Analytical solutions for two inhomogeneous cosmological models with energy flow and dynamical curvature

Abstract

Recently we have introduced a nonrelativistic cosmological model (NRCM) exhibiting a dynamical spatial curvature. For this model the present day cosmic acceleration is not attributed to a negative pressure (dark energy), but it is driven by a nontrivial energy flow leading to a negative spatial curvature. In this paper, we generalize the NRCM in two different ways to the relativistic regime and present analytical solutions of the corresponding Einstein equations. These relativistic models are characterized by two inequivalent extensions of the Friedmann-Lema\^{\i}tre-Robertson-Walker metric with a time-dependent curvature function $K(t)$ and an expansion scalar $a(t)$. The fluid flow is supposed to be geodesic. The model V1 is shear free with isotropic pressure and therefore conformally flat. It shows some common properties with the spherically symmetric Stephani models, but it exhibits also some specific differences. In contrast to V1, the second model V2 shows a nontrivial shear and an anisotropic pressure. For both models, the inhomogeneous solutions of the corresponding Einstein equations will agree in leading order at small distances with the NRCM if $a(t)$ and $K(t)$ are each identical with those determined in the NRCM. This will be achieved by the demand of vanishing isotropic pressure and its first derivative with respect to ${r}^{2}$ at the coordinate origin $r=0$. Then the metric is completely fixed by three constants. The arising energy momentum tensor contains a nontrivial energy flow vector. Our models violate locally the weak energy condition. As this may be caused by some averaging, we speculate about to view each of our models as a local average of some other more fundamental model. Global volume averaging leads to explicit expressions for the effective scale factor and the expansion rate $H(z)$. Backreaction effects cancel each other for the model V2 but they are nonzero and proportional to the square of the magnitude of the energy flow for the model V1. The large-scale (relativistic) corrections to the NRCM results are small for the model V2 for a small-sized energy flow. We have reproduced a corresponding adjustment of the three free constants from [1] to cosmic chronometer data, leading to the prediction of an almost constant, negative value for the dimensionless curvature function $k(z)\ensuremath{\sim}\ensuremath{-}1$ for redshifts $z<2$.