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.