Note c^2(area of future horizon)^-1/2 ~ 10^17/10^26 ~ 10^-9 meters/sec^2

Hence their conclusion below is puzzling in my opinion.

REVIEWS OF MODERN PHYSICS, VOLUME 82, JANUARY–MARCH 2010

Influence of global cosmological expansion on local dynamics and

kinematics

Matteo Carrera*

Institute of Physics, University of Freiburg, Hermann-Herder-Straße 3, D-79104 Freiburg,

Germany

Domenico Giulini†

Institute for Theoretical Physics, University of Hanover, Appelstraße 2, D-30167 Hannover,

Germany

Published 28 January 2010

"Attempts to estimate the influence of global cosmological expansion on local systems are reviewed. Here “local” is taken to mean that the sizes of the considered systems are much smaller than cosmologically relevant scales. For example, such influences can affect orbital motions as well as configurations of compact objects, like black holes. Also discussed are how measurements based on the exchange of electromagnetic signals of distances, velocities, etc. of moving objects are influenced. As an application, orders of magnitude of such effects are compared with the scale set by the apparently anomalous acceleration of the Pioneer 10 and 11 spacecrafts, which is 10^−9 m/s2. There is no reason to believe that the latter is of cosmological origin. However, the general problem of gaining a qualitative and quantitative understanding of how the cosmological dynamics influences local systems remains challenging, with only partial clues being so far provided by exact solutions to the field equations of general relativity."

DOI: 10.1103/RevModPhys.82.169

This paper is not easy reading, but goes into the nitty gritty details on the complex kinematics of how to relate the abstract formalism of GR to real life NASA radar, Doppler measurements.

**"VII. SUMMARY AND OUTLOOK**

**We think it is fair to say that there are no theoretical**

**hints that point towards a**

*dynamical*influence of cosmological**expansion comparable in size to, say, that of the**

**anomalous acceleration of the Pioneer spacecrafts.**

**There seems to be no controversy over this point,**

**though for completeness it should be mentioned that**

**there exist speculations**

**Palle, 2005**

**according to which**

**it might become relevant for future missions. But such**

**speculations are often based on models which are not**

**easily related to the intended physical situation, like that**

**of**

**Gautreau 1984**

**. Rather, as the**

*a¨*/*a*-improved Newtonian**analysis in Sec.**

**III**

**together with its justification**

**given in the subsequent sections shows, there is no genuine**

**relativistic effect coming from cosmological expansion**

**at the levels of precision envisaged here.**

**On the other hand, as regards**

*kinematical*effects, the**situation is less unanimous. It is important to unambiguously**

**understand what is meant by “mapping out a trajectory,”**

**i.e., how to assign “times” and “distances.”**

**Eventually we compare a functional relation between**

**distance and time with observed data. That relation is**

**obtained by solving some equations of motion and it has**

**to be carefully checked whether the methods by which**

**the tracking data are obtained match the interpretation**

**of the coordinates in which the analytical problem is**

**solved. In our way of speaking, dynamical effects really**

**influence the worldline of the object in question whereas**

**kinematical effects change the way in which one and the**

**same worldline is mapped out from another worldline**

**representing the observer. Here we have derived exact**

**results concerning the influence of cosmic expansion on**

**this mapping procedure, which allow one to reliably estimate**

**upper bounds on their magnitude. They turn out**

**to be too small to be of any relevance in current satellite**

**trackings, which is in accord with naive expectation but**

**in contrast to some statements found in the literature."**