Wave Effects in the Gravitational Lensingof Gravitational Waves from Chirping
Binaries

Ryuichi Takahashi

Department of Physics, Kyoto University, Kyoto 606-8502, Japan
January 2004

Abstract
In the gravitational lensing of gravitational waves, wave optics should be usedinstead of geometrical optics when the wavelength λ of the gravitational waves islarger than the Schwarzschild radius of the lens mass ML . For example, for thewave length of the gravitational waves for the space interferometer λ ∼ 1 AU, thewave effects become important for a lens mass smaller than ∼ 108 M . In thisthesis, we discuss the wave optics in the gravitational lensing and its application
to the gravitational wave observations in the near future.
The wave optics is based on the diffraction integral which is the amplificationof the wave amplitude by the lensing. We studied the asymptotic expansion of
the diffraction integral in powers of the wavelength λ. The first term, arising fromthe short wavelength limit λ → 0, corresponds to the geometrical optics limit.
The second term, being of the order of λ/ML , is the first correction term arisingfrom the diffraction effect. By analyzing this correction term, we find that (i)the lensing magnification µ is modified to µ(1 + δ), where δ is of the order of(λ/ML )2 , and (ii) if the lens has cuspy (or singular) density profile at the center
ρ(r) ∝ r −α (0 < α ≤ 2), the diffracted image is formed at the lens center with themagnification µ ∼ (λ/ML )3−α .
We consider the gravitational lensing of chirp signals from the coalescense of
supermassive black holes at redshift z ∼ 1 relative to the Laser InterferometerSpace Antenna. For such cases, we compute how accurately we can extract themass of the lens and the source position from the lensed signal. We consider two
simple lens models: the point mass lens and the SIS (Singular Isothermal Sphere).
We find that the lens mass and the source position can be determined within ∼0.1% [(S/N )/103 ]−1 for the lens mass larger than 108 M , where (S/N ) is the signal
to noise ratio of the unlensed chirp signals. For the SIS model, if the source positionis outside the Einstein radius, only a single image exists in the geometrical optics
approximation so that the lens parameters can not be determined. While in the
wave optics cases we find that the lens mass can be determined for ML ∼ 108 M .
For the point mass lens, one can extract the lens parameters even if the source
position is far outside the Einstein radius. As a result, the lensing cross section isan order of magnitude larger than that for the usual strong lensing of light.

iii

Contents
Abstractiii

1 Introduction
1.1 Wave Effects in Gravitational Lensing . . . . . . . . . . . . . . . . .
1.1.1 Double Slit . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Chirp Signal . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Application to Gravitational Wave Observations . . . . . . . . . . .
1.3 Differences between the Gravitational Lensing of Gravitational Waves
and Gravitational Lensing of Light . . . . . . . . . . . . . . . . . .
1.4 Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . .

1
1
2
4
4

2 Wave Optics in Gravitational Lensing
2.1 Gravitational Waves Propagating through the Curved Spacetime . .
2.1.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Kirchhoff diffraction integral . . . . . . . . . . . . . . . . . .
2.1.3 Amplification factor . . . . . . . . . . . . . . . . . . . . . .
2.1.4 Geometrical Optics Approximation . . . . . . . . . . . . . .
2.2 Amplification Factor for Various Lens Models . . . . . . . . . . . .
2.2.1 Point Mass Lens . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Singular Isothermal Sphere . . . . . . . . . . . . . . . . . . .
2.2.3 Navarro-Frenk-White lens . . . . . . . . . . . . . . . . . . .
2.2.4 Binary lens . . . . . . . . . . . . . . . . . . . . . . . . . . .

7
7
7
9
11
13
14
14
16
17
20

5
6

3 Quasi-geometrical Optics Approximation in Gravitational Lensing
25
3.1 Effect on the Magnifications of the Images . . . . . . . . . . . . . . 26
3.2 Contributions from the Non-stationary Points . . . . . . . . . . . . 28
3.3 Central Cusp of the Lens . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 Results for Specific Lens Models . . . . . . . . . . . . . . . . . . . . 32
3.4.1 Point Mass Lens . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.2 Singular Isothermal Sphere . . . . . . . . . . . . . . . . . . . 32
3.4.3 Isothermal Sphere with a Finite Core . . . . . . . . . . . . . 35
3.4.4 NFW lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36v

4 Wave Effects in the Gravitational Lensing of Gravitational Waves
from Chirping Binaries
39
4.1 Gravitationally Lensed Waveform and Parameter Estimation . . . . 40
4.1.1 Gravitational Wave Measurement with LISA . . . . . . . . . 40
4.1.2 Gravitationally Lensed Signal Measured by LISA . . . . . . 41
4.1.3 Parameter Extraction . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.1 Lensing Effects on the Signal to Noise Ratio . . . . . . . . . 46
4.2.2 Parameter Estimation for the Lens Objects . . . . . . . . . . 46
4.2.3 Lensing Effects on the Estimation Errors of the Binary Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.4 Results for Various SMBH Masses and Redshifts . . . . . . . 53
4.3 Lensing Event Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Summary57

A Path Integral Formula for the Wave Optics
61
A.1 Basic Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 61
A.2 Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
A.3 Application to Multi-Lens System . . . . . . . . . . . . . . . . . . . 64
B Numerical Computation for the Amplification Factor
67
B.1 Axially Symmetric Lens Model . . . . . . . . . . . . . . . . . . . . 67
B.2 Non-axially Symmetric Lens Model . . . . . . . . . . . . . . . . . . 68
C Asymptotic Expansion of the Amplification Factor for Non-axially
Symmetric Lens Models
69
D Estimation Errors in the Geometrical Optics Limit73

Chapter 1
Introduction
Inspirals and mergers of compact binaries are the most promising gravitational
wave sources and will be detected by the ground based as well as the space based
detectors in the near future (e.g. Cutler & Thorne 2002). Laser interferometers are
now coming on-line or planned on broad frequency bands: for the high frequency
band 10 − 104 Hz, the ground based interferometers such as TAMA300, LIGO,
VIRGO and GEO600 will be operated; for the low frequency band 10−4 − 10−1
Hz, the Laser Interferometer Space Antenna1 (LISA) will be in operation; forthe intermediate frequency band 10−2 − 1 Hz, the space based interferometerssuch as the Decihertz Interferometer Gravitational Wave Observatory (DECIGO;
Seto, Kawamura & Nakamura 2001), and Big Bang Observer2 (BBO) are planned.
For the templates of the chirp signals from coalescing compact binaries, the postNewtonian computations of the waveforms have been done by many authors. Using
the matched filter techniques with the template, we can obtain the binary parameters such as the mass and the spatial position of the source (e.g. Cutler & Flanagan
1994). The detection rate for the compact binary mergers is also estimated for thevarious sources such as neutron star binaries and massive black hole binaries (seeCutler & Thorne 2002 and references therein). Thus, it is important to investigate
the various possibilities that alter these theoretical calculations. One of these possibilities is the gravitational lensing which affects the template (e.g. Thorne 1987),
the angular resolution of the detector (Seto 2003), and the event rate (Wang,
Stebbins, & Turner 1996; Varvella, Angonin, & Tourrenc 2003). We study thegravitational lensing of gravitational waves in this thesis.

1.1

Wave Effects in Gravitational Lensing

Gravitational lensing of light is a well known phenomenon in the astrophysics.
If the light from distant sources pass near the massive objects, the light ray is
1
2

See http://lisa.jpl.nasa.gov/index.html
See http://universe.gsfc.nasa.gov/be/roadmap.html.

1Figure 1.1: The gravitational lensing configuration. ML is the lens mass, D is the
distance to the lens, and rE is the impact parameter.
deflected. We show the lensing configuration in Fig.1.1, the lens is located at
the distance D from the observer. The deflection angle is α ∼ ML /rE  1,where ML is the lens mass and rE is the impact parameter. From the deflection
angle α ∼ ML /rE and the geometrical relation rE ∼ αD in the Fig.1.1, we obtain the Einstein radius as, rE ∼ (ML D)1/2 , which is the characteristic length of
the impact parameter. The Einstein radius rE is typically given by rE ∼ 10kpc(ML /1011 M )1/2 , where we set D = H0−1 (the Hubble distance).
In the gravitational lensing of light, the lensing quantities (the deflection angle,
the image positions, its brightness, the number of images, etc.) are calculated in
the geometrical optics, which is valid in all observational situations (Schneider,
Ehlers & Falco 1992; Nakamura & Deguchi 1999). However for the gravitational
lensing of gravitational waves, the wavelength is long so that the geometrical optics
approximation is not valid in some cases. For example, the wavelength λ of thegravitational waves for the space interferometer is ∼ 1 AU which is extremely
larger than that of a visible light (λ ∼ 1µ m).

1.1.1

Double Slit

As shown by several authors (Ohanian 1974, Bliokh & Minakov 1975, Bontz &Haugan 1981, Thorne 1983, Deguchi & Watson 1986a), if the wavelength λ islarger than the Schwarzschild radius of the lens mass ML , the diffraction effect isimportant and the magnification is small. To see the reason why the ratio ML /λdetermines the significance of the diffraction, we consider a double slit with the slit
width comparable to the Einstein radius rE ∼ (ML D)1/2 where D is the distancefrom the screen to the slit (Nakamura 1998). The illustration of the double slit
is shown in Fig.1.2. When monochromatic waves with the wavelength λ passthrough the slit, the interference pattern is produced on the screen. Denoting theFigure 1.2: Illustration of the double slit with the slit width rE ∼ (ML D)1/2(Einstein radius). λ is the wavelength of the monochromatic wave, D is the distance
between the screen and the slit, and l± is the distance from each of the slit to theobserver.
path length l± as the distance from each of the slit to the observer as shown in
Fig.1.2, l± is given by,p(1.1)
l± = (x ± rE /2)2 + D 2 ,where x is the position of the observer on the screen. Denoting path length difference as ∆l ≡ |l+ − l− |, we obtain from equation (1.1) as,
rE x
∆l '
,
(1.2)
D
where we assume D  rE , x. The width ∆x of the central peak of the interference
pattern is obtained from equation (1.2) with setting ∆l ∼ λ as, ∆x ∼ (D/rE )λ.
Then the maximum magnification of the wave flux is of the order ∼ rE /∆x ∼ML /λ. Thus the diffraction effect is important for
−1

f
8
,
(1.3)
ML . 10 M
mHz
where f is the frequency of the gravitational waves. However as suggested by Ruffa(1999), the focused region by the gravitational lensing would have a relatively large
area because of the diffraction, so that the lensing probability will increase.
Since the gravitational waves from the compact binaries are coherent, the interference is also important (Mandzhos 1981, Ohanian 1983, Schneider & SchmidBurgk 1985, Deguchi & Watson 1986b, Peterson & Falk 1991). The length of theinterference pattern on the screen ∆x is given by,
∆x ∼

D
λ,
rE

∼ 0.1AU



ML
106 M

−1/2 D
10kpc

1/2 

f
kHz

−1

.

(1.4)

Here, we assume that the lens is the suppermassive black hole of mass ML =106 M at our galactic center (D ∼ 10 kpc) and the detector is the ground-based
interferometer (f ∼ kHz). Thus it may be possible to detect the interference
pattern by the orbital motion of the earth around the Sun (Ruffa 1999).

1.1.2

Chirp Signal

We consider the inspiraling binaries as the gravitational wave sources. The inspirals
of compact binaries are the most promising gravitational wave sources for thelaser interferometers. As the binary system loses its energy by the gravitationalradiation, the orbital separation decreases and the orbital frequency increases.
Thus, the frequency of the gravitational wave increases with time (df /dt > 0).
This is called a chirp signal. Hence we can see wave effects for different frequencies
in the lensed chirp signals.

1.2

Application to Gravitational Wave Observations

The coalescence of supermassive black holes (SMBHs) of mass 104 − 107 M isthe one of the most promising sources for LISA and will be detected with a very
high signal to noise ratio S/N ∼ 103 (Bender et al. 2000). The coalescence rateof SMBHs binaries detected by LISA is estimated to be 0.1 − 102 events yr−1
(Haehnelt 1994,1998; Menou, Haiman, & Narayanan 2001; Jaffe & Backer 2003;
Wyithe & Loeb 2003). Recently, Wyithe & Loeb (2003) suggested that the event
rate was up to some hundred per year, considering the merger rate as exceedingly
high redshift (z > 5 − 10). Since the merging SMBH events will be detected for
extremely high redshift (z > 5), the lensing probability is relatively high (it could
reach several percent, see Turner, Ostriker & Gott 1984), and hence some lensing
events are expected. From the LISA frequency band (f ∼ mHz), the diffraction
effect become important for the lens mass smaller than ∼ 108 M from equation(1.3). We will discuss the wave effects in gravitational lensing of the gravitational
waves detected by LISA in §4.
Similarly, neutron star binaries are the most promising sources for the groundbased interferometers (TAMA300,LIGO,VIRGO,GEO). The diffraction effect isimportant for the lens mass smaller than 103 M for the ground based detectors(f ∼ 102 Hz). The detection rates for the neutron star mergers are at least several per year for advanced LIGO (Phinney 1991; Narayan, Piran & Shemi 1991;
Kalogera et al. 2001,2003). But since the source redshifts are relatively small(z < 1) for the ground based detectors (e.g. Finn 1996), the lensing probability issmall (it would be less than 0.1 percent), and hence the lensing event rate would besmall (e.g. Wang, Stebbins & Turner 1996). Hence, we do not consider the gravitational lensing of gravitational waves detected by the ground based detectors.

1.3

Differences between the Gravitational Lensing of Gravitational Waves and Gravitational
Lensing of Light

We discuss the differences between the gravitational lensing of gravitational waves
and gravitational lensing of light. The wave effects (diffraction and interference)are some of these differences, which is discussed in the previous section. Except
for the wave effects, there are some differences as follows:• Gravitational waves propagate through surrounding matter without absorption and scattering. Hence, lens objects even in optically thick region can bedetected by lensing.
• In the gravitational lensing of light, in order to detect multiple images, angular resolution θmin of detector should be smaller than the image separation
which is roughly equal to the Einstein angle θE = rE /D ∼ (M/D)1/2 . Thiscondition θmin . θE is rewritten as,
11

M & 10 M



D
Gpc



θmin
1arcsec

2

.

(1.5)

Thus, the lens object should be more massive than galactic mass scale. But
the case of gravitational waves, the observable is time delay between theimages. If the time delay τ ∼ 103 sec(M/108 M ) is larger than than the
period of the gravitational waves 1/f , the two signals are detected. Thiscondition τ & 1/f is the same as the equation (1.3), and hence the lens
object lighter than the galactic mass (∼ 1011 M ) can be detected in the caseof the gravitational waves.
• The observable for the light is the flux f of the source, while that for thegravitational waves is the amplitude A. Then, the flux is changed in proportion to the magnification µ by lensing, while the amplitude is changed in√proportion to the square root µ.
• In the gravitational lensing of gravitational waves, the lensing cross section
to extract the information on the lens object (its mass, its position, etc.) isan order of magnitude larger than that for the usual strong lensing of light(Ruffa 1999; Takahashi & Nakamura 2003).

1.4

Organization of this thesis

We outline the organization of this thesis. In §2 we review the wave optics ingravitational lensing of gravitational waves. We show the results for the various
lens models: the point mass lens, the SIS model, the NFW model, and the binarylens model. In §3 we discuss the gravitational lensing in quasi-geometrical optics
approximation which is the geometrical optics including corrections arising fromthe effects of the finite wavelength. In §4 we discuss the gravitationally lensed
waveform detected by LISA and mention the parameter estimation based on thematched filtering analysis. We numerically evaluate the signal-to-noise ratio andthe parameter estimation errors. We also estimate the lensing event rate. Section
5 is devoted to summary and discussions. We assume a (ΩM , ΩΛ ) = (0.3, 0.7)
cosmology and a Hubble parameter H0 = 70kms−1 Mpc−1 and use units of c =G = 1.

Chapter 2
Wave Optics in Gravitational
Lensing
In this chapter, we review the wave optics in gravitational lensing of gravitational
waves (Schneider, Ehlers & Falco 1992; Nakamura & Deguchi 1999). The wave
optics is more fundamental than the geometrical optics, and it has been used forthe gravitational lensing of gravitational waves (see Nakamura & Deguchi 1999 andreference therein).

2.1

Gravitational Waves Propagating through theCurved Spacetime

2.1.1

Basic equations

We consider gravitational waves propagating under the gravitational potential ofa lens object (e.g. Misner, Thorne & Wheeler 1973). The background metric isgiven by,(B)
ds2 = − (1 + 2U ) dt2 + (1 − 2U ) dr2 ≡ gµν
dxµ dxν ,(2.1)
where U (r) ( 1) is the gravitational potential of the lens object. Let us consider(B)
the linear perturbation hµν in the background metric tensor gµν as(B)gµν = gµν
+ hµν .

(2.2)

Under the transverse traceless Lorentz gauge condition of hνµ;ν = 0 and hµ µ = 0 wehave(B)hµν;α ;α + 2Rαµβν hαβ = 0,
(2.3)
(B)

(B)

where the semicolon is the covariant derivative with respect to gµν and Rαµβν isthe background Riemann tensor. If the wavelength λ is much smaller than the
typical radius of the curvature of the background, we havehµν;α ;α = 0.

7

(2.4)

Figure 2.1: Gravitational lens geometry for the source, the lens and the observer.
Here DL , DS and DLS are the distances between them, η is the displacement of
the source from the line of sight to the lens and ξ is an impact parameter. We use
the thin lens approximation in which the gravitational waves are scattered only at
the thin-lens plane.
Following the eikonal approximation to the above equation by Baraldo, Hosoya
and Nakamura (1999), we express the gravitational wave ashµν = φ eµν ,(2.5)

where eµν is the the polarization tensor of the gravitational wave (eµ µ = 0, eµν eµν =2) and φ is a scalar. The polarization tensor eµν is parallel-transported along thenull geodesic (eµν;α k α = 0, where k α is a wave vector) (Misner, Thorne & Wheeler
1973). Then, the change of the polarization tensor by gravitational lensing is of
the order of U ( 1) which is very small in the present situation, and hence we
can regard the polarization tensor as a constant. Thus, we treat the scalar wave φ,instead of the gravitational wave hµν , propagating through the curved space-time.
The propagation equation of the scalar wave isp∂µ ( −g (B) g (B)µν ∂ν φ) = 0.
(2.6)
For the scalar wave in the frequency domain φ̃(ω, r), the above equation (2.6) withthe metric (2.1) is rewritten as,
(2.7)∇2 + ω 2 φ̃ = 4ω 2 U φ̃.

Figure 2.2: Illustration of the volume V which is inside of the dashed line. This V
is enclosed with the lens plane and a sphere at the observer of radius R. Note that
this V does not include the lens plane. n is the normal vector to the lens plane.

2.1.2

Kirchhoff diffraction integral

We derive the Kirchhoff diffraction integral in order to solve the above equation(2.7).1 In Fig.2.1, we show the gravitational lens geometry of the source, the lensand the observer. DL , DS and DLS are the distances to the lens, the source andfrom the lens to the source, respectively. η is a position vector of the source in thesource plane while ξ is the impact parameter in the lens plane. We use the thinlens approximation in which the lens is characterized by the surface mass density
Σ(ξ) and the gravitational waves are scattered only at the thin lens plane.
We define a volume V which is inside of the dashed line in Fig.2.2. This V is
enclosed by both the lens plane and sphere at the observer of radius R. Note thatthe boundary of V is very near the lens plane, but V does not include the lensplane (see Fig.2.2). Since we assume the thin lens approximation, U = 0 in V .
Then the equation (2.7) is reduced to the Helmholtz equation (∇2 + ω 2 )φ̃ = 0 in
V . The Green’s function for the Helmholtz equation is eiωr /r and satisfies,∇2 + ω 2 eiωr= −4πδ 3 (r)
r

(2.8)

where δ(r) is the delta function. With equations (2.7) and (2.8), the scalar field at
1

For detailed discussions about the diffraction integral, see Born & Wolf (1997), Chapter 8.

Figure 2.3: The point I is the intersection point of the ray path and the lens plane.
r is the distance from I to the observer.
the observer φ̃Lobs is written as,iωr
eiωr 22edV φ̃ ∇−∇ φ̃ ,
4π Vrr

1
φ̃Lobs = −

Z

(2.9)

where r denotes the distance from the observer. The volume integral in the above
equation is reduced to the surface integral by Green’s theorem:Z∂ eiωr eiωr ∂
1
2
L
−
φ̃ ,
(2.10)d ξ φ̃
φ̃obs =
4π S
∂n r
r ∂n
where n is normal vector to the lens plane (see Fig.2.2). Here, we set R → ∞ in
Fig.2.2 and assume that φ̃ and ∂ φ̃/∂n vanish at the surface of the sphere. Thus
the surface integral in (2.10) is done on the lens plane.
We use the eikonal approximation to the equation (2.7) in order to obtain thefield φ̃ and its derivative ∂ φ̃/∂n on the boundary (the lens plane) in equation(2.10). The field is written as φ̃ = AeiSP , where the amplitude A and the phase
SP satisfy the following conditions: |∇A/A|  |∇SP /SP | and |∇2 SP |  |∇SP |2 .
Then, we have the eikonal equation from the equation (2.7): (∇SP )2 = ω 2 (1−2U )2 .
The
R phase is obtained by integrating this equation along the ray path as, SP =
ω d`(1 − 2U ). Denoting a point I which is the intersection point of the ray pathand the lens plane as shown in Fig.2.3, we have the phase at I as,
Z obs.
Z I
Z obs. 
SP = ωd` −d`(1 − 2U ) = ωd` (1 − 2U ).
(2.11)
sourcesourceI

Since the wave propagates along the null geodesic, we obtain dt = d`(1 − 2U ) fromthe metric (2.1). Thus, the first term of the integral in equation (2.11) is the arrival
time td from the source to the observer. The second term is the distance r from I
to the observer as shown in Fig.2.3. Then, the phase (2.11) is rewritten as,
SP = ω(td − r).

(2.12)

The arrival time td at the observer from the source position η through ξ is givenby (Schneider, Ehlers & Falco 1992),
DL DS
td (ξ, η) =
2DLS



ξ
η
−
DL DS

2

− ψ̂(ξ) + φ̂m (η),(2.13)

where DL , DS and DLS are the distances to the lens, the source and from thelens to the source, respectively, as shown in Fig.2.1. The first and second term inequation (2.13) is the time delay relative to the unlensed ray. The third term φ̂m isthe arrival time for the unlensed ray. The deflection potential ψ̂(ξ) is determinedby,∇2ξ ψ̂ = 8πΣ,(2.14)
where ∇2ξ denotes the two-dimensional Laplacian with respect to ξ, and Σ(ξ) is
the surface mass density of the lens.
The derivatives in equation (2.10) are given by,eiωr∂ eiωr= −iω cos θ
,
∂n r
r
∂
φ̃ = iω cos θ 0 φ̃,∂n(2.15)(2.16)

where the angles θ and θ 0 are shown in Fig.2.3 and θ, θ 0  1. Then, the lensed
field at the observer φ̃Lobs in equation (2.10) is finally given by,ZωAL
d2 ξ exp [iωtd (ξ, η)] ,(2.17)
φ̃obs (ω, η) =2πiDLwhere we assume r ' DL .
We used the Kirchhoff diffraction formula to derive the lensed waveform (2.17).
We discuss the path integral formula which is another method to drive the solution(2.17) in the Appendix A.

2.1.3

Amplification factor

It is convenient to define the amplification factor as
F (ω, η) =φ̃Lobs (ω, η),φ̃obs (ω, η)(2.18)

where φ̃Lobs and φ̃obs are the lensed (given in equation (2.17)) and unlensed (U = 0
in Eq.(2.7)) gravitational wave amplitudes at the observer, respectively. Then, theamplification factor F is given by (Schneider, Ehlers & Falco 1992),ZωDS
F (ω, η) =d2 ξ exp [iωtd (ξ, η)] ,(2.19)DL DLS 2πiwhere F is normalized such that |F | = 1 in no lens limit (U = ψ̂ = 0).
Though we do not take account of the cosmological expansion in the metric, inEq.(2.1), the results can be applied to the cosmological situation since the wavelength of the gravitational waves is much smaller than the horizon scale. What
we should do is 1) take the angular diameter distances and 2) replace ω withω(1 + zL ) where zL is the redshift of the lens (Baraldo, Hosoya and Nakamura
1999). Then, the amplification factor F in the above equation (2.19) is rewritten
in the cosmological situation as,Z
DS ω(1 + zL )F (ω, η) =d2 ξ exp [iω(1 + zL )td (ξ, η)] .
(2.20)DL DLS
2πi
where DL , DS and DLS denote the angular diameter distances.

It is useful to rewrite the amplification factor F in terms of dimensionlessquantities. We introduce ξ0 as the normalization constant of the length in the lensplane. The impact parameter ξ and the source position η (in Fig.2.1) are rewritten
in dimensionless form,ξDL
x= ; y=η.
(2.21)
ξ0ξ 0 DS
Similarly, we define dimensionless frequency w by,w=DSξ 2 (1 + zL )ω.
DLS DL 0(2.22)

Then, the dimensionless time delay is given by,
DL DLS −2
ξ0 td (ξ, η)(2.23)
DS1|x − y|2 − ψ(x) + φm (y),(2.24)=
2
where ψ(x) and φm (y) correspond to ψ̂(ξ) and φˆm (η) in equation (2.13): ψ =DL DLS /(DS ξ02 )ψ̂ and φm = DL DLS /(DS ξ02 )φ̂m , respectively. We choose φm (y) sothat the minimum value of the time delay is zero.2 The dimensionless deflection
potential ψ(x) satisfies,2Σ
∇2x ψ =,
(2.25)
Σcr
where ∇2x denotes the two-dimensional Laplacian with respect to x, Σ is the surface
mass density of the lens, and Σcr = DS /(4πDL DLS ). Using the above dimensionlessquantities, the amplification factor is rewritten as,
Zw
F (w, y) =d2 x exp [iwT (x, y)] .
(2.26)
2πi
T (x, y) =2

φm (y) corresponds to the arrival time in the unlensed case in Eq.(2.13) originally. But,φm (y) is also the additional phase in F , which can be chosen freely (see the sentence after
equation (2.19)). Hence, we choose a convenient value for later calculation.

For axially symmetric lens models, the integral of the amplification factor F inequation (2.26) becomes a relatively simple form, since the deflection potential
ψ(x) depends only on x = |x|. F is expressed as (Nakamura & Deguchi 1999),
 

Z ∞
1 2
iwy 2 /2
dx x J0 (wxy) exp iw
F (w, y) = −iwex − ψ(x) + φm (y) , (2.27)
20where J0 is the Bessel function of zeroth order. It takes a long time to calculate
F numerically in equations (2.26) and (2.27), because the integrand is rapidly oscillating function especially for large w. In the Appendix B, we present a method
for numerical computation to shorten the computing time.
√
We use the Einstein radius (∼ M D) as a arbitrary scale length ξ0 (except for
the NFW lens), for convenience. The Einstein radius is the typical scale length of
the impact parameter in gravitational lensing, and hence the dimensionless quantities x, y become of the order of one. The dimensionless potential ψ(x) and timedelay T (x, y) are also of the order of one. Then, the dimensionless frequency isroughly w ∼ M/λ from equation (2.22).

2.1.4

Geometrical Optics Approximation

In the geometrical optics limit (w  1), the stationary points of the T (x, y)
contribute to the integral of Eq.(2.26) so that the image positions xj are determinedby the lens equation, ∇x T (x, y) = 0 or
y = x − ∇x ψ(x).

(2.28)

This is just the Fermat’s principle. We expand T (x, y) around the j-th image
position xj as,
T (x, y) = T (xj , y) +X

∂a T (xj , y)x̃a +a1X∂a ∂b T (xj , y)x̃a x̃b + O(x̃3 ) (2.29)
2 a,bwhere x̃ = x − xj and the indices a,b,... run from 1 to 2. The second term inEq.(2.29) vanishes because xj is the stationary point of T (x, y). Inserting Eq.(2.29)
to (2.26) with the Gaussian integral,3 we obtain the amplification factor in geometrical optics limit as (Nakamura & Deguchi 1999),
X
Fgeo (w, y) =|µj |1/2 exp [ iwTj − iπnj ] ,(2.30)jwhere the magnification of the j-th image is µj = 1/ det (∂y/∂xj ), Tj = T (xj , y)and nj = 0, 1/2, 1 when xj is a minimum, saddle, maximum point of T (x, y),
3

R∞

−∞

2

dxeiax =q

π iπ/4×sign(a)|a| erespectively. In the time domain the lensed wave is expressed as
X
φLgeo (t, r) =|µj |1/2 e−iπnj φ(t − td,j , r),(2.31)jwhere td,j is the time delay of the j-th image (td = DS ξ02 /(DL DLS ) × T ). Thisshows that the oscillatory behavior of F in high frequency w is essential to obtainthe time delay among the images.

2.2

Amplification Factor for Various Lens Models

In this section, we show the behavior of the amplification factor F for the various
lens models. We consider the following lens models: point mass lens, singularisothermal sphere (SIS) lens, Navarro Frenk White (NFW) lens, and binary lens.

2.2.1

Point Mass Lens

The surface mass density is expressed as Σ(ξ) = ML δ 2 (ξ) where ML is the lensmass. As the normalization constant ξ0 we adopt the Einstein radius given byξ0 = (4ML DL DLS /DS )1/2 while the dimensionless deflection potential is given as
ψ(x) = ln x. In this case, Eq.(2.27) is analytically integrated as (Peters 1974),h πw
oiw n w
F (w, y) = exp+iln− 2φm (y)
4
2 
2

i
i 2
i
× Γ 1 − w 1 F1
w, 1; wy ,(2.32)222pwhere w = 4MLz ω; φm (y) = (xm − y)2 /2 − ln xm with xm = (y + y 2 + 4)/2;
MLz = ML (1 + zL ) is the redshifted lens mass and 1 F1 is the confluent hypergeometric function. Thus, the amplification factor F depends on two parameters; the(dimensionless) frequency w and the source position y. In the geometrical optics
limit (w  1) from equation (2.30) we have
Fgeo (w, y) = |µ+ |1/2 − i |µ− |1/2 eiw∆T ,(2.33)p
2 + 4) andwhere the magnification of each image is µ± = 1/2 ±p(y 2 + 2)/(2y yp
2
2
the time
p delay between the double images is ∆T = y y + 4/2 + ln(( y + 4 +y)/( y 2 + 4 − y)). The time delay is typically ∆td = 4MLz × ∆T = 2 × 103 sec
×(MLz /108 M ).
The point mass lens model has been used for lensing by compact objects suchas black holes or stars. Even for the extended lens, this model can be used if thelens size is much smaller than the Einstein radius, because of Birkhoff’s theorem.

Figure 2.4: The amplification factor |F | (left) and the phase θF = −i ln[F/|F |](right) for point mass lens as a function of w with the fixed source position y =0.1, 0.3, 1 and 3. For w . 1, the amplification is very small due to the diffraction
effect. For w & 1, the oscillatory behavior appears due to the interference between
the double images.
Hence, this model is most frequently used in wave optics in gravitational lensingof gravitational waves (Nakamura 1998; Ruffa 1999; De Paolis et al. 2001,2002;
Zakharov & Baryshev 2002; Takahashi & Nakamura 2003). As far as we know, theanalytical solution of F in Eq.(2.26) is known only for this model.
In the left panel of Fig.2.4, the amplification factor |F | for the point mass lensis shown as a function of w with a fixed source position y = 0.1, 0.3, 1 and 3. For
w . 1, the amplification is very small due to the diffraction effect (e.g., Bontz &Haugan 1980). Since in this case the wave length is so long that the wave does
not feel the existence of the lens. For w & 1, |F | asymptotically converges to thegeometrical optics limit (Eq.(2.33)),|Fgeo |2 = |µ+ | + |µ− | + 2 |µ+ µ− |1/2 sin(w∆T ).

(2.34)

The first and second terms in Eq.(2.34), |µ| = |µ+ | + |µ− |, represent the total
magnification in the geometrical optics. The third term represents the interference
between the double images. The oscillatory behavior (in Fig.2.4) is due to thisinterference. The amplitude and the period of this oscillation are approximately
equal to 2|µ+ µ− |1/2 and w∆T in the third term of Eq.(2.34), respectively. As
the source position y increases, the total magnification |µ| (= |µ+ | + |µ− |) and theamplitude of the oscillation 2|µ+ µ− |1/2 decrease. This is because each magnification|µ± (y)| decreases as y increases.
The right panel of Fig.2.4 is the same as the left panel, but we show the phaseof the amplification factor θF = −i ln[F/|F |]. The behavior is similar to that of
the amplitude (in left panel), and the wave effects appear in the phase θF as wellas the amplitude |F |. For w & 1, θF asymptotically converges to the geometrical
optics limit (Eq.(2.33)),−|µ− |1/2 cos(w∆T )θF = arctan,(2.35)|µ+ |1/2 + |µ− |1/2 sin(w∆T )From the above equation (2.35), the phase θF oscillates between − arctan(|µ− /µ+ |1/2 )and arctan(|µ− /µ+ |1/2 ) with the period of w∆T . As the source position y increases,
the magnification ratio |µ− /µ+ | and the amplitude of the oscillation arctan(|µ− /µ+ |1/2 )decrease.

2.2.2

Singular Isothermal Sphere

The surface density of the SIS (Singular Isothermal Sphere) is characterized by
the velocity dispersion v as, Σ(ξ) = v 2 /(2ξ). As the normalization constant we
adopt the Einstein radius ξ0 = 4πv 2 DL DLS /DS and the dimensionless deflection
potential is ψ(x) = x. In this case F in Eq.(2.27) is expressed as,

 
Z ∞
1 2
iwy 2 /2x − x + φm (y) , (2.36)
dx x J0 (wxy) exp iw
F (w, y) = −iwe
20where J0 is the Bessel function of zeroth order; φm (y) = y + 1/2 and w = 4MLz ω
where MLz is defined as the mass inside the Einstein radius given by MLz =4π 2 v 4 (1 + zL )DL DLS /DS . Then, F depends on the two parameters w and y. We
computed the above integral numerically for various parameters. In the geometrical
optics limit (w  1), F is given by,
Fgeo (w, y) = |µ+ |1/2 − i |µ− |1/2 eiw∆T
1/2= |µ+ |for y < 1,
for y ≥ 1,

(2.37)

where µ± = ±1 + 1/y and ∆T = 2y. For y < 1 double images are formed, while
for y ≥ 1 single image is formed.
The SIS model is used for more realistic lens objects than the point mass lens,such as galaxies, star clusters and dark halos (Takahashi & Nakamura 2003).
Fig.2.5 is the same as Fig.2.4, but for the SIS lens. The behavior is similar
to that for the point mass lens. In the left panel, for w & 1, |F | asymptoticallyconverges to the geometrical optics limit (Eq.(2.37)),|Fgeo |2 = |µ+ | + |µ− | + 2 |µ+ µ− |1/2 sin(w∆T ) for y < 1,
= |µ+ |for y ≥ 1.

(2.38)For y < 1, the oscillatory behavior in the left panel of Fig.2.5 is because of theinterference between the double images. The amplitude and period are given inFigure 2.5: Same as Fig.2.4, but for SIS lens. We note that even if y ≥ 1 (a single
image is formed in the geometrical optics limit), the dumped oscillatory behavior
appears.
the above equation (2.38) as, 2|µ+ µ− |1/2 and w∆T , respectively. As y increases,
the magnifications |µ± | and the amplitude of the oscillation 2|µ+ µ− |1/2 decrease.
Even for y ≥ 1 (y = 3 in Fig.2.5), the damped oscillatory behavior appears, whichlooks like the time delay factor of sin(w∆T ) although only a single image exists in
the geometrical optics limit (see equation (2.38)). In the right panel, for w & 1,
θF converges to the geometrical optics limit (Eq.(2.37)),θF−|µ− |1/2 cos(w∆T )= arctan|µ+ |1/2 + |µ− |1/2 sin(w∆T )
= 0




for y ≤ 1,for y ≥ 1.

(2.39)For y < 1, θF oscillates with the amplitude of arctan(|µ− /µ+ |1/2 ) and the periodof w∆T as shown in right panel. We note for y ≥ 1 (y = 3 in the right panel) thatthe dumped oscillation is seen in θF as well as in |F | (left panel). We discuss thereason for this dumped oscillations in the next chapter.

2.2.3

Navarro-Frenk-White lens

The NFW profile was proposed from numerical simulations of cold dark matter(CDM) halos by Navarro, Frenk & White (1996,1997). They showed that thedensity profile of the dark halos has the “universal” form,
ρ(r) =

ρs,(r/rs )(r/rs + 1)2(2.40)

where rs is a scale length and ρs is a characteristic density. The scale length rs
is typically 10 (100) kpc on a galactic (cluster) halo scale. The NFW lens is used
for lensing by galactic halos and clusters of galaxies. The surface mass density is
written as (Bartelmann 1996),s#"2
1 − ξ/rs
1−(2.41)
Σ(ξ) = 2ρs rs
arctanh
1 + ξ/rs 1 − (ξ/rs )2{1 − (ξ/rs )2 }3/2
= 2ρs rs"−2{(ξ/rs )2 − 1}

arctan3/2sfor ξ ≤ rs ,#ξ/rs − 11+
ξ/rs + 1 (ξ/rs )2 − 1(2.42)for ξ ≥ rs .

The deflection potential is also written as (Bartelmann 1996; Keeton 2001),2 √
κs  x  2 
for x ≤ 1,
ψ(x) =
ln− arctanh 1 − x2
2
2


2
√
κs
x 2 
2+ arctan x − 1=
for x ≥ 1,
(2.43)ln22where κs = 16πρs (DL DLS /DS )rs is the dimensionless surface density and we adopt
the scale radius, not the Einstein radius, as the normalization length: ξ0 = rs . With
the above equation (2.43), we numerically integrate the integral of F in equation(2.27). F depends on three parameters; w, y and κs . We show the results forκs = 1 and 10, since κs is of the order of 1 suggested by the numerical simulations(e.g. Bullock et al. 2001).
We show the lens equation, y = x − ψ 0 (x), for the NFW lens with κs = 1 (left)
and 10 (right) in Fig.2.6. For |y| < ycrit three images are formed, while for |y| > ycrita single image is formed. The tangential (radial) caustic is y = 0 (y = ycr ) in
which the magnification becomes infinite. The image positions xj , magnificationsµj and time delays Tj are numerically obtained from the lens equation. Fgeo is also
numerically calculated.
The amplification factor |F | and the phase θF for the NFW lens are shownfor κs = 1 in Fig.2.7 and for κs = 10 in Fig.2.8. The source position is fixed asy/ycr = 0.1, 0.3 and 2 in both figures. The values of w in the horizontal axis of theFig.2.7 (Fig.2.8) is about 100 (0.1) times larger (smaller) than that in the previous
figures (see Fig.2.4 and 2.5). This is because that w is proportional to ξ02 and weadopt the scale radius, not the Einstein radius (which is adopted in the previous
cases), as the normalization length: ξ0 = rs . From the left (right) panel of Fig.2.6,
the Einstein radius is x ∼ 0.2 (x ∼ 3), which is determined by y = 0 = x − ψ 0 (x).
Then rs is about 5 (0.3) times larger (smaller) than the Einstein radius, and hence
the dimensionless frequency w is about 30 (0.1) times larger (smaller) than thatin the case of using the Einstein radius (since w ∝ ξ02 , see Eq.(2.22)). Thus, asFigure 2.6: The lens equation for the NFW lens with κs = 1 (left) and κs = 10(right). For |y| < ycrit three images are formed, while for |y| > ycrit single image isformed.

Figure 2.7: Same as Fig.2.4, but for NFW lens with κs = 1. We note that even
if y > ycr (a single image is formed in the geometrical optics limit), the dumped
oscillatory behavior appears similar to the SIS model.

Figure 2.8: Same as Fig.2.7, but for κs = 10.
shown in Fig.2.7, the diffraction effect is important for w . 102 . For w & 102
in y/ycr = 0.1 and 0.3, the three images are formed, and hence the interference
patterns between the three images are relatively complicated. We note that for
y/ycr = 2 the dumped oscillatory behavior is appeared similar to the case in the
SIS model with y = 3 as shown in Fig.2.5. The reason for this dumped oscillation
is discussed in the next chapter.
Fig.2.8 is the same as Fig.2.7, but for κs = 10. The behavior of F and θF
is similar to that for κs = 1. The diffraction effect is important for w . 1. The
amplitude of the oscillation in |F | is smaller than that for κs = 1 in Fig.2.7, because
the magnification µj is large in the case of κs = 1.

2.2.4

Binary lens

We consider the two point mass lenses with mass M1 and M2 in the lens plane. We
assume the two lens positions are fixed with the separation 2L. The surface densityof the binary lens is, Σ(ξ) = M1 δ 2 (ξ − L) + M2 δ 2 (ξ + L), where we set L = (L, 0)
in the lens plane. As the normalization constant we adopt the Einstein radius
ξ0 = [4(M1 + M2 )DL DLS /DS ]1/2 , and the (dimensionless) deflection potential is(Schneider & Weiss 1986),
ψ(x) = ν1 ln |x − `| + ν2 ln |x + `| .

(2.44)The mass ratio ν1,2 and the separation vector ` are defined as,
ν1,2 =

M1,2
L, `= .
M1 + M 2ξ0(2.45)

With the above equation (2.44), we numerically calculate the amplification factor
F in Eq.(2.26). Since the binary lens is non-axially symmetric, it takes a longtime to numerically calculate the double integral in Eq.(2.26). Thus, we consider
the equal mass lens (ν1,2 = 1/2) for simplicity. Then, the amplification factor F
depends on w, y and `.
The lens equation for the binary lens is obtained from Eq.(2.44) as, y =x − ∇x ψ(x). The image positions xj , magnifications µj , and time delays Tj are
numerically obtained.4 In√Fig.2.9, we show the lensing configurations with fixedsource position y = (1, 3)/10 and the separation ` = 1.2 (upper panel) and` = 0.5 (lower panel). The upper part of each panel displays the source position(star) and the caustics (solid line) in the source plane. The lower part of each panel
displays the two point mass lenses (filled circle), the image positions (open circles),and the critical curves (dotted line) in the lens plane. The caustics (critical curves)
are defined as the curves on which the magnification is infinite in the source (lens)plane. Thus, if the source is near the caustics the amplification factor F is verylarge. The three images are formed in the upper panel, while the five images areformed in the lower panel. In the binary lens, if the source is inside the caustics thefive images are formed, while if the source is outside the caustics the three images
are formed (see Schneider & Weiss 1986).
In Fig.2.10, we show the amplification factor |F | and the phase θF for the binarylens in the case shown in Fig.2.9. In this case, since the three or five images areformed, the interference patterns are quit complicated. For ` = 0.5, since the sourceis near the caustics in Fig.2.9 and the five images are formed, the amplification F
is larger than that for ` = 1.2.
In Fig.2.11, we show the amplification factor |F | with the separation ` =0.3, 0.1, and√ 0 (this corresponds to the√point mass lens). The source position
is y = (1, 3)/10 (left panel) and y = ( 3, 1) × 0.15 (right panel). In both thesepanels, the shape for ` = 0.3 (dashed line) is different from that for the point masslens (solid line). But, the shape for ` = 0.1 (dotted line) is very similar to thatfor the point mass lens. Thus, if the separation L is about 0.1 times smaller thanthe Einstein radius ` . 0.1 or L . 0.1 × ξ0 , then the amplification factor is quit
similar to that in the point mass lens.

4

We use the method developed by Asada (Asada 2002; Asada, Kasai & Kasai 2002) to obtain
xj , µj , and Tj . They showed that the lens equation for the binary lens, which is a simultaneous
equation, can be reduced to a real fifth-order equation.

Figure 2.9: The lensing configurations with the separation ` = 1.2 (upper) and` = 0.5 (lower)
for the binary lens with equal mass. The source position is fixed√
as y = (1, 3)/10. In the upper part of each panel, the source position (star) andthe caustics (solid line) are shown in the source plane. In the lower part of eachpanel, the two point mass lenses (filled circle), the image positions (open circles),and the critical curves (dotted line) are shown in the lens plane.

Figure 2.10: Same as Fig.2.4, but for the binary lens with equal mass. The separation of the two lenses is ` =√ 1.2 (solid line) and ` = 0.5 (dashed line) with thefixed source position y = (1, 3)/10.

Figure 2.11: The amplification factor |F | for the binary lens with the separation√
3)/10` = 0.3, 0.1, and 0 (the
point
mass
lens).
The
sourceposition
is
y
=
(1,
√
(left panel) and y = ( 3, 1) × 0.15 (right panel).

Chapter 3
Quasi-geometrical Optics
Approximation in Gravitational
Lensing
In the previous chapter, we reviewed the wave optics in the gravitational lensingof gravitational waves. We showed that for λ & ML (where λ is the wavelength ofgravitational waves and ML is the Schwarzschild radius of the lens) the diffraction
effect is important and the magnification is small, and for λ  ML the geometrical
optics approximation is valid. In this chapter, we consider the case for λ . ML ,i.e. the quasi-geometrical optics approximation which is the geometrical optics
including corrections arising from the effects of the finite wavelength. We canobtain these correction terms by an asymptotic expansion of the diffraction integral(discussed in §2.1) in powers of the wavelength λ.1 We note that these terms canbe obtained analytically.
It is important to derive these correction terms for the following two reasons:(i) calculations in the wave optics are based on the amplification factor F (inEq.(2.26)), but it is time consuming to numerically calculate F especially for highfrequency ω  1 (see the sentences after equation (2.27)). Hence, it is a greatsaving of computing time to use the analytical expressions. (ii) We can understandclearly the difference between the wave optics and the geometrical optics.
In this chapter, we explore the asymptotic expansion of F (w, y) (in Eq.(2.26))
in powers of the inverse of the dimensionless frequency 1/w. Here, 1/w is roughly
equal to λ/ML from the discussion at the end of §2.1.3. The first term, arisingfrom the high frequency limit w → ∞, corresponds to the geometrical optics limit
Fgeo in the previous chapter. The second term, being of the order of 1/w, is thefirst correction term arising from diffraction effect. We study this first correction
term, and do not consider the higher order terms, for simplicity.
1

The asymptotic expansion of the diffraction integral has been studied in optics. See thefollowing text books for detailed discussion: Kline & Kay (1965), Ch.XII; Mandel & Wolf (1995),
Ch.3.3; Born & Wolf (1997), App.III, and references therein.

25

3.1

Effect on the Magnifications of the Images

We expand the amplification factor F (w, y) in powers of 1/w ( 1) and discuss
the behavior of the order of 1/w term. Here, we only consider axially symmetriclens models because the basic formulae are relatively simple, while the case of
the non-axially symmetric lens models is discussed in the Appendix C. In theaxially symmetric lens, the deflection potential is a function of |x| as ψ(x) = ψ(|x|)
where x = (x1 , x2 ); the source position and the image position are y = (y, 0) andxj = (xj , 0), respectively. The lens equation reduces to a one-dimensional form,y = x1 − ∂1 ψ(x). In this case, the expansion of T (x, y) around the image position
xj in Eq.(2.29) is simply rewritten as,
T (x, y) = Tj + αj x̃21 + βj x̃22 + O(x̃3 ),(3.1)

where x̃ = x − xj . Tj , αj and βj are defined by,1(xj − y)2 − ψj + φm
2

1
1 − ψj00
=
2

ψj0
1
1−
=
2
|xj |

Tj =

(3.2)

αj

(3.3)

βj

(3.4)

(n)

where ψj = dn ψ(|xj |)/dxn . The magnification µj and the coefficient nj in Fgeo of
equation (2.30) are also rewritten as, µj = 1/(4αj βj ) = 1/[(1 − ψj00 )(1 − ψj0 /|xj |)]and nj = 1/2 − sign(αj )/4 − sign(βj )/4.
We expand T (x, y) in Eq.(3.1) up to the fourth order of x̃ as,
T (x, y) = Tj + αj x̃21 + βj x̃22 ++1X∂a ∂b ∂c T (xj , y)x̃a x̃b x̃c
6 a,b,c

1 X∂a ∂b ∂c ∂d T (xj , y)x̃a x̃b x̃c x̃d + O(x̃5 ).
24 a,b,c,d(3.5)

Inserting the above equation (3.5) into (2.26), we obtain,Z
1 X iwTj
F (w, y) =e
d2 x0 exp i αj x01 2 + βj x02 2
2πi j
1 X
+ √
∂a ∂b ∂c T (xj , y)x0a x0b x0c
6 w a,b,c
1 X+∂a ∂b ∂c ∂d T (xj , y)x0a x0b x0c x0d + O(w −3/2 )24w a,b,c,d

!#

.

(3.6)

√
√
Here, we change the integral variable from x to x0 = wx̃ = w(x − xj ). We
expand the above equation (3.6) in powers of 1/w as,
"
Z
X
i X
1
02
02
F (w, y) =eiwTj d2 x0 ei(αj x1 +βj x2 ) 1 + √∂a ∂b ∂c T (xj , y)x0a x0b x0c
2πi j
6 w a,b,c
!2
1 1 X+−∂a ∂b ∂c T (xj , y)x0a x0b x0c
w  72 a,b,c
)
#
i X
(3.7)
∂a ∂b ∂c ∂d T (xj , y)x0a x0b x0c x0d + O(w −3/2 ) .
+24 a,b,c,d
The first term of the above equation (3.7) is the amplification factor in the geometrical optics limit Fgeo in equation (2.30). The integral in the second term vanishes
because the integrand is an odd function of x0a . The third term is the correction
term, being proportional to 1/w, arising from the diffraction effect. Thus the deviation from the geometrical optics is of the order of 1/w ∼ λ/ML . Inserting T (x, y)
in Eq.(2.24) into (3.7), we obtain F as,


X
i
1/2
F (w, y) =
|µj |
(3.8)
1 + ∆j eiwTj −iπnj + O(w −2 ),
w
j
where

"
#
(3)
ψ
2
1
5
α
−
β
1 (4)
1
1
j
j
(3)
j
+ 2
,
ψ +
ψ
+
∆j =
2 j
3 j
16 2αj12αj
αj |xj |αj βj |xj |2(3.9)

and ∆j is a real number. We denote dFm as the second term of the above equation(3.8) as,i X
dFm (w, y) ≡∆j |µj |1/2 eiwTj −iπnj .
(3.10)
w j
Since dFm is the correction term arising near the image positions, this term represents the corrections to the properties of these images such as its magnifications,and the time delays. We rewrite F in above equation (3.8) as, 2  1/2
X∆j
eiwTj +iδϕ−iπnj + O(w −2 ),(3.11)F (w, y) =µj 1 +wj

where δϕ = arctan(∆j /w). Thus in the quasi-geometrical optics approximation,
the magnification µj is modified to µj [1+(∆j /w)2 ], where (∆j /w)2 is of the order of(λ/ML )2 . That is, the magnification is slightly larger than that in the geometrical
optics limit. The phase is also changed by δϕ, which is of the order of λ/ML .
The lensed wave in the time domain is given by,
Z t−td,jX
LL
1/2 −iπnj
˜
dt0 φ(t0 , r),(3.12)φ (t, r) = φgeo (t, r) +∆j |µj | ej˜ j = (DL DLS /DS )ξ0−2 (1 + zL )−1 ∆j and φL is the lensed wave in thewhere ∆
geo
geometrical optics limit in equation (2.31). The second term is the deviation fromthe geometrical optics limit. For example, we consider the monochromatic wave
φ(t) with the frequency ω as the unlensed waveform: φ(t, r) = A sin(ωt + ϕ0 ),where A is the amplitude and ϕ0 is the phase. Inserting this monochromatic wave
into Eq.(2.31) and (3.12), we obtain the lensed wave φL as,L

φ (t, r) = AXj ˜ 2  1/2∆jµj 1 +
sin [ ω(t − td,j ) + ϕ0 + δϕ] e−iπnj ,ω(3.13)˜ j /ω), and we note that ∆˜ j /ω = ∆j /w. From the above
where δϕ = arctan(∆˜ j /ω)2 ], and the phaseequation, the each lensed image is magnified by µj [1 + (∆is changed by δϕ. These results are consistent with that in the frequency domain(see the sentences after equation (3.11)).

3.2

Contributions from the Non-stationary Points

In the previous section, we showed that the contributions to the diffraction integral
F arise from the stationary points (or image positions). In this section, we discuss
the contributions from the non-stationary points. We denote xns as the nonstationary point, at which the condition |∇T | 6= 0 satisfies. If T (x, y) can be
expanded at xns , we obtain the series of T similar to Eq.(3.1) as,
T (x, y) = Tns + T10 x̃1 + T20 x̃2 + αns x̃21 + βns x̃22 + O(x̃3 ),(3.14)

0where x̃ = x − xns , and T1,2
= ∂1,2 T (xns , y). Note that either T10 or T20 does notvanish because of |∇T | 6= 0 at xns . Tns , αns , and βns are defined same as equations
from (3.2) to (3.4), but at the non-stationary point xns .
Inserting the above equation (3.14) to (2.26), we obtain,
 

Z

1
eiwTns
2 0
0 0
0 0
0 2
0 2
−2
d x exp i T1 x1 + T2 x2 +αj x1 + βj x2 + O(w ) ,F (w, y) =
2πiw
w
(3.15)
0where x = wx̃. We expand the integrand of the equation (3.15) in powers of 1/w
as,


Z

eiwTns1
2 0 i(T10 x01 +T20 x02 )
0 2
0 2
−2
F (w, y) =dx e
1+
αns x1 + βns x2 + O(w ) .
2πiw
w(3.16)
The above equation can be integrated as,
eiwTns
∂2
∂2
1
0
0
0
0
−2
F (w, y) = 2π
δ(T1 )δ(T2 ) −αns 02 + βns 02 δ(T1 )δ(T2 ) + O(w ) .
iw
w
∂T1
∂T2
(3.17)

Thus, if both T10 6= 0 and T20 6= 0, the above equation vanishes. Even if either
T10 = 0 or T20 = 0, it is easy to show that F vanishes doing the same calculations
from equations (3.15) to (3.17). Thus, the contributions to the amplification factor
F at the non-stationary points are negligible.
In the above discussion, we assume that T (x, y) has the derivatives at the nonstationary point xns (see the sentences before the equation (3.14)). But, if thederivatives of T are not defined at xns , the result in Eq.(3.17) should be reconsidered. If the lens has the cuspy (or singular) density profile at the center, thederivatives of T are not defined at the lens center. We will discuss this case in thenext section.

3.3

Central Cusp of the Lens

We consider the correction terms in the amplification factor F arising at the central
cusp of the lens. For the inner density profile of the lens ρ ∝ r −α (0 < α ≤ 2),2
the surface density and the deflection potential at small radius are given by,Σ(ξ) ∝ ξ −α+1
∝ ln ξ

for α 6= 1,
for α = 1,

(3.18)

ψ(x) ∝ x−α+3∝ x2 ln x

for α 6= 1,
for α = 1.

(3.19)

We note that the Taylor series of ψ(x) around x = 0 like Eq.(3.14) cannot be
obtained from the above equation (3.19). For example, in the case of α = 2, thedeflection potential is ψ ∝ x, but the derivative of |x| is discontinuous at x = 0.
Hence we use ψ in Eq.(3.19) directly, not the Taylor series in equation (3.14). Let
us calculate the correction terms in the amplification factor contributed from thelens center for following four cases; α = 2, 1 < α < 2, α = 1, and 0 < α < 1.
Case of α = 2 If the inner density profile is ρ ∝ r −2 (e.g. the singular isothermal
sphere model), the deflection potential is given by ψ(x) = ψ0 x (ψ0 is a constant)from equation (3.19). Inserting this potential ψ into equation (2.26), we obtain,
 

Z
2
q
eiw[y /2+φm (y)]
02
0
02
02
F (w, y) =dx exp −i yx1 + ψ0 x1 + x2 + O(1/w) ,2πiw(3.20)
where we changed the integral variable from x to x0 = wx. We denote dFc (w, y)
as the leading term of the above integral which is proportional to 1/w. dFc is
2

We do not consider the steeper profile α > 2, since the mass at the lens center is infinite.

obtained by integrating the above equation as,
2

eiw[y /2+φm (y)]1
dFc (w, y) =2w
(y − ψ02 )3/2

for |y| > |ψ0 |,(3.21)

for |y| < |ψ0 |.

(3.22)

2ieiw[y /2+φm (y)]=2w
(ψ0 − y 2 )3/2

Thus, the contributions to the amplification factor F at the lens center is of theorder of 1/w ∼ λ/ML . This is because of the singularity in the density profile atthe lens center. The correction terms dFc in equation (3.21) and (3.22) represent
a diffracted image which is formed at the lens center by the diffraction effect. The
magnification of this image is of the order of ∼ λ/ML .
Case of 1 < α < 2 The deflection potential at the small radius is ψ(x) =ψ0 x−α+3 (ψ0 is a constant) from equation (3.19). We insert ψ into equation(2.27),
Zi iw[y2 /2+φm (y)] ∞ 0 0
02
α−2
0−α+3
dx x J0 (yx0 ) ei/(2w)x e−iw ψ0 x
, (3.23)F (w, y) = − ew0where x0 = wx. We expand the integrand in powers of 1/w as:Z
i iw[y2 /2+φm (y)] ∞ 0 0
F (w, y) = − edx x J0 (yx0 ) 1 − iw α−2 ψ0 x0−α+3 + O(w 2(α−2) ) .
w
0(3.24)
The above equation can be integrated analytically (e.g. Gradshteyn & Ryzhik2000). 3 The first term in Eq.(3.24) vanishes, and the second term is the leading
term. We denote the leading term as dFc (w, y):
1  y α−5Γ((5 − α)/2)2,dFc (w, y) = −ψ0 w α−3 eiw[y /2+φm (y)]2 2Γ((α − 3)/2)(3.25)

where Γ is the gamma function. Thus, the diffracted image is formed at the lenscenter similar to the case of α = 2. The magnification is roughly ∼ (λ/ML )3−α .
Case of α = 1 If the inner density profile is ρ ∝ r −1 (e.g. the Navarro Frenk
White model), the deflection potential is given by ψ(x) = ψ0 x2 ln x (ψ0 is a constant) from equation (3.19). Inserting this ψ into equation (2.27), we have,Z
i iw[y2 /2+φm (y)] ∞ 0 0
020
F (w, y) = − edx x J0 (yx0 ) ei/(2w)x [1−2ψ0 ln(x /w)] ,(3.26)w
0
3

A formula for the Bessel function, xn J0 (yx) = [d/(ydy)]n (y n Jn (yx)), where n is a integer(Abramowitz & Stegun 1970), is useful to integrate F .

where x0 = wx. We expand the integrand of Eq.(3.26) in powers of 1/w:



Z
i 02
i iw[y2 /2+φm (y)] ∞ 0 0x0
0dx x J0 (yx ) 1 +x
F (w, y) = − e
1 − 2ψ0 ln
w
2w
w
0

+O(w −2 ) .
(3.27)
The above equation can be integrated analytically (e.g. Gradshteyn & Ryzhik2000), similar to the previous case of 1 < α < 2. The first term in Eq.(3.27)
vanishes and the second term becomes the leading term. Denoting dFc (w, y) as
this leading term, dFc is written as,dFc (w, y) =−4ψ0 iw[y2 /2+φm (y)]e.
(wy 2 )2

(3.28)

Thus, the diffracted image is formed at the lens center similar to the previous cases.
The magnification is roughly ∼ (λ/ML )2 .
Case of 0 < α < 1 The diffraction potential is the same as that in the previous
case 1 < α < 2, and the amplification factor is similarly given in equation (3.23).
We expand the integrand of Eq.(3.23) in powers of 1/w as:Zi 02i iw[y2 /2+φm (y)] ∞ 0 00x
dx x J0 (yx ) 1 +F (w, y) = − e
w
2w
0

α−2
0−α+3
2
−iw ψ0 x+ O(1/w ) ,(3.29)
where we note that the leading correction term in the integrand is ∼ 1/w, not∼ w α−2 (this is the case for equation (3.24)), since 0 < α < 1. The above equation
can be integrated similar to the previous cases. The first and the second term inEq.(3.29) vanish, and the leading term is the third term. Denoting dFc (w, y) as
this leading term, we have dFc as,
1  y α−5Γ((5 − α)/2)2dFc (w, y) = −ψ0 w α−3 eiw[y /2+φm (y)].
2 2Γ((α − 3)/2)(3.30)

This is the same as dFc in Eq.(3.25).
From the discussion for the four cases, the diffracted image is always formed at
the lens center for the inner density profile ρ ∝ r −α (0 < α ≤ 2). The magnificationof this central image is roughly given by, µ ∼ (λ/ML )3−α .

3.4Results for Specific Lens Models

We apply the quasi-geometrical optics approximation to simple lens models. We
consider the following axially symmetric lens models: point mass lens, singularisothermal sphere (SIS) lens, isothermal sphere lens with a finite core, and NavarroFrenk-White (NFW) lens. Since the amplification factor F and that in the geometrical optics Fgeo are discussed in the previous chapter, we derive only dFm (in§3.1) and dFc (in §3.3) for the above lens models. We define dF for convenienceas the sum of dFm and dFc :dF (w, y) ≡ dFm (w, y) + dFc (w, y).

3.4.1(3.31)

Point Mass Lens

The surface mass density is given by, Σ(ξ) = ML δ 2 (ξ), where ML is the lens mass.
F and Fgeo are given in equation (2.32) and (2.33), respectively. In this modeldFc = 0, and dF (= dFm ) is given from equation (3.10) by,
4x2+ − 1
4x2− − 1
i
1
1/2
|µ
|
+
|µ− |1/2 eiw∆T ,
+
2
2
2
2
3
3
3w (x+ + 1) (x+ − 1)
3w (x− + 1) (x− − 1)
(3.32)p2
1/2 ± (y 2 +where px± = (y ± y + 4)/2 is the position of each image, µ± =p22)/(2ythe magnification of each image, and ∆T = y y 2 + 4/2 +p y + 4) is pln(( y 2 + 4 + y)/( y 2 + 4 − y)) is the time delay between the double images. The
first and second terms in Eq.(3.32) are the correction terms for the magnificationsof the two images as discussed in §3.1.
In Fig.3.1(a), the amplification factor is shown as a function of w with a fixedsource position y = 0.3. The solid line is the full result |F | in Eq.(2.32); the dottedline is the geometrical optics approximation |Fgeo | in Eq.(2.33); the dashed line is
the quasi-geometrical optics approximation |Fgeo +dF | in Eq.(2.33) and (3.32). The
oscillatory behavior (in Fig.3.1(a)) is due to the interference between the double
images (see also the discussion in §2.2.1). For large w (& 10), these three lines
asymptotically converge.
In Fig.3.1(b), the differences between F , Fgeo and Fgeo + dF are shown as afunction of w with y = 0.3. The thin solid line is |F − Fgeo |, and the thin dashedline is |F − (Fgeo + dF )|. The thick solid (dashed) line represents the power ofw −1 (w −2 ). From this figure, for larger w ( 1) F converges to Fgeo with the errorof O(1/w) and converges to Fgeo + dF with the error of O(1/w 2 ). These results
are consistent with the analytical calculation in section 3.1.
dF (w, y) =3.4.2

Singular Isothermal Sphere

The surface density of the SIS model is characterized by the velocity dispersionv as, Σ(ξ) = v 2 /(2ξ), and the deflection potential is given by, ψ(x) = x. F andFigure 3.1: (a) The amplification factor |F | for a point mass lens as a functionof w with a fixed source position y = 0.3. The solid line is the full result F ; thedotted line is the geometrical optics approximation Fgeo ; the dashed line is thequasi-geometrical optics approximation Fgeo + dF . (b) The differences between F ,Fgeo and Fgeo + dF for a point mass lens as a function of w with y = 0.3. The thinsolid line is |F − Fgeo |, and the thin dashed line is |F − (Fgeo + dF )|. The thicksolid (dashed) line represents the power of w −1 (w −2 ).

Figure 3.2: Same as Fig.3.1, but for SIS lens with a source position y = 0.3.
Fgeo are given in equation (2.36) and (2.37), respectively. In the quasi-geometrical
optics approximation, dF is given by,
i |µ+ |1/2
1
1 |µ− |1/2 iw∆T
i
2
−
e
+
eiw[y /2+φm (y)] ,
2
2
2
3/2
8w y(y + 1)
8w y(1 − y)w (1 − y )for y < 1,1/2
11i |µ+ |
2+eiw[y /2+φm (y)]for y ≥ 1,
=
2
2
3/2
8w y(y + 1)
w (y − 1)(3.33)

dF (w, y) =where µ± = ±1 + 1/y, ∆T = 2y, and φm (y) = y + 1/2. For y < 1, the first and thesecond term in Eq.(3.33) are correction terms for the magnifications of the images
formed in the geometrical optics (i.e. dFm in §3.1), and the third term corresponds
to the diffracted image at the lens center (i.e. dFc in §3.2). For y ≥ 1, the first term
is correction term for the magnification dFm , and the second term corresponds tothe diffracted image at the lens center dFc . Thus, in the quasi-geometrical optics
approximation, for y < 1 the three images are formed, while for y ≥ 1 the double
images are formed in the SIS model.
Fig.3.2 is the same as Fig.3.1, but for the SIS lens with a source position y = 0.3.
In Fig.3.2(a), the behavior is similar to that in the point mass lens (in Fig.3.1(a)).
The oscillation of |Fgeo | is because of the interference between the double images,
while the oscillation of |Fgeo + dF | is because of that among the three images. But,
since the diffracted image is fainter than the others, the difference between |Fgeo |and |Fgeo + dF | is very small for w & 10 as shown in Fig.3.2(a). As shown in
Fig.3.2(b), the errors decrease as w increases, and the results are consistent withthe analytical calculation.

Figure 3.3: Same as Fig.3.2, but for y = 2.
Fig.3.3 is the same as Fig.3.2, but for the source position y = 2. In Fig.3.3(a),the damped oscillatory behavior of |F | appears, which looks like the interference
between the images, although only a single image exists in the geometrical optics(see the behavior of |Fgeo |). For large w ( 1), |F | converges to |Fgeo + dF | which
is given from Eq.(2.37) and (3.33) by,|Fgeo + dF | = |µ+ |1/2 +

hw
i
1
2
cos
(y
+
1)
+ O(1/w 2 ).
w(y 2 − 1)3/2
2(3.34)

The first term is the magnification of the single image formed in the geometrical
optics. The second term is the interference between this image and the diffracted
image formed at the lens center in the quasi-geometrical optics approximation.
Hence, the damped oscillatory behavior is due to the interference between theseimages. The amplitude and the period of this oscillation are 1/[w(y 2 − 1)3/2 ]and w(y + 1)2 /2 (in equation (3.34)), respectively. As shown in Fig.3.3(b), theerrors decrease as w increases, and the results are consistent with the analytical
calculation.

3.4.3

Isothermal Sphere with a Finite Core

We investigate the effect of a finite core at lens center on the amplification factor.
We consider the isothermal sphere having a finite core. The deflection potential isψ(x) = (x2 + x2c )1/2 where xc is a dimensionless core radius, and we set xc = 0.2.
The amplification factor F is numerically calculated in equation (2.27).
Fig.3.4 is the same as Fig.3.1, but for the isothermal sphere with a core model(core radius is xc = 0.2). The source position is y = 2, and the single image

Figure 3.4: Same as Fig.3.1, but for the isothermal sphere with a core lens. The
core radius is xc = 0.2 and the source position is y = 2.
is formed in the geometrical optics. In this model, the central core of the lenscontributes the integral of F in Eq.(2.26). Denoting dFc as the contribution of F
at the lens center, we obtain,

 
Z
2
q
eiwy /2
02
0
02
02
2
dFc (w, y) =dx exp −i yx1 + x1 + x2 + (wxc ) + O(1/w) .
2πiw(3.35)
For wxc . 1 the above equation is the same as dFc in the SIS model (see equation(3.20)) and dFc ∝ 1/w, but for wxc & 1 dFc exponentially decreases as w increases.
Thus, for the small core xc . 1/w the wave does not feel the existence of the core,
and the behavior of dFc is similar to that for the lens without the core. The resultsfor F , Fgeo , and dF are consistent with the analytical calculation as shown in
Fig.3.4(b).

3.4.4

NFW lens

The density profile is given in Eq.(2.40) as, ρ(r) = ρs (r/rs )−1 (r/rs + 1)−2 wherers is the scale length and ρs is the characteristic density. The deflection potential
is given in Eq.(2.43), and the amplification factor F is numerically obtained with
Eq.(2.27). In the quasi-geometrical optics approximation, dF is given by,i X
κs2eiw[y /2+φm (y)](3.36)
dF (w, y) =∆j |µj |1/2 eiwTj −iπnj +
2
2
w j
(wy )
where κs = 16πρs (DL DLS /DS )rs is the dimensionless surface density. The firstterm in Eq.(3.36) is the corrections for the magnifications of the images (i.e. dFm

Figure 3.5: Same as Fig.3.1, but for the NFW lens for κs = 10 with y/ycr = 2.
in §3.1). µj , Tj , and nj are numerically obtained from the lens equation. The
second term corresponds to the diffracted image at the lens center (i.e. dFc in§3.2).
Fig.3.5 is the same as Fig.3.1, but for the NFW model for κs = 10 with thesource position y/ycr = 2. A single image is formed in the geometrical optics. In
Fig.3.5(a), the dumped oscillatory behavior of |F | is appeared, similar to the caseof the SIS. For large w, |F | asymptotically converges to |Fgeo + dF |, which is given
from Eq.(3.36) as,|Fgeo + dF | ' |µ1 |1/2 +



∆1
κs
|µ1 |1/2 +cos w T1 − y 2 /2 − φm (y)
2
2
2
2w
(wy )

(3.37)

where we set j = 1 in Eq.(3.36) since the single image is formed in the geometrical
optics. The first and the second term in Eq.(3.37) correspond to the image formedin the quasi-geometrical optics approximation. The third term is the interference
between the single image formed in the geometrical optics and the diffracted image
formed at the lens center. The amplitude and the period of the oscillation areκs /(wy 2 )2 and w (T1 − y 2 /2 − φm (y)), respectively. As shown in Fig.3.3(b), theresults are consistent with the analytical calculation.

Chapter 4
Wave Effects in the Gravitational
Lensing of Gravitational Waves
from Chirping Binaries
In the previous chapters 2 and 3, we discussed the physical aspect of the wave
optics in the gravitational lensing. In this chapter, we apply the wave optics to thereal observational situations. We consider the gravitational lensing of gravitational
waves from chirping binaries. We take the coalescence of the super massive black
holes (SMBHs) of mass 104 −107 M at the redshift z = 1−10 as the sources. SMBH
binary is one of the most promising sources for LISA and will be detected withvery high signal to noise ratio, S/N ∼ 103 (Bender et al. 2000). Since the merging
SMBHs events will be detected for extremely high redshift (z > 5), the lensing
probability is relatively high and hence some lensing events are expected. We
consider two cases for lens models: 1) the point mass lens for compact objects (suchas black holes), and 2) the SIS (Singular Isothermal Sphere) lens for galaxies, starclusters and CDM (Cold Dark Matter) halos. The wave effects become important
for the lens mass 106 − 109 M which is determined by the LISA band, 10−4 to 10−1
Hz from Eq.(1.3). The frequency of the gravitational waves from the coalescing
SMBH binary chirps so that we could see wave effects for different frequency inthe lensed chirp signals.
We calculate the gravitationally lensed waveform using the wave optics for thetwo lens models: the point mass lens and the SIS. Then, we investigate how accurately we can extract the information on the lens object from the gravitational
lensed signals detected by LISA using the Fisher-matrix formalism (e.g. Cutler& Flanagan 1994). Cutler (1998) studied the estimation errors for the merging
SMBHs by LISA (see also Vecchio & Cutler 1998; Hughes 2002; Moore & Hellings
2002; Hellings & Moore 2003; Seto 2002; Vecchio 2003). Following Cutler (1998),
we calculate the estimation errors especially for the lens mass and the source position. We also calculate the lensing effects on the signal-to-noise ratio. We assume
the 1 yr observation before the final merging and consider the lens mass in the39

Earth5x106 km

LISA

20 o

o
60

Sun
Mercury
Venus

Figure 4.1: Orbital configuration of the LISA antenna. From Hughes (2003).
range 106 − 109 M . Then the typical time delay between the double images is
10 − 104 sec which is much smaller than 1 yr. Throughout this chapter, we assume
a (ΩM , ΩΛ ) = (0.3, 0.7) cosmology and a Hubble parameter H0 = 70kms−1 Mpc−1 .

4.1

Gravitationally Lensed Waveform and Parameter Estimation

4.1.1

Gravitational Wave Measurement with LISA

We briefly discuss the gravitational wave measurement with LISA (see Cutler 1998;
Bender et al. 2000). LISA consists of three spacecrafts forming an equilateraltriangle and orbits around the Sun, trailing 20◦ behind the Earth (see Fig.4.1).
The sides of the triangle are L = 5 × 106 km in length, and the plane of the triangle
is inclined at 60◦ with respect to the ecliptic. The triangle rotates annually. The
gravitational wave signal is reconstructed from three data streams that effectively
correspond to three time-varying armlengths (δL1 , δL2 , δL3 ). We basically analyze
two data streams given by sI (t) = [δL1 (t) − δL2 (t)]/L and sII (t) = [δL1 (t) +δL2 (t) − 2δL3 (t)]/(3)1/2 L. These data can be regarded as the response of two 90◦
interferometers rotated by 45◦ to one another (Cutler 1998). The data sI,II (t)contain both gravitational waves signals hI,II (t) to be fitted by matched filteringand noises nI,II (t). The latter is constituted by the detector’s noise and binary
confusion noise. As in Cutler (1998), we assume that the noises are stationary,Gaussian and uncorrelated with each other.
The gravitational wave signals hI,II (t) from a binary are written as
√

3 +
×
hI,II (t) =FI,II (t)h+ (t) + FI,II(t)h× (t) ,
(4.1)
2

+,×
where FI,II(t) are the pattern functions which depend on the source’s angular
position of the binary, its orientation and detector’s configuration. The quantities
h+,× (t) are the two polarization modes of gravitational radiation from the binary.
The direction and the orientation of the binary and the direction of the lens are
assumed to be constant during the observation in a fixed barycenter frame of thesolar system. Further discussion and details about the pattern functions are shownin Cutler (1998).

4.1.2

Gravitationally Lensed Signal Measured by LISA

We consider the SMBH binaries at redshift zS as the sources. We use a restricted
post-Newtonian approximation for the in-spiral waveform (Cutler & Flanagan
1994). The coalescing time for circular orbit is typically tc = 0.1yr (Mz /106 M )−5/3
×(f /10−4 Hz)−8/3 where Mz = (M1 M2 )3/5 (M1 + M2 )−1/5 (1 + zS ) is the redshiftedchirp mass. At the solar system barycenter, the unlensed waveforms h̃+,× (f ) in
the frequency domain are given byh̃+ (f ) = A 1 + (L · n)2 f −7/6 eiΨ(f ) ,h̃× (f ) = −2iA (L · n) f −7/6 eiΨ(f ) ,(4.2)

where L (given by θ̄L , φ̄L ) is the unit vector in the direction of the binary’s orbital
angular momentum and n (given by θ̄S , φ̄S ) is the unit vector toward the binary.
These vectors are defined in a fixed barycenter frame of the solar system. The
amplitude A and the phase Ψ(f ) depend on six parameters; the redshifted chirp
mass Mz and reduced mass µz = M1 M2 (1+zS )/(M1 +M2 ); the spin-orbit coupling
constant β; a coalescence time tc and phase φc ; the angular diameter distance tothe source DS . The amplitude isr
5/6
5 π −2/3 Mz
A=
.
(4.3)
96 DS (1 + zS )2
where DS (1 + zS )2 is the luminosity distance to the source. The phase Ψ(f ) is
given by,




π 3
20 743 11µz
−5/3
3/2
Ψ(f ) = 2πf tc −φc − + (8πMz f )1+
x + (4β − 16π) x,+
4 4
9 336 4Mz
(4.4)
where Mz = (M1 + M2 )(1 + zS ) is the redshifted total mass, and x = (πMt,z f )2/3
is the post-Newtonian expansion parameter which is roughly O(v 2 /c2 ).
The gravitationally lensed waveforms h̃L+,× (f ) in the frequency domain are givenby the product of the amplification factor F (f ) and the unlensed waveforms h̃+,× (f )(see §2):h̃L+,× (f ) = F (f ) h̃+,× (f ).
(4.5)

where the function F (f ) is given in Eq.(2.32) for the point mass lens and inEq.(2.36) for the SIS lens. The dimensionless frequency w in these equations (2.32)
and (2.36) is related to the frequency f as, w = 8πMLz f , where MLz = ML (1 + zL )is the redshifted lens mass. We note that the amplification factor F (f ) includes
the two lens parameters: the redshifted lens mass MLz and the source positiony 1 as discussed in §2.2.1 for the point mass lens and §2.2.2 for the SIS. Using
Eq.(4.1),(4.2) and (4.5), the observed lensed signals h̃Lα (f ) (α = I, II) with LISA
are given in the stationary phase approximation as,h̃Lα (f )=√Z
3 DS ξ02 (1 + zL ) f
d2 x Λα (t + td (x, y)) e−i(φD +φp,α )(t+td (x,y))2
DL DLSi× Af −7/6 eiΨ(f ) e2πif td (x,y) ,
(4.6)

where
φp,α (t) = tan−1

2(L · n)Fα× (t)
1 + (L · n)2 Fα+ (t)




1/2
Λα (t) = (2 L · n)2 Fα× 2 (t) + {1 + (L · n)2 }2 Fα+ 2 (t).

The Doppler phase isφD (t) = 2πf (t)R sin θ̄S cos φ̄(t) − φ̄S ,where R = 1 AU, φ̄(t) = 2πt/T (T = 1 yr), and t = t(f ) is given by,t(f ) = tc − 5 (8πf )

−8/3

Mz



4
1+
3



743 11µz+
336 4Mz
32π 3/2
x−
x
.
5

(4.7)

In the no lens limit of |F | = 1 in Eq.(4.5), the lensed signals h̃Lα (f ) in Eq.(4.6)
agree with the unlensed ones h̃α (f ) in Cutler (1998). We assume the source position y is constant during the observation, since the characteristic scale of the interference pattern, ∼ 107 AU(MLz /108 M )−1/2 (f /mHz)−1 [(DS DL /DLS )/Gpc]1/2 ,is extremely larger than the LISA’s orbital radius (1 AU).
Since the lensed signals h̃Lα (f ) in Eq.(4.6) are given by double integral, we
approximate h̃Lα (f ) in the two limiting cases; 1) geometrical optics limit (f  t−1d )and 2) the time delay being much smaller than LISA’s orbital period of (td  1yr). In the geometrical optics limit, from Eq.(2.30) we obtain,h̃Lα (f ) =√3X|µj |1/2 Λα (t + td,j )e2πif td,j −iπnj e−i(φD +φp,α )(t+td,j ) × Af −7/6 eiΨ(f ) .
2 j

(4.8)

1

The source position y is defined in Eq.(2.21) and Fig.2.1.

If the time delay is much smaller than LISA’s orbital period (td  1 yr), we expandΛα , φD and φp,α around td = 0 as,√

3Λα (t)e−i(φD +φp,α )(t) × Af −7/6 eiΨ(f )
2





d
f d F (f )2× F (f ) + {ln Λα − i (φD + φp,α )}. (4.9)+ O (td /1yr)dt
2πi dffh̃Lα (f ) =Since we consider the lens mass MLz = 106 −109 M , the time delay is much smaller
than 1 yr. Thus we use the above equation (4.9) as the lensed waveforms for thefollowing calculations.
In Fig.4.2, the lensed signals |h̃Lα (f )| (α = I, II) and the unlensed ones |h̃α (f )|are shown. We show the results from one year before the final merging to theinner most stable circular orbit (the binary separation is r = 6(M1 + M2 )). We
set typical parameters at the SMBH binary masses M1,2z = 106 M , the lens mass
MLz = 108 M and the source position y = 1 for the point mass lens. The angularparameters are cos θ̄S = 0.3, φ̄S = 5.0, cos θ̄L = 0.8, φ̄L = 2.0, and the source
redshift is zS = 1 (the angular diameter distance is H0 DS = 0.386). Therefore thefrequency range is from 5×10−5 to 2×10−3 Hz and the time delay is 4×103 sec. The
strange behavior in the lower frequency f . 10−4 Hz is due to the LISA’s orbital
motion. In this frequency region, however, the difference between the lensed signal
and the unlensed one is small due to the diffraction (see Fig.2.4). On the other
hand, the oscillatory behavior appears in the higher frequency region f & 10−4 Hz.
This critical frequency is determined by the inverse of the lens mass 8πMLz (seeFig.2.4). The oscillatory amplitude andp
the period are determined by the product1/2of the magnifications 2|µ+ µ− | = 2/(y y 2 + 4) and the inverse of the time delay,
1/∆td (see the third term of Eq.(2.34)).

4.1.3

Parameter Extraction

We briefly mention the matched filtering analysis and the parameter estimation
errors (Finn 1992; Cutler & Flanagan 1994). We assume that the signal h̃Lα (f ) is
characterized by some unknown parameters γi . In the present case, there are tensource parameters (Mz , µz , β, φc , tc , DS , θ̄S , φ̄S , θ̄L , φ̄L ) and two lens parameters(MLz , y). In the matched filtering analysis the variance-covariance matrix of theparameter estimation error ∆γi is given by inverse of the Fisher information matrix
Γij as h∆γi ∆γj i = (Γ−1 )ij . The Fisher matrix becomes
Γij = 4X

α=I,IIReZ

L
df ∂ h̃L∗α (f ) ∂ h̃α (f ),Sn(f )∂γi∂γj(4.10)

where Sn(f ) is the noise spectrum. The noise spectrum Sn(f ) is the sum of the
instrumental and the confusion noise, and we adopt the same noise spectrum asFigure 4.2: The lensed signals |h̃Lα (f )| (α = I, II) (solid line) and unlensed ones|h̃α (f )| (dashed line) measured by LISA. The signals are shown from 1yr before
coalescence to ISCO of r = 6(M1 +M2 ). The redshifted masses of the SMBH binaryis M1,2z = 106 M , the redshifted lens mass is MLz = 108 M and the source positiony = 1. The angular parameters are cos θ̄S = 0.3, φ̄S = 5.0, cos θ̄L = 0.8, φ̄L = 2.0,
and the source redshift is zS = 1 (distance is H0 DS = 0.386). The strange behavior
for f . 10−4 Hz is due to the LISA orbital motion, and the difference between thetwo signals is small due to the diffraction. On the other hand, the oscillatory
behavior appears for f & 10−4 Hz which is determined by the inverse of the lensmass 8πMLz (see Fig.2.4). This oscillation is due to the interference between the
double images.

Figure 4.3: The noise spectrum for LISA. The solid line is the instrumental noise,and the dotted line is the confusion noise. The total noise is the sum of them.
that in Cutler (1998). We show Sn(f ) in Fig.4.3, and the solid (dotted) line isinstrumental (confusion) noise. The signal to noise ratio (S/N ) is given by
X Z
2
df
2
(S/N ) = 4h̃Lα (f ) .
(4.11)Sn(f)α=I,II
We computed the variance-covariance matrix Γij for a wide range of the lens parameters (MLz , y), using the lensed waveform in Eq.(4.9). Since the S/N is very
high for the SMBH merger, the Fisher matrix approach to calculate the estimation errors is valid (Cutler 1998). We integrate gravitational lensed waveform (inEq.(4.10) and Eq.(4.11)) from 1 yr before the final merging to the cut-off frequency
fcut when the binary separation becomes r = 6(M1 + M2 ). Then, the initial frequency is given by,
finit = 4.1 × 10

−5



Mz
106 M

−5/8

Hz,

(4.12)

−1

Hz.

(4.13)and the cut-off frequency is
fcut = 4.4 × 10

−3

M1z + M2z
106 M

4.2

Results

In this section, we present numerical results to compute the signal to noise ratio(S/N) and the errors in estimation parameters. We randomly distribute 100 binaries over various directions and orientations on celestial spheres at zS = 1 (thedistance is H0 DS = 0.386). We present the mean value averaged for 100 binaries.

4.2.1

Lensing Effects on the Signal to Noise Ratio

We demonstrate the gravitational lensing effect on the signal to noise ratio (S/N ).
In Fig.4.4, the increasing factor of S/N by the gravitational lensing for the point
mass lens is shown for the fixed source position2 y = 0.1, 0.3, 1, and 3 as a functionof the lens mass MLz . The vertical axis is the S/N with the gravitational lensing
divided by the unlensed S/N . Four panels are shown for the various SMBH binary
masses M1,2z = 104 , 105 , 106 , and 107 M . We only show the mean value averaged
for 100 binaries, but the dispersion is negligibly small (less than 5 %). For the lensmass smaller than 106 M the magnification is very small irrespective of the SMBH
binary masses due to the diffraction effects. In this case the Schwarzschild radiusof the lens mass MLz is smaller than the wavelength of gravitational waves λ ∼ 1
AU, and the waves are not magnified by lensing. This critical lens mass (106 M )is mainly determined by the inverse of the knee frequency of the LISA’s noise
spectrum, 1/(8πf ) ∼ 8×106 M (f /mHz)−1 (see Fig.2.4). But for 107 +107 M , the
SMBH binary coalescences at the lower frequency (f ∼ 10−4 Hz), thus the criticallens mass is shifted for a larger mass (107 M ) as shown in the right bottom panel of
Fig.4.4. This tells us that if the lens mass is smaller than 106 M , the effect of thelens is very small. If the lens mass is larger than 107 M , the damped oscillatory
behavior appears due to the interference between the two images, and the S/Nconverges to the geometrical optics limit, |µ|1/2 = (y 2 + 2)1/2 /[y 1/2 (y 2 + 4)1/4 ],which is independent of the lens mass. As y increases from 0.1 (solid line) to3 (dashed line), the amplification decreases since the magnifications of the two
images (|µ± (y)|) decrease as y increases (see also Fig.2.4).
Fig.4.5 is the same as Fig.4.4, but for the SIS lens model. The behavior isvery similar to that in the point mass lens. For the lens mass larger than 107 M ,the S/N converges to the geometrical optics limit, |µ|1/2 = (2/y)1/2 for y ≤ 1 and|µ|1/2 = (1 + 1/y)1/2 for y ≥ 1. As y increases from 0.1 (solid line) to 3 (dashedline), the amplification decreases (see also Fig.2.5).

4.2.2

Parameter Estimation for the Lens Objects

In this section, we show the parameter estimation for the lens objects. We show theresults for the SMBH binary with masses 106 + 106 M , because we found S/N ishigher than the other binary masses (M1,2z = 104 , 105 and 107 M ). We distribute2

The source position y is defined in Eq.(2.21), and see also Fig.2.1.

Figure 4.4: The increasing factor in S/N due to the gravitational lensing by the
point mass lens for the various SMBH binary masses M1,2z = 104 , 105 , 106 , and
107 M . The horizontal axis is the redshifted lens mass; the vertical axis is thelensed S/N divided by the unlensed S/N . The source position is fixed at y =0.1, 0.3, 1, and 3. For MLz . 106 M , the magnification is very small due to thediffraction effect irrespective of the SMBH binary masses. For MLz & 107 M ,the damped oscillatory patterns appear due to the interference between the two
images, and this behavior converge in the geometrical optics limit, |µ|1/2 = (y 2 +2)1/2 /[y 1/2 (y 2 + 4)1/4 ].

Figure 4.5: Same as Fig.4.4, but for the SIS lens model. For MLz & 107 M , theresults converge in the geometrical optics limit, |µ|1/2 = (2/y)1/2 for y ≤ 1 and|µ|1/2 = (1 + 1/y)1/2 for y ≥ 1.

Figure 4.6: The estimation errors for the redshifted lens mass ∆MLz (left panel)and the source position ∆y (right panel) for the point mass lens. The results arepresented for the SMBH binary of masses 106 + 106 M at zS = 1. The errors
are normalized by S/N = 103 and simply scale as (S/N )−1 . For MLz . 107 M
the errors are relatively large, since the effect of lensing is very small due to thediffraction. For MLz & 108 M the geometrical optics approximation is valid, anderrors converge to constants.
the 100 binaries over the various directions and the orientations at zS = 1, and themean value of the S/N without lensing is 2600 in these 100 binaries. We show themean value of errors averaged for 100 binaries, for MLz . 107 M the dispersion isrelatively large (. 40%), but for MLz & 107 M the results converge to that in thegeometrical optics limit and the dispersion is negligibly small.
In Fig.4.6, the estimation errors for the redshifted lens mass ∆MLz (left panel)and the source position ∆y (right panel) are shown as a function of MLz with
y = 0.1, 0.3, 1, and 3 for the point mass lens. We use the units of S/N = 103 , andthe results (∆MLz , ∆y) scale as (S/N )−1 . For MLz . 107 M the estimation errors
are relatively large & 10%, since the effect of lensing on the signals is very small
due to the diffraction. For MLz & 108 M the geometrical optics approximation isvalid, and the errors converge to a constant in Fig.4.6. The redshifted lens massand the source position can be determined up to the accuracy of ∼ 0.1%, as shownin Fig.4.6. The errors in the geometrical optics limit are well fitted by (see the
Appendix D),
∆MLz
MLz

1
=
×
S/N

p
y(y 2 + 2)(y 2 + 4)5/4,
2τ

Figure 4.7: Same as Fig.4.6, but as a function of y. We note that even for y & 10
we can extract the lens information. Then the lensing cross section (∝ y 2 ) increases
an order of magnitude larger than that for the usual strong lensing of light (y = 1).
∆y
1
=
×
y
S/N

p

y 2 + 2(y 2 + 4)3/4,√
2 y(4.14)pp
2 + 4/2+ln((
where
S/N
is
in
the
unlensed
case,
and
τ
=
∆t
/4M
=
y
y
y 2 + 4+dLzpy)/( y 2 + 4 − y)). Thus, one could determine the lens parameters, the redshifted
lens mass and the source position, up to the accuracy of ∼ (S/N )−1 . The above
equations (4.14) are valid if the time delay ∆td is much smaller than the LISA’s
orbital period 1 yr. If the time delay ∆td becomes comparable to 1 yr, the LISA’s
orbital motion affects the results.
Fig.4.7 is the same as Fig.4.6, but as a function of y. For y & 1, the errors
are convergent to the geometrical optics limit of Eq.(4.14) irrespective of the lensmass. As y increases, the time delay td increases, and the geometrical optics
limit (f td  1) is valid. We note that even for y & 10 one can extract the lensinformation. For light the observable is the lensed flux which is proportional to themagnifications, ∝ |µ± |, while for gravitational waves the observable is the lensed
amplitude which is proportional to the square root of the magnifications, ∝ |µ± |1/2 .
Thus, it is possible to detect the faint image |µ|  1 by the gravitational waves.
For example, let us consider the case where the flux ratio of a brighter image to a
fainter one is 100 : 1. Then the amplitude ratio is 10 : 1 so that the fainter image
can be observed even if the source position is far from the Einstein radius in thecase of gravitational waves. Denoting the largest source position for which one canextract the lens parameters as ycr , we approximate the errors in Eq.(4.14) for theFigure 4.8: The estimation errors for the redshifted lens mass ∆MLz (left) andthe source position ∆y (right) for the SIS model. The results are presented for
the SMBH binary of masses 106 + 106 M at zS = 1. The errors are normalized
by S/N = 103 and simply scale as (S/N )−1 . Even for y = 3 (a single image isformed in the geometrical optics limit), the lens parameters can be extracted at
MLz ∼ 106 − 108 M due to the wave effects.
large y limit; ∆γ/γ ' (S/N )−1 y 2 , where γ = MLz , y. Then we obtain
ycr ' 10



∆γ/γ
0.1

1/2 

S/N
103

1/2

.

(4.15)

2
Thus the lensing cross section (∝ ycr) increases an order of magnitude larger thanthat for the usual strong lensing of light (ycr = 1) (e.g. Turner, Ostriker & Gott
1984).
In Fig.4.8, the estimation errors for the SIS model are shown. For MLz . 107 M
the behavior is similar for the point mass lens. But for MLz & 108 M the behavior
strongly depends on y. The errors in the quasi-geometrical optics approximation
are well fitted by (see the Appendix D),p∆y1
2 1 − y2
∆MLz
=
=
×
,
(4.16)MLzyS/N
2y + 1for y < 1. We note that even for y = 3 the lens parameters can be extracted for
MLz ∼ 106 − 108 M as shown in Fig.4.8. This is because the diffracted image
is newly formed at the lens center by the wave effects (see the discussion §4.3).
Hence the double images are formed even for y > 1, and the information on thelens can be extracted. The errors for y > 1 are given in the quasi-geometricalFigure 4.9: Same as Fig.4.8, but as a function of y. We note that even for y > 1 wecan determine the lens parameters. Then the lensing cross section (∝ y 2 ) becomeslarger than that in the geometrical optics approximation (y = 1).
optics approximation as (see the Appendix D),p∆MLz
1
32πy(y − 1)2 y 2 − 1=
×MLz hf i,MLz
S/N
2y + 1∆y
116π(y 2 − 1)3/2 (y − 1)=×
MLz hf i,y
S/N
2y + 1(4.17)(4.18)

where hf i is defined by,

hf i2 = (S/N )2 × 4

X Z

α=I,II

df
Sn(f )h̃Lα (f )
f

2

−1


,

and the mean value of hf i averaged for 100 binaries is 1.2 mHz. For y = 1,
the errors decrease as lens mass increases as shown in Fig.4.8, because the errors
converge to the results in the quasi-geometrical optics limit of Eq.(4.16) - (4.18)
which vanish at y = 1. As a result, if y > 1, the errors asymptotically increase
with the increase of the lens mass, but if y < 1, they asymptotically converge toconstants.
Fig.4.9 is the same as Fig.4.8, but as a function of y. We note that even forlarger y & 1 we can extract the lens information. Thus the lensing probability(∝ y 2 ) to determine the lens parameters increases as compared with the results ingeometrical optics limit for the lens objects in the mass range 106 − 108 M .

4.2.3

Lensing Effects on the Estimation Errors of the Binary Parameters

We discuss the gravitational lensing effects on the estimation errors of the SMBH
binary parameters. We study five binary parameters; the redshifted chirp mass
Mz , the reduced mass µz , the distance to the source DS and the angular resolution(θ̄S , φ̄S ). We find that the estimation errors of these parameters decrease because
S/N increases by lensing (see Fig.4.4 and 4.5). The error ∆γ is roughly proportional
to the inverse of the S/N as ∆γ ∝ (S/N )−1 (see Eq.(4.10) and (4.11)).

4.2.4

Results for Various SMBH Masses and Redshifts

So far we presented the results for equal mass SMBH binaries with redshift zS = 1.
In this section, we comment on the results for the case of various (unequal) SMBH
masses 104 − 107 M and redshifts zS = 1 − 10.
The critical lens mass in which the wave effects become important (106 −108 M )is mainly determined by the inverse of the knee frequency of the LISA’s noise
spectrum, ∼ 8 × 106 M (f /mHz)−1 , independent of the binary mass. But forthe massive total mass binary (M1z + M2z ) & 107 M , the binary coalescences atthe lower frequency (∼ 10−4 Hz), thus the critical lens mass is shifted for larger
mass (107 − 109 M ). For the larger lens mass MLz & 108 M , the results (the S/Nincrease and the estimation errors) converge to that in the geometrical optics limit
irrespective of the binary mass. The estimation errors in Fig.4.6-4.9 are the caseof 106 − 106 M binary at redshift zS = 1 and are normalized to S/N = 103 andsimply scale as (S/N )−1 . In order to translate the results into the various unequal
SMBH binaries, we present the S/N for binary masses M1,2z = 104 − 107 M with
redshifts zS = 1, 3, 5, 10 in Table.4.1. We assume 1 yr observation of in-spiralphase before final merging. The results are the mean value of 100 binaries which
are randomly distributed at each redshift, and the dispersion is relatively large∼ 50%. From Table 4.1, one could translate the results in Fig.4.6-4.9 into errors
in real situations.
We also comment on the results for the case with only hI data available, as
opposed to the combination of hI and hII data that we used. In this case, theS/N increase in Fig.4.4 and 4.5 is not changed, but the estimation errors are
7slightly larger (∼ 30%) than that in Fig.4.6-4.9 for M√Lz . 10 M if the errors are3normalized to S/N = 10 . We note that the S/N is 2 times smaller than that in
the case of the two data available in Table 4.1.

4.3

Lensing Event Rate

We discuss the event rate of merging SMBHs and estimate the lensing probability
and the lensing event rate. The expected rate of merging SMBHs detected by LISA
is in the range 0.1 − 102 events per year (Haehnelt 1994,1998). Recently, Wyithe

Binary Masses (M )107 + 107
107 + 106
107 + 105
107 + 104
106 + 106
106 + 105
106 + 104
105 + 105
105 + 104
104 + 104zS = 1
1038
519
175
52
2575
1517
508
877
310
132

zS = 3
270
135
46
14
669
394
132
228
81
34

zS = 5
147
74
25
7
365
215
72
124
44
19

zS = 10
66
33
11
3
164
97
32
56
20
8

Table 4.1: The signal to noise ratio (S/N) for the various binary masses 104 −107 M
with redshift zS = 1, 3, 5, 10. We assume 1 yr observation of in-spiral phase before
final merging.
& Loeb (2003) suggested that some hundreds detectable events per year could be
expected, considering the merger rate at exceedingly high redshift (z > 5 − 10).
Thus we take ∼ 300 events per year as the merging event rate.

We consider the lens objects distributed over the universe and calculate thelensing probability for each lens model. For the point mass lens, we take thecompact objects (106 − 109 M ) such as black holes as lens. Denoting the massdensity parameter of compact objects as Ωco , the lensing probability for a source
at redshift zS is (Schneider, Ehlers & Falco 1992),
3
2
P (zS ) = Ωco ycr
2

Z zS
0

dzL

(1 + zL )2 H0 DLS (zL , zS ) H0 DL (zL ),H(zL )/H0
H0 DS (zS )(4.19)

where H(z) is the Hubble parameter at redshift z. The cosmological abundance ofthe compact objects in the mass range 106 −109 M is limited by Ωco ≤ 0.01 by thesearch for multiple images in radio sources (Wilkinson et al. 2001; see also Nemiroff
et al. 2001). In Table 4.2, we show the upper limit on the lensing probability forthe point mass lens. Since we set ycr = 10 (Eq.(4.15)), the lensing probability
is one hundred times larger than that normally assumed for the strong lensing oflight (ycr = 1). As shown in Table 4.2, the upper limit of the lensing probability isvery high (almost 1) and is typically ∼ (Ωco /10−2 ). The lensing event rate is the
product of the merging rate (∼ 300 per year) and the lensing probability, so thatthe lensing events will be 1 event per year if Ωco = 10−4 .
For the SIS model we take CDM halos (106 − 109 M ) as the lens objects (e.g.

Lens Model
Point mass lens
SIS

zS = 1< 0.21
7.2 × 10−5zS = 3< 1.1
8.1 × 10−4

zS = 5< 2.0
2.0 × 10−3

zS = 10< 3.9
4.7 × 10−3

Table 4.2: The lensing probability by the lens mass in the range 106 − 109 M withthe source redshift zS = 1, 3, 5, 10. For the point mass lens, we give the upper
limit which is determined by the observational constraint on the abundance of thecompact objects. When the lensing probability is more than one, the lensing occurs
sometimes. For the SIS, CDM halos are assumed to be lenses. The presented values
are for the case of S/N = 103 , and hence the results are somewhat overestimatedfor the binaries of S/N < 103 in Table 4.1. If the expected rate of merging SMBHs
is ∼ 300 per year (Wyithe & Loeb 2003), then the lensing events will be detected
1 event per year.
Narayan & White 1988). The lensing probability is
Z zS
Z 9(1 + zL )2 H0 DLS (zL , zS ) H0 DL (zL ) 10 M
2dML vNv (v, zL ),P (zS ) = πycr
dzL
H(zL )/H0
H0 DS (zS )106 M
0(4.20)
where Nv is the comoving number density of the lens and is assumed to be givenby the Press-Schechter velocity function (Press & Schechter 1974) with σ8 = 1.
In Table 4.2, we show the lensing probability for the SIS model. We set ycr = 3(see Fig.4.8) and hence the lensing probability is almost ten times larger than thatfor light (ycr = 1). As shown in Table 4.2, the lensing probability is typically∼ 10−4 − 10−3 . The merger rate is ∼ 300 events per year at high redshift (z > 5),then the lensing events would be 1 event per year.
We note that the results in Table 4.2 are for the case of the S/N = 103 , and aresomewhat overestimated for the binaries of S/N < 103 in Table 4.1. For example,
the lensing probability is proportional to (S/N ) from Eq.(4.15) for the point masslens, and it is appropriate to use ycr = 1 for S/N < 103 in the SIS. In the caseof the high event rate (∼ 300 events/year), many fainter signals (S/N  10 3 ) are
expected, and the errors in Fig.4.6-4.9 are worse for these binaries.
Next, we discuss how we can identify the lensing signal. If the lensing event
occurs, the amplitude and the arrival time of the gravitational waves are changed
by lensing. But the other features (such as binary mass) are not changed. Thus, if
the two signals have the same binary parameters (such as chirp mass) except for theamplitude and the arrival time, that would be a signature of gravitational lensing in
the geometrical optics limit. More generally, oscillatory behavior in the waveform|h̃L (f )| is a signature of gravitational lensing (see Fig.4.2). However it will bedifficult to identify the source and the lens objects in the sky, since the angularresolution of the LISA is ∼ 1 deg (see Cutler 1998). Furthermore, the gravitational
wave amplitude is changed by the lensing magnification and hence one must assume

the lens model in order to determine the distance to the source. (Effect of lensing
on measuring the distance is recently discussed in Holz & Hughes (2002).) As
one determines the distance to the source DS (zS ), The redshift zS (DS ) could bedetermined if the distance to the source DS (zS ) could be measured assuming thecosmological parameters (see Hughes 2002).

Chapter 5
Summary
In this thesis, we discussed the wave optics in gravitational lensing of gravitational
waves and its application to the gravitational wave observations in the near future.
We review the wave optics in the gravitational lensing, which is more fundamental than the geometrical optics, in §2. If the wavelength λ of the gravitational
waves is larger than the Schwarzschild radius of the lens mass ML , the wave optics
should be used. In the short wavelength limit λ → 0, the wave optics is reduced
to the geometrical optics limit. We use the diffraction integral formula to obtain
the gravitationally lensed waveform. We show the wave optics for the various lensmodels: the point mass lens, the SIS model, the NFW model, and the binary lens.
In §3, we studied the gravitational lensing in the quasi-geometrical optics approximation which is the geometrical optics including the corrections arising fromthe effects of the finite wavelength. Theses correction terms can be obtained analytically by the asymptotic expansion of the diffraction integral in powers of the
wavelength λ. The first term, arising from the short wavelength limit λ → 0,
corresponds to the geometrical optics limit. The second term, being of the orderof λ/ML (ML is the Schwarzschild radius of the lens), is the first correction termarising from the diffraction effect. By analyzing this correction term, we obtainthe following results:• The lensing magnification µ is modified to µ(1 + δ), where δ is of the orderof (λ/ML )2 .
• If the lens has cuspy (or singular) density profile at the center ρ(r) ∝ r −α(0 < α ≤ 2) the diffracted image is formed at the lens center with themagnification µ ∼ (λ/ML )3−α . Thus if we observe this diffracted image by
the various wavelength (e.g. the chirp signal), the slope α can be determined.
In §4, we discussed the gravitational lensing of gravitational waves from chirpingbinaries. The SMBH binary is taken as the source detected by LISA, and the two57

simple lens models are considered: the point mass lens and the SIS model. We
calculate the lensing effects on the signal to noise ratio (S/N) and how accurately
the information of the lens object, its mass, can be extracted from the lensed
signal. As expected, for the lens mass smaller than 108 M , the wave effects arevery important to calculate the S/N and the errors in the estimation parameters.
We obtain the following results:• For the lens mass smaller than 106 M the signals are not magnified by lensing
due to the diffraction effect. For the lens mass larger than 108 M the lensparameters can be determined within (very roughly) ∼ 0.1% [(S/N )/103 ]−1 .
• The lensing cross section to determine the lens parameters is order of magnitude larger than that for light.

Cutler & Thorne (2003) suggest that the mergers of compact binaries will
be detected before ∼ 2010 or sooner by the ground based interferometers. Thisis because in about 2007 the first generation interferometers (TAMA300, LIGO,
VIRGO, GEO) will be upgraded to produce the second generation detectors (advanced LIGO, LCGT), and the neutron star mergers will be detected by these
upgraded detectors. The space interferometer LISA will be launched in ∼ 2011.
There are some known Galactic binaries which LISA is guaranteed to observe. The
mergers of SMBHs are also some of the promising candidates. Furthermore, thespace interferometer DECIGO/BBO is planed to construct in ∼ 2020, and will detect ∼ 105 events yr−1 of the neutron star mergers. Hence, the gravitational wave
astronomy will be established at the broad frequency band in the near future, andthe lensing events will also be expected for these various detectors.

Acknowledgements
I would like to thank Prof. Takashi Nakamura for variable discussion and encouragement. We would like to thank Takeshi Chiba, Naoki Seto and Kazuhiro
Yamamoto for useful comments and discussions. I also wish to thank all the members of theoretical astrophysics group at Kyoto University and Yukawa Institutefor Theoretical Physics.

Appendix A
Path Integral Formula for the
Wave Optics
We present the path integral formula which was developed by Nakamura & Deguchi(1999) to derive the diffraction integral.

A.1

Basic Wave Equation

We consider the scalar wave φ(t, r) propagating under the curved space-time dueto the gravitational potential U (r) of the lens objects (see also §2.1.1). The wave
equation in the frequency domain is given in equation (2.7) as,
∇2 + ω 2 φ̃ = 4ω 2 U φ̃.
(A.1)

where ω is the frequency.
We show the lensing configuration in Fig.A.1 for the source, the lens, and theobserver. The source is located at the center of the coordinate system. rl , ro , andrlo are distances to the lens, to the observer, and from the lens to the observer,
respectively. We use the spherical coordinate system (r, θ, φ), then the observer islocated at ro = (ro , θo , φo ) with θo  1.
In the no lens limit (U = 0), the solution of the wave equation (A.1) is,
eiωr
,
(A.2)
r
where A is the amplitude. With the above equation (A.2) and the amplification
factor F of the wave amplitude due to the lensing, the gravitationally lensed waveφ̃ is written as,
eiωr
.
(A.3)
φ̃(ω, r) = F (ω, r)r
Inserting this φ̃ into the equation (A.1), we obtain,φ̃0 = A1
∂2F∂F
+ 2 ∇2θ F = 4ω 2 U F,
+ 2iω
2
∂r
∂r
r

61

(A.4)

Figure A.1: Lensing configuration for the source, the lens, and the observer. rl , ro ,and rlo are the distances between them. The angles are very small: θo , θs  1.
where
1
∇2θ =

1
θ ∂θ



1
θ
∂θ



+

∂2
.
∂φ2

We assume that the wave length ∼ 1/ω is much smaller than the scale on which
F varies, then we have |∂ 2 F/∂r 2 |  |ω ∂F/∂r| and the first term in Eq.(A.4) is
negligible. The equation (A.4) is rewritten as,
i

1
∂F
=−
∇2 F + 2ωU F∂r
2ωr 2 θ(A.5)

The above equation (A.5) is the same as the Schrödinger equation, if we considerr as the time, ωr 2 as the mass, 2ωU as the potential, and F as the wave function.

A.2

Path Integral

The wave equation (A.5) can be solved by using the path integral (e.g. Sakurai 1994). From the analogy of Eq.(A.5) with the Schrödinger equation, the Lagrangian is given by,
1
L(r, θ, θ̇) = ωr 2 |θ̇|2 − 2ωU (r, θ)
2

(A.6)

where θ = θ(cos φ, sin φ) and θ̇ = dθ/dr. Then the solution of the wave equation(A.5) at the observer ro is obtained with Eq.(A.6) as, Z roZF (ω, ro ) = Dθ(r) exp i
drL .
(A.7)
0

Figure A.2: The thick solid line is the path in the integration of Eq.(A.7). The
dashed line is the lens plane on which the wave is scattered.
We assume the thin lens approximation in which the gravitational potential of thelens U does not vanish only on the lens plane. Then, the potential U is rewritten
as,
1
(A.8)
U (r, θ) = δ(r − rl )ψ̂(θ),
2where ψ̂ is the deflection potential as defined in Eq.(2.14).
We calculate the integration in Eq.(A.7) along the path denoted the thick solidline from the source to the observer as shown in Fig.A.2. The wave is scattered only
on the lens plane at r = rl . We note that θ̇ in the Lagrangian (A.6) corresponds
to the slope of this thick solid line. Hence we obtain,θ̇ = 0,
for 0 ≤ r < rl ,1 rl ro= 2(θ o − θ l ) , for rl < r ≤ ro .
r rlo

(A.9)

From Eqs.(A.6), (A.8) and (A.9), the integrand of Eq.(A.7) is written as,
Z ro
rl ro
|θ l − θ o |2 − ω ψ̂.
(A.10)
drL = ω2rlo
0Thus, the amplification factor F at the observer ro is obtained from Eqs.(A.7) and(A.10) as, Zω rl rorl ro22|θ l − θ o | − ψ̂(θ l ) ,(A.11)F (ω, ro ) =d θ l exp iω
2πi rlo2rlowhere F is normalized such that F = 1 in the no-lens limit (ψ̂ = 0). The result(A.11) is coincident with the amplification factor in Eq.(2.19) which is derived byusing the Kirchhoff diffraction integral.

Figure A.3: Same as Fig.A.2, but for the multi-lens system with N lens planes(dashed lines).

A.3

Application to Multi-Lens System

We present the application of the path integral formula to the multi-lens system
which was developed by Yamamoto (2003). We consider N lens planes which arelocated at r = rlj (j = 1, 2, ···, N ) with the deflection potential ψ̂j (θ) (see Fig.A.3).
Then, the gravitational potential U in Eq.(A.8) is rewritten as,
N

1X
U (r, θ) =δ(r − rlj )ψ̂j (θ).
2 j=1

(A.12)

θ̇ in Eq.(A.9) is also rewritten as,θ̇ = 0,
1 rlj rl(j+1)= 2θ l(j+1) − θ lj ,r rlj,l(j+1)
1 rlN ro(θ o − θ lN ) ,= 2
r rlN,ofor 0 ≤ r < rl1 ,for rlj < r < rl(j+1) ,for rlN < r ≤ ro ,(A.13)

where rlj,l(j+1) = rl(j+1) − rlj for j = 1, 2, · · ·, N − 1, and rlN,o = ro − rlN . From
above equations (A.12) and (A.13), the equation (A.10) is rewritten as,
Z ro0

drL = ω"N −1
X rlj rl(j+1)

2rlj,l(j+1)
j=1

θ l(j+1) − θ lj

2

#
N
X
rlN ro2ψ̂j (θ lj ).
|θ o − θ lN | − ω+
2rlN,oj=1(A.14)

The amplification factor is obtained from Eqs.(A.7) and (A.14) as, Zω rl1 rl2
rl1 rl222
F (ω, ro ) =d θ l1 exp iω
|θ l1 − θ l2 | − ψ̂1 (θ l1 )2πi rl1,l2
2rl1,l2 Zω rl2 rl3
rl2 rl32
2
×d θ l2 exp iω|θ l2 − θ l3 | − ψ̂2 (θ l2 )2πi rl2,l3
2rl2,l3···
ω rlN ro×
2πi rlN,o

Z

 

rlN ro2d θ lN exp iω|θ lN − θ o | − ψ̂N (θ lN ) ,2rlN,o2where F is normalized such that F = 1 in no lens limit (ψ̂j = 0 for all j).

Appendix B
Numerical Computation for the
Amplification Factor
We present the method for the numerical integration of the amplification factor F
discussed in §2.1.3. The amplification factor is given in Eq.(2.26) as,Zw
d2 x exp[iwT (x, y)],(B.1)
F (w, y) =2πiwhere w is the frequency and y is the source position. The time delay T is given
in Eq.(2.24) as,1
T (x, y) = |x − y|2 − ψ(x) + φm (y),(B.2)
2where ψ is the deflection potential of the lens and φm is the additional phase in F .
Since the integrand of F is rapidly oscillating function especially for large w, it istime consuming to numerically calculate F .

B.1

Axially Symmetric Lens Model

If the lens is axially symmetric, the potential ψ(x) is a function of x = |x| and theamplification factor F is reduced to the simple form given in Eq.(2.27):
Z ∞2
iw[y 2 /2+φm (y)]F (w, y) = −iwedx x J0 (wxy)eiw[x /2−ψ(x)](B.3)
0where J0 is the Bessel function of zeroth order. Changing the integral variablefrom x to z = x2 /2 in the above equation (B.3), we obtain,Z ∞F (w, y) =dzf (z : w, y) eiwz
(B.4)
0

where the function f is defined by,√√
2
f (z : w, y) ≡ −iweiw[y /2+φm (y)] J0 (wy 2z)e−iwψ( 2z) .

67

(B.5)

Thus, we obtain the Fourier integral (B.4). We present the method for computing
Fourier integral in Numerical Recipes (Press et al. 1992). The equation (B.4) isrewritten as,
Z ∞
Z b
iwz
dzf (z : w, y) eiwz ,dzf (z : w, y) e +F (w, y) =b
0Z bf (b : w, y)eiwb f 0 (b : w, y)eiwb=
dzf (z : w, y) eiwz −+− · · ·,
iw
(iw)2
0
(B.6)
where f 0 = ∂f /∂b. The first term of the above equation (B.6) is evaluated bynumerical integration directly. For the series of the above equation, we use theintegration by parts. Since f (b : w, y) ∝ b−1/4 for b  1 from Eq.(B.5),1 theasymptotic expansion in Eq.(B.6) converges for large b.

B.2

Non-axially Symmetric Lens Model

For the non-axially symmetric lens model, the amplification factor F in Eq.(B.1)
is rewritten as,
Z 2πZ
w iw[y2 /2+φm (y)] ∞iwx2 /2dθ e−iw[xy cos θ+ψ(x,θ)] , (B.7)
F (w, y) =dx x ee
2πi00where θ is defined as x · y = xy cos θ. Similarly for the previous case, we use theintegral variable z = x2 /2 in the above equation (B.7), we obtain,Z ∞F (w, y) =dzf (z : w, y) eiwz
(B.8)
0where f is defined by,
w iw[y2 /2+φm (y)]f (z : w, y) ≡
e
2πi

Z 2π

√

√

dθ e−iw[y 2z cos θ+ψ( 2z,θ)] .

(B.9)

0

Thus we can evaluate the above integral (B.8) using the method in the previous
section.

1

Note that J0 (x) ∝ x−1/2 for x  1 (e.g. Abramowitz & Stegun 1970).

Appendix C
Asymptotic Expansion of the
Amplification Factor for
Non-axially Symmetric Lens
ModelsIn §3.1, we considered the asymptotic expansion of the amplification factor F in
powers of the inverse of the frequency 1/w for the axially symmetric lens model for
simplicity. In this Appendix, we consider the expansion of F for the non-axiallysymmetric lens model.
In this case, the expansion of T (x, y) around the j-th image position xj inEq.(2.29) is rewritten as,
T (x, y) = Tj +1X
Tab x̃a x̃b + O(x̃3 ),
2

(C.1)

a,b

where x̃ = x − xj , Tj = T (xj , y), andPTab = ∂a ∂b T (xj , y) is a 2 × 2 matrix. Wechange the variable from x̃a to za = b Aab x̃b in order to diagonalize the matrix
Tab in the above equation (C.1). Here, Aab satisfies,X

Tab Aac Abd = λc δcd ,

(C.2)

a,b

where λc is the eigenvalue of Tab . Thus, the equation (C.1) is simply rewritten as,
T (x, y) = Tj +

1
λ1 z12 + λ2 z22 + O(z 3 ).
2

(C.3)

The above result is similar to the equation (3.1) in the case of the axially symmetriclens. The magnification µj and the coefficient nj in Fgeo of equation (2.30) are also
rewritten as, µj = 1/|λ1 λ2 | and nj = 1/2 − sign(λ1 )/4 − sign(λ2 )/4.
69

We expand T (x, y) in Eq.(C.1) up to the fourth order of x̃ as,
1X
1X
Tab x̃a x̃b +∂a ∂b ∂c T (xj , y)x̃a x̃b x̃c
2 a,b
6 a,b,c
1 X∂a ∂b ∂c ∂d T (xj , y)x̃a x̃b x̃c x̃d + O(x̃5 ).
+24

T (x, y) = Tj +(C.4)

a,b,c,dChanging the variable x̃a to za , the above equation (C.4) is rewritten as,
T (x, y) = Tj +X
 X1
λ1 z12 + λ2 z22 +Mabc za zb zc +Nabcd za zb zc zd + O(z 5 ),2a,b,ca,b,c,d(C.5)

where Mabc and Nabcd are defined by,
1X∂d ∂e ∂f T (xj , y) Aad Abe Acf ,6 d,e,f
1 X∂e ∂f ∂g ∂h T (xj , y) Aae Abf Acg Adh .
=24

Mabc =
Nabcd

e,f,g,hInserting the above equation (C.5) to (2.26), we obtain,
" (
Z

1
1 X iwTj
1 X
d2 z 0 exp i
F (w, y) =e
Mabc za0 zb0 zc0
λ1 z102 + λ2 z202 + √2πi j
2w a,b,c)#
1 X
Nabcd za0 zb0 zc0 zd0 + O(w −3/2 )
,
(C.6)
+
w a,b,c,d√where we change the integral variable from za to za0 = wza and we use d2 x =det A d2 z = d2 z. We expand the above equation (C.6) in powers of 1/w as,
"
Z
X
i X
02
1
02
F (w, y) =eiwTj d2 z 0 ei(λ1 z1 +λ2 z2 )/2 1 + √Mabc za0 zb0 zc0
2πi jw a,b,c



!2


X
X
1
1
−
Mabc za0 zb0 zc0++i
Nabcd za0 zb0 zc0 zd0 + O(w −3/2 ) .

w 2
a,b,c

a,b,c,d(C.7)

The first term of the above equation (C.7) is the amplification factor in the geometrical optics limit Fgeo in equation (2.30). The integral in the second term
vanishes because the integrand is an odd function of za0 . The third term is theleading correction term, being proportional to 1/w, arising from diffraction effect.
Integrating the above equation (C.7), we obtain,


X
i
1/2
(C.8)
F (w, y) =|µj |1 + ∆j eiwTj −iπnj + O(w −2 ),
w
j
where
∆j

15
=
2



2
2
M111 M122
M222
M111
M112 M222+
3++
3
λ1 |λ1 |2|λ1 |2 λ2
λ1 |λ2 |2λ2 |λ2 |2
N1111
N1122 N2222
−3
+2
+
2
|λ1 |
λ1 λ2
|λ2 |2and ∆j is the real number.


(C.9)

Appendix D
Estimation Errors in theGeometrical Optics Limit
To evaluate the estimation errors ∆MLz , ∆y in the geometrical optics limit, weconsider the simple waveform


(D.1)
h̃L (f ) = |µ+ |1/2 − i |µ− |1/2 e2πif ∆td × h̃(f ),where h̃ ∝ A is the unlensed signal and ∆td ∝ MLz , with three parameters γi =(ln MLz , y, ln A). Then, the Fisher matrix Γij (i, j = 1, 2, 3) in Eq.(4.10) can beanalytically obtained as,Γ11 = (2π∆td )2 |µ− | (f h̃|f h̃),∂∆td|µ− | (f h̃|f h̃),
Γ12 = 4π 2 ∆td∂yΓ13 = 0,
"

2

2

2 #
∂∆td
1
∂|µ+ |∂|µ− |
11
Γ22 =
2π(h̃|h̃),|µ− | (f h̃|f h̃) ++∂y
4 |µ+ |
∂y
|µ− |
∂y
1 ∂
(|µ+ | + |µ− |) (h̃|h̃),2 ∂y= (|µ+ | + |µ− |) (h̃|h̃),

Γ23 =
Γ33

(D.2)

and Γji = Γij . (h̃|h̃) and (f h̃|f h̃) in the above equation (D.2) are,Z2
df2(h̃|h̃) = (S/N ) = 4h̃(f ) ,Sn(f )
Z2
dff h̃(f ) .
(f h̃|f h̃) = 4
Sn(f )(D.3)

The S/N is the signal to noise ratio for the unlensed signal h̃. The estimation errors can be analytically obtained by the inverse of the Fisher matrix, ∆MLz /MLz =73[(Γ−1 )11 ]1/2 and ∆y/y = [(Γ−1 )22 ]1/2 /y. Using the geometrical optics approximation, f ∆td  1, we obtain the errors with Eq.(D.2) and (D.3) as
s

∂
µ+
∆MLz
|µ+ | + |µ− | ∂1,×ln
2 ln ∆td=
MLz
S/N|µ+ µ− |
∂y
∂y
µ−
s

∆y
|µ+ | + |µ− |
1
∂
µ+
=
×
2 y ln
.
(D.4)
y
S/N
|µ+ µ− |∂yµ−The above equations (D.4) are used for the general lens model when the double
images form.
Estimation errors in the quasi-geometrical optics approximation
We evaluate the estimation errors in the quasi-geometrical optics approximation
for the SIS lens. We take into account the effect of the diffracted image formed atlens center on the estimation errors. Then, the lensed waveform (D.1) is rewritten
as,


δc
1/2
1/2 2πif ∆td
2πif tcLe× h̃(f ),(D.5)
h̃ (f ) = |µ+ | − i |µ− | e+MLz fwhere µ− = 0 for y > 1. δc is defined as,

1i
for y < 1,
8π (1 − y 2 )3/2
1
1=
for y > 1,2
8π (y − 1)3/2

δc =and tc = 2MLz (y + 1)2 for the SIS lens (see §4.4.2). The third term in Eq.(D.5)
corresponds to the diffracted image. The Fisher matrix Γij (i, j = 1, 2, 3) can beobtained similar to equations (D.2) as,2|δc |2 −1 −1tc
2
22(h̃|h̃) + 2 (f h̃|f h̃),
Γ11 = (2π∆td ) |µ− | (f h̃|f h̃) + (2π) |δc |MLz
MLz
tc ∂tc∂∆td|µ− | (f h̃|f h̃) + (2π|δc |)2 2(h̃|h̃)Γ12 = 4π 2 ∆td∂yMLz ∂yδc ∂δc∗ −1 −1− 2(f h̃|f h̃),MLz ∂y|δc |2
Γ13 = − 2 (f −1 h̃|f −1 h̃),
MLz
"
2

2

2 #

1
∂|µ+ |∂|µ− |
11∂∆td(h̃|h̃)|µ− | (f h̃|f h̃) +
+
Γ22 =
2π
∂y
4 |µ+ |
∂y
|µ− |
∂y

2
2
|δc | ∂tc
1 ∂δc
+ 2π
(f −1 h̃|f −1 h̃),(h̃|h̃) + 2
MLz ∂yMLz ∂y1 ∂δc ∂δc∗ −1 −1(|µ+ | + |µ− |) (h̃|h̃) + 2(f h̃|f h̃),2 ∂yMLz ∂y|δc |2= (|µ+ | + |µ− |) (h̃|h̃) + 2 (f −1 h̃|f −1 h̃),
MLz

Γ23 =
Γ33

(D.6)

where (f −1 h̃|f −1 h̃) is defined as,
(f

−1

h̃|f

−1

h̃) = 4Z2
df
f −1 h̃(f ) .
Sn(f )(D.7)

The estimation errors ∆MLz /MLz = [(Γ−1 )11 ]1/2 and ∆y/y = [(Γ−1 )22 ]1/2 /y are
obtained from Eq.(D.6) as,
∆MLz
MLz

1
=
×

S/N
|µ µ |∂+ −|µ+ |+|µ− |

∂y

1
∆y
=
×

y
S/N
|µ µ |∂+ −|µ+ |+|µ− |∂y∂ln ∆td
2 ∂y

ln

ln

µ+
µ−

µ+
µ−

2

2
+ 16π 2 |δc |2 Mtc2
Lz

2

2
+ 16π 2 |δc |2 Mtc2
Lz

2/y



tc
∂
ln ∆t
∂y
d

2 1/2 ,



∂
tc
ln ∆t
∂y
d

2 1/2 ,(D.8)

for y < 1. If we set δc = 0, the results in Eq.(D.8) coincide with the previous
results in Eq.(D.4). For y > 1, the errors is also obtained from Eq.(D.6) as,
∆MLz
MLz

=

∆y
=
y

2
MLz
(∂/∂y) ln tc
×,δc (∂/∂y) ln (tc δc /|µ+ |1/2 )(f −1 h̃|f −1 h̃)
2
MLz
1
×
.
δc (∂/∂y) ln (tc δc /|µ+ |1/2 )(f −1 h̃|f −1 h̃)(D.9)

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