Hidden invariance in Gurzadyan-Xue cosmological models

The dark energy formula derived by Gurzadyan and Xue which leads to a value fitting the SN data, provides a scaling relation between the physical constants and cosmological parameters and defines a set of cosmological models. In previous works we have considered several of those models and derived the cosmological equations for each case. In this letter, we present the phase portrait analysis of those models. Surprisingly we found, first, that the separatrix in the phase space which determines the character of solutions depends solely on the value of the current matter density. Namely, at Ωm > 2/3 the equations describe Friedmannian Universe with the classical singularity at the beginning. While at Ωm < 2/3 all solutions for all models start with zero density and non vanishing scale factor. Secondly, more remarkable, the value Ωsep = 2/3 defining the separatrix is the same for all models, which reveales an underlying invariance hidden in the models, possibly, due to the basic nature of the GX-scaling. Among the variety of models for dark energy proposed in the literature, a remarkable fit to the SN data is given by Gurzadyan-Xue formula [1] for the vacuum fluctuations energy density ρGX = π 8 c G 1 a , (1) where c is the speed of light, G is the gravitational constant and a is the scale factor of the Universe. Taking Hubble length for the value of the scale factor today, we find the value of the density parameter for dark energy ΩΛ ≃ 3.29. (2) The formula (1) opens interesting possibilities related to the scaling between physical constants and the scale of the Universe. In the previous papers [2, 3] we have derived cosmological equations assuming variation of the physical constants such as the speed of light and the gravitational constant. Certain observational aspects of the GX-formula (1) are discussed in [5]. In this Letter we present phase portrait analysis (c.f. [4]) of these cosmological equations. This method . Surprisingly, we found that although equations look very differently, the character of solutions for two pairs of models is similar. At the same time, even more striking feature of the models came out: the position of the separatrix in the phase space of dynamical variables is determined by the unique value of the density parameter Ωm, moreover, its value is the same for all models! This invariance should be deeply connected to the scaling, containing world constants and the scale factor of the Universe. Following [3], we consider 4 models and present equations for them: ∗Corresponding author. Email: [email protected] A factor 1/2 was missing in our previous papers, so this value of ΩΛ is even closer to the observational one. 1 Model I. If neither the speed of light nor the gravitational constant vary with time, then the cosmological constant scales as Λ ∝ a as can be seen from (1). Cosmological equations for this model are (