Expansion of effective action in curved spacetime with large anisotropy

Abstract

A simple perturbative method is used to evaluate the renormalized effective action of a free scalar field in the Bianchi type I spacetime with large anisotropy. With the help of the zeta function regularization method the action is expanded to the second order in the Schwinger perturbative formula. We apply the method of iterative reduction to reduce the generalized Einstein equation to a second-order equation to obtain the self-consistent solution of the semiclassical gravity theory. The reduced equation shows that the space anisotropy, which will be smoothed out during the evolution of Universe, may play an important role in the dynamics of early Universe. We thus quantize the corresponding minisuperspace model to investigate the behavior of space anisotropy in the early Universe. From the wavefunction of the Wheeler–De Witt equation we see that the state probability of the Bianchi type I spacetime with large anisotropy is less than that with a small anisotropy.