A&A 438, L5–L8 (2005) Astronomy DOI: 10.1051/0004-6361:200500140 & c ESO 2005 Astrophysics Editor Scattering of gravitational waves by the weak gravitational fields the of lens objects to R. Takahashi1, T. Suyama2, and S. Michikoshi2 1 Division of Theoretical Astronomy, National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan Letter 2 Department of Physics, Kyoto University, Kyoto 606-8502, Japan Received 17 March 2005 / Accepted 7 June 2005 Abstract. We consider the scattering of gravitational waves by the weak gravitational fields of lens objects. We obtain the scattered gravitational waveform by treating the gravitational potential of the lens to first order, i.e. using the Born approxima- tion. We find that the effect of scattering on the waveform is roughly given by the Schwarzschild radius of the lens divided by the wavelength of gravitational wave for a compact lens object. If the lenses are smoothly distributed, the effect of scattering is of the order of the convergence field κ along the line of sight to the source. In the short wavelength limit, the amplitude is magnified by 1 + κ, which is consistent with the result in weak gravitational lensing. Key words. gravitational lensing – gravitational waves – scattering 1. Introduction which is the ratio of surface density of lens Σ to a critical density Σ ∼ c2/GD ,whereD is the distance to the lens Ground-based laser interferometric detectors of gravitational cr L L (Kaiser 1992; Bartelmann & Schneider 2001). In the strong waves such as LIGO, VIRGO, TAMA and GEO are currently lensing regime, κ ∼> 1, the multiple images of distant source in operation to search for astrophysical sources such as neutron are formed. But the strong lensing probability is small, ∼0.1%, star binaries, black hole binaries and supernovae (e.g. Cutler for high redshift sources. Hence, the weak lensing approxima- & Thorne 2002). The gravitational wave signals from these tion, κ 1, is valid for most sources. We use the weak field binaries are extracted from the data using matched filtering approximation in the wave optics. with a gravitational waveform template. If the gravitational In the past, Peters (1974) studied the scattering by a point waves pass near massive compact objects or pass through in- mass lens and a thin sheet of matter in the weak field ap- tervening inhomogeneous mass distribution, the gravitational proximation, and obtained the scattered waveform for these waveform is changed due to the scattering (or the gravitational lens models. The gravitationally lensed waveform (which is lensing) by the gravitational potential of these objects. The the solution of wave Eq. (1)) was given in Schneider et al. gravitational waves do not directly interact with matter (e.g. (1992), Sects. 4.7 and 7, using the diffraction integral under the Thorne 1987), but the gravitational lensing occurs in the same thin lens approximation. Recently, several authors have been way as it does for electromagnetic waves. In this letter, we in- studying wave optics in the gravitational lensing of gravita- vestigate the effects of scattering by lens objects on the gravi- tional waves using this integral (Nakamura 1998; Nakamura tational waveform. & Deguchi 1999; Ruffa 1999; Takahashi & Nakamura 2003; In the gravitational lensing of light, the scattering is dis- Yamamoto 2003; Macquart 2004; Takahashi 2004). In this let- cussed in terms of gravitational lensing under the geometrical ter, we present another method to derive the solution of Eq. (1) optics approximation, which is valid because the wavelength in the weak-field limit. We treat the gravitational field of lens is much smaller than the typical size of lens objects. But in to first order, i.e. using the Born approximation, and discuss its the case of gravitational waves, since the wavelength is much validity. We use units of c = G = 1. larger than that of light, geometrical optics is not valid in some cases. If the wavelength is larger than the Schwarzschild radius of the lens, wave optics should be used (Peters 1974; Ohanian 2. Gravitational waves propagating through weak 1974; Bontz & Haugan 1981; Thorne 1983; Deguchi & Watson 3 2 −1 gravitational fields 1986). This condition is rewritten as M ∼< 10 M( f /10 Hz) , where 102 Hz is the typical frequency of gravitational waves We consider the scattering of gravitational waves by the weak for ground-based detectors. Hence, we use wave optics in this gravitational fields of lens objects. We consider the scattering letter. of scalar waves, instead of gravitational waves, since the ba- In the geometrical optics for the lensing of light, the strong sic equation is the same as the scalar field wave equation (see and weak lensing are distinguished by the convergence field κ Sect. IIC of Peters 1974). The gravitational fields of the lens Article published by EDP Sciences and available at http://www.edpsciences.org/aa or http://dx.doi.org/10.1051/0004-6361:200500140 L6 R. Takahashi et al.: Scattering of gravitational waves by lens objects z observer Using the second-order Taylor series1 for these small quanti- 1 ro=( s o ,DL ) ties r and so, φ˜ is reduced to 0 1 1 2 φ +φ φ˜ (ro) ω DS = − d3r U(s , z )eiωtd(s ,so), (4) Editor 0 lens φ˜ (r ) π DLDLS o where D = D + D and we set r = (s, z). x,y S L LS the Here, td is the geometrical time delay which is given by 2 0 td(s , so) = DS/(2DLDLS)×[s − (DLS/DS) so] .Usingthetwo- φ to s = z U s z dimensional gravitational potential ψ( ) 2 d ( , ), the result in Eq. (4) is reduced to source r =( 0 ,−D ) S LS 1 2 φ˜ (ro) ω DS = − d2 s ψ(s )eiωtd(s ,so). (5) Fig. 1. Gravitational lens geometry for the source, the lens and the 0 φ˜ (ro) 2π DLDLS Letter observer. The lens is distributed around the origin. The source position = − = is rS (0, DLS), while the observer position is ro (so, DL)where The above equation is also derived by expanding ψ to first order | | so is a two-dimensional vector with so DL. DL and DLS are the in the diffraction integral (see Schneider et al. 1992). The sur- distances from the lens to the observer and to the source, respectively. face density of the lens is defined as Σ(s) = dzρ(s, z), where φ0 is the incident wave, while φ0 + φ1 is the scattered wave. ρ is the mass density. Since the potential ψ is written using the surface density as ψ(s) = 4 d2 sΣ(s)ln|s − s|,there- 2 µ ν objects are described by the metric gµν as ds = gµνdx dx = sult (5) is rewritten as (Appendix A), − (1 + 2U) dt2 + (1 − 2U) dr2,whereU(r) is the gravitational φ˜1(r ) potential of the lens. √ Since the propagation equation of the o = 2iω d2 sΣ(s) − µν = φ˜0(r ) scalar wave is ∂µ gg ∂νφ 0, we have the wave equa- o tion as 2 DLS × Ei iωtd(s , so) − ln s − so , (6) ∇2 + ω2 φ˜(r) = 4ω2U(r)φ˜(r), (1) DS where E is the exponential integral function: E (ix) = where ω is the frequency of the gravitational waves and we set ∞ i i iωt − it φ(t, r) = φ˜(r)e . x dt e /t (e.g. Abramowitz & Stegun 1970). If we shift the 1 0 We show the lens geometry of the source, the lens and the potential ψ by a constant value ψ0, φ˜ /φ˜ in Eq. (5) is shifted 2 observer in Fig. 1. The lens is distributed around the origin of by −iωψ0 . Hence we can choose ψ0 so that the second term of the coordinate axes. The source position is rS = (0, −DLS), Eq. (6) vanishes, and we eliminate this term. while the observer position is ro = (so, DL)whereso is a Let us discuss the validity of the Born approximation. If 0 1 two-dimensional vector with |so|DL. DL and DLS are the the scattered wave φ˜ + φ˜ is not too different from the incident distances from the lens to the observer and to the source, wave φ˜0, this approximation is valid. We discuss the condition respectively. of |φ˜1/φ˜0|1 for the two lens models, point mass lens and In the unlensed case U = 0inEq.(1),wewriteφ˜0(r)as smoothly distributed lens, in the following subsections. the solution of this equation. Including the effect of U to first order, the scattered wave at the observer is written as 2.1.1. Point mass lens φ˜(r ) = φ˜0(r ) + φ˜1(r ), (2) o o o The surface density is Σ(s) = Mδ2(s)whereM is the lens where φ˜1 represents the effect of scattering. Using Green’s mass. Then, φ˜1 in Eq. (6) is rewritten as (see also Peters 1974, function for the Helmholtz equation, eiω|r−r |/ |r − r|, we obtain Sect. III A) 1 φ˜ (ro) from Eqs. (1) and (2) as φ˜1(r ) iωD | − | o LS 2 2 iω ro r = 2iMω E s . (7) ω e 0 i o ˜1 = − 3 ˜0 φ˜ (r ) 2DLDS φ (ro) d r U(r )φ (r ). (3) o π |ro − r | In Fig. 2, we show E (ix) as a function of x. The solid line is Note that the above result (3) can be used if the lenses are i the absolute value of E , the dashed line is the real part, and the broadly distributed between the source and the observer since i dotted line is the imaginary part.