  Growing amount of data show evidence for statistical and apparent physical
association between low-redshift galaxies and high-redshift quasi-stellar
objects, suggesting noncosmological origin of their redshift and failure of
classical quasar explanation. Here we find an analytical solution of Einstein
equations describing bubbles made from axions with periodic interaction
potential. Such particles are currently considered as one of the leading dark
matter candidate. The bubble interior has equal gravitational potential and,
hence, photons emitted from the interior possess identical gravitational
redshift. The redshift depends on the bubble mass and can have any value
between zero and infinity. Quantum pressure supports the bubble against
collapse and yields states stable on the scale of the Universe age. Our results
explain the observed quantization of quasar redshift and suggest that
intrinsically faint point-like quasars associated with nearby galaxies (a few %
of known objects) are axionic bubbles with masses 10^8-10^9 M_{Sun} and radii
10^3-10^4 R_{Sun}. They are born in active galaxies and ejected into
surrounding space. Properties of such quasars unambiguously indicate presence
of axion dark matter in the Universe and yield the axion mass m=0.4-3 meV,
which fits in the open axion mass window constrained by astrophysical and
cosmological arguments. We also found that tachyons, another dark matter
candidate, can form objects with galactic size, negligible mass and any value
of the gravitational redshift. Such finding combined with quasar observations
suggests that bright quasars 3C 48, 3C 273 and 3C 279 are nuclei of forming
nearby small galaxies embedded into tachyonic clots and possess pure
gravitational redshift.
