Can we measure acceleration of galaxies and stars?

[elcaro]

TL;DR Summary

We can (with increasing precission) measure distances of far away object in space, and velocities. But is there a method of measuring acceleration (so, in fact changes of velocity, or the time derivative of velocities) of far away galaxies.?

As for example we see a large void, the Great Repeller, which in fact is an underdense region, and with respect to this region, matter seems to be repelled by this region. The explenation for that is that matter outside that regions pulls on the matter inside it. But if that is really the explenation (attracation from surrounding matter, instead of repulsion from the center of that region) we would expect that closer to the center of that region, the acceleration falls down, whereas if there would be repulsion from this region, we would expect the accleration to increase when going further to the center. But can cosmologists/astrophysicist actually measure accelerations?

 

[kimbyd]

The short answer is not for most astrophysical systems, not directly.

For the most part we can only accurately measure the instantaneous velocity along the line of sight. The rest comes from modeling. You assume some sort of average velocity distribution model for lots of objects, and then measure the line-of-sight velocities to fit the parameters of that model. If you get the model wrong, this can lead to biases in the imputed accelerations, which is one complication in interpreting data about galaxies and galaxy clusters.

A typical model that is used is a virial model: this one assumes that the object being observed is in a "steady state" configuration, and that its velocities can be described using the virial theorem. The virial theorem gives a relationship between the average kinetic and potential energy of a cloud of particles, and so by measuring their velocities, we get a measure of their kinetic energy, which tells us about their potential energy, which tells us about the gravitational field if the system is static.

If we have an independent way of measuring the mass of the system, such as gravitational lensing, then measuring higher velocities than predicted by the virial theorem may suggest the system is expanding over time. Lower velocities would mean the system is collapsing.