Analytical Estimate of the Effect of Spherical Inhomogeneities on Luminosity Distance and Redshift

We provide an analytical estimate of the effect of a spherical inhomogeneity on light beams that travel through it. We model the interior of the inhomogeneity in terms of the Lemaitre-Tolman-Bondi metric. We assume that the beam source is located outside the inhomogeneity. We study the relative deviations of travelling time, redshift, beam area and luminosity distance from their values in a homogeneous cosmology. They depend on the ratio Hb=H r_0 of the radius r_0 of the inhomogeneity to the horizon distance 1/H. For an observer located at the center, the deviations are of order Hb^2. For an observer outside the inhomogeneity, the deviations of crossing time and redshift are of order Hb^3. The deviations of beam area and luminosity distance are of order Hb^2. However, when averaged over all possible locations of the observer outside the inhomogeneity, they also become of order Hb^3. We discuss the implications for the possibility of attributing the observed cosmological acceleration to the emergence of large-scale structure.