Lambda does not Lens: Deflection of Light in the Schwarzschild-de Sitter Spacetime

Abstract

Debate persists as to whether the cosmological constant $\Lambda$ can directly modify the power of a gravitational lens. With the aim of reestablishing a consensus on this issue, I conduct a comprehensive analysis of gravitational lensing in the Schwarzschild--de Sitter spacetime, wherein the effects of $\Lambda$ should be most apparent. The effective lensing law is found to be in precise agreement with the $\Lambda =0$ result: ${\alpha }_{\mathrm{eff}}=4m/b_{\mathrm{eff}}+15\pi m^2/4b_{\mathrm{eff}}^2+O(m^3/b_{\mathrm{eff}}^3)$, where the effective bending angle ${\alpha }_{\mathrm{eff}}$ and impact parameter $b_{\mathrm{eff}}$ are defined by the angles and angular diameter distances measured by a comoving cosmological observer. [These observers follow the timelike geodesic congruence which (i) respects the continuous symmetries of the spacetime and (ii) approaches local isotropy most rapidly at large distance from the lens.] The effective lensing law can be derived using lensed or unlensed angular diameter distances, although the inherent ambiguity of unlensed distances generates an additional uncertainty $O(m^5/\Lambda b_{\mathrm{eff}}^7)$. I conclude that the cosmological constant does not interfere with the standard gravitational lensing formalism.

11 pages, 7 figures

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