The Born approximation in wave optics gravitational lensing revisited

Gravitational lensing of gravitational waves provides us with much information about the Universe, including the dark matter distribution at small scales. The information about lensed gravitational waves is encapsulated by an amplification factor, which is calculated by an integration of an oscillatory function. The Born approximation, which has been studied in terms of wave optics in gravitational lensing, may provide a means of overcoming the difficulty in evaluating the oscillating function and better understanding the connection between the amplification factor and the lens mass distribution. In this paper, we revisit the Born approximation for a single lens plane. We find that the distortion of gravitational waves induced by wave optics gravitational lensing is in general connected with the mass distribution of the lens object through a convolution integral, where the scale of the kernel is determined by the Fresnel scale. We then study the validivity and accuracy of the Born approximation specifically for the case of a point mass lens for which the exact analytical expression of the amplicfication factor is available. Using the dimensionless parameter $y$, which represents the normalized impact parameter, and the dimensionless parameter $w$, which denotes the normalized frequency, we show that the $n$-th term of the Born approximation scales as $y^{-2}w^{n-1}$. This indicates that, for the case of a point mass lens, the Born approximation is valid when $w$ is less than 1, with its accuracy scaling as $wy^{-2}$ in this regime.