3 + 1 approach to the long-wavelength iteration scheme

Abstract

Large-scale inhomogeneities and anisotropies are modelled using the long-wavelength iteration scheme. In this scheme solutions are obtained as expansions in spatial gradients, which are taken to be small. It is shown that the choice of foliation for spacetime can make the iteration scheme more effective in two respects: (i) the shift vector can be chosen so as to dilute the effect of anisotropy on the late-time value of the extrinsic curvature of the spacelike hypersurfaces of the foliation; and (ii) pure gauge solutions present in a similar calculation using the synchronous gauge vanish when the spacelike hypersurfaces have extrinsic curvature with constant trace. We furthermore verify the main conclusion of the synchronous gauge calculation which is that large-scale inhomogeneity decays if the matter - considered to be that of a perfect fluid with a barotropic equation of state - violates the strong-energy condition. Finally, we obtain the solution for the lapse function and discuss its late-time behaviour. It is found that the lapse function is well behaved when the matter violates the strong-energy condition.