Asymptotical flatness for galaxies - Why? During the construction of solutions for General Relativity (when leaving out Lambda), asymptotical flatness is always assumed. Why?
I fully understand that it fits the solar system, as it results in Newtons law here.
However, why is it assumed to be the boundary condition for galaxies as well?
The solar system is moving in free fall - condition around the centre of the galaxy and therefore, asymptotical flatness would be the boundary condition of the solar system regardless of the gravitational law and the boundary condition in galaxies. Galaxies aren't moving in free fall around bigger objects, are they?
Spacetime is curved at the centres of gravity, so why should it be flat in the infinity?
There is an analogy between curvature of spacetime and electromagnetic fields. And electromagnetic fields are not flat in the infinity, instead they are vanishing in the infinity.
Is there an explanation why galaxies have to be asymptotically flat?
Is there an experiment which confirmed asymptotical flatness for galaxies?
 A: Asymptotical flatness is not assumed in GR in the sense of assumed to hold in fact. Rather, various interesting and useful results hold if the asymptotic regime is flat, and it is good to find those facts (e.g. conservation laws and things like that). Meanwhile what the condition really is at long distances is an empirical issue. It is studied in cosmology, and as it happens the evidence suggests spatial flatness or near-flatness at large scales. But this was not assumed, it was tested by observation of background radiation, galaxy surveys, supernovae and things like that (and it is thought to be implied by inflationary scenarios of early-universe physics).
A: Asymptotic flatness is not an assumption of GR, but an assumption of the solutions that are constructed. In a lot of circumstances (as you have pointed out, like the solar system) it make sense to assume that the universe above a certain scale is flat -- after all in a flat universe geodesics are just straight lines. And since people are often interested in localised matter distributions, where we know from Newton that far away they produce a negligible gravitational force, this seems reasonable to assume.
However, there are solutions to the Einstein Equations that are not asymptotically flat, for example a de Sitter or an anti de Sitter universe. In this type of universe, the Ricci scalar is a (non-zero) constant over the entire space time. Also the $\Lambda$CDM universe (which we think describes our universe, although there are some problems like the Hubble tension) with a curvature term that is non-zero is another example for a solution where we do not impose asymptotic flatness.
There even seems to be evidence that we are living in a non-flat universe from observations of the cosmic microwave background.
