Probability distribution function for inclinations of merging compact binaries detected by gravitational wave interferometers
aa r X i v : . [ a s t r o - ph . C O ] O c t Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 4 October 2018 (MN L A TEX style file v2.2)
Probability distribution function for inclinations of mergingcompact binaries detected by gravitational wave interferometers
Naoki Seto
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
ABSTRACT
We analytically discuss probability distribution function (PDF) for inclinations of mergingcompact binaries whose gravitational waves are coherently detected by a network of groundbased interferometers. The PDF would be useful for studying prospects of (1) simultaneouslydetecting electromagnetic signals (such as gamma-ray-bursts) associated with binary mergersand (2) statistically constraining the related theoretical models from the actual observationaldata of multi-messenger astronomy. Our approach is similar to Schutz (2011), but we explic-itly include the dependence of the polarization angles of the binaries, based on the conciseformulation given in Cutler and Flanagan (1994). We find that the overall profiles of the PDFsare similar for any networks composed by the second generation detectors (Advanced-LIGO,Advanced-Virgo, KAGRA, LIGO-India). For example, . of detected binaries would haveinclination angle less than ◦ with at most . differences between the potential networks.A perturbative expression is also provided for generating the PDFs with a small number ofparameters given by directional averages of the quantity ǫ that characterises the asymmetry ofnetwork sensitivities to incoming two orthogonal polarization modes. Key words: gravitational waves—binaries: close
Gravitational waves (GWs) from merging neutron star binaries(NS-NSs) are the most promising targets of ground-based detec-tors. For the upcoming second generation interferometers, the esti-mated detection rate of NS-NSs is 1-100/yr, and it is likely that wecan succeed to directly detect their GWs within five years (Abadie2010). This estimated rate for NS-NSs is an order of magnitudehigher than that for black hole-neutron star binaries (BH-NSs),which also have relevance to this paper.Meanwhile, merging NS-NSs (and BH-NSs) are strong candi-dates for progenitors of short gamma ray bursts (SGRBs) (see e.g.
Nakar 2007; Berger 2013). Reflecting geometry of the precedentinspiral phase, a merger product would have nearly axisymmet-ric profile around the direction l of the orbital angular momentumof the binary (Metzger & Berger 2012). If the progenitor scenariofor SGRBs is the case, jet like structures would be launched soonafter the merger, toward the polar directions ± l , and they wouldbe responsible for the observed gamma ray emissions. Later, moreisotropic electromagnetic (EM) radiation might be emitted at lowerenergy band as recently discovered for GRB130603B (Tanvir et al.2013; Berger, Fong & Chornock 2013; Hotokezaka et al. 2013).Therefore, searches for EM signals triggered by GW detec-tions of compact binary inspirals would become an exciting field ofastronomy, and various possibilities have been actively discussedthese days (see e.g. Fairhurst 2011; Schutz 2011; Cannon et al.2012; Evans et al. 2012; LIGO Scientific Collaboration 2013; Nis- sanke, Kasliwal & Georgieva 2013; Dietz et al. 2013; Kelley, Man-del & Ramirez-Ruiz 2013; Piran, Nakar & Rosswog 2013; Ghosh& Bose 2013; Kyutoku, Ioka & Shibata 2014; Arun et al. 2014;Kyutoku & Seto 2014). Given the expected axisymmetric profile ofthe merger products, it would be meaningful to evaluate the prob-ability distribution function (PDF) of inclinations for compact bi-naries whose GWs are detected by the second generation detectors.Using the expected PDF, we can make statistical arguments aboutthe future prospects for simultaneously detecting EM signals andconstraining theoretical models based on observational data.For a network of GW interferometers, the SNR of a binarydepends on its sky direction n and orientation l (Cutler & Flana-gan 1994; Sathyaprakash & Schutz 2009). Numerical studies byMonte Carlo simulations have been performed to properly dealwith these multi-dimensional angular parameters (see e.g. Nissankeet al. 2010, 2013).In this paper, we analytically study the PDFs of inclinations.Our underlying approach is similar to Seto (2014) in which the rel-ative detection rates of merging binaries were formally examinedfor general networks of detectors, but with no attention to the PDFsof inclinations. Schutz (2011) discussed these two issues together,by introducing certain approximation to the dependence of the po-larization angles ψ (explained in the next section) of binaries. But,for the PDFs of inclinations, the accuracy of this approximationhas not been clarified so far. With the help of a concise expressionprovided by Cutler and Flanagan (1994), our analysis does not relyon the approximation and thus can be used to study validity of the c (cid:13) N. Seto convenient method by Schutz (2011), as demonstrated below. Herethe key quantity is ǫ ( n ) which characterizes relative sensitivities ofa network to two orthogonal polarization modes of incoming GWs.Our analytical expressions derived in this paper are easily ap-plicable to any networks of ground-based interferometers. We showthat, in general, the PDFs depend weakly on networks, especiallyfor nearly face-on binaries. This is because the emitted GW poweris strongest to the face-on direction for which the quantity ǫ ( n ) be-comes less important, since the amplitudes of the two orthogonalpolarization modes are nearly the same.In contrast, for edge-on binaries, the PDFs depend strongly on ǫ ( n ) and have largest scatters, when comparing different networks.However, the emitted GW power (and the detectable volume) issmallest for the edge-on binaries. Therefore, among the sample ofthe detected merging binaries, the relative fraction of the edge-onbinaries is much smaller than the face-on binaries.This paper is organized as follows. In Sec.2, we explain ourbasic formulation, assuming a coherent signal analysis for GWsfrom compact binary inspirals. We relate the total SNR and the ex-pected detection rate of binaries. In Sec.3, we evaluate the PDF forinclinations of binaries at a given sky direction. In Sec.4, we discussthe full PDFs, including the angular averages with respect to skydirections. We also evaluate the PDFs concretely for the plannedsecond-generation interferometers. Then we mention relative de-tection rates of merging binaries, in relation to Seto (2014). Sec.5is devoted to a brief summary of this paper. Let us consider a binary at a sky direction n . We use the unit vec-tor l for the orientation of its orbital angular momentum. This ori-entation vector is geometrically characterized by the two parame-ters θ and ψ . Here the inclination angle θ is the angle between n and l , and the polarization angle ψ fixes the rotational degree offreedom of l around the line-of-sight n (Cutler & Flanagan 1994;Sathyaprakash & Schutz 2009).In the principle polarization frame of the binary, the two po-larization modes + and × of the mass-quadrupole waveform areproportional to d + and d × given by d + ( I ) = I + 12 , d × ( I ) = I (1)with I ≡ cos θ ( I = 1 for face-on and I = 0 for edge-on; Peters &Mathews 1963). Below, we simply term I inclination. In astrophys-ical context, we are not interested in the sign of I and hereafter con-sider its absolute value (namely I ). Correspondingly, theinclination angle θ is limited to the range θ ◦ (identifying π − θ → θ ). Throughout this paper, we neglect the precessionsof orbital planes of binaries due to their spins. This would be a rea-sonable approximation for NS-NSs whose orbital angular momentawould dominate the spin angular momenta, due to their compara-ble masses and expected spin parameters much smaller than thoseof black holes (Cutler & Flanagan 1994; Apostolatos et al. 1994).For detecting GWs from binaries, we consider to make coher-ent signal analysis using totally m ground-based interferometerswith no correlated detector noises. Reflecting the spin-2 nature ofGWs, the responses of each interferometer (labeled with i ) to thetwo polarization modes are written by c i + ( n , ψ ) = a i ( n ) cos 2 ψ + b i ( n ) sin 2 ψ, (2) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:5)(cid:8)(cid:9)(cid:3)(cid:2)(cid:10)(cid:11)(cid:7)(cid:12)(cid:13)(cid:9)(cid:12)(cid:14)(cid:15)(cid:16)(cid:13)(cid:12)(cid:2)(cid:9)(cid:17)(cid:3)(cid:17)(cid:5)(cid:14)(cid:7)(cid:16)(cid:17)(cid:3)(cid:18)(cid:9)(cid:12)(cid:9)(cid:10)(cid:11)(cid:14)(cid:12)(cid:2)(cid:19)(cid:20)(cid:5)(cid:14)(cid:20)(cid:11)(cid:7)(cid:11)(cid:21)(cid:11)(cid:7)(cid:11)(cid:5)(cid:20)(cid:9)(cid:7)(cid:3)(cid:9)(cid:7)(cid:22)(cid:5)(cid:9)(cid:7)(cid:23)(cid:3)(cid:9)(cid:3)(cid:2)(cid:7)(cid:22)(cid:3)(cid:15)(cid:3)(cid:14)(cid:12)(cid:13)(cid:1)(cid:3)(cid:13)(cid:12)(cid:2)(cid:11)(cid:24)(cid:12)(cid:7)(cid:11)(cid:3)(cid:14)(cid:9)(cid:17)(cid:3)(cid:8)(cid:5)(cid:20) Figure 1.
The geometric interpretation of Eq.(5) for incoming GW froma sky direction n . (Left panel) In the plane normal to n , the networkhas two orthogonal polarization bases at specific orientations, and measurethese two modes with sensitivities proportional to p σ ( n )(1 + ǫ ( n )) and p σ ( n )(1 − ǫ ( n )) . Here the parameter σ ( n ) represents the total sensitiv-ity to the two modes and ǫ ( n ) shows the asymmetry between them. (Rightpanel) The orbital angular momentum of the binary is projected to the nor-mal plane. Its orientation is characterised by the angle ψ ′ measured fromthe better sensitivity mode in the left panel. The original amplitudes (1) aregiven for the polarization modes symmetric to this projected vector. c i × ( n , ψ ) = − a i ( n ) sin 2 ψ + b i ( n ) cos 2 ψ. (3)The explicit forms of the functions a i ( n ) and b i ( n ) can be foundin Schutz (2011) (see Eqs.(19) and (20) therein). Here the overallamplitude of ( a i , b i ) is proportional to the so-called horizon dis-tance of the detector i . But, below, we simply assume that all theinterferometers have an identical noise curve with the same horizondistance. In practice, it is straightforward to take into account thedifferences of the horizon distances by setting appropriate weightsfor the functions ( a i , b i ) .For the coherent signal analysis, the total signal-to-noise ratio(SNR) is obtained from Eqs.(1)-(3) and depends on the geometricalparameters ( n , I, ψ ) as SNR ∝ m X i =1 (cid:2) ( c i + d + ) + ( c i × d × ) (cid:3) ≡ f ( n , I, ψ ) (4)(Cutler & Flanagan 1994; Dietz et al. 2013). Note that we onlyincluded the lowest quadrupole mode (1) for estimating the totalSNR. This would be a good approximation for NS-NSs, since thenext order correction is proportional to the mass difference andNS-NSs are expected to have similar masses, as commented earlier(Van Den Broeck & Sengupta 2007; Blanchet et al. 2008; Tagoshiet al. 2014). In Eq.(4), SNR is inversely proportional to the distanceto the binary, while we omitted its explicit dependence.With trigonometric relations (Cutler & Flanagan 1994), the ψ -dependence of f is simplified as f ( n , ψ, I ) = σ ( n ) (cid:2) D + ǫ ( n ) D cos 4 ψ ′ (cid:3) , (5)where the new polarization angle ψ ′ = ψ − δ ( n ) is related tothe original ones ψ with an offset δ ( n ) that satisfies the followingrelation The horizon distance is the detectable range of a gravitational wavesource that is optimally located and oriented. For a NS-NS binary of . M ⊙ + 1 . M ⊙ with the detection threshold of SNR = 8 , eachadvanced-LIGO interferometer is planned to have the horizon distance of445Mpc (Abadie et al. 2010). Advance-Virgo and KAGRA would have sim-ilar values. c (cid:13) , 000–000
DF for inclinations of merging compact binaries tan δ ( n ) = 2 P mi =1 a i ( n ) b i ( n ) P mi =1 ( a i ( n ) − b i ( n ) ) . (6)In Eq.(5), the functions D and D are defined by D ( I ) ≡ ( d + d × ) = I + 6 I + 14 (7) D ( I ) ≡ ( d − d × ) = ( I − (8)and the parameters σ and ǫ are defined by σ ( n ) ≡ m X i =1 a i + b i (9) ǫ ( n ) ≡ q(cid:2)P mi =1 ( a i − b i ) (cid:3) + 4( P mi =1 a i b i ) σ ( n ) . (10)From Cauchy-Schwartz inequality, we have ǫ ( n ) . (11)The equality ǫ ( n ) = 1 holds only when the vector ( a ( n ) , · · · , a m ( n )) is parallel to ( b ( n ) , · · · , b m ( n )) , includingthe case for a single detector network (with identity ǫ ( n ) = 1 forall the directions n ).The geometric meaning of Eq.(5) is explained in Fig.1. Theparameter ǫ ( n ) characterizes the asymmetry of network sensitivityto the two orthogonal polarization modes given for each direction n (Cutler & Flanagan 1994). This parameter plays an importantrole in this paper. With respect to the polarization decompositionshown in the left panel, the amplitudes of the quadrupole waves ofthe binary (with the projected angle ψ ′ ) are given by (cid:0) d cos ψ ′ + d × sin ψ ′ (cid:1) / (12)and (cid:0) d sin ψ ′ + d × cos ψ ′ (cid:1) / (13)with d + and d × defined by Eq.(1). Then we have SNR ∝ σ ( n )(1 + ǫ ( n )) (cid:0) d cos ψ ′ + d × sin ψ ′ (cid:1) + σ ( n )(1 − ǫ ( n )) (cid:0) d sin ψ ′ + d × cos ψ ′ (cid:1) . The right-hand-side of this relation is identical to that of Eq.(5).For a face-on binary I = 1 , we have d + = d × (thus D =0 ) and this expression does not depend on the angle ψ ′ (and ǫ ).On the other hand, edge-on binaries ( I = 0 ) emit linearlypolarized GWs and Eq.(5) depends strongly on ǫ ( n ) and ψ ′ with D = D = 1 / . We define r max as the maximum distances to the binaries de-tectable above a given SNR threshold. Then, from Eq.(4), we havea scaling relation r max ∝ f ( n , I, ψ ) / (14)(Finn & Chernoff 1993; Schutz 2011; Dietz et al. 2013). There-fore, assuming that merging binaries have random orientations andspatial distributions, the expected number of detectable ones in aparameter range d n dIdψ is proportional to f ( n , I, ψ ) / d n dψdI. (15) Here we neglected cosmological effects that would be unimportantat least for NS-NSs observed with second generation detectors. Inthis paper, we study the PDFs in appropriately normalized forms.Therefore the actual values of the horizon distance and the comov-ing merger rate are irrelevant to our results.Next we integrate out the less interesting polarization param-eter ψ and define the new function α ( n , I ) by α ( n , I ) ≡ π Z π/ f ( n , I, ψ ) / dψ. (16)As we initially integrate the polarization angle ψ (or equiv-alently ψ ′ ) before integrating the sky direction n , we actually donot need to directly handle the complicated offset δ ( n ) . This is anadvantageous point of our approach, and simplifies the actual eval-uation of PDFs.From Eq.(5), the integral α ( n , I ) can be formally expressedas α ( n , I ) = σ ( n ) / D ( I ) / γ [ ǫ ( n ) R ( I )] , (17)where we define R ( I ) ≡ (cid:16) D D (cid:17) = ( I − I + 6 I + 1 (18)and γ ( x ) ≡ π Z π/ (1 + x cos 4 ψ ) / dψ. (19)The integral γ ( x ) is given as follows γ ( x ) = 2(1 − x ) / π h E (cid:16) xx − (cid:17) − (1 + x ) K (cid:16) xx − (cid:17)i (20)with the incomplete elliptic integral of the second kind E ( x ) andthe complete elliptic integral of the first kind K ( x ) defined respec-tively by E ( x ) ≡ Z π/ (1 − x sin θ ) / dθ (21) K ( x ) ≡ Z π/ (1 − x sin θ ) − / dθ (22)(see also Dietz et al. 2013).Around x = 0 , the integral γ ( x ) is expanded as follows; γ ( x ) = 1 + 316 x + 91024 x + 3516384 x + 34654194304 x + 2702767108864 x + 9699694294967296 x + O ( x ) . (23)We use this expression later in Sec.4.Finally, after integrating the sky direction n of binaries, thePDF for a network can be formally expressed as P net ( I ) = Z π d n α ( n , I ) Z dI Z π d n α ( n , I ) . (24)Here the denominator is a normalization factor to realize Z P net ( I ) dI = 1 . (25) c (cid:13) , 000–000 N. Seto
Our formulation up to Eq.(15) is similar to Seto (2014) in which therelative detection rates of binaries were examined by integrating allthe angular variables including I , without paying attention to itsPDF.In this subsection, we define the following quantity X ≡ Z dI Z π d n α ( n , I ) , (26)and briefly summarize the arguments in Seto (2014). Here we onlyextracted geometrical information relevant for the relative detectionrates, considering comparisons between different networks. Actu-ally, the integral (26) for the relative rates is identical to the denom-inator in Eq.(24).For a hypothetical network with ǫ ( n ) = 0 , the function α ( n , I ) becomes a separable form as σ ( n ) / D ( I ) / and wehave X ≡ Z dID ( I ) / Z π d n σ ( n ) / = N Z π d n σ ( n ) / (27)with the parameter N ≡ . . This expression can be easilyevaluated and we do not need to directly deal with the dependenceon the orientation angles ( I, ψ ) of binaries. Therefore, as a conve-nient approximation to the original complicated one X , we mightuse X for general networks with ǫ ( n ) = 0 . Indeed, the expres-sion X is essentially the same as that proposed by Schutz (2011)for estimating the relative rates.The question here is how well the original integral X is repro-duced by the approximation X . In order to check this, we definethe ratio Y ≡ XX . (28)The main result in Seto (2014) is the following relation Y = R π d n σ ( n ) / G [ ǫ ( n )] R π d n σ ( n ) / (29)where G ( x ) is a monotonically increasing function of x and per-turbatively expanded as G ( x ) = 1 + 0 . x + 0 . x + O ( x ) (30)with G (0) = 1 and G (1) = C ≡ . .Given the inequality σ ( n ) > , we generally have the bounds Y . . (31)Therefore the simple expression X is an excellent approximationto X . These inequalities would be practically sufficient for astro-nomical arguments, but, we can actually evaluate the ratio Y , as abyproduct of our perturbative formulation. This will be discussedin Sec.4.5. In this section, we discuss the PDFs of inclinations I for a fixedparameter ǫ , without taking the sky average as in Eq.(24). FromEq.(17), we define the function P ( I, ǫ ) as follows P ( I, ǫ ) = D ( I ) / γ [ ǫR ( I )] N ǫ (32)with the normalization factor I = cos Θ P H I L , P H I L Figure 2.
The functions P ( I, ǫ ) for ǫ = 0 and 1. The solid curve represents P ( I ) = P ( I, and the dashed one is for P ( I ) = P ( I, . The latter isidentical to the sky averaged PDF for a single detector. N ǫ = Z dID ( I ) / γ [ ǫR ( I )] . (33)The function P ( I, ǫ ) (0 ǫ can be regarded as the PDF fora given sky direction n with ǫ ( n ) = ǫ . In addition, for the specialvalue ǫ = 1 , it corresponds to the full (sky averaged) PDF for anetwork composed by a single interferometer that identically has ǫ ( n ) = 1 , as mentioned earlier. Our primary task in this sectionis to explicitly demonstrate that the function P ( I, ǫ ) does not havestrong dependence on ǫ .To begin with, we introduce the notations P ( I ) and P ( I ) forthe two boundary parameters ǫ = 0 and 1 by P ( I ) ≡ P ( I,
0) = D ( I ) / N (34) P ( I ) ≡ P ( I,
1) = D ( I ) / N γ [ R ( I )] (35)with the normalization factors N = 0 . (already appearedin Eq.(27)) and N = 0 . . We have N /N = C =1 . .Schutz (2011) studied the PDF of inclinations for detected bi-naries. He used an approximation in which the explicit ψ depen-dence was not included for the effective volume (5). In our lan-guage, this treatment corresponds to commute the order of the fol-lowing two operations in Eq.(16); (i) the nonlinear manipulation [ · · · ] / and (ii) the ψ -averaging. It is equivalent to taking ǫ ( n ) = 0 in Eq.(17). Consequently, his PDF is identical to P ( I ) defined inEq.(34). In this paper, we can analytically show that this PDF gen-erally serves as a good approximation, irrespective of the details ofa network.In Fig.2 we present P ( I ) (solid curve) and P ( I ) (dashedcurve). The two curves show similar shapes. In order to enhance thedifferences between them, we show the ratio P ( I ) /P ( I ) (dashedcurve) in Fig.3, together with P ( I, ǫ ) /P ( I ) at the intermediatevalues ǫ = 0 . , . , · · · , . (solid curves).For a given ǫ , the function P ( I, ǫ ) becomes minimum at I =0 , reflecting the smallest amplitude at the edge-on configuration. Atthe same time, as shown in Fig.3, the ratios P ( I, ǫ ) /P ( I ) showthe largest scatter at I = 0 . This is because the emitted waves are100% linearly polarized and the effects of the asymmetry parameter ǫ become significant. c (cid:13) , 000–000 DF for inclinations of merging compact binaries Table 1.
The cumulative PDF: [ P cum ( θ,
0) + P cum ( θ, / at samplepoints. θ ◦ ◦ ◦ ◦ ◦ ◦ cumulative PDF . × − In contrast, at the face-on configuration I = 1 , we obtain R ( I ) = 0 and γ ( ǫR ( I )) = 1 . Therefore, around I ∼ , we ap-proximately have P ( I, ǫ ) ≃ D ( I ) / N ǫ (36)with P ( I, ǫ ) /P ( I ) ≃ N /N ǫ that is now a decreasing function of ǫ with the minimum value N /N = 1 /C = 0 . at ǫ = 1 . Thisshows that, around I ∼ , the relative difference between P ( I, ǫ ) is at most ∼ .As shown in Fig.3, the two functions P ( I ) and P ( I ) inter-sect at I = 0 . where the family P ( I, ǫ ) depends very weaklyon ǫ . Except the tiny region around this intersection, the function P ( I, ǫ ) ( ǫ ) is bounded by the two curves P ( I ) and P ( I ) . In the next section, we apply this result for discussing theoverall profile of the sky-averaged function P net ( I ) .So far, we have studied the PDFs for I only in a differentialform. Here we examine the cumulative PDFs P cum ( θ, ǫ ) for theinclination angle θ = cos − I defined by P cum ( θ, ǫ ) ≡ Z θ P ( I, ǫ ) dI (37)with θ ◦ . This function represents the probability that adetected binary has a viewing angle less than θ , from its symme-try axis l . In this cumulative form, we rigidly have the followingbounds P cum ( θ, P cum ( θ, ǫ ) P cum ( θ, (38)and the two boundaries have small relative differences P cum ( θ, / P cum ( θ, C = 1 . . (39)We can confirm their similarity in Fig.4. The tight confinement (38)would become useful in the next section.For conveniences at astronomical studies, we provide a fittingfunction for P cum ( θ, ǫ ) P cum,f ( θ ) = 4 . (cid:16) θ ◦ (cid:17) − . (cid:16) θ ◦ (cid:17) − . (cid:16) θ ◦ (cid:17) (40)which reproduces the functions P cum ( θ, ǫ ) ( ǫ ) with rel-ative error less than 1% in the range θ ◦ . In Table.1,we also evaluate the mean [ P cum ( θ,
0) + P cum ( θ, / for somerepresentative angles θ . In this section, we discuss the full (sky averaged) functions P net ( I ) defined in Eq.(24) for various networks of ground-based interfer-ometers. In Sec.4.1 we first mention their overall profiles basedon the results shown in the previous section. Then, in Sec.4.2, weuse the perturbative expansion (23) and derive an expression formore preciously evaluating P net ( I ) . The validity of our pertur-bative method is examined in Sec.4.3. In Sec.4.4, we apply our I P H I , Ε L (cid:144) P H I L Figure 3.
The ratios P ( I ) /P ( I ) (dashed curve) and P ( I, ǫ ) /P ( I ) with ǫ = 0 . , . , · · · , . (solid curves from bottom to top). At I = 0 , the ratio P ( I, ǫ ) /P ( I ) is an increasing function of ǫ . The two function P ( I ) and P ( I ) intersect at I = 0 . . viewing angle Θ @ degree D c u m u l a ti v e d i s t r i bu ti on P c u m H Θ , Ε L Figure 4.
The cumulative functions P cum ( θ, ǫ ) for ǫ = 0 (solid curve) and1 (dashed curve). Their relative difference is only ∼ . For ǫ ,the function P cum ( θ, ǫ ) is tightly bounded by these two curves. method for networks composed by second generation interferome-ters. In Sec.4.5, we mention the relative detection rates of mergingbinaries, in relation to Seto (2014) and Sec.2.3. From Eqs.(17) and (24), the function P net ( I ) is obtained by takingan average of P [ I, ǫ ( n )] with the following relative weights d n σ ( n ) / N ǫ ( n ) . (41)Therefore, similar to the previous one P ( I, ǫ ) , the averaged one P net ( I ) should be bounded by the two functions P ( I ) and P ( I ) except the tiny region around their intersection at I = 0 . ,as mentioned earlier in Fig.3. This means that the overall profileof P net ( I ) can be approximately understood from the shapes ofthe two functions P ( I ) and P ( I ) . Around I ∼ , the function P net ( I ) weakly depends on the details of a network (see Fig.3).Among the binaries detected by a single interferometer, the frac-tion of nearly edge-on ones ( I ∼ ) could be at most ∼ largerthan a network with multiple interferometers.Next, we discuss the cumulative PDFs for networks. As inEq.(37), we define P cum,net ( θ ) by c (cid:13) , 000–000 N. Seto P cum,net ( θ ) ≡ Z θ P net ( I ) dI. (42)By changing the order of the integrals d n and dI , we can un-derstand that the function P cum,net ( θ ) is obtained by averaging P cum [ θ, ǫ ( n )] again with the weight (41). Since the cumulativePDFs P cum [ θ, ǫ ( n )] are tightly bounded by the two functions P cum ( θ, and P cum ( θ, , the sky averaged one P cum,net ( θ ) must be also bounded by them. Therefore, with relative error lessthan ∼ , we can apply the previous fitting formula (40) for thesky averaged one P cum,net ( θ ) in the range θ ◦ , irrespec-tive of the details of networks. Similarly, we can apply Table.1 forgiven networks.For example, ∼ of detected binaries have viewing angle θ less than ◦ . The fraction becomes ∼ . for θ ◦ . Inother words, if one hundred binaries are detected by a network, theminimum inclination angle would be θ ∼ ◦ and we will have ∼ binaries with θ less than ◦ . Now we move to develop a perturbative method for evaluating P net ( I ) more precisely. First we rewrite P net ( I ) as follows P net ( I ) = Q net ( I ) M net , (43)where the numerator and the denominator are non-dimensionalquantities defined by Q net ( I ) = Z π d n α ( n , I ) Z π d n σ ( n ) / , M net = Z dI Z π d n α ( n , I ) Z π d n σ ( n ) / . (44)Here we introduced the common factor (cid:2)R π d n σ ( n ) / (cid:3) − tomake our analysis comprehensive. Applying the expansion (23) for Q net ( I ) , we obtain Q net ( I ) = D ( I ) / (cid:18) R s
16 + 9 R s R s · · · (cid:19) (45)with a function R ( I ) defined in Eq.(18) and the coefficients s j given by s j ≡ Z π σ ( n ) / ǫ ( n ) j d n Z π σ ( n ) / d n . (46)From the inequalities ǫ ( n ) , we have s j +1 s j (47)with the equality s j = s j +1 only for s j = 0 (identically ǫ ( n ) = 0 )or s j = 1 (identically ǫ ( n ) = 1 ). In our perturbative approach,all the information of a network is projected into the sequence ofnumbers ( s , s , s , · · · ) . We thus call them network parameters.In the same manner, the normalization factor M net can be per-turbatively evaluated as M net = N + 316 u s + 91024 u s + 3516384 u s + · · · (48)where we define the parameters u j given by the following integrals u j ≡ Z D ( I ) / R ( I ) j dI. (49) Table 2.
The parameters defined in Eq.(49). u u u u u u I P H I L (cid:144) P H I L Figure 5.
The ratio P ( I ) /P ( I ) (dashed curve) and its perturbative ex-pansions according to Eq.(50) (0th, 2nd and 4th order approximations: solidcurves from bottom). Convergence of the perturbative expansion is fast. These are constants and do not depend on networks. In Table.2, wepresent them up to u . In the previous subsection, we explained how to perturbativelyevaluate the sky averaged function P net ( I ) . Our expression (43)is characterized by the network parameters ( s , s , · · · ) with s j . From Eqs.(45) and (48), we will have better convergencefor smaller s j . On the other hand, the convergence would becomeworst for the maximum value s j = 1 , corresponding to a sin-gle detector network. But, for this case, we actually have the non-perturbative result P ( I ) given in Eq.(35). Therefore, we can testthe validity of our perturbative expansion by comparing the tworesults.For s j = 1 , our perturbative expression is given by P ( I ) = D ( I ) / R + R + R + · · · N + u + u + u + · · · . (50)In Fig.5, we show the non-perturbative results (dashed curve) andthe 0th, 2nd and 4th order approximations (solid curves). This fig-ure shows that, even in the worst case s j = 1 , the convergence isfast and the relative error is at most ∼ . with the 4th order ap-proximation. Therefore, our perturbative method would be efficientto reproduce the function P net ( I ) . Now we concretely evaluate the averaged function P net ( I ) for net-works of ground-based interferometers. We consider the followingfive second-generation interferometers; LIGO-Hanford (H), LIGO-Livingston (L), Virgo (V), KAGRA (K) and LIGO-India (I). Fortheir locations and orientations, we use Table.2 in Schutz (2011).But, for KAGRA, we apply the updated data; the geographical po-sition ( . ◦ E, . ◦ N) and the orientation angle . ◦ for the c (cid:13)000
The ratio P ( I ) /P ( I ) (dashed curve) and its perturbative ex-pansions according to Eq.(50) (0th, 2nd and 4th order approximations: solidcurves from bottom). Convergence of the perturbative expansion is fast. These are constants and do not depend on networks. In Table.2, wepresent them up to u . In the previous subsection, we explained how to perturbativelyevaluate the sky averaged function P net ( I ) . Our expression (43)is characterized by the network parameters ( s , s , · · · ) with s j . From Eqs.(45) and (48), we will have better convergencefor smaller s j . On the other hand, the convergence would becomeworst for the maximum value s j = 1 , corresponding to a sin-gle detector network. But, for this case, we actually have the non-perturbative result P ( I ) given in Eq.(35). Therefore, we can testthe validity of our perturbative expansion by comparing the tworesults.For s j = 1 , our perturbative expression is given by P ( I ) = D ( I ) / R + R + R + · · · N + u + u + u + · · · . (50)In Fig.5, we show the non-perturbative results (dashed curve) andthe 0th, 2nd and 4th order approximations (solid curves). This fig-ure shows that, even in the worst case s j = 1 , the convergence isfast and the relative error is at most ∼ . with the 4th order ap-proximation. Therefore, our perturbative method would be efficientto reproduce the function P net ( I ) . Now we concretely evaluate the averaged function P net ( I ) for net-works of ground-based interferometers. We consider the followingfive second-generation interferometers; LIGO-Hanford (H), LIGO-Livingston (L), Virgo (V), KAGRA (K) and LIGO-India (I). Fortheir locations and orientations, we use Table.2 in Schutz (2011).But, for KAGRA, we apply the updated data; the geographical po-sition ( . ◦ E, . ◦ N) and the orientation angle . ◦ for the c (cid:13)000 , 000–000 DF for inclinations of merging compact binaries Table 3.
The network parameters s j for various networks of ground-basedinterferometers. We consider up to five interferometers (H: LIGO-Hanford,L: LIGO-Livingston, V: Virgo, K: KAGRA, I: LIGO-India). All of them areassumed to have an identical noise curve. The networks with bold letters arethose shown in Fig.6.network s s s s s single HL HLV
HLVK
HLVKI bisector of its two arms measured counter-clock wise from the lo-cal East direction. All the detectors are assumed to have identicalnoise spectrum (and thus the identical horizon distance).In Table.3, we present the network parameters s j for vari-ous potential networks composed by the five interferometers. Wehave the identities s j = 1 for single interferometer, as mentionedearlier. The two LIGO interferometers H and L are separated by ∼ km but configured to realize large overlaps for incomingGW signals (Cutler & Flanagan 1994). To this end, their orienta-tions are nearly aligned. This results in larger network parameters s j , compared with other two-detector networks such as HV or VK.The five-detector network HLVKI has the smallest networkparameters s j in Table.3, indicating that due to the randomness ofdetector configurations, the degree of asymmetry ǫ decreases.In Fig.5, we show the full functions P net ( I ) for a single in-terferometer (dashed curve) as well as the HL, HLV, HLVK andHLVKI networks (solid curves from top to bottom). We use the12th order approximation for the perturbative expansion. Using Mathematica , we can straightforwardly calculate the network pa-rameters s j and evaluate the perturbative expressions. As we in-crease the number of interferometers, the PDF moves from P ( I ) (for a single interferometer) to P ( I ) , decreasing fraction of edge-on binaries.The PDF for the HL network is close to that of a single inter-ferometer, as easily expected from the relatively large network pa-rameters in Table.3. For nearly edge-on binaries ( I ∼ ), the num-ber of detectable volume depends on ψ ′ as ∝ [1+ ǫ ( n ) cos 4 ψ ′ ] / (see Eq.(5)), and the detected binaries are likely to have polariza-tion angles around ψ ′ = 0 (mod π/ ) for the HL network. ThePDFs for the HLVKI network is reproduced by Schutz’s approxi-mation P ( I ) with error less than , even around I ∼ .The fraction of nearly edge-on binaries detected by the HLVKInetwork would be ∼ smaller than that of the HL network. Butwe should recall that the emitted GW power (thus the detectablerange) is smallest to the edge-on direction I ∼ . Indeed, we havethe ratio of the emitted powers D ( I = 0) /D ( I = 1) = 1 / ,compared with face-on binaries I = 1 . As shown in Fig.2, thenearly edge-on binaries would be a minor component in the wholedetected sample. So far, we have studied PDFs of inclinations I (and θ ). In this sub-section, we go back to § I P n e t H I L (cid:144) P H I L Figure 6.
The ratios P net ( I ) /P ( I ) for various networks. Dashed curve isgiven for a single detector network and four solid curves are for HL, HLV,HLVK and HLVKI (from top to bottom at I = 0 ). Table 4.
The ratio Y for various networkssingle HL HLV HLVK HLVKI1.010125 1.00919 1.00588 1.00495 1.00424 tive formulation for the ratio Y = X/X defined in Eq.(28). Thisratio represents validity of Schutz’s approximation X for estimat-ing the relative detection rates X .From Eqs.(44) and (48) we can easily obtain Y = M net N = 1+ 316 u s N + 91024 u s N + 3516384 u s N + · · · , (51)and the ratio Y can be directly evaluated, as actual numbers. InTable.4, we provide them for various networks of detectors, againassuming that all the component detectors have the same sensitivity.As expected from Table 3, the HL network has the deviation . close to the maximum value . for a single detector(see also inequalities (39)). This deviation would be sufficientlysmall for astronomical arguments, but the deviation for the HLVKInetwork is further smaller and ∼ . . In this paper, we discussed the probability distribution function P net ( I ) of inclinations I = cos θ ( θ : inclination angle) for com-pact binaries that are detected by a coherent signal analysis with anetwork of ground-based GW interferometers. In a coherent signalanalysis, the SNR of a binary depends not only on its sky direction n and inclination I , but also on its polarization angle ψ . We haveextensively used the simple form (5) given by Cutler and Flana-gan (1994) to properly include the ψ -dependence. Here we have animportant parameter ǫ ( n ) that characterizes the asymmetry of thenetwork sensitivities to two orthogonal polarization modes fromdirection n . This parameter has the identity ǫ ( n ) = 1 for a singleinterferometer and an asymptotic behaviour ǫ ( n ) → for largenumber of randomly placed interferometers. One of the central is-sues in this paper was how to deal with the effects of the parameter ǫ ( n ) .Schutz (2011) derived a PDF under a simplification equivalentto setting ǫ ( n ) = 0 in this paper. This simplified PDF correspondsto P ( I ) defined in Eq.(34), and we showed that it works well for c (cid:13) , 000–000 N. Seto face-on binaries ( I = 1 ) with errors less than ∼ . On the otherhand, for edge-on binaries ( I = 0 ), this function is ∼ smallerthan P ( I ) defined for a single interferometer.In the cumulative form defined in Eq.(42), the PDF for agiven network is reproduced by the simple expression P cum ( θ, at ∼ accuracy. Therefore, the fitting formula (40) and Table.1would be useful for astronomical arguments such as prospects ofEM counterpart searches triggered by GW detections.We also developed a perturbative method to evaluate the func-tion P net ( I ) by introducing the network parameters s j ( j =2 , , , · · · ). These parameters are given by certain angular averagesof the moments ǫ ( n ) j . Convergence of our expansion is fast, andexpressions including the first few correction terms of Eqs.(45) and(48) would be sufficient in practice. Even if the horizon distancesof individual interferometers are different, we can easily apply ourmethod for arbitrary networks, by introducing appropriate weightsfor detectors.We generated the PDFs concretely for the potential networkscomposed by the second generation detectors. The network withthe two LIGO interferometers (HL) has relatively large values s j ,due to their nearly aligned configurations, and the function P net ( I ) is similar to P ( I ) defined for a single interferometer. On the otherhand, the PDF of the network composed by the five interferometers(HLVKI) is closer to P ( I ) with smaller network parameters s j .The author thanks to H.Tagoshi and K.Kyutoku for helpfulconversations. This work was supported by JSPS (24540269) andMEXT (24103006). REFERENCES
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