GEOMETRICAL OPTICS IN AN INHOMOGENEOUS UNIVERSI

In comparison with a homogeneous universe, a clumpy universe possess two types of effects on geometrical optics: 1. the convergence due to the gravititational-lens effects of the clumps; 2. the divergence due to the less mass density outside the clumps. On the other hand, thermodynamics requires that the total effects are double offset. This give us an effective method to judge the rationality of various calculations of the lensing effect in a clumpy universe.