Angular Diameter Distances in Clumpy Friedmann Universes

Solving null-geodesic equations, behavior of angular diameter distances is studied in inhomogeneous cosmological models, which are given by performing N-body simulations with the CDM spectrum. The distances depend on the separation angle θ of ray pairs, the mass m and the radius rs of particles consisting of galaxies and dark matter balls, and cosmological model parameters. The calculated distances are compared with the Dyer-Roeder angular diameter distance with the clumpiness parameter α (= 0 − 1), and for each ray pair the corresponding α is determined. Shooting many ray pairs, we derive statistical quantities such as the average value (α), the dispersion (σα), and the distribution of α. It is found in four cosmological models with (Ω0, λ0)=(1, 0), (0.2, 0), (0.4, 0), (0.2,0.8) for the particle parameters m ≃2 ×1011 M⊙ and rs = (10 − 40)h−1 kpc that (1) α is nearly equal to 1 or the Friedmann distance is best fitted, (2) σα decreases with the increase of the redshift z and radius rs, (3) for θ≫1 arcsec,σα is very small, but for θ< 1 arcsec it is so large that the use of only the the Friedmann distance (α= 1) may cause some errors for quantitative analysis of cosmological lensing, (4) the distribution of α is symmetric and asymmetric for σα< 0.5 and > 0.8, respectively, and (5) σα is smallest and largest in models with (1, 0) and (0.2, 0), respectively. The cosmological constant has the role of decreasing σα.