Research Note

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

^{1}
Tuorla Observatory, Department of Physics and Astronomy, University of Turku, 21500 Piikkiö, Finland e-mail: pekkatee@utu.fi

^{2}
Sternberg Astronomical Institute, Moscow University, Moscow 119899, Russia

Received:
3
March
2010

Accepted:
18
April
2010

*Aims. *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.

*Methods. *We calculate the present locus of test particles injected a fixed time ago (~the age of
the universe), in the standard *Λ* cosmology and for different values of the system parameters (the model includes a central point mass *M* and a local dark energy density *ρ*_{loc}) and
discuss the position of the zero-gravity distance *R*_{v} in the Hubble diagram.

*Results. *Our main conclusion are: 1) when the local DE density *ρ*_{loc} is equal to the global DE density *ρ*_{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 (*≈*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 *ρ*_{loc}/*ρ*_{v},
the local flow reaches the known global rate (the Hubble constant) at a distance *R*_{2} 1.5 × *R*_{v}. Hence, *R*_{v} will be between *R*_{2}/2 and *R*_{2}, giving
upper and lower limits to *ρ*_{loc}/*M*. For the Local Group, this supports the view that the local density is near the global one.

Key words: Local Group / dark energy / cosmological parameters

*© ESO, 2010*