Consider an extended congruence propagating through an otherwise homogeneous universe, from which all matter has been removed. Show that the affine distance functions as an effective angular diameter distance in this congruence. Now reinstate the matter as compact galaxies and modify Eq. (28.79) for the convergence and shear. Bookkeep the average density of matter as purely positive density perturbations. Explain qualitatively how the convergence and shear are changed when the ray passes through one or more galaxies, when it misses all of them, and on average.

Equation 28.79.

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(x,y;}') = - [ ²^ dx' (x² - ^-^) √ (277³ x«*²@ {{x^4} = a^*^x} | J {k², kjk² — kjk¹8 Þe¹k-x'. (28.79)