  It is believed that soon after the Planck time, Einstein's general relativity
theory should be corrected to an effective quadratic theory. In this work we
present the 3+1 decomposition for the zero vorticity case for arbitrary
spatially homogenous spaces. We specialize for the particular Bianchi $I$
diagonal case. The 3- curvature can be understood as a generalized potential,
and the Bianchi $I$ case is a limiting case where this potential is negligible
to the dynamics. The spirit should be analogous, in some sense to the BKL
solution. In this sense, a better understanding of the Bianchi $I$ case could
shed some light into the general Bianchi case.
