Volume Expansion Rate and The Age of The Universe

Abstract

Under four assumptions such that 1) Einstein's theory of gravity is correct, 2) existence of foliation by geodesic slicing, 3) the trace of the extrinsic curvature, $K$, is negative at the present time in observed region, that is, the observed universe is now expanding, 4) the strong energy condition is satisfied, we show that $H_{v0} t_{0} \leq 1$, where $H_v \equiv - K/3$ agrees with the Hubble parameter in the case of a homogeneous and isotropic universe, and $t_0$ is the age of the Universe. If $H_{v0} t_0 > 1$ is confirmed observationally, at least one of the four assumptions is incorrect.