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.