ARRIVAL TIME DIFFERENCES BETWEEN GRAVITATIONAL WAVES AND ELECTROMAGNETIC SIGNALS DUE TO GRAVITATIONAL LENSING

First, we briefly discuss previous studies on wave optics for the gravitational lensing of GWs (see also a review by Schneider et al. 1992; Nakamura & Deguchi 1999). Peters (1974) was the first to study GW propagation in the weak gravitational field of a lens object. He derived a basic wave equation in curved spacetime and solved it exactly for a point mass lens. Ohanian (1974) also derived a lensed waveform using the Kirchhoff diffraction integral, which can be used for general lens mass distributions (see also Bliokh & Minakov 1975; Bontz & Haugan 1981). Nakamura (1998) discussed the diffraction effect on lensing magnifications for inspiraling binaries induced by intervening compact objects. Wave effects in the gravitational lensing of GWs have been discussed linked to several topics (e.g., Baraldo et al. 1999; Takahashi & Nakamura 2003; Takahashi 2004; Takahashi et al. 2005; Suyama et al. 2005, 2006; Yoo et al. 2007, 2013; Sereno et al. 2010; Cao et al. 2014).

Figure 1 shows a lensing configuration with an observer, a lens, and a source in a nearly straight line. The angular diameter distances to the lens and the source are D_L and D_S, respectively. The distance between them is D_LS. The lens mass distribution is projected on the lens plane (the thin-lens approximation), and the lens center is located at the origin (${\boldsymbol{\theta }}=0$) on the plane. The waves from the source are scattered on the plane at a point ${\boldsymbol{\xi }}$. Then, the incoming direction of the wave is ${\boldsymbol{\theta }}(={\boldsymbol{\xi }}/{D}_{{\rm{L}}})$, while the angular source position is ${{\boldsymbol{\theta }}}_{{\rm{s}}}$. In geometrical optics, the light ray paths are uniquely determined by Fermat's principle (or the lens equation), while in wave optics, the GWs are described by the superposition (or integration) of many paths. Throughout this study, we will discuss the lensing of light using geometrical optics and the lensing of GWs using wave optics. In this section, we assume that the GWs are monochromatic with a frequency of f. In Section 4, we will discuss a non-monochromatic case (a chirp signal).

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The gravitationally lensed waveform of GWs is usually studied in the frequency domain (Fourier space) because the waveform has a simpler form than in the time domain. The waveform is a complex (real) function in the frequency (time) domain. According to the previous studies, the lensed waveform ${\tilde{h}}_{+,\times }^{{\rm{L}}}(f)$ in the frequency domain is the product of a complex function $F(f,{{\boldsymbol{\theta }}}_{{\rm{s}}})$ and the unlensed waveform ${\tilde{h}}_{+,\times }(f)$,

$$\begin{eqnarray}&&{\tilde{h}}_{+,\times }^{{\rm{L}}}(f)=F(f,{{\boldsymbol{\theta }}}_{{\rm{s}}}){\tilde{h}}_{+,\times }(f).\end{eqnarray} \tag{ 1 }$$

The function F is called the amplification factor or the transmission factor and is given by the diffraction integral (Schneider et al. 1992),

$$\begin{eqnarray}&&F(f,{{\boldsymbol{\theta }}}_{{\rm{s}}})=\displaystyle \frac{{D}_{{\rm{L}}}{D}_{{\rm{S}}}}{{{cD}}_{\mathrm{LS}}}\displaystyle \frac{(1+{z}_{{\rm{L}}})f}{i}\int {d}^{2}\theta \exp [2\pi {{ift}}_{{\rm{d}}}({\boldsymbol{\theta }},{{\boldsymbol{\theta }}}_{{\rm{s}}})],\end{eqnarray} \tag{ 2 }$$

with the time delay,

$$\begin{eqnarray}&&{t}_{{\rm{d}}}({\boldsymbol{\theta }},{{\boldsymbol{\theta }}}_{{\rm{s}}})=\displaystyle \frac{(1+{z}_{{\rm{L}}})}{c}\left[\displaystyle \frac{{D}_{{\rm{L}}}{D}_{{\rm{S}}}}{2{D}_{\mathrm{LS}}}{| {\boldsymbol{\theta }}-{{\boldsymbol{\theta }}}_{{\rm{s}}}| }^{2}-\displaystyle \frac{\hat{\psi }({\boldsymbol{\theta }})}{{c}^{2}}\right],\end{eqnarray} \tag{ 3 }$$

relative to the unlensed case. The first term is the geometrical time delay and the second term is the Shapiro time delay (or the potential time delay). The overall factor $(1+{z}_{{\rm{L}}})$ is the cosmological time dilation, where z_L is the lens redshift. The two-dimensional lens potential $\hat{\psi }({\boldsymbol{\theta }})$ is determined by the surface mass density of the lens ${\rm{\Sigma }}({\boldsymbol{\theta }})$ via the Poisson equation, ${{\rm{\nabla }}}_{\theta }^{2}\hat{\psi }({\boldsymbol{\theta }})=8\pi {{GD}}_{{\rm{L}}}^{2}{\rm{\Sigma }}({\boldsymbol{\theta }})$. The amplification factor F is normalized to unity for the unlensed case $\hat{\psi }=0$.

We can rewrite the equations in their dimensionless forms by introducing a characteristic angular scale θ_*, which will be chosen to be the angular Einstein radius. We can define the scaled angle ${\boldsymbol{x}}={\boldsymbol{\theta }}/{\theta }_{* }$, source position ${\boldsymbol{y}}={{\boldsymbol{\theta }}}_{{\rm{s}}}/{\theta }_{* }$, the dimensionless frequency $w=[{D}_{{\rm{S}}}{D}_{{\rm{L}}}/({{cD}}_{\mathrm{LS}})]{\theta }_{* }^{2}(1+{z}_{{\rm{L}}})2\pi f$, and time delay $T{({\boldsymbol{x}},{\boldsymbol{y}})=[{{cD}}_{\mathrm{LS}}/({D}_{{\rm{S}}}{D}_{{\rm{L}}})]{\theta }_{* }^{-2}(1+{z}_{{\rm{L}}})}^{-1}{t}_{{\rm{d}}}({\boldsymbol{\theta }},{{\boldsymbol{\theta }}}_{{\rm{s}}})$. Then, the amplification factor has a simple form:

$$\begin{eqnarray}&&F(w,{\boldsymbol{y}})=\displaystyle \frac{w}{2\pi i}\int {d}^{2}x\exp [{iwT}({\boldsymbol{x}},{\boldsymbol{y}})].\end{eqnarray} \tag{ 4 }$$

In the low-frequency limit ($w\to 0$), the factor F approaches unity. In the high-frequency limit ($w\to \infty$), the stationary points of $T({\boldsymbol{x}},{\boldsymbol{y}})$ contribute to the integral in Equation (4). Then, multiple images can form and these angular positions can be determined by solving the lens equation, ${{\rm{\nabla }}}_{x}T({\boldsymbol{x}},{\boldsymbol{y}})=0$. Given the jth image position ${{\boldsymbol{x}}}_{j}$, its time delay and magnification are ${T}_{j}=T({{\boldsymbol{x}}}_{j},{\boldsymbol{y}})$ and ${\mu }_{j}={[\det (\partial {\boldsymbol{y}}/\partial {{\boldsymbol{x}}}_{j})]}^{-1}$, respectively. Hereafter, ${T}_{\mathrm{EM},j}$ indicates the time delay of the jth EM image.

When the wavelength is larger than the path difference between the multiple images, the geometrical-optics approximation breaks down (e.g., Schneider et al. 1992). This condition can be rewritten as $w({T}_{i}-{T}_{j})\lesssim 1$ for the ith and jth images. Throughout this paper, we take $w({T}_{i}-{T}_{j})=1$ to be the border between the geometrical-optics regime and the wave-optics regime. We assume that the GW wavelength is large enough to be in the wave-optics regime. Then, there is always only one GW image. We define the time delay of the GW signal from the phase of the amplification factor,

$$\begin{eqnarray}&&{T}_{\mathrm{GW}}(w,{\boldsymbol{y}})\equiv -\displaystyle \frac{i}{w}\mathrm{ln}\left(\displaystyle \frac{F(w,{\boldsymbol{y}})}{| F(w,{\boldsymbol{y}})| }\right).\end{eqnarray} \tag{ 5 }$$

Note that the GW time delay depends on the frequency.

We can define the arrival time difference of the jth EM image signal relative to the GW signal as

$$\begin{eqnarray}&&{\rm{\Delta }}{T}_{\mathrm{EM},j-\mathrm{GW}}(w,{\boldsymbol{y}})\equiv {T}_{\mathrm{EM},j}({\boldsymbol{y}})-{T}_{\mathrm{GW}}(w,{\boldsymbol{y}}).\end{eqnarray} \tag{ 6 }$$

The time difference ${\rm{\Delta }}{T}_{\mathrm{EM},j-\mathrm{GW}}$ definitely approaches zero in the high-frequency limit ($w\to \infty$) when there is a single EM image. However, it does not happen when there are multiple EM images. This is because as the GW frequency increases into the geometrical-optics regime, multiple GW images appear. Then, the T_GW contains the time delays of several images, not just the jth image.

In the following subsections, we will calculate the arrival time differences in Equation (6) for specific lens models: a point mass (Section 2.1) and a singular isothermal sphere (SIS; Section 2.2). We will basically discuss in dimensionless form for convenience but comment for specific lens systems (sources, lenses, and detectors) in dimensional form at the end of Sections 2.2 and 3.

First, we consider lensing by a compact object (for instance, a star, planet, or black hole) with mass M. The surface density profile and the potential are ${\rm{\Sigma }}(\theta )=M{\delta }^{2}({D}_{{\rm{L}}}{\boldsymbol{\theta }})$ and $\hat{\psi }(\theta )=4{GM}\mathrm{ln}\theta$, respectively. We chose the angular Einstein radius to be the characteristic scale

$$\begin{eqnarray*}&&{\theta }_{* }=\sqrt{\displaystyle \frac{4{{GMD}}_{\mathrm{LS}}}{{c}^{2}{D}_{{\rm{L}}}{D}_{{\rm{S}}}}}.\end{eqnarray*}$$

Then, the source position, the dimensionless GW frequency, and time delay are

$$\begin{eqnarray}&&y=\displaystyle \frac{{\theta }_{{\rm{s}}}}{{\theta }_{* }},\,w=8\pi \displaystyle \frac{{{GM}}_{z}}{{c}^{3}}f,\,\mathrm{and}\,T({\boldsymbol{x}},{\boldsymbol{y}})=\displaystyle \frac{{c}^{3}}{4{{GM}}_{z}}{t}_{{\rm{d}}}({\boldsymbol{\theta }},{{\boldsymbol{\theta }}}_{{\rm{s}}}),\end{eqnarray} \tag{ 7 }$$

respectively. Here, ${M}_{z}=(1+{z}_{{\rm{L}}})M$ is the redshifted lens mass. The four parameters of the lens mass M_z, source position θ_s, GW frequency f, and time delay t_d reduce to the three dimensionless parameters of y, w, and T. The amplification factor is $F(w,y)=\exp [\pi w/4+({iw}/2)\mathrm{ln}(w/2)]$ ${\rm{\Gamma }}(1-{iw}/2){}_{1}{F}_{1}({iw}/2,1;{{iwy}}^{2}/2)$, where ${}_{1}{F}_{1}$ is the the confluent hypergeometric function (Peters 1974). For very low frequencies (w ≪ 1) we can obtain the GW time delay in Equation (5) as a series in w,

$$\begin{eqnarray}&&{T}_{\mathrm{GW}}(w,y)=\displaystyle \frac{1}{2}\left[\mathrm{ln}\left(\displaystyle \frac{w}{2}\right)+\gamma \right]+{ \mathcal O }({w}^{2}),\end{eqnarray} \tag{ 8 }$$

where γ = 0.5772⋯ is the Euler constant. This GW time delay is independent of the source position y.

In the geometrical-optics limit, the two EM images always form a bright (+) image and a faint (−) image. Their time delays and magnifications are ${T}_{\mathrm{EM},\pm }(y)\,$ $=\,({y}^{2}+2\mp y\sqrt{{y}^{2}+4})/4-\mathrm{ln}|$ $(y\pm \sqrt{{y}^{2}+4})/2|$ and ${\mu }_{\mathrm{EM},\pm }=1/2\pm ({y}^{2}+2)/(2y\sqrt{{y}^{2}+4})$, respectively.

Figure 2 (left panel) shows the arrival time differences ${\rm{\Delta }}{T}_{\mathrm{EM},\pm -\mathrm{GW}}(w,y)$ in Equation (6) as a contour map. The horizontal axis is the dimensionless GW frequency $w=8\pi ({{GM}}_{z}/{c}^{3})f\,$ $=\,1.2(f/\mathrm{Hz})({M}_{z}/{10}^{4}\,{M}_{\odot })$, while the vertical axis is the source position y = θ_s/θ_*. The solid red and dashed blue curves show the results for the bright (+) and faint (−) EM images, respectively. The results simply scale in proportion to the lens mass M_z via $w\propto {M}_{z}f$ and ${\rm{\Delta }}{t}_{{\rm{d}}}\propto {M}_{z}{\rm{\Delta }}T$. The dotted black curve denotes the boundary between the geometrical- and the wave-optics regimes at $w({T}_{+}-{T}_{-})=1$. The top-right (bottom-left) side of the dotted curve is the geometrical-(wave-)optics regime. In plotting the figure, we used the exact formula in Equation (5), not the approximate one in Equation (8). As expected, the time differences are larger for lower frequencies. As the dotted curve (the boundary) is approached, the solid red contours approach zero; however, the dashed blue contours do not because the GW time delay contains the contributions of both images and the bright image is dominant. For a very small source position y ≪ 1, the time delays of the bright and faint images are asymptotically the same; therefore, the solid red and dashed blue curves converge.

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As seen in the figure, the time differences are always positive, which means that the GW signal arrives earlier than both of the EM signals. This may seem surprising and counterintuitive because the bright EM image corresponds to the minimum time delay via Fermat's principle. To understand why the GWs arrive earlier, we can rewrite the amplification factor, Equation (2), with the time delay, Equation (3), by replacing the variables ${{\boldsymbol{\theta }}}^{\prime }=\sqrt{(1+{z}_{{\rm{L}}})f}{\boldsymbol{\theta }}$ and ${{\boldsymbol{\theta }}}_{{\rm{s}}}^{\prime }=\sqrt{(1+{z}_{{\rm{L}}})f}{{\boldsymbol{\theta }}}_{{\rm{s}}}$,

$$\begin{eqnarray}F(f,{{\boldsymbol{\theta }}}_{{\rm{s}}}) & = & \displaystyle \frac{{D}_{{\rm{L}}}{D}_{{\rm{S}}}}{{{icD}}_{\mathrm{LS}}}\displaystyle \int {d}^{2}{{\boldsymbol{\theta }}}^{\prime }\exp \left[\displaystyle \frac{2\pi i}{c}\left\{\displaystyle \frac{{D}_{{\rm{L}}}{D}_{{\rm{S}}}}{2{D}_{\mathrm{LS}}}{| {{\boldsymbol{\theta }}}^{\prime }-{{\boldsymbol{\theta }}}_{{\rm{s}}}^{\prime }| }^{2}\right.\right.\\ & & -\left.\left.\displaystyle \frac{(1+{z}_{{\rm{L}}})f}{{c}^{2}}\hat{\psi }\left(\displaystyle \frac{{{\boldsymbol{\theta }}}^{\prime }}{\sqrt{(1+{z}_{{\rm{L}}})f}}\right)\right\}\right].\end{eqnarray} \tag{ 9 }$$

Therefore, in the low-frequency limit $(f\to 0)$, the potential term is negligibly small. Then, a point near ${{\boldsymbol{\theta }}}^{\prime }={{\boldsymbol{\theta }}}_{{\rm{s}}}^{\prime }$ contributes to the integral, which means that the GWs pass in a straight line from the source to the observer and therefore the geometrical time delay (the first term in Equation (3)) is also small, and the GW time delay is smaller than both of the EM time delays.

Even for a relatively high source position y > 1, an arrival time difference arises for the bright image, as seen in Figure 2 (left panel). This is because ${\rm{\Delta }}{T}_{\mathrm{EM},\pm -\mathrm{GW}}$ is not sensitive to y because (i) the Shapiro time delay is proportional to the logarithm of y and (ii) the GW time delay is independent of y in the low-frequency limit from Equation (8).

In GW measurements, the time difference can be measured from the phase difference. Therefore, we plot the phase differences defined by $w{\rm{\Delta }}{T}_{\mathrm{EM},\pm -\mathrm{GW}}$ in Figure 2 (right panel). As seen in the figure, the phase differences are largest at w ≃ 1 with y ≲ 0.3. The maximum phase differences are $w{\rm{\Delta }}{T}_{\mathrm{EM},\pm -\mathrm{GW}}^{\max }\simeq 0.1,$ 0.5, and 0.7 rad for y = 1, 0.1, and 0.01, respectively. In a matched filtering analysis, the phase of the waveform can be roughly measured within the accuracy of the inverse signal-to-noise ratio $\approx {({\rm{S}}/{\rm{N}})}^{-1}$ (Cutler & Flanagan 1994).² For example, we can measure the phase differences for a lensed GW signal with S/N = 10(100) if the parameters w and y are in the region where $w{\rm{\Delta }}{T}_{\mathrm{EM},\pm -\mathrm{GW}}\gtrsim {10}^{-1}({10}^{-2})$ rad in the right panel. The arrival time differences can be most accurately determined near w ≃ 1.

Finally, let us comment on the "dimensional" arrival time difference. The dimensional time delay and frequency are proportional to the dimensionless quantities via t_d ∝ MT and f ∝ w/M. Therefore, we obtain t_d ∝ wT for the fixed frequency f. From the phase relation $2\pi f{\rm{\Delta }}{t}_{{\rm{d}},\mathrm{EM},\pm -\mathrm{GW}}=w{\rm{\Delta }}{T}_{\mathrm{EM},\pm -\mathrm{GW}}$, we can estimate the dimensional time difference as

$$\begin{eqnarray}{\rm{\Delta }}{t}_{{\rm{d}},\mathrm{EM},\pm -\mathrm{GW}} & = & \displaystyle \frac{1}{2\pi f}w{\rm{\Delta }}{T}_{\mathrm{EM},\pm -\mathrm{GW}},\\ & = & 0.16\,{\rm{s}}{\left(\displaystyle \frac{f}{\mathrm{Hz}}\right)}^{-1}\left(\displaystyle \frac{w{\rm{\Delta }}{T}_{\mathrm{EM},\pm -\mathrm{GW}}}{1}\right).\end{eqnarray} \tag{ 10 }$$

Therefore, the dimensional time difference is more prominent for lower GW frequencies and larger phase differences. The phase difference is largest at w ≃ 1 and the corresponding lens mass is

$$\begin{eqnarray}{M}_{z} & = & \displaystyle \frac{{c}^{3}}{8\pi G}\displaystyle \frac{w}{f},\\ & = & 8100\,{M}_{\odot }{\left(\displaystyle \frac{f}{\mathrm{Hz}}\right)}^{-1}\left(\displaystyle \frac{w}{1}\right),\end{eqnarray} \tag{ 11 }$$

from Equation (7). Because the maximum phase difference is $w{\rm{\Delta }}{T}_{\mathrm{EM},\pm -\mathrm{GW}}^{\max }\simeq 0.7$, the maximum time difference is

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{{\rm{d}},\mathrm{EM},\pm -\mathrm{GW}}^{\max }\simeq 0.11\,{\rm{s}}{\left(\displaystyle \frac{f}{\mathrm{Hz}}\right)}^{-1},\end{eqnarray} \tag{ 12 }$$

from Equation (10). This is typically ∼1 ms (f/100 Hz)⁻¹ for ground-based detectors, ∼2 minutes (f/mHz)⁻¹ for space-based interferometers, and ∼4 months ${(f/{10}^{-8}\mathrm{Hz})}^{-1}$ for pulsar timing arrays.

For the ground-based detectors, extra-galactic compact binaries (NS–NS/NS–BH) and nearby supernovae are promising sources that may be associated with EM signals. Similarly, for the planned space interferometer eLISA (the Evolved Laser Interferometer Space Antenna), galactic white dwarf binaries are the most secure sources. The orbital phase differences between the GW/EM signals in these binaries can be used to measure the arrival time differences. Here let us consider that the sources are lensed by galactic stars as a plausible case. Then, the lensing-induced time lag is typically $\sim ({GM}/{c}^{3})\sim {10}^{-5}\,{\rm{s}}\,(M/{M}_{\odot })$ but it is too small to be resolved by optical detectors. The angular separation between the bright/faint images is approximately 10⁻⁶ deg, which is also very small. The lensing probability that the GW/EM signals from the source pass near the star within the Einstein radius is ≈10⁻⁶(≲10⁻⁸) for the direction to the Galactic bulge (perpendicular to the Galactic disk; e.g., Wang & Smith 2011). Therefore, detecting the time lags in such lens systems in the near future is unlikely.

Next, we consider lensing by an SIS, which approximately represents the density profile of a galaxy, dark halo, or star cluster. The surface density is characterized by the velocity dispersion v as ${\rm{\Sigma }}(\theta )={v}^{2}/(2{{GD}}_{{\rm{L}}}\theta )$. The potential is $\hat{\psi }(\theta )=4\pi {v}^{2}{D}_{{\rm{L}}}\theta$. We adopt the Einstein radius as the characteristic scale, ${\theta }_{* }=4\pi {(v/c)}^{2}{D}_{\mathrm{LS}}/{D}_{{\rm{S}}}$. Then, the dimensionless GW frequency and time delays are the same as in the point mass lens with a mass of ${M}_{z}=4{\pi }^{2}({v}^{4}/({c}^{2}G))(1+{z}_{{\rm{L}}}){D}_{{\rm{L}}}{D}_{\mathrm{LS}}/{D}_{{\rm{S}}}$, which is the enclosed mass within the Einstein radius. Next, we can obtain the amplification factor F(w, y) via numerical integration (e.g., Takahashi & Nakamura 2003; Moylan et al. 2008). The GW time delay for w ≪ 1 is

$$\begin{eqnarray}&&{T}_{\mathrm{GW}}(w,y)=-\displaystyle \frac{\sqrt{\pi }}{2}{w}^{-1/2}-\left(1-\displaystyle \frac{\pi }{4}\right)+{ \mathcal O }({w}^{1/2}),\end{eqnarray} \tag{ 13 }$$

which is independent of y.

In geometrical optics, two images or one image forms for y < 1 or y ≥ 1, respectively. The time delays and magnifications are ${T}_{\pm }=\mp y-1/2$ and ${\mu }_{\pm }=\pm 1+1/y$, respectively, for the bright (+) and faint (−) images. The faint image appears only for y < 1.

Figure 3 shows the arrival time differences and the phase differences for the SIS lens. The dotted line is extrapolated to y ≥ 1. Similar to the results of Figure 2, the GWs can arrive before both EM images. The time differences are larger than those of the point mass lens.

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In this subsection, we present our detailed calculation of the lensing probability that a distant SMBHB is lensed by a foreground galaxy. We suppose that the future PTAs will detect the GW signals. Here, we basically follow the theoretical model in Oguri et al. (2012) to evaluate the lensing probability. They calculated the probability that a distant quasar is lensed by a foreground galaxy to form multiple images. Their model agrees well with observational results in the SDSS Quasar Lens Search (SQLS). We assume that the galaxy density profile is described by the SIS model and the source at redshift z_S emits the GWs with frequency f. Then, the lensing probability that the time lag between the EM/GW signals is larger than Δt_d is written as

$$\begin{eqnarray}&&P(\gt {\rm{\Delta }}{t}_{{\rm{d}}};f,{z}_{{\rm{S}}})=\int {cdt}\int d\sigma ({\rm{\Delta }}{t}_{{\rm{d}}},f,{z}_{{\rm{S}}})\int {dn},\end{eqnarray} \tag{ 15 }$$

where ${cdt}=c({dt}/{{dz}}_{{\rm{L}}}){{dz}}_{{\rm{L}}}=-{{cdz}}_{{\rm{L}}}/(H({z}_{{\rm{L}}})(1+{z}_{{\rm{L}}}))$, where H(z_L) is the Hubble expansion rate at the lens redshift z_L. We adopt the cosmological parameters of the Hubble parameter ${H}_{0}=70\,\mathrm{km}\ {{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$, the matter density Ω_m = 0.27, and the cosmological constant Ω_Λ = 0.73. The cross section is

$$\begin{eqnarray}d\sigma ({\rm{\Delta }}{t}_{{\rm{d}}},f,{z}_{{\rm{S}}}) & = & 2\pi {r}_{{\rm{E}}}^{2}{dyy}\\ & & \times {\rm{\Theta }}({\rm{\Delta }}{t}_{{\rm{d}}}-{\rm{\Delta }}{t}_{{\rm{d}},\mathrm{EM},\pm -\mathrm{GW}}(w,y)),\end{eqnarray} \tag{ 16 }$$

where the Einstein radius is ${r}_{{\rm{E}}}=4\pi {(v/c)}^{2}{D}_{{\rm{L}}}{D}_{\mathrm{LS}}/{D}_{{\rm{S}}}$ and the step function is Θ(x) = 1(0) for x ≥ 0 (x < 0). The arrival time difference between the GW signal and the bright/faint EM image (±) signals ${\rm{\Delta }}{t}_{{\rm{d}},\mathrm{EM},\pm -\mathrm{GW}}(w,y)$ is numerically evaluated as in Section 2.3. The number density of the lens is dn = (dn/dv)dv where dn/dv is the velocity-dispersion function of galaxies. Suppose that the galaxy velocity function does not evolve in time, then it has a simpler form, ${dn}{(v,{z}_{{\rm{L}}})/{dv}=(1+{z}_{{\rm{L}}})}^{3}{{dn}}_{0}/{dv}(v)$, where dn₀/dv is the present velocity function. We adopt the velocity function in SDSS data (Bernardi et al. 2010). They provided a distribution function of velocity dispersion for all types (both early and late) galaxies using ∼2.5 × 10⁵ samples:

$$\begin{eqnarray}&&\displaystyle \frac{{{dn}}_{0}}{{dv}}(v)=\displaystyle \frac{{\phi }_{* }}{v}\displaystyle \frac{\beta }{{\rm{\Gamma }}(\alpha /\beta )}{\left(\displaystyle \frac{v}{{v}_{* }}\right)}^{\alpha }\exp \left[-{\left(\displaystyle \frac{v}{{v}_{* }}\right)}^{\beta }\right],\end{eqnarray} \tag{ 17 }$$

with α = 0.07, β = 2.22, v_* = 141 km s⁻¹, and ${\phi }_{* }=2.47\times {10}^{-1}\,{\mathrm{Mpc}}^{-3}$. Here, Γ(x) is the Gamma function.

Let us estimate the typical velocity dispersion, which causes the maximum time lag for the PTA frequency band (f = 10⁻⁹–10⁻⁷ Hz). Using Equation (11) with the mass enclosed within the Einstein radius (${M}_{z}=4{\pi }^{2}({v}^{4}/({c}^{2}G))(1+{z}_{{\rm{L}}}){D}_{{\rm{L}}}{D}_{\mathrm{LS}}/{D}_{{\rm{S}}}$), the typical velocity dispersion is

$$\begin{eqnarray}v & = & 298\,\mathrm{km}\,{{\rm{s}}}^{-1}{(1+{z}_{{\rm{L}}})}^{-1/4}{\left(\displaystyle \frac{{D}_{{\rm{L}}}{D}_{\mathrm{LS}}/{D}_{{\rm{S}}}}{\mathrm{Gpc}}\right)}^{-1/4}\\ & & \times {\left(\displaystyle \frac{f}{{10}^{-8}\mathrm{Hz}}\right)}^{-1/4}{\left(\displaystyle \frac{w}{1}\right)}^{1/4}.\end{eqnarray} \tag{ 18 }$$

The above velocity dispersion is comparable to v_* in the velocity function. Therefore, the galaxies are the suitable lenses for the PTAs.

Finally, we have the lensing probability as

$$\begin{eqnarray}P(\gt {\rm{\Delta }}{t}_{{\rm{d}}};f,{z}_{{\rm{S}}}) & = & 32{\pi }^{3}{\displaystyle \int }_{0}^{{z}_{{\rm{s}}}}\displaystyle \frac{{{cdz}}_{{\rm{L}}}}{H({z}_{{\rm{L}}})}{(1+{z}_{{\rm{L}}})}^{2}{\left(\displaystyle \frac{{D}_{{\rm{L}}}{D}_{\mathrm{LS}}}{{D}_{{\rm{S}}}}\right)}^{2}\\ & & \times {\displaystyle \int }_{0}^{\infty }{dyy}{\displaystyle \int }_{0}^{\infty }{dv}\displaystyle \frac{{{dn}}_{0}}{{dv}}(v){\left(\displaystyle \frac{v}{c}\right)}^{4}\\ & & \times {\rm{\Theta }}({\rm{\Delta }}{t}_{{\rm{d}}}-{\rm{\Delta }}{t}_{{\rm{d}},\mathrm{EM},\pm -\mathrm{GW}}(w,y)).\end{eqnarray} \tag{ 19 }$$

When calculating the probability to form the double source images, we simply replace the step function to ${\rm{\Theta }}(1-y)$ in the above equation.

Figure 4 shows the probability that the arrival time difference is larger than Δt_d. The lower x axes are the time lags Δt_d between the bright EM and the GW signals, while the upper x axes are the S/N necessary to measure the time lags (which is simply ${\rm{S}}/{\rm{N}}={(2\pi f{\rm{\Delta }}{t}_{{\rm{d}}})}^{-1}$). The GW frequencies are f = 10⁻⁹ Hz to f = 10⁻⁷ Hz from left to right. The corresponding orbital periods are T = 2/f = 63 year, 6.3 year, and 0.63 year, respectively. The source redshifts are z_s = 2, 1, and 0.5 from top to bottom. The horizontal dashed lines are the probabilities that the double source images form (note again that they form when y < 1). As shown in the figure, the solid curves are much higher than the dashed lines for smaller Δt_d. This is because for high S/N the smaller phase difference can be resolved even for y > 1 (see right panel of Figure 3). Then, the cross section is proportional to y² and thus the lensing probability becomes larger. For instance, the maximum y to resolve the signal is about 2(20) for S/N = 10(100). In such a high y(>1), only the bright EM image appears. From the figure, for f = 10⁻⁸ Hz at z_s = 1, the lensing probability of Δt_d > 2(10) days is 1 × 10⁻² (2 × 10⁻³) with the necessary S/N of 100(20). Therefore, if some hundred sources will be resolved, as suggested in Sesana et al. (2012), some lensing events will be detected.

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At present, over 140 lensed quasar-galaxy systems are known.³ In the near future, the Large Synoptic Survey Telescope (LSST)⁴ will find ∼8000 lensed quasars (Oguri & Marshall 2010). If we find SMBHBs at the centers of such lensed quasars, we will possibly detect the time lags in such systems.