Effect of the cosmological constant on the bending of light and the cosmological lens equation

We revisit the effect of cosmological constant $\Lambda$ on the light deflection and its role in the cosmological lens equation. First, we reexamine the motion of photon in the Schwarzschild spacetime, and explicitly describe the trajectory of photon and deflection angle $\alpha$ up to the second order in $G$. Then the discussion is extended to the contribution of the cosmological constant $\Lambda$ in the Schwarzschild-de Sitter or Kottler spacetime. Contrary to the previous arguments, we emphasize the following points: (a) the cosmological constant $\Lambda$ does appear in the orbital equation of light, (b) nevertheless the bending angle of light $\alpha$ does not change its form even if $\Lambda \ne 0$ since the contribution of $\Lambda$ is thoroughly absorbed into the definition of the impact parameter, and (c) the effect of $\Lambda$ is completely involved in the angular diameter distance $D_A$.

Figure 1

Light trajectories on Schwarzschild geometry and on the flat space which is tangent to the asymptotically flat region of the Schwarzschild geometry. The coordinates of the point $P$ is $(r(\varphi ),\varphi ))=(r(0),0)$.

Figure 2

Light trajectory on the flat space with the metric $d{\ell }^2=d{r}^2+r^{2}d{\varphi }^2$. The impact parameter $b$ and the bending angle $\alpha$ are defined on this space.

Figure 3

The angle $\psi$ between the radial direction and the light trajectory at $P^{\prime }$ on Schwarzschild-de Sitter geometry.