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.

Figure 1

A depiction of the motion of a photon (red curve) from a point $P$ on the surface of last scattering $\Sigma$ (black ellipse) to an observer $O$. Without loss in generality, the observer’s position can be taken as the origin of the coordinate system. The anisotropic curvature causes $\Sigma$ to deviate from sphericity and warps the photon trajectories. ${\theta }_0$ is the angle between the photon’s trajectory and the observer $O$’s $z$ axis. ${\theta }_P$ is the angle between the photon’s trajectory and the $z$ axis at $P$.

Figure 2

The effect of the anisotropic curvature on a measurement of the angular size of standard rulers. The black ellipse is the surface of recombination and the red lines are photon paths from standard rulers on this surface to the observer at $O$. The standard rulers are depicted by the thick straight lines. The angular size varies depending upon the location and orientation of the ruler.