  The zeta-function regularization method is used to evaluate the renormalized
effective action for massless conformally coupling scalar field propagating in
a closed Friedman spacetime perturbed by a small rotation. To the second order
of the rotational parameter in the model spacetime the analytic form of the
effective action is obtained with the help of the Schwinger perturbation
formula. After investigating the time evolution of the rotational parameter we
find that the quantum field effect can produce an effect which damps the
cosmological rotational in the early universe.
