A search for nontoroidal topological lensing in the Sloan Digital Sky Survey quasar catalog

Flat space models with multiply connected topology, which have compact dimensions, are tested against the distribution of high-redshift ($z \geq 4$) quasars of the Sloan Digital Sky Survey (SDSS). When the compact dimensions are smaller in size than the observed universe, topological lensing occurs, in which multiple images of single objects (ghost images) are observed. We improve on the recently introduced method to identify ghost images by means of four-point statistics. Our method is valid for any of the 17 multiply connected flat models, including nontoroial ones that are compactified by screw motions or glide reflection. Applying the method to the data revealed one possible case of topological lensing caused by sixth-turn screw motion, however, it is consistent with the simply connected model by this test alone. Moreover, simulations suggest that we cannot exclude the other space models despite the absence of their signatures. This uncertainty mainly originates from the patchy coverage of SDSS in the South Galactic cap, and the situation will be improved by future wide-field spectroscopic surveys.