Quasi-geometrical optics approximation in gravitational lensing

Abstract

Abstract. The gravitational lensing of gravitational waves should be treated in wave optics instead of geometrical optics when the wave length λ of the gravitational waves is larger than the Schwarzschild radius of the lens mass M. Wave optics is based on the diffraction integral which represents the amplification of the wave amplitude by lensing. We study the asymptotic expansion of the diffraction integral in the powers of the wave length λ. The first term, arising from the short wavelength limit λ → 0, corresponds to the geometrical optics limit. The second term, being of the order of λ/M, is the leading correction term arising from the diffraction effect. Analysing this correction term, we find that (1) the lensing magnification µ is modified to µ (1 + δ), where δ is of the order of (λ/M) 2 , and (2) if the lens has a cuspy (or singular) density profile at center ρ(r) ∝ r −α (0 < α ≤ 2), the diffracted image is formed at the lens center with magnification µ ∼ (λ/M) 3−α .

Arrival Time Differences Between Gravitational Waves and Electromagnetic Signals Due to Gravitational Lensing

“…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 2005Suyama et al , 2006Yoo et al 2007;Sereno et al 2010;Yoo et al 2013;Cao et al 2014). Figure 1 shows a lensing configuration with an observer, a lens, and a source in a nearly straight line.…”

Gravitational lensing effects on parameter estimation in gravitational wave detection with advanced detectors

“…There have been few motivations for a long time to consider the lensing effect on gravitational waves. Until recently, the authors in [3,4] realized that the gravitational lensing effect on gravitational waves should be described by wave dynamics instead of optics approximation. The features of wave dynamics make the fraction of lensed events larger than that optics approximation has ever predicted.…”

“…This fact motivated many authors to study lensing effects on gravitational waves in the past decade [3][4][5][6][7][8][9][10][11][12][13][14][15][16]. As noted in [4], the lensing effect is very important for the space-based gravitational wave detector LISA [17]. Even for the coming generations of ground-based gravitational wave detectors such as Advanced LIGO and Einstein telescope (ET), lensing effects are also nonnegligible.…”

Gravitational Wave Enhancement as a Tool to Distinguish Dark Matter Halo Profiles

“…This integral is carried out over the optical axis. Due to the fact that we limit our analysis to the case in which the wave length, λ, of the gravitational wave is larger than the Schwarzschild radius of the thin lens, we must use wave optics instead of geometric optics, as in Takahashi (2004). In this framework, the change in the amplitude is quantified via the enhancement factor or complex amplification factor (Nakamura & Deguchi 1999):…”