Oscillating gravitational potential due to ultralight axion: linear
  theory
  We derive oscillating gravitational potential caused by an ultralight axion
as the cosmic dark matter, assuming the Compton wavelength smaller than the
horizon scale. A new oscillatory term is found which dominates the previously
known one on scales larger than the quantum Jeans-scale. In the context of
linear perturbations in cosmology this new term might be more relevant in
future observation. Both the quantum stress in density perturbation and the
quantum oscillation of the gravitational potential are derived in the
zero-shear gauge. We show that the axion fluid in relativistic analysis is
quite different from the relativistic and Newtonian zero-pressure fluids,
corresponding to Newtonian fluid only in the density perturbation. The
consistency demands the relativistic analysis valid on scales larger than the
Compton wavelength.
