HOW DOES INFLATION ISOTROPIZE THE UNIVERSE

The lack of a preferred frame in the de Sitter metric leads one to argue that the end of inflation should result in an inhomogeneous rather than a homogeneous universe. Statements to the contrary have relied on scalar perturbation theory around an isotropic background. We show that to analyze properly the problem requires the introduction of velocities, implying an inhomogeneous scalar field phi and minimally an anisotropic background. We find by both analytic and numerical calculations that inflation does not isotropize the Universe in the short-wavelength limit lambda-R << H-1. (R is a typical scale equal to the radius of the Universe.) However, if inflation persists lambda-R becomes larger than the "horizon" H-1 and the wave behavior freezes in. At this point all velocities go to zero pointwise at a rate tanh-beta approximately 1/R, where beta is the relativistic tilt angle, a measure of velocity. For periodic boundary conditions the periodicity forces the average tanh-beta to zero faster, approximately 1/R4. The results are purely classical, showing that the invocation of quantum fluctuations is not the relevant answer to the problem.