The Hubble diagram for a system within dark energy: the location of the zero-gravity radius and the global Hubble rate

Abstract

Here we continue to discuss the principle of the local measurement of dark energy using the normalized Hubble diagram describing the environment of a system of galaxies. We calculate the present locus of test particles injected a fixed time ago (\sim the age of the universe), in the standard \Lambda -cosmology and for different values of the system parameters (the model includes a central point mass M and a local dark energy density \rho_{loc}) and discuss the position of the zero-gravity distance R_v in the Hubble diagram. Our main conclusions are: 1) When the local DE density \rho_{loc} is equal to the global DE density \rho_v, the outflow reaches the global Hubble rate at the distance R_2 = (1+z_v)R_v, where z_v is the global zero-acceleration redshift (\approx 0.7 for the standard model). This is also the radius of the ideal Einstein-Straus vacuole. 2) For a wide range of the local-to-global dark energy ratio \rho_{loc}/\rho_v, the local flow reaches the known global rate (the Hubble constant) at a distance R_2 \ga 1.5 \times R_v. Hence, R_v will be between R_2/2 and R_2, giving upper and lower limits to \rho_{loc}/M. For the Local Group, this supports the view that the local density is near the global one.