Intrinsic torsion, extrinsic torsion, and the Hubble parameter

We study the intrinsic and extrinsic torsions (defined by analogy with the intrinsic and extrinsic curvatures) of the spatial sections of torsional spacetimes. We consider two possibilities. First, that the intrinsic torsion might prove to be directly observable. Second, that it is not observable, having been ‘inflated away’ in the early Universe. We argue that, even in this second case, the extrinsic torsion may grow during the inflationary era and be non-negligible at reheating and thereafter. Even if the spatial intrinsic curvature and torsion are too small to be detected directly, then, the extrinsic torsion might not be. We point out that, if its presence is not recognised, the extrinsic torsion could lead to anomalies in the theoretical estimate of the Hubble parameter—a result with obvious potential applications. We stress that extrinsic torsion is by far the most natural way to produce such anomalies, simply because it mixes naturally with the Hubble parameter; that is, the second fundamental form of a spacelike section depends on a sum of two terms, one determined by the Hubble parameter, the other by the extrinsic torsion.