Observing the Dimensionality of Our Parent Vacuum
  It seems generic to have vacua with lower dimensionality than ours. We
consider the possibility that the observable universe originated in a
transition from one of these vacua. Such a universe has anisotropic spatial
curvature. This may be directly observable through its late-time effects on the
CMB if the last period of slow-roll inflation was not too long. These affect
the entire sky, leading to correlations which persist up to the highest CMB
multipoles, thus allowing a conclusive detection above cosmic variance.
Further, this anisotropic curvature causes different dimensions to expand at
different rates. This leads to other potentially observable signals including a
quadrupolar anisotropy in the CMB which limits the size of the curvature.
Conversely, if isotropic curvature is observed it may be evidence that our
parent vacuum was at least 3+1 dimensional. Such signals could reveal our
history of decompactification, providing evidence for the existence of vastly
different vacua.
