Volume expansion rate of the Lorentzian manifold based on integral Ricci curvature over a timelike geodesic

Recent astronomical observations show that the universe is not only expanding but also undergoing accelerated expansion [A.G. Riess, et al., The farthest known supernova, Astrophys. J. 560 (2001) 49–71; P.K. Townsend, M.N.R. Wohlfarth, Accelerating cosmologies from compactification, Phys. Rev. Lett. 91 (2003) 061302]. Then the timelike convergence condition does not hold every time, i.e. the Ricci curvature Ric(v,v) cannot be nonnegative for every timelike vector vv. We obtain the volume expansion rate of the universe based on the integral norm of negative part of the Ricci curvature along a timelike geodesic.