Exponential Growth of Distance between Nearby Rays due to Multiple Gravitational Scatterings

Abstract

We give an estimate of the relative error in the angular measurement of observations for high redshift objects induced by gravitational scatterings (lensing). Gunn (1967) concluded that the gravitational scatterings by galaxies induce the relative error of a few percent in the observations for objects at $z=1$. This estimate has been considered as a fundamental limitation of accuracy of the angular measurements in the observational cosmology. In multiple graviational scatterings, bending angle of single ray grows through the random work process. Gunn (1967) assumed that the difference of nearby rays also grows through the random walk process. However, distance between nearby photons grows exponentially because the two rays suffer coherent scatterings by the same scattering object. This exponential growth continues as long as the scattering is coherent. In the case of scattering by individual galaxies, the exponential growth continues until the angular distance reaches an arcminute or so. The relative error of the angular measurements under an arcminute due to the exponential growth is $\sim 30\%$ at $z=1$ and exceeds $100\%$ at $z=3$, in the case that the density parameter of galaxies is 0.2. The effects of clusters of galaxies or superclusters are more difficult to estimate accurately, but might be significant. In the case of supercluster the angular measurements up to a few degrees could be affected.