There's the usual stuff about galaxy rotation curves, there's the http://en.wikipedia.org/wiki/Bullet_Cluster which people consider to be more of a smoking gun, but then there's also evidence from http://en.wikipedia.org/wiki/Baryon_acoustic_oscillations visible in the CMB. That last one's slightly over my head, but apparently cosmologists find it extremely convincing, especially when combined with the other evidence. The wiki page talks about dark energy, but the subject has significance for dark matter too. Let me see if I can find the relevant plot...
At the end of the day it's just another weakly interacting particle. Last time we discovered one of those was the neutrino in the 50s, it's not an outlandish thing to think about.
EDIT: here's the BAO stuff: http://ned.ipac.caltech.edu/level5/Sept09/Einasto/Einasto6.h...
Of course, quite a bit of it might turn out to be exotic new types of particles, but that's a question for theory and experiment.
The precedent here is quite strong, incidentally, on a few counts.
We have high confidence in our understanding of gravity on the macro scale, and that understanding has survived some crises. For example, the orbit of the planet Uranus, as originally calculated, was "wrong" -- it did not appear to be conforming to the understood behavior of gravity. One theory which was developed to explain this, and which could at the time have been criticized in the style of your comment, was that there must be another planet further out, whose gravity was perturbing Uranus' orbit. Calculating backwards from Uranus' behavior, this theory predicted the location in the sky of the additional planet... which was then promptly observed via telescope (this was the discovery of Neptune).
Similarly, we have high confidence that weakly-interacting particles -- undiscovered variants of which are a popular candidate for examples of dark matter -- exist. The neutrino is the classic example: a particle which was posited by theory as "this is the only thing that makes sense given what we see", but which would necessarily be incredibly difficult to detect on account of barely if ever interacting with anything (and thus, again, would be open to your criticism). Of course, the neutrino was eventually detected and its existence confirmed.
So there is nothing wrong or arbitrary or unusual, in terms of the history of physics, in positing something like dark matter.
I've often wondered about this(and don't know any physicists so would really love some insight on this), why do we assume that there's additional matter rather than question whether our models are correct at that scale?
There are also physicists that question the models. Some scientists follow a path of inquiry based on assumptions that Einstein's theory is mostly correct -- hence a postulation of "dark matter" which leads to searching for the existence of it(1). But there are other physicists that assume there's something missing in the accepted equations that would explain the galaxy rotation speeds without any need for "dark matter".
Each competing theory tries to accumulate more and more evidence for their case until one "wins" (scientific consensus)(2). Right now, the "dark matter" line of inquiry has more scientists, more papers, and more press coverage.
(1)direct instead of indirect observation
(2)Some might say Dark Matter theory has already "won" in the marketplace of ideas
As for dismissing God, I don't see why, we just need to come up with some signature that would differ between a designed universe and an accidental one and apply for observation time on a suitable instrument.
1. Huh, those are some weird observations. What could explain it? 2. Maybe it's X? That could explain it. 3. So could Y? 4. Here's an experiment we could do to rule out Y! 5. Hmm, actually the result of that experiment was consistent with Y, but constrains some of its free parameters. And Y could still be false.
And so on.
Science doesn't require that you have something in a lab, just that you make hypotheses which are testable by some sort of observation. Otherwise we'd have to throw out nearly all of cosmology.
There are competing hypotheses still with respect to dark matter (e.g. that the force of gravity works differently than we thought over long distances), precisely because we haven't yet figured out how to do the experiments to rule out all but one hypothesis. But that's why it's called a hypothesis. We haven't yet figured out the answer! That's the beauty of science.
http://www.lsst.org/lsst/science/scientist_dark_matter
http://scienceblogs.com/startswithabang/2011/04/20/how-gravi...
http://www.ifa.hawaii.edu/~ger/ASTRO-110_sp08/Lecture28_Dark...
Closest references I could find: http://phys.org/news/2013-02-dark-galaxies-ways-collision-he...
"In the Bullet Cluster, a collision between two galaxy clusters appears to have caused a separation of dark matter and baryonic matter." - https://en.wikipedia.org/wiki/Dark_Matter
Also, despite the name "dark energy" has no relation to dark matter and the evidence for either does not overlap. Moreover, the inability of one theory to explain everything else in the universe does not invalidate that theory. The theory of biological evolution fails to explain the existence of neutron stars, yet it is still a valid theory.