Effective action of a scalar field in a curved spacetime with a small inhomogeneity

The effective action of a quantum field in an inhomogeneous spacetime is studied. We extend the Hartle-Hu method, which was developed to investigate the model in a homogeneous spacetime with a small anisotropy, to evaluate the one-loop effective action for a scalar field in a curved spacetime with a small inhomogeneity. Through a rather lengthy calculation we obtain the final expression which is then applied to numerically investigate the quantum-field effect on the dissipation of the space inhomogeneity in the early Universe.