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.