑦 )) .In the left panel of Fig. 1, we show the normalised (cid:101) 𝐹 ( 𝑡 ) curves,computed using Eq. (14), for different source positions. The x-axisrepresents the time delay measured with respect to the minimum(situated at 𝑡 = − Hz, we do not need to obtain (cid:101) 𝐹 ( 𝑡 ) for all the timedelay values but instead for only a selected interval, like ∼ − -1 sec such that we cover the regime where 𝑓 𝑡 d (cid:46)
1. The curvesare normalised such that its value approaches one in the no lenslimit (large time delays). Since the time delay surface approachesa paraboloid at large values, this normalization is done by dividingthe obtained curves by 2 𝜋 (since Eq. (14) yields 2 𝜋 in the no lenslimit, i.e., for circular contours). In this way, the curves start froma value corresponding to the amplification of the image formedat the minimum, i.e., √ 𝜇 + , and eventually approach unity at largetime delay. Between these two expected behaviors, it encounters alogarithmic divergence corresponding to the saddle point (imagewith negative parity). The time delay at the point of this divergenceis the time delay between the two images (since the first image occursat 𝑡 = 𝑦 =
0) while the amplitude of the logarithmic pulseincreases, increasing the magnification of the saddle image.In the middle and right panels of Fig. 1, we show the com-parison between the analytical results (obtained using Eq. (8)) andthe numerical results for the computation of the amplification fac-tor 𝐹 ( 𝑓 ) = | 𝐹 | exp ( 𝑖𝜃 𝐹 ) . The black-dotted lines represent the | 𝐹 ( 𝑓 )| and 𝜃 𝐹 calculated using the analytical formula given by Eq. (8),respectively. The different solid-coloured lines represent the | 𝐹 ( 𝑓 )| and 𝜃 𝐹 values, which have been computed numerically using ourcode. For the numerical computation, we first generate (cid:101) 𝐹 ( 𝑡 ) valuesusing Eq. (14) and substitute it in Eq. (12) along with a suitableapodization function (because of the finite width of (cid:101) 𝐹 ( 𝑡 ) ). Withoutthis apodization, the computed values for both | 𝐹 | and 𝜃 𝐹 are signif-icantly imprecise and show oscillatory behavior at low frequencies,especially below 100Hz, due to the erroneous contribution from theedges. In our analysis, we have used a cosine window function (e.g., MNRAS , 1–16 (2021)
L of GWs: Effect of Microlens Population y = = = = - - - - - - t ( s ) F ∼ ( t ) AnalyticalNumerical Frequency ( Hz ) | F | - - Frequency ( Hz ) θ F Figure 1.
Test of numerical code shown for a point mass microlens of 100 M (cid:12) . The numerically computed (cid:101) 𝐹 ( 𝑡 ) and frequency dependent amplification factor 𝐹 ( 𝑓 ) = | 𝐹 | 𝑒 𝑖𝜃 𝐹 are shown for four different values of the impact parameter 𝑦 = 𝛽 / 𝜃 . The solid coloured curves have been computed numerically using themethod described in subsection 3.1, whereas the dotted black curves (the middle and the right panel) denote analytical results (see Eq. (8) ). The (cid:101) 𝐹 ( 𝑡 ) curveshave been normalised by the factor of 2 𝜋 so that its value approaches one in the no − lens limit (large time delays). see D19) that removes these irregularities and produces an excellentoutput, as shown in the above plots.As one can see from Fig. 1, the agreement between analyti-cal and numerical values is excellent. In all cases, the factor | 𝐹 | approaches unity and the phase factor 𝜃 𝐹 approaches zero as wego lower in the frequency ( 𝑓 (cid:28) 𝑡 − ), which means that the lensis invisible for signals with large wavelengths compared to theSchwarzschild radius of the lens ( 𝜆 (cid:29) 𝑅 s0 ). The wave effectsstart to appear when 𝜆 ∼ 𝑅 s0 ( 𝑓 𝑡 ∼ 𝜆 (cid:28) 𝑅 s0 ),wave optics approaches ray optics, and the average magnificationover a frequency range becomes independent of the frequency (asin strong lensing). In the frequency range shown in the figure, onlythe gray curve (y=1.0) has been able to approach the ray optics limitapproximately. In general, for a point lens, the average values of | 𝐹 | and 𝜃 𝐹 at high frequencies approach √ 𝜇 + and zero, respectively, inaccordance with Eq. (9). We notice that our numerical code recoversall the features mentioned for (cid:101) 𝐹 ( 𝑡 ) and 𝐹 ( 𝑓 ) very well. In this section, we test our code for a slightly complicated casewhere we place a point lens of 100M (cid:12) close to a type-I macroimage(source) in the presence of an external shear ( 𝛾 ) with no convergence( 𝜅 = 𝛾 = | 𝛾 | = | 𝛾 | . Forsuch a case, the effective lens potential in Eq. (10) can be written as 𝜓 total = ln (cid:18)√︃ 𝑥 + 𝑥 (cid:19) + 𝛾 (cid:16) 𝑥 − 𝑥 (cid:17) . (15)For the potential written above, we do not have an analytic solutionfor the diffraction integral, unlike in the case of point mass lens.Therefore, to perform the numerical test for this case, we directlyevaluate the double integral in Eq. (5) numerically and compare itwith the one obtained via our code. However, the direct evaluationis slow and does not work well for higher frequencies where theintegrand becomes too oscillatory.For the computation of the amplification factor, the comparisonbetween the direct numerical evaluation (dotted-black lines) andthe one obtained via our code (solid-coloured lines) is shown in themiddle and right panel of Fig. 2. The analysis has been done for threedifferent values of shear, 𝛾 = {− . − . − . } , keeping the sourceposition fixed at ( 𝑦 , 𝑦 ) = . ( cos ( 𝜋 / ) , sin ( 𝜋 / )) . Again, weobserve that there is excellent agreement between direct numericalintegration and the adopted numerical method of UG95. Also, the amplification curves approach the strong lensing amplification valuefor low frequency values (no − microlensing limit) and the phase shiftcurves approach zero, as expected.The corresponding calculations of (cid:101) 𝐹 ( 𝑡 ) are shown in the leftpanel of Fig. 2. We have again normalised the plots by dividingthe originally obtained ones by a factor of 2 𝜋 . This normalizationensures that the curves approach to value √ 𝜇 = ( − 𝛾 ) − / in theirno − microlensing (strong lensing) limit at large time delays. Thetime delay function, in this case, includes four stationary points, atleast two of which are always real. In (cid:101) 𝐹 ( 𝑡 ) plots in Fig. 2, we observethat two microimages form in the case of 𝛾 = − . 𝑡 = 𝑡 ∼ . 𝐹 ( 𝑓 ) , using our code, throughoutour analysis in the paper. In this subsection, we describe the numerical scheme that is used tocompute the amplification factor for a type-II (saddle point) image.Unlike in the case of a type-I (minima) image, here contours neitherclose locally nor have global minima, as they are hyperbolic innature rather than elliptical. However, if one chooses a sufficientlylarge region, the contribution from the neighborhood of a saddlepoint is given by (cid:101) 𝐹 ( 𝑡 ) = − √︁ | 𝜇 − | log | 𝑡 − 𝑡 𝑖 | + non-singular part + constant (16)where 𝑡 𝑖 and 𝜇 − denote the time delay and magnification valuecorresponding to the saddle point, respectively, and the constantdepends on the size of the region. By sufficiently large, we meanthe size of the region should be such that | 𝑢 − 𝜇 − | − / (cid:29) 𝑢 denotes the arc parameter of the contour(the reader is referred to Appendix B of UG95 for further de-tails). The presence of a constant does not affect the computationof 𝐹 ( 𝑓 ) , especially when the integration range is chosen carefully, MNRAS000
Test of numerical code shown for a point mass microlens of 100 M (cid:12) . The numerically computed (cid:101) 𝐹 ( 𝑡 ) and frequency dependent amplification factor 𝐹 ( 𝑓 ) = | 𝐹 | 𝑒 𝑖𝜃 𝐹 are shown for four different values of the impact parameter 𝑦 = 𝛽 / 𝜃 . The solid coloured curves have been computed numerically using themethod described in subsection 3.1, whereas the dotted black curves (the middle and the right panel) denote analytical results (see Eq. (8) ). The (cid:101) 𝐹 ( 𝑡 ) curveshave been normalised by the factor of 2 𝜋 so that its value approaches one in the no − lens limit (large time delays). see D19) that removes these irregularities and produces an excellentoutput, as shown in the above plots.As one can see from Fig. 1, the agreement between analyti-cal and numerical values is excellent. In all cases, the factor | 𝐹 | approaches unity and the phase factor 𝜃 𝐹 approaches zero as wego lower in the frequency ( 𝑓 (cid:28) 𝑡 − ), which means that the lensis invisible for signals with large wavelengths compared to theSchwarzschild radius of the lens ( 𝜆 (cid:29) 𝑅 s0 ). The wave effectsstart to appear when 𝜆 ∼ 𝑅 s0 ( 𝑓 𝑡 ∼ 𝜆 (cid:28) 𝑅 s0 ),wave optics approaches ray optics, and the average magnificationover a frequency range becomes independent of the frequency (asin strong lensing). In the frequency range shown in the figure, onlythe gray curve (y=1.0) has been able to approach the ray optics limitapproximately. In general, for a point lens, the average values of | 𝐹 | and 𝜃 𝐹 at high frequencies approach √ 𝜇 + and zero, respectively, inaccordance with Eq. (9). We notice that our numerical code recoversall the features mentioned for (cid:101) 𝐹 ( 𝑡 ) and 𝐹 ( 𝑓 ) very well. In this section, we test our code for a slightly complicated casewhere we place a point lens of 100M (cid:12) close to a type-I macroimage(source) in the presence of an external shear ( 𝛾 ) with no convergence( 𝜅 = 𝛾 = | 𝛾 | = | 𝛾 | . Forsuch a case, the effective lens potential in Eq. (10) can be written as 𝜓 total = ln (cid:18)√︃ 𝑥 + 𝑥 (cid:19) + 𝛾 (cid:16) 𝑥 − 𝑥 (cid:17) . (15)For the potential written above, we do not have an analytic solutionfor the diffraction integral, unlike in the case of point mass lens.Therefore, to perform the numerical test for this case, we directlyevaluate the double integral in Eq. (5) numerically and compare itwith the one obtained via our code. However, the direct evaluationis slow and does not work well for higher frequencies where theintegrand becomes too oscillatory.For the computation of the amplification factor, the comparisonbetween the direct numerical evaluation (dotted-black lines) andthe one obtained via our code (solid-coloured lines) is shown in themiddle and right panel of Fig. 2. The analysis has been done for threedifferent values of shear, 𝛾 = {− . − . − . } , keeping the sourceposition fixed at ( 𝑦 , 𝑦 ) = . ( cos ( 𝜋 / ) , sin ( 𝜋 / )) . Again, weobserve that there is excellent agreement between direct numericalintegration and the adopted numerical method of UG95. Also, the amplification curves approach the strong lensing amplification valuefor low frequency values (no − microlensing limit) and the phase shiftcurves approach zero, as expected.The corresponding calculations of (cid:101) 𝐹 ( 𝑡 ) are shown in the leftpanel of Fig. 2. We have again normalised the plots by dividingthe originally obtained ones by a factor of 2 𝜋 . This normalizationensures that the curves approach to value √ 𝜇 = ( − 𝛾 ) − / in theirno − microlensing (strong lensing) limit at large time delays. Thetime delay function, in this case, includes four stationary points, atleast two of which are always real. In (cid:101) 𝐹 ( 𝑡 ) plots in Fig. 2, we observethat two microimages form in the case of 𝛾 = − . 𝑡 = 𝑡 ∼ . 𝐹 ( 𝑓 ) , using our code, throughoutour analysis in the paper. In this subsection, we describe the numerical scheme that is used tocompute the amplification factor for a type-II (saddle point) image.Unlike in the case of a type-I (minima) image, here contours neitherclose locally nor have global minima, as they are hyperbolic innature rather than elliptical. However, if one chooses a sufficientlylarge region, the contribution from the neighborhood of a saddlepoint is given by (cid:101) 𝐹 ( 𝑡 ) = − √︁ | 𝜇 − | log | 𝑡 − 𝑡 𝑖 | + non-singular part + constant (16)where 𝑡 𝑖 and 𝜇 − denote the time delay and magnification valuecorresponding to the saddle point, respectively, and the constantdepends on the size of the region. By sufficiently large, we meanthe size of the region should be such that | 𝑢 − 𝜇 − | − / (cid:29) 𝑢 denotes the arc parameter of the contour(the reader is referred to Appendix B of UG95 for further de-tails). The presence of a constant does not affect the computationof 𝐹 ( 𝑓 ) , especially when the integration range is chosen carefully, MNRAS000 , 1–16 (2021)
Anuj Mishra et al. γ = - γ = - γ = - - - - - - - t ( s ) F ∼ ( t ) - × - NumericalDirect Integration Frequency ( Hz ) | F | - - - Frequency ( Hz ) θ F Figure 2.
Test of our numerical code for a 100 M (cid:12) point mass microlens in presence of shear. We show our numerically computed (cid:101) 𝐹 ( 𝑡 ) and frequencydependent amplification factor 𝐹 ( 𝑓 ) = | 𝐹 | 𝑒 𝑖𝜃 𝐹 for three different values of the shear 𝛾 = 𝛾 ∈ {− . , − . , − . } . For comparison, we also calculate 𝐹 ( 𝑓 ) via direct (numerical) integration of Eq. (5) (dotted black lines) and compare it with the one obtained numerically using the method described insubsection 3.1 (solid coloured lines). The lens and the source are placed at redshifts 𝑧 d = . 𝑧 s =
2, respectively, and the source position is fixed to ( 𝑦 , 𝑦 ) = . ( cos ( 𝜋 / ) , sin ( 𝜋 / )) . The (cid:101) 𝐹 ( 𝑡 ) curves have been normalised by the factor 2 𝜋 , so that they approach their no-microlens (strong lensing) limit( √ 𝜇 = ( − 𝛾 ) − / ) at large time delay values. such that Re (cid:8)∫ d 𝑡 𝑒 𝑖 𝜋 𝑓 𝑡 (cid:9) =
0, and at higher frequencies whereIm (cid:8)∫ d 𝑡 𝑒 𝑖 𝜋 𝑓 𝑡 (cid:9) (cid:28) (cid:101) 𝐹 ( 𝑡 ) and 𝐹 ( 𝑓 ) . In our simulation, we first find this image and measure thetime delay values relative to this image, i.e., we fix the arrival timeof the dominant saddle image at 𝑡 = (cid:101) 𝐹 ( 𝑡 ) values symmetrically about 𝑡 = 𝐹 ( 𝑓 ) . Atime range of the order O( ) s is sufficient for most cases since onlythe region closer to the divergence would mainly contribute. This isdue to the fact that the contribution of the nearly flat part of (cid:101) 𝐹 ( 𝑡 ) inthe inverse Fourier transform will mostly be averaged out.We test our numerical recipe for two cases, namely, in theabsence of microlens and in the presence of a 100 M (cid:12) microlensleading to a four − microimage configuration. For both cases, wefix our macro-magnification value to 𝜇 = − .
4. In the case ofno microlens, one expects to recover strong lensing values for theamplification factor, i.e., | 𝐹 | = √︁ | 𝜇 | and 𝜃 𝐹 = − 𝜋 /
2, since there willbe no interference in the absence of microlenses. We indeed recoverthese values as shown in the middle and right panel of Fig. 3, where | 𝐹 | = √ . 𝜃 𝐹 = − 𝜋 / 𝐹 ( 𝑓 ) values asobtained through the code numerically.In the presence of a 100 M (cid:12) microlens, we keep the sourceinside the caustic to get a four − microimage geometry. The timedelay and magnification of these microimages are (cid:16) 𝑡 − s , 𝜇 (cid:17) ∈{(− . , . ) , (− . , − . ) , ( . , − . ) , ( ., − . )} .These microimages correspond to the discontinuity and spikes inthe (cid:101) 𝐹 ( 𝑡 ) as shown in the left panel of Fig. 3. Using Eq. (7), onecan then find the 𝐹 ( 𝑓 ) in the geometrical optics limit ( 𝑓 𝑡 d (cid:29) ) and then compare it with the computed 𝐹 ( 𝑓 ) at high frequencies.The comparison is shown in the middle and right panel of Fig. 3,where the dotted black curve represents the geometrical optics limit of 𝐹 ( 𝑓 ) obtained using Eq. (7) and solid orange curve shows thenumerically computed 𝐹 ( 𝑓 ) using the (cid:101) 𝐹 ( 𝑡 ) curve as shown in Fig. 3.As we can see, for both cases, the numerically computed 𝐹 ( 𝑓 ) and the expected geometrical optics limit values are in excellentagreement. Furthermore, in Fig. 3, the (cid:101) 𝐹 ( 𝑡 ) curves in both casescontain a dominant logarithmic divergence at 𝑡 = 𝐹 ( 𝑓 ) below (cid:46) Hz is mainly dueto the diffraction effects and clearly demonstrates why one needs toincorporate wave optics in such cases.
We first define our macromodel which is kept fixed throughout ouranalysis while other model parameters are varied. We consider anisolated elliptical galaxy, as a lens, at redshift 𝑧 d =0.5, and modelledthe smooth matter fraction with a singular isothermal ellipsoid (SIE)density profile and a velocity dispersion ( 𝜎 vd ) of 230 km s − (takenas a rough mean 𝜎 vd from lens sample of Sonnenfeld et al. 2013).Next, we calculate the surface density of compact objects at anygiven position in the macromodel. The density profile of the pop-ulation of compact objects is modeled using the Sérsic profile (seeEquation 8 in Vernardos 2019). Next, to determine the mass functionof the compact objects, we use the Chabrier IMF Function (Chabrier2003) with the mass range 0.01 M (cid:12) to 200 M (cid:12) . We then evolve thisinitial population for ∼ − final mass relation from the Binary Population And SpectralSynthesis (BPASS, Eldridge et al. 2017) code. Before the evolu-tion, all compact objects are assumed to be stars. However, once weevolve the population, the stars above 1.2 M (cid:12) become remnants astheir life span is much shorter than the low − mass stars. As a result,the final population of single objects comprise of stars and stellarremnants (i.e., white dwarfs, neutron stars and black holes). Thetotal fraction of the mass in the mass range 𝑚 ∈ (0.01, 0.08) M (cid:12) isaround 5%. This mass range predominantly affects the frequenciesabove the higher end of the LIGO frequency range and the relativeerror due to the removal of this mass range is about ∼ O( ) % inthe 𝐹 ( 𝑓 ) curve for typical strong lensing amplification values. Asa result, for computational efficiency, we remove the stars below MNRAS , 1–16 (2021)
L of GWs: Effect of Microlens Population - - - - t ( s ) F ∼ ( t ) - - - - - Geometric Optics limit Frequency ( Hz ) | F | Numerical - - - Frequency ( Hz ) θ F Figure 3.
Test of our numerical code for the case of microlensing of a saddle-like macroimage. We show the comparison of 𝐹 ( 𝑓 ) in the geometrical opticslimit ( 𝑓 𝑡 d (cid:29)
1) for two scenarios, namely, in the absence and the presence of a 100 M (cid:12) microlens leading to a four-microimage configuration (see orangecurve in the inset of the left-most panel). In the middle and the right panel, the black dotted curves denote geomtrical optics regime values (Eq. (7)) whilethe solid colored ones have been obtained numerically by inverse Fourier transforming the computed (cid:101) 𝐹 ( 𝑡 ) . For no microlens case, geometric optics limit isequivalent to the strong lens limit ( 𝑓 𝑡 d (cid:28) (cid:38) Hz where 𝐹 ( 𝑓 ) and 𝐹 ( 𝑓 ) (cid:12)(cid:12) geo match (dotted black and orange curve at high frequencies). Table 1.
Lens parameter values for minima and saddle points used insimulations. The ( 𝜅, 𝛾 ) are the local convergence and shear values due tothe (smooth) macrolens mass distribution. The 𝜅 star is the local convergencedue to the mass in compact objects. The 𝜇 represents macro − magnification,and Σ • represents the microlens density. 𝜅 𝛾 𝜅 𝑠 tar √ 𝜇 Σ • (M (cid:12) pc − )Minima0.276 0.276 0.013 1.49 270.354 0.354 0.024 1.85 500.412 0.412 0.035 2.40 720.467 0.467 0.046 3.87 950.495 0.495 0.052 10.01 108Saddle points0.504 0.504 0.11 11.05 1130.546 0.546 0.12 3.21 1350.722 0.722 0.16 1.50 239 (cid:12) from our population. Since galaxies also comprise of astellar population in binary systems, we account for these in a sta-tistical manner based on Duchêne & Kraus (2013). Thus, the finaldistribution of the mass of microlenses is confined to a narrowerrange than the initial population, 𝑚 • ∈ ( . , ) M (cid:12) . The averagemass of a microlens in our simulation is ∼ .
44 M (cid:12) . In other words,the number of microlenses present per solar mass is ∼ .
28, wherewe are treating each binary system as a single microlens.Since type I (minima) and type II (saddle points) are the mostcommon types of lensed images seen in galaxy − scale lenses, weconsider the microlensing effects for these two images in our analy-ses. For an SIE lens, the values of 𝜅 and 𝛾 due to the macrolens, at theposition of lensed images, are equal to each other, i.e., | 𝜅 | = | 𝛾 | (e.g.,Vernardos 2019). Owing to the azimuthal symmetry, it suffices toconsider varying positions in the radial direction for the minima andsaddle point images which then correspond to varying microlensdensities, Σ • , and macro − magnifications, 𝜇 ( 𝜅, 𝛾 ) . Thus, we samplea wide range in ( √ 𝜇, Σ • ) for both types of images without having tosimulate a large number of strong lenses with varied lensed imageconfigurations (e.g., see Figure 2 of Vernardos 2019 for typical 𝜅, 𝛾 covered by large sample of mock lenses). We generate simulationsfor five cases of ( √ 𝜇, Σ • ) for the minima and and three cases for thesaddle points which are listed in Table 1. For each ( √ 𝜇, Σ • ), we first simulate a large box with an area of about100 pc . We then populate this box with microlenses of surfacemass density, Σ • such that they follow the evolved mass function.We draw 36 patches from the large box such that each patch has anarea of ∼ . . In this way, we generate 36 realisations of theamplification curve, 𝐹 ( 𝑓 ) corresponding to each ( √ 𝜇, Σ • ).Dividing the larger box of density Σ • into smaller patches al-lows us to have density variations. As a result, the patches can havedensities Σ • ± 𝜎 d where 𝜎 d is the scatter. To obtain the amplifi-cation curves, we first have to generate (cid:101) 𝐹 ( 𝑡 ) curves, as explainedin §3. Calculating the (cid:101) 𝐹 ( 𝑡 ) is the most computationally expensivepart of the whole simulation, and depends strongly on the numberof microlenses (as it increases the number of microimages). Ad-ditionally, the macro-magnification also affects the computationalcost because with increasing values, we get larger and more compli-cated time − delay contour structures since the macro − magnificationamplifies the microlensing effects.As discussed in §3, once we generate a sufficiently large num-ber of points for (cid:101) 𝐹 ( 𝑡 ) , we then interpolate it using Hermite in-terpolation method to obtain the function (cid:101) 𝐹 ( 𝑡 ) . By taking the in-verse Fourier transform, we get the required amplification factor, 𝐹 ( 𝑓 ) = | 𝐹 ( 𝑓 )| 𝑒 𝑖𝜃 𝐹 ( 𝑓 ) , in a given frequency range. For caseswhere Σ • (cid:38)
100 M (cid:12) , we perform an optimisation where we re-move microlenses 𝑚 l (cid:46) . (cid:12) that are present outside an areaof ∼ . from the center. In this way, we increase the com-putational efficiency substantially at the cost of introducing only ∼ O( ) % relative error at low macro − magnifications. However, forhigh macro − magnifications ( 𝜇 (cid:38) (cid:38) Hz). Therefore, ourresults for high values of ( √ 𝜇, Σ • ) may underestimate the effects ofmicrolensing. In this section, we present the results of our microlensing analysis.For this, we draw realisations of the amplification factor 𝐹 ( 𝑓 ) ,using the formalism discussed in §3, for cases where a microlenspopulation is embedded in a macromodel, as discussed in §4. Wethen study the broad effects of microlensing on GW waveforms viamismatch analysis between the lensed and unlensed waveforms. MNRAS000
100 M (cid:12) , we perform an optimisation where we re-move microlenses 𝑚 l (cid:46) . (cid:12) that are present outside an areaof ∼ . from the center. In this way, we increase the com-putational efficiency substantially at the cost of introducing only ∼ O( ) % relative error at low macro − magnifications. However, forhigh macro − magnifications ( 𝜇 (cid:38) (cid:38) Hz). Therefore, ourresults for high values of ( √ 𝜇, Σ • ) may underestimate the effects ofmicrolensing. In this section, we present the results of our microlensing analysis.For this, we draw realisations of the amplification factor 𝐹 ( 𝑓 ) ,using the formalism discussed in §3, for cases where a microlenspopulation is embedded in a macromodel, as discussed in §4. Wethen study the broad effects of microlensing on GW waveforms viamismatch analysis between the lensed and unlensed waveforms. MNRAS000 , 1–16 (2021)
Anuj Mishra et al. | F | ( ) - θ F | F | ( ) - - θ F | F | ( ) - - - θ F | F | ( ) - - - θ F Frequency ( Hz ) | F | ( ) - - - Frequency ( Hz ) θ F Figure 4.
Effect of microlens population on the minima images. The left column represents the absolute value of the amplification factor (dimensionless) andthe right column represents the corresponding phase shift factor (in radians). The x-axis shows frequencies specific to the LIGO detector. Each row correspondsto a simulation at a given strong lensing amplification and microlens density (see Table 1). The curves in each panel correspond to all of the 36 realisationsfor each simulation. MNRAS , 1–16 (2021)
L of GWs: Effect of Microlens Population | F | ( ) - - - θ F | F | ( ) - - - - - - - θ F Frequency ( Hz ) | F | ( ) - - - - - Frequency ( Hz ) θ F Figure 5.
Effect of microlens population on the saddle point images. The left column represents the absolute value of the amplification factor (dimensionless)and the right column represents the corresponding phase shift factor (in radians). The x-axis shows frequencies specific to the LIGO detector. Each rowcorresponds to a simulation at a given strong lensing amplification and microlens density (see Table 1). The curves in each panel correspond to all of the 36realisations for each simulation.
As discussed in §4, we generate realisations for five cases by varyingthe microlens surface densities, Σ • , and the macro-magnifications, 𝜇 , found typically at the location of minima (see Table 1). Fur-thermore, we draw 36 realizations for each case to understand thetypical and extreme situations, if any. The effects of microlens pop-ulation on type-I (minima) images are shown in Fig. 4. Each rowshows all of the 36 realisations for each case. In each row, the abso-lute value of amplification (dimensionless) and the correspondingphase shift (in radians) due to the combined effect of strong lensingand microlensing are shown in the left and right panels, respectively.When inspecting any row, we find that, at low GW frequencies( ∼ √ 𝜇 )and a phase shift of zero, since the oscillations due to microlens-ing are minimal to none. This behavior is expected, as explainedin §2 and §3. Towards higher frequencies, as the effects due to mi-crolensing become significant, the curves begin to show strongeroscillations. This is owing to the formation of many microimages with sufficiently long time-delays (such that 𝑓 𝑡 𝑑 ∼
1) with respectto the macrominimum (global minima of the time delay surface).Some realisations show extreme excursions from the rest of theset. Interestingly, the blue and red curve in the top-most panel area consequence of several low-mass stars (cid:46) (cid:12) located in theneighbourhood of the source, conspiring together to mimic the ef-fect of a heavy microlens placed close to the source, thereby causingdramatic effects in those realisations. From our analysis, we inferthat our choice of drawing 36 realisations is reasonable enough toproduce such extreme cases.As we go to higher frequencies, the distortions increase and be-come more significant. Since the microimages from low mass starshave smaller time delay values, a greater number of microimagesstart contributing to the diffraction integral 𝐹 ( 𝑓 ) at higher frequen-cies. As a result, the amplification factor 𝐹 ( 𝑓 ) is more random andchaotic at higher frequencies for low macro-amplification values( √ 𝜇 ). However, as √ 𝜇 increases, the amplification of microimagesincreases and can lead to the formation of multiple dominant mi-croimages (microimages having high amplification) contributing MNRAS000
1) with respectto the macrominimum (global minima of the time delay surface).Some realisations show extreme excursions from the rest of theset. Interestingly, the blue and red curve in the top-most panel area consequence of several low-mass stars (cid:46) (cid:12) located in theneighbourhood of the source, conspiring together to mimic the ef-fect of a heavy microlens placed close to the source, thereby causingdramatic effects in those realisations. From our analysis, we inferthat our choice of drawing 36 realisations is reasonable enough toproduce such extreme cases.As we go to higher frequencies, the distortions increase and be-come more significant. Since the microimages from low mass starshave smaller time delay values, a greater number of microimagesstart contributing to the diffraction integral 𝐹 ( 𝑓 ) at higher frequen-cies. As a result, the amplification factor 𝐹 ( 𝑓 ) is more random andchaotic at higher frequencies for low macro-amplification values( √ 𝜇 ). However, as √ 𝜇 increases, the amplification of microimagesincreases and can lead to the formation of multiple dominant mi-croimages (microimages having high amplification) contributing MNRAS000 , 1–16 (2021) Anuj Mishra et al. to 𝐹 ( 𝑓 ) . Such contributions cause a relatively lesser chaotic butstronger modulations (see bottom two rows of Fig. 4).Although not visible in the LIGO frequency range, at suffi-ciently high frequencies, we approach the geometrical optics regimewhere again the amplification becomes independent of frequencywhen averaged over a frequency range. The value of the magnifica-tion in the geometrical optics regime can be obtained, in principle,from the (cid:101) 𝐹 ( 𝑡 ) curves along with the rough estimate of the frequencyafter which this limit is reached. In Fig. B1, the (cid:101) 𝐹 ( 𝑡 ) at very lowtime delays ( ∼ − ) s, where the curve is almost flat, gives theamplification at geometrical optics limit, as opposed to the macro-amplification at large (cid:101) 𝐹 ( 𝑡 ) (where all curves tend after ∼ ∼ − Hz with varying amplification. This ampli-fication at the geometric optics limit tells us whether microlensingcaused the overall amplification or deamplification of our signal.We notice that except for the case of ( . , ) , almost 50% ofthe realisations had overall amplification.However, since our chosen area of the patch is not sufficient forthe computation of the amplification factor at such high frequencies( (cid:38) Hz), we have mostly underestimated our results in the ge-ometrical optics limit. Therefore, in real case scenarios, we shouldexpect a net amplification for most of the type-I (minima) images inthe geometric optic limit, which is consistent with our observationin the LIGO frequency range. The fact that minima tend to be ampli-fied, on average, due to microlensing is consistent with microlensingstudies in the EM domain (e.g., Schechter & Wambsganss 2002).
Contrary to minima, saddle points (or images) are formed at a rel-atively higher microlens densities and their macro-magnificationvalues decrease with increasing microlens densities (see Table 1).Owing to this and the fact that the microlensing effects are not assignificant for low macro-amplification values, we generate realisa-tions for three cases only, thereby reducing the computational cost.As in the case of minima, every row of Fig. 5 shows the amplifica-tion factor and phase shifts in the left and right panels, respectivelyfor all of the 36 realisations. It is evident from Fig. 5 that the mi-crolensing effects are more significant at higher √ 𝜇 even when thedensities are comparatively low.For any panel, the curves converge to the macro-magnificationat lower frequencies and the scatter increases at higher frequenciessimilar to the case of minima. Moreover we do recover the Morsephase in our simulations which causes an overall phase shift of − 𝜋 / √ 𝜇 , we observe very strong modulations evenat low frequencies ( ∼ −
100 Hz). As many of the GW signalstend to have a longer inspiral in the low-frequency regime, such amodulation at low frequencies is likely to affect the match filteringand parameter estimation significantly (see §5.6).In general, we find saddle points tend to be deamplified onaverage, as opposed to minima, which is also consistent with mi-crolensing studies in the EM domain.Although the microlens densities of 113 M (cid:12) pc − for saddleand 108 M (cid:12) pc − for minima and the corresponding strong lensamplification are similar to each other, the corresponding amplifi-cation and phase shift plots (top-most row in Fig. 5 and bottom-mostrow in Fig. 4, respectively) show significant differences. In this subsection, we discuss the effect of varying microlens density, Σ • , on the amplification factor 𝐹 ( 𝑓 ) . For this, we have fixed ourmacro-amplification value to √ 𝜇 = .
40 and the comparison is doneby drawing 36 realisations for each of the three sets of (√ 𝜇, Σ • ) ∈{( . , ) , ( . , ) , ( . , )} . The results are shown in Fig. 6.From the figure, it is clear that, on average, distortions andscatter are proportional to the microlens density, i.e., microlensingeffects are most significant for the highest density case of 72 M (cid:12) pc − and least for 24 M (cid:12) pc − . Also, as we increase the density,these distortions start becoming significant from relatively lowerfrequencies (see top panel). This behavior is expected from the factthat different microlens population affects the potential in differentways. With increasing microlens density / number, the microlensingconfiguration also gets complicated and leads to an increase in thenumber density around the source. The region with higher numberdensity acts as a heavier microlens and leads to the formation ofa relatively greater number of significantly amplified microimages,thereby contributing more to the amplification factor. The reasoningcan be understood from Eq. (11), which implies that if there are 𝑁 number of microlenses of mass 𝑚 present in a small neighborhoodaround some 𝑥 o , i.e., if the position of the microlenses, 𝑥 i , is suchthat | 𝑥 i − 𝑥 | (cid:28)
1, then the net effect of those microlenses will besimilar to that of a single microlens of mass 𝑁𝑚 present at 𝑥 o . Whenstudying the microlensing effect of a population, the other crucialfactors apart from population density are macro-model potential inwhich it is embedded and the frequency range under consideration.As explained in §2, the microlensing effect from a point lens of mass 𝑚 is significant at a frequency 𝑓 only when the order of the timedelay between the two images 𝑡 d (cid:38) 𝑓 − . This feature is translatedto the case of populations as well. Due to this, in lower densitypopulations where the abundance of heavy mass stars, as well asdensity fluctuations, is low, like in the case of 24 M (cid:12) pc − in ourpopulation, the microlensing effects become significant at relativelyhigher frequencies (cid:38) Hz.In the bottom row of Fig. 6, we quantify the microlensing effectby computing the normalised standard deviation relative to the nomicrolens limit ( | 𝐹 | = √ 𝜇 and 𝜃 F =
0) for each realisation, i.e., 𝜎 = (cid:104)∫ 𝑑𝑓 ( 𝐺 ( 𝑓 )) (cid:105) / √︃∫ 𝑑𝑓 , (17)where 𝐺 ( 𝑓 ) = (cid:0) | 𝐹 ( 𝑓 )|/√ 𝜇 − (cid:1) for 𝜎 [| 𝐹 |] and 𝐺 ( 𝑓 ) = ( 𝜃 𝐹 ( 𝑓 ) − ) for 𝜎 [ 𝜃 𝐹 ] . The median over the realisations corre-sponding to 𝜎 [| 𝐹 |] are 10 . .
0% and 28 . 𝜎 [ 𝜃 𝐹 ] the values are 3 . .
9% and 7 . 𝐹 ( 𝑓 ) curves as shown in the top row.The colors of the markers help us study the effect of differenttypes of population in our realisation. It is worth noting that the real-isations corresponding to the extreme behaviours are not necessarilythe ones containing a heavy star. This implies that the distribution ofstars around the source is a crucial factor for studying microlensingdue to a population and, therefore, the inverse problem of find-ing the microlensing configuration from lensed GWs is a highlynonlinear and multivariate problem and would require further in-vestigation. This inference also shows that in a typical case of GWlensing, microlenses of low mass O( ) 𝑀 (cid:12) cannot be neglected MNRAS , 1–16 (2021)
L of GWs: Effect of Microlens Population ( )( )( ) Frequency ( Hz ) | F | - - - Frequency ( Hz ) θ F { m. } max ≥
10 M ☉ { m. } max ∈ [
5, 10 ) M ☉ { m. } max ∈ [ ) M ☉ { m. } max < ☉
20. 40. 60. 80.0.0.10.20.30.4
Microlens Density ( ∑ . ) σ [ | F | ]
20. 40. 60. 80.0.0.10.20.30.4
Microlens Density ( ∑ . ) σ [ θ F ] Figure 6.
Effect of varying microlens density. We show comparison between the amplification curves corresponding to three different microlens densities, Σ • ∈ { , , } M (cid:12) pc − , for a fixed macro-amplification value of √ 𝜇 = .
40. The top row shows the amplification factor, | 𝐹 | , and the phase shift, 𝜃 F , forall the three cases. In the bottom row, we study the overall scatter due to microlensing and the effect of different kinds of population. 𝜎 [ | 𝐹 |] and 𝜎 [ 𝜃 𝐹 ] denote the normalised standard deviation of a realisation in | 𝐹 ( 𝑓 ) | and in 𝜃 𝐹 ( 𝑓 ) relative to the strong lensing values (no-microlens limit) in the LIGOfrequency range (see Eq. (17)). Here, each marker represents a specific realisation while its color represents the mass range in which the maximum mass ofthat realisation, { 𝑚 • } max , lies. in general, especially for significantly higher values of microlensdensities ( (cid:38) 𝑀 (cid:12) pc ) and macro-magnification values ( 𝜇 (cid:38) √ 𝜇 ) value In this subsection, we discuss the effect of varying macro-amplification value, √ 𝜇 , on the amplification factor 𝐹 ( 𝑓 ) . For this,the microlens density has been fixed to Σ • =
27 M (cid:12) pc − while wevary √ 𝜇 for four different values as √ 𝜇 ∈ { . , . , . , . } . Thecomparison is done by drawing 36 realisations for each of the foursets of (√ 𝜇, Σ • ) . The results are shown in Fig. 7.In the top panel of Fig. 7, we clearly observe that the over-all scatter width among realisations increase significantly as weincrease √ 𝜇 , which implies that microlensing effects are stronglydependent upon √ 𝜇 . For example, for the realisations correspondingto √ 𝜇 ∈ { . , . } , the microlensing effects are not much signifi-cant (i.e. | 𝐹 | ∼ √ 𝜇 and 𝜃 𝐹 ∼
0) throughout the observed frequencyrange, especially, for 𝑓 (cid:46) Hz,as compared to higher √ 𝜇 val-ues. This fact becomes even more evident from the bottom panelof Fig. 7, where we show the standard deviation 𝜎 of realisationsrelative to the strong lens limit ( | 𝐹 | = √ 𝜇 and 𝜃 𝐹 =
0) as a function of frequency, i.e., 𝜎
2, 3 ( 𝑓 ) = (cid:20) 𝑛𝑟 = (cid:205) 𝑖 ( 𝐺 ( 𝑓 )) (cid:21) / √︃(cid:205) 𝑖 𝑖 , (18)where 𝐺 ( 𝑓 ) = (cid:0) | 𝐹 𝑖 ( 𝑓 )|/√ 𝜇 − (cid:1) for 𝜎 [| 𝐹 |]( 𝑓 ) and 𝐺 ( 𝑓 ) = (cid:0) 𝜃 𝐹 𝑖 ( 𝑓 ) − (cid:1) for 𝜎 [ 𝜃 𝐹 ]( 𝑓 ) and the summation is over all 36 re-alisations. The quantity thus computed is purely dependent uponthe microlensing effects. As we can see, both 𝜎 [| 𝐹 |] and 𝜎 [ 𝜃 𝐹 ] rise with the increasing frequency, which signifies that, overall, themicrolensing effects rise as we move toward higher frequencies.Furthermore, as we vary √ 𝜇 from 1 . . 𝐹 ( 𝑓 ) athigh frequencies, we observe that 𝜎 [| 𝐹 |] rises from ∼
6% to 60%,while 𝜎 [ 𝜃 𝐹 ] shows an increase from ∼
4% to 40%. Such a steeprise clearly demonstrates that the microlensing effects are stronglydependent upon the macro-amplification value √ 𝜇 . Typically, lensing galaxies are of early-type with old stellar popula-tion. Knowledge of the stellar Initial Mass Function (IMF) is an im-portant pursuit to understand galaxy formation and evolution. Manystudies aim to understand if there is a universal IMF that can describeall early-type galaxies and if it evolves as a function of redshift (e.g.,Bastian et al. 2010; Smith 2020). Thus far, there is no consensus
MNRAS000
MNRAS000 , 1–16 (2021) Anuj Mishra et al. ( )( )( )( ) | F | - - - θ F ( )( )( )( ) Frequency ( Hz ) σ [ | F | ] Frequency ( Hz ) σ [ θ F ] Figure 7.
Effect of varying macro-amplification ( √ 𝜇 ). We show the comparison between the amplification curves corresponding to four different √ 𝜇 ∈{ . , . , . , . } for a fixed microlens density of √ Σ • =
27 M (cid:12) pc . The top row shows the amplification factor, | 𝐹 | , and the phase shift, 𝜃 F , for all thefour cases. In the bottom row, we study the microlensing effects by computing the standard deviation of realisations relative to the strong lens limit ( | 𝐹 | = √ 𝜇 and 𝜃 𝐹 =
0) as a function of frequency (see Eq. (18)). on a single universal form for the IMF. The Chabrier (Chabrier2003) and Salpeter (Salpeter 1955) IMFs are some of the standardIMFs known to fit early-type galaxies well (e.g., Lagattuta et al.2017; Vaughan et al. 2018) including those from strong lensingstudies (e.g., Treu et al. 2010; Ferreras et al. 2010; Smith et al.2015; Leier et al. 2016; Sonnenfeld et al. 2019).To investigate the effects of stellar IMF on the amplification,we compare Salpeter IMF with our default choice of IMF in thiswork, i.e., Chabrier. The comparison is done at minima by drawing36 realisations for two cases of ( √ 𝜇 , Σ • ), namely, (1.85, 50) and(2.40, 72). The choice of these numbers is based on the fact thatthe corresponding strong lensing magnification ( 𝜇 ) values cover therange of magnification that we find in typical strong lens systems.The results are shown in Fig. 8.On closer inspection, it is evident from the plots that the effectof microlensing in the case of Salpeter IMF becomes prominent at arelatively higher frequency than that of Chabrier IMF. For example,in the top two panels of Fig. 8, observing the lower frequencyrange (cid:46)
200 Hz in both | 𝐹 | and 𝜃 𝐹 , we find that the scatter in thecase of Chabrier IMF is more, while for Salpeter IMF the curvesare mostly following an average trend. This observation is furtherconfirmed by the bottom-most panel where we explicitly show thestandard deviation among the realisations as a function of frequency,defined by Eq. (18), where 𝐺 ( 𝑓 ) = (| 𝐹 𝑖 ( 𝑓 )| − avg . [| 𝐹 𝑖 ( 𝑓 )|)) for 𝜎 [| 𝐹 |]( 𝑓 ) and 𝐺 ( 𝑓 ) = (cid:0) 𝜃 𝐹 𝑖 ( 𝑓 ) − avg . (cid:2) 𝜃 𝐹 𝑖 ( 𝑓 ) (cid:3)(cid:1) for 𝜎 [ 𝜃 𝐹 ]( 𝑓 ) and the summation is over all 36 realisations. The quantity thuscomputed is purely dependent upon the microlensing effects. wheresummation is over all 36 realisations, and avg.[ | 𝐹 𝑖 | ] and avg.[ 𝜃 𝐹 𝑖 ]denote the average value of 𝐹 ( 𝑓 ) obtained among our realisationsat frequency 𝑓 . Such behavior is expected because the Salpeter IMF is comparatively more bottom-heavy than the Chabrier IMF, i.e., itcontains a significantly high number of low mass stars compared tothe Chabrier IMF. Due to this, in the case of Salpeter IMF, the timedelay of microimages (relative to macrominima), with significantlyhigh amplification, is, on an average, relatively smaller which is thereason distortions coming later in 𝐹 ( 𝑓 ) for Salpeter IMF.Since the differences are not significant in the LIGO frequencyrange, the possible constraints on the IMF from strongly lensedGWs will require further investigation. Here, we quantify the effects of microlensing in the case of stronglylensed GW signals from compact binary coalescences (CBCs) bycomputing the match (noise-weighted normalised inner product)between the unlensed and the corresponding lensed waveforms (see§A). For generating the unlensed waveforms and computing thematch, we have used PyCBC package (Biwer et al. 2019, Usmanet al. 2016). The lensed GW waveforms have been obtained bymodifying the unlensed waveforms using the realisations of theamplification factor 𝐹 ( 𝑓 ) for ( 𝜇, Σ • ) as shown in Figs. 4 and 5.Due to normalisation, match does not contain any information aboutthe distance of the source or the strong lensing amplification value.Therefore, it is a pure measure of the microlensing effects. We firstdiscuss the results for some extreme cases and then for the typicalcases. We only focus on non-spinning and non-eccentric waveformsin this paper. For the case of (√ 𝜇, Σ • ) corresponding to saddlepoints, we have applied a low pass filter to the realisations beforecomputing the match in order to remove the small-scale oscillations MNRAS , 1–16 (2021)
L of GWs: Effect of Microlens Population Chabrier ( ) Salpeter ( ) Chabrier ( ) Salpeter ( ) Chabrier ( ) Salpeter ( ) Frequency ( Hz ) σ [ | F | ] Frequency ( Hz ) σ [ θ F ] Figure 8.
Effect of different stellar IMFs. The comparison between Chabrier and Salpeter IMFs is done for three cases, (√ 𝜇, Σ • ) ∈{( . , ) , ( . , ) , ( . , ) } . 𝜎 [ | 𝐹 |] and 𝜎 [ 𝜃 𝐹 ] denote the scatter among realisations in the absolute value | 𝐹 | of the amplification factor (left)and the phase shift 𝜃 F (right) as defined in Eq. (18), where 𝐺 ( 𝑓 ) = ( | 𝐹 𝑖 ( 𝑓 ) | − avg . [ | 𝐹 𝑖 ( 𝑓 ) |)) for 𝜎 [ | 𝐹 |] ( 𝑓 ) and 𝐺 ( 𝑓 ) = (cid:0) 𝜃 𝐹 𝑖 ( 𝑓 ) − avg . (cid:2) 𝜃 𝐹 𝑖 ( 𝑓 ) (cid:3)(cid:1) for 𝜎 [ 𝜃 𝐹 ] ( 𝑓 ) . The result is based upon 36 realisations of 𝐹 ( 𝑓 ) computed for each of the six cases. introduced due to numerical error and, therefore, to not overestimateour mismatch results. In Fig. 9, we show the match values for two cases of ( 𝜇, Σ • ) , namely, ( . , ) and ( . , ) (top and bottom panels, respectively).These cases correspond to the extreme ones that we have covered forminima and saddle type images, respectively, as they correspond toa large macro-amplification value ( ∼ ) compared to the typicalvalues ( 𝜇 (cid:46)
10, see Table 1).In Fig. 9, M p and M s represent the primary and secondarymass of a binary, respectively, such that M p ≥ M s and mass ratio 𝑞 = M s / M p ∈ ( , ] . The first column shows the match as a functionof the total mass of the (non-spinning) binary for 𝑞 =
1, while secondand third column shows match values plotted against varying massratio parameter for fixed M p and M s values of M p =
50 M (cid:12) andM s = . (cid:12) , respectively.We notice that, in extreme cases, the mismatch can even go upto ∼ 𝑞 =
1, on average,the mismatch increases as we lower the mass of the binary (first col-umn). The reason can be understood from the fact that the smallermass binaries have longer GW signals and have higher chirp fre-quencies ( 𝑓 ( 𝑡 ) ∝∼ 𝑞 − / 𝑀 − / , where M is the total mass of binary(Abbott et al. 2017)). Microlensing effects are mostly significant athigher frequencies and hence, introduced mainly by the distortionsin 𝐹 ( 𝑓 ) near the frequency with maximum strain amplitude 𝑓 max ,which is roughly twice the orbital frequency at ISCO (inner-moststable circular orbit). However, since most of the power of the signalis contained in the lower frequencies, the nature of 𝐹 ( 𝑓 ) at lowerfrequencies becomes important to estimate the overall microlensingeffect on the GW signal. In case distortions are not as significantat low frequencies, then the mismatch value may as well decreaseowing to the greater length of the signal invisible to the lens. Ingeneral, there will be trade-off between these quantities, namely,nature of 𝐹 ( 𝑓 ) at lower frequencies and near 𝑓 max , and the lengthof the signal which is related to its chirp evolution. Even thoughthe mismatch should overall increase with increasing 𝑓 max , for aparticular realisation, oscillations could be observed because of thestrong dependency of mismatch on the modulation of 𝐹 ( 𝑓 ) near 𝑓 max , rather than 𝑓 max alone. This is the reason we see some os-cillations initially and not a monotonic increase in match for the case of ( . , ) . The match rises with increasing binary massbecause 𝑓 max decreases, and so does the microlensing effects. How-ever, in case of (11 .
05, 113), we observe, roughly, a monotonic risein match because in addition to its dependency on 𝑓 max , mismatchis introduced mainly by strong modulations in 𝐹 ( 𝑓 ) at lower fre-quencies (top row in Fig. 5), which are common for all waveforms.Therefore, mismatch is expected to increase with increasing lengthof the signal (a gradual frequency evolution). Analysing the fre-quency range ∼ −
100 Hz in Figs. 4 and 5 and studying theaverage behaviour, we see that, | 𝐹 | varies from ∼ − . − ( . , ) , whereas for ( . , ) case it variesfrom ∼ . − − ( . , ) .In the second column, we observe different trends betweenthe top and the bottom panel. For this case, GW signal lengthwill be inversely proportional to the mass ratio. Therefore, when themodulations in 𝐹 ( 𝑓 ) are significant at lower frequencies ( ∼ − ( . , ) case, 𝑓 max seems to be a dominant factor leading to a decreasing trend of thematch with increasing mass ratio. Whereas for saddle, as before, thetrade-off between the effects is strong, leading to an overall smoothbehaviour with some exceptions on either side. In Fig. 11, we show the minimum match values found among therealisations corresponding to the typical cases of ( 𝜇, Σ • ) observedin lensing galaxies (see Figs. 4 and 5), as we vary the total mass ofthe binary.It is worth noting that for typical cases, the mismatch hardly ex-ceeds one percent and is mostly within sub-percent levels. The case ( . , ) can be considered to lie within typical and extreme cases(as 𝜇 ∼ ∼ .
5% de-
MNRAS , 1–16 (2021) Anuj Mishra et al.
50 100 150 20093949596979899100
Total Mass ( M ☉ ) M a t c h ( % ) ( ) M p = M s Mass Ratio ( q = M s / M p ) M a t c h ( % ) ( ) M p = ⊙ Mass Ratio ( q = M s / M p ) M a t c h ( % ) ( ) M s = ⊙
50 100 150 20093949596979899100
Total Mass ( M ☉ ) M a t c h ( % ) ( ) M p = M s Mass Ratio ( q = M s / M p ) M a t c h ( % ) ( ) M p = ⊙ Mass Ratio ( q = M s / M p ) M a t c h ( % ) ( ) M s = ⊙ Figure 9.
Match between unlensed and the corresponding lensed waveforms for highly magnified minima (top row) and saddle point (bottom row) macroimages.The lensed GW waveforms have been obtained by modifying the unlensed waveforms using the respective realisations of the amplification factor 𝐹 ( 𝑓 ) asshown in Figs. 4 and 5 (keeping the same coloring scheme). In all the plots, 𝑀 p and 𝑀 s represent the primary and secondary mass of a binary, respectively,such that 𝑀 p ≥ 𝑀 s and mass ratio 𝑞 = 𝑀 s / 𝑀 p ∈ ( , ] . The first column shows match as a function of the total mass of the (non-spinning) binaries. Thesecond column shows match as a function of the mass ratio 𝑞 of the binary for a fixed value of 𝑀 p =
50 M (cid:12) , while the third column shows the same for a fixedvalue of 𝑀 s = . (cid:12) . Time (s) S t r a i n [ h ( t )]
1e 17
SLSL+ML Frequency (Hz) | h ( f ) | SLSL+ML
Figure 10.
Effect of microlensing on a strongly lensed GW waveform of 45 M (cid:12) +45 M (cid:12) =90 M (cid:12) binary in extreme case of ( . , ) corresponding asaddle point. ‘SL’ represents strong lensing, while ‘SL+ML’ stands for microlensing of strongly lensed waveform. In the left, we show effect in the time domainwaveform ℎ ( 𝑡 ) , while on the right, we show the same in frequency domain by plotting absolute value of frequency domain waveform | ˜ ℎ ( 𝑓 ) | as function offrequency. The mismatch value obtained in this case is ∼ . pending upon the source parameters. Therefore, while microlensingwill not affect the detectability of GW signals in typical situations,it might still affect the estimation of the source parameters in asignificant way, as its influence on the waveform can be degeneratewith source parameters. We investigated the effects of microlensing in the LIGO/Virgo fre-quency range by a population of point mass objects such as stars andstellar remnants embedded in the potential of a macrolens. In partic-ular, we calculated the microlensing effects for various combinationsof microlens density and macro − magnification, typically, found at the location of minima and saddle points in galaxy-scale lenses. Ad-ditionally, we individually explored the three most crucial factorsresponsible for microlensing effects due to a population, namely,macro − magnification value, the microlens density, and IMF. Foreach of the investigated cases, we generated 36 realisations to ro-bustly infer general trends and the effects of microlensing. Lastly,to understand the impact of microlensing on CBC signals, we com-puted the match between the unlensed and lensed waveforms. Themismatch thus obtained is a pure measure of microlensing effectsas it is independent of macro-magnification value and the extrinsicGW source parameters.Our main conclusions from these investigations are as follows. • The most important factor for microlensing to be significant
MNRAS , 1–16 (2021)
L of GWs: Effect of Microlens Population ( )( )( )( )( )( ) Total Mass ( M ☉ ) M i n ( m a t c h )( % ) Figure 11.
Minimum match (or maximum mismatch) among realisations(shown in Figs. 4 and 5), corresponding to typical values of ( 𝜇, Σ • ) , as wevary the total mass of the binary. The result is based upon 36 realisations of 𝐹 ( 𝑓 ) computed for each of the six cases. is the strong lensing amplification value ( √ 𝜇 ) regardless of otherparameters, such as the stellar density, type of images or IMF.This happens due to the fact that the image plane gets compressedby a factor of 𝜇 in the source plane leading to high density ofmicrocaustics in the source plane, which leads to the overlap of moreand more caustics. Moreover, higher 𝜇 allows relatively larger timedelays between sufficiently amplified microimages, which causesmodulation even at lower frequencies (e.g., Diego 2019; Diego2020). From our analysis, we observed that for sufficiently highmacrolensing magnifications ( 𝜇 (cid:38) • On average, the microlensing population tends to introducefurther amplification (de-amplification) for minima (saddle points)in the LIGO frequency range. Similar behaviour is also seen in thegeometric optics limit for minima and saddle points (e.g., Schechter& Wambsganss 2002; Foxley-Marrable et al. 2018). • Overall, we observed the microlensing effects to be more pro-nounced in the case of saddle point-like macroimages, especially atlower frequencies. The presence of microlenses becomes importantin the case of type-II macroimages as the probability of the sourcelying in the region of low magnification is significantly higher, un-like in case of type-I macroimages (e.g., Diego et al. 2018; see Figs.5 and 10 in Diego et al. 2019). • With increasing microlens density, we find an overall rise inscatter in 𝐹 ( 𝑓 ) and the distortions also become more significantfrom relatively lower frequencies. This could be attributed to thepresence of multiple microlenses at a location which can mimicthe effect of a heavier microlens, thereby increasing the probabilityof significantly amplified microimages with sufficiently high timedelay values to have interference. • The microlens population generated using the Chabrier IMFgives rise to an overall scatter in 𝐹 ( 𝑓 ) larger than the Salpeter IMF,at typical values of ( √ 𝜇, Σ • ). However, the effect of a bottom-heavyIMF, like Salpeter, increases at higher frequencies and becomescomparable to IMFs like Chabrier, which has a relatively largenumber of high mass stars. Since the impact of both IMFs, on the 𝐹 ( 𝑓 ) , are found to be nearly indistinguishable, the exact choice ofIMF model will probably not impact the accuracy of microlensingeffects. In other words, any microlensing signatures detected in thedata may not be useful in constraining the IMF and needs furtherinvestigation. • For the mismatch analysis, in general, microlensing effectsshould increase with increasing frequency as the time delay between microimages ranges from roughly O( − ) 𝜇 s. For a given signal,the mismatch introduced due to microlensing depends mainly onthe behaviour of 𝐹 ( 𝑓 ) near the frequency 𝑓 max where the GWstrain reaches maximum amplitude. However, when the distortionsin 𝐹 ( 𝑓 ) are significant even at lower frequencies ( (cid:46)
100 Hz), longerGW signals with gradual frequency evolution also develop a highermismatch. In general, the mismatch mainly depends on the natureof 𝐹 ( 𝑓 ) at lower frequencies and near 𝑓 max , and the length of thesignal which is related to its chirp evolution. • We find that in typical cases of ( √ 𝜇, Σ • ), the mismatch be-tween unlensed and lensed waveforms is mostly within sub-percentlevels ( (cid:46) 𝜇 (cid:38) 𝜇 ∼ O( − ) and hence, such possibilities cannot be ignored.In fact, our mismatch analysis suggests that the microlensing effectscannot be neglected for 𝜇 (cid:38)
15 when measuring the GW sourceparameters. This will be explored in more detail in the future.
ACKNOWLEDGEMENTS
We would like to thank S. More and V. Prasad for helpful discus-sions; A. Ganguly and Sudhagar S. for their help with the mismatchcalculation. We would also like to thank D. Rana, K. Soni and S.Banerjee for their help during this work. AKM would like to thankCouncil of Scientific & Industrial Research (CSIR) for financialsupport through research fellowship No. 524007. A. Mishra wouldlike to thank the University Grants Commission (UGC), India, forfinancial support as a research fellow. We gratefully acknowledgethe use of high performance computing facilities at IUCAA, Pune.
DATA AVAILABILITY
The data underlying this article will be shared on reasonable requestto the corresponding authors.
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APPENDIX A: MISMATCH BETWEEN WAVEFORMS
In the presence of intervening stellar-mass microlenses, the timedelay between the microimages can range from ∼ − 𝜇 s. As aconsequence, GW signals from chirping binaries travelling throughsuch a region would undergo microlensing (since 𝑓 𝑡 𝑑 (cid:46) 𝐹 ( 𝑓 ) (defined in §2), the un-lensed GW signal, ℎ 𝑢 ( 𝑡 ) , and the corresponding lensed waveform, ℎ 𝑙 ( 𝑡 ) , are related by the expression˜ ℎ 𝑙 ( 𝑓 ) = 𝐹 ( 𝑓 ) ˜ ℎ 𝑢 ( 𝑓 ) . (A1)where ˜ ℎ 𝑙 and ˜ ℎ 𝑢 are the Fourier transforms of the timeseries ℎ 𝑙 and ℎ 𝑢 , respectively. To quantify the effect of microlensing, onecan compute the mismatch ( M ) between the unlensed and lensedwaveform, defined as (Usman et al. 2016) M = − match ( ℎ 𝑙 , ℎ 𝑢 ) = − max 𝑡 ,𝜙 (cid:104) ℎ 𝑙 | ℎ 𝑢 (cid:105) √︁ (cid:104) ℎ 𝑙 | ℎ 𝑙 (cid:105) (cid:104) ℎ 𝑢 | ℎ 𝑢 (cid:105) , (A2)where 𝑡 and 𝜙 are, respectively, the arrival time and phase of, say, ℎ 𝑢 and (cid:104) . | . (cid:105) is the noise-weighted inner product, defined as (cid:104) ℎ | ℎ (cid:105) = ∫ 𝑓 high 𝑓 low 𝑑𝑓 ˜ ℎ ∗ ( 𝑓 ) ˜ ℎ ( 𝑓 ) 𝑆 𝑛 ( 𝑓 ) , (A3)where ℎ 𝑖 and ˜ ℎ 𝑖 are, respectively, the timeseries signal and itsFourier transform, and 𝑆 𝑛 ( 𝑓 ) is the single-sided power spectraldensity (PSD) of the detector noise. APPENDIX B: (cid:101) 𝐹 ( 𝑇 ) CURVES
Here we show the (cid:101) 𝐹 ( 𝑡 ) curves that have been used to obtain theamplification factors 𝐹 ( 𝑓 ) shown in Fig. 4 and Fig. 5. This paper has been typeset from a TEX/L A TEX file prepared by the author.MNRAS , 1–16 (2021)
L of GWs: Effect of Microlens Population - - - - - - - t ( s ) F ~ ( t ) ( ) - - - - - - - - - - - t ( s ) F ~ ( t ) ( ) - - - - - - - - - - - t ( s ) F ~ ( t ) ( ) - - - - - - - - - - - t ( s ) F ~ ( t ) ( ) - - - - - - - - - - - t ( s ) F ~ ( t ) ( ) - - - - t ( s ) F ~ ( t ) ( ) - -
0. 5x10 - - - - - t ( s ) F ~ ( t ) ( ) - -
0. 5x10 - - - - - t ( s ) F ~ ( t ) ( ) - -
0. 5x10 - Figure B1.
The figure shows realisations of the (cid:101) 𝐹 ( 𝑡 ) curves for the ( √ 𝜇, Σ • ) cases given in Table 1 corresponding to type-I (minima) and type-II images.Horizontal axis, denoted by ‘t’, represents time delay values and is measured relative to arrival of the first image in case of type-I macroimages and relativeto the arrival of dominant saddle microimage in case of type-II (negative parity) macroimages. The inset focuses on a smaller region where we can observemicroimages (discontinuities and spikes). In case of minima, the value of (cid:101) 𝐹 ( 𝑡 ) at large time delays ( ∼ 𝑡 ∼ 𝐹 ( 𝑓 ) curves, obtained via inverse Fourier transforming the above curves (with suitable window function), areshown in Fig. 4 and Fig. 5.MNRAS000