My (possibly poor) understanding of the argument for Dark Matter's existence is that stars in a large galaxy move more slowly than "they should" (presumably due to either some simplified model of the massively multi-body gravitational system of a large galaxy, to many long hours of numerical computations on supercomputers, or both) in their orbit about the "center" of a galaxy.
I have a hard time with accepting this as a proof of existence of something we cannot otherwise detect, given the known issues with the unbelievably complex math involved.
Do either the models, or the massive numerical computations, used take into account the fact that - assuming gravitational effects are almost certainly limited in propagation to light-speed - the effective location of mass on a galactic scale is where it was anything up to a very long time ago, as opposed to where it might be by the time it has an effect on other masses in the galaxy?
Intuitively, given that we can probably assume that galaxies are themselves moving at relativisticly significant speeds, it seems likely that the sheer size of large galaxies would tend to diminish the effect of the very remote mass locations on the "orbits" of individual stars - especially with respect the effects of closer masses, that are still many years removed from where they seem to be based on their gravitational impact on each other.
And what is the effect of gravity itself between masses that are moving at relativistic speeds with respect to each other?
Given the massive complexity of the problem, I would just like to know how we can have so much confidence in our modeling of gravitational influences on that scale that we have a fairly large percentage of our scientists today who are utterly convinced that Dark Matter exists on the basis of the degree to which these predictive results differ from our observations of reality.