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 FLWR 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 conformal 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 NCRM. This will be achieved by the demand of vanishing isotropic pressure and its first derivative w.r.t. 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 NCRM 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) ∼ −1 for redshifts z < 2.

Dynamical spatial curvature as a fit to Type Ia supernovae

“…where q i is the number of parameters and L i is the maximum likelihood of model i, where p i is the probability that model i minimizes the (estimated) information loss, and where the two models are labelled i = 1, 2, respectively. The AIC relative likelihood measure (25) can be viewed as a generalization of the likelihood ratio to non-nested models. The interpretation of the relative numerical estimates of the AIC measure for different models is context-dependent.…”

“…17 for further motivations for introducing the template metric, where it is discussed how constant-curvature metrics can be obtained via Ricci flow smoothing of Riemannian hypersurfaces. 24 Even though the template metric described in this section is not solution to Einstein's equations, local metrics of the same form have been studied as solutions to the Einstein equations (see the recent paper by Stichel 25 and references therein).…”

Does relativistic cosmology software handle emergent volume evolution?

“…Accurate modelling of non-linear 3-Ricci (spatial) curvature is a key challenge that all of these software packages need to meet if they are to be used to test the hypothesis that structure formation leads to recently emerging negative mean curvature that physically explains 'dark energy' (Räsänen 2006;Nambu & Tanimoto 2005;Kai et al 2007;Räsänen 2008;Larena et al 2009;Chiesa et al 2014;Wiegand & Buchert 2010;Buchert & Räsänen 2012;Wiltshire 2009;Duley et al 2013;Nazer & Wiltshire 2015;Roukema et al 2013;Barbosa et al 2016;Bolejko & Célérier 2010;Lavinto et al 2013;Roukema 2018;Sussman et al 2015;Chirinos Isidro et al 2017;Bolejko 2017b,a;Krasinski 1981Krasinski , 1982Krasinski , 1983Stichel 2016Stichel , 2018Coley 2010;Kašpar & Svítek 2014;Rácz et al 2017).…”

Does relativistic cosmology software handle emergent volume evolution?

“…Accurate modelling of non-linear 3-Ricci (spatial) curvature is a key challenge that all of these software packages need to meet if they are to be used to test the hypothesis that structure formation leads to recently emerging negative mean curvature that physically explains 'dark energy' (Räsänen 2006;Nambu & Tanimoto 2005;Kai et al 2007;Räsänen 2008;Larena et al 2009;Chiesa et al 2014;Wiegand & Buchert 2010;Buchert & Räsänen 2012;Wiltshire 2009;Duley et al 2013;Nazer & Wiltshire 2015;Roukema et al 2013;Barbosa et al 2016;Bolejko & Célérier 2010;Lavinto et al 2013;Roukema 2018;Sussman et al 2015;Chirinos Isidro et al 2017;Bolejko 2017b,a;Krasinski 1981Krasinski , 1982Krasinski , 1983Stichel 2016Stichel , 2018Coley 2010;Kašpar & Svítek 2014;Rácz et al 2017).…”