Sample incompleteness in the study of the quasar-galaxy associations as a result of gravitational lensing

For decades, the angular associations between faint galaxies and high redshift quasars have remained to be a well-known puzzle, which is often cited as an evidence for the noncosmological origin of quasar redshifts. This happens because the attempt at interpreting the phenomenon as being due to gravitational lensing fails in producing the reported high amplitude of the quasar-galaxy associations unless some unconventional hypotheses are made such as an extremely large velocity dispersion for galaxies, a serious contamination of the overall quasar number counts or a strong matter clustering on galaxy scale. While there are good reasons to believe that the magnification bias should be a natural cause for the quasar-galaxy associations, we re-address the question by taking into account an important factor that has been overlooked in previous studies: The expected amplitude of the association in the framework of gravitational lensing depends sensitively on the quasar limiting magnitude, whereas none of the quasar samples in the measurements is actually complete. Taking this correction into account our predictions of the galaxy enhancement factors based on a simple lensing model are essentially consistent with observations, thus providing a helpful clue to eventually resolving the longstanding puzzle in astrophysics.