5 v 1 2 2 Ju l 1 99 5 Yukawa Institute Kyoto YITP / K 1103 KUNS - 1328 astro - ph / 9507085 July 1995 Volume Expansion Rate and The Age of The Universe

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 Hv0t0 ≤ 1, where Hv ≡ −K/3 agrees with the Hubble parameter in the case of a homogeneous and isotropic universe, and t0 is the age of the Universe. If Hv0t0 > 1 is confirmed observationally, at least one of the four assumptions is incorrect. Key word: Hubble parameter Recent observations of Cepheid variables in NGC4571 (Pierce et al. 1994) and M100 (Freedman et al. 1994) suggest that the Hubble parameter (H0) is ∼ 80km/s/Mpc (Fukugita et al. 1993; Jacoby et al. 1992; van den Bergh 1992). Such a high value of H0 may contradict the age estimate of our universe using, for example, globular clusters (Demanqul et al. 1991; Renzini 1991) so that H0t0 > 1. However one may consider an inhomogeneous universe such thatH0 in our neighborhood is high (Turner et al. 1992) but globalH0 is low enough to agree with t0. In the previous paper (Nakao et al. 1995) we considered a simple inhomogeneous model in which we are in a void expressed by an open Friedmann universe and showed that H0t0 ≤ 1 even in such an inhomogeneous model as long as the peculiar velocity correction is perfect. In this paper, we extend our argument to a more general inhomogeneous universe. We shall put four assumptions on the evolution of the universe. 1) First we assume that Einstein’s theory of gravity is the correct theory to describe the evolution of the universe after the Planck time. Then the basic equations in (3+1)-formalism become: the constraint equations are given by R +K = Kij K ij + 16πρH , (1) K i|j −K|i = 8πJi, (2) the evolution equations ∂K i ∂t = α(R i +KK j i )− α|i − 8π ( S i + 1 2 δ i (ρH − S l) ) − K i β |m +K mβm |i +K j i|mβ , (3) ∂γij ∂t = −2αKij + βi|j + βj|i, (4) where α and βi are the lapse function and the shift vector, respectively, and the vertical bar means the covariant derivative with respect to the 3-metric γij . ρH , Ji and Sij are defined by