Cosmological Gravitational Lens Equation: Its Validity and Limitation

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Abstract
Based on the geometric optics in curved spacetime, we investigate the validity and limitation of the cosmological gravitational lens equation in detail. We first review the conventional derivation of the lens equation which relies on an intuitive construction of the lens geometry, and point out ambiguities in the resulting formula when applied to cosmological situations. We then consider the geometric optics equations in curved spacetime, and derive the cosmological lens equation by integrating the deviation equation for null geodesics under plausible assumptions. Finally, we remove these assumptions and formally integrate the deviation equation, to clarify the limitation on the use of the cosmological lens equation. Some implications of the results are also discussed.

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