Oscillating gravitational potential in cosmological scalar field perturbation.

We derive oscillating gravitational potential caused by a massive scalar field in cosmological perturbation theory, 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. The quantum stress term in density perturbation is known to be available only in the comoving gauge. In contrast, the quantum oscillation of the gravitational potential is derived in the zero-shear gauge. We derive these two quantum effects using the mixed gauge-invariant notation. We show the full consistency of this mixed formulation in handling the two quantum effects; the consistency demands the relativistic analysis valid on scales larger than the Compton wavelength.