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 through 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 H¯=Hr0 of the radius r0 of the inhomogeneity to the horizon distance 1/H. For an observer located at the center, the deviations are of order H¯2. For an observer outside the inhomogeneity, the deviations of crossing time and redshift are of order H¯3. The deviations of beam area and luminosity distance are of order H¯2. However, when averaged over all possible locations of the observer outside the inhomogeneity, they also become of order H¯3. We discuss the implications for the possibility of attributing the observed cosmological acceleration to the emergence of large-scale structure.