From what I understand the expansion of the universe has no "center". If we're flying through space away from the "center of the big bang", there's basically no way to tell. Every two given points in space gets farther away from each other, and we can pick any point as center if we like.
I also understand that the speed of light is not relative to the speed of the source emitting the light. If I go on a train in $c/2$, turn on a flash light pointing forward, the light emitted from the flashlight will still travel at the speed of $c$.
Now here's my question: Why can't we set up a sphere with photodetectors with synchronized clocks on the inner walls, turn on a light in the center, record the exact time at which each photodetector detects the light, and compare the times to figure out if the sphere was traveling in some certain direction?
I mean if we turn on a lightbulb and light travels with the speed of $c$ in all directions at the same time, my intuition tells me that we should be able to figure out some form of "reference stand still".
(Tagging this with general relativity because I suspect that it's impossible to set up the experiment the way I like due to relativity.)