Gravity pulls galaxies together into groups and into clusters of galaxies. We can look at the mass of these by looking at the motions of individual galaxies in the clusters. The more mass in the cluster, the bigger the velocities of the galaxies. Here is a nice java animation of motions of galaxies in clusters. Also, clusters of galaxies are a big gravitational potential well, and trap gas inside. The gas falls into the big gravitational potential of the cluster and so heats up - to X-ray temperatures! The temperature tells us about how much gravitational energy it has, and so how much mass. More on X-ray hot gas in clusters.
All these assume that the masses are bound and in orbits - a fairly good assumption as if galaxies were not gravitationally bound then the material would dissipate. But it'd be nice to have a way to measure mass without having to make this assumption. Mass = curvature of space, and curving space will bend light. This is gravitational lensing. The amount of lensing tells us the mass. A cluster is a big mass concentration, and its gravity will distort the images of background galaxies - see this nice animation . Cluster masses derived in this way agree with the masses derived above from galaxy velocities and the X-ray hot gas and rotation curves of spiral galaxies in saying that there is much more mass than we can see in the stars, - something like 10-100x more! and gives Omega=0.3
another way to look at the value of omega is to look at the way mass is distributed now. Suppose we find all the galaxies down to a certain brightness and look at their distribution on the sky, and then get spectra of all of them to get their redshift distance. But the sky is big, so its easiest to pick a slice of sky and study that. And galaxies are distributed in a very lumpy fashion as in the CFA survey which goes out to about 1 billion light years from our galaxy. But it has been done for the whole sky as well so we have a 3D picture of our local Universe. It is centred on the Milky Way, and shows the density of galaxies within a sphere of radius about 1 billion light years. Again its can see its rather lumpy. But we expect this - galaxies clump together because of gravity into structures called clusters of galaxies, and even clusters of galaxies clump together into superclusters. The big lumps on the 3D picture are superclusters containing about 1000 bright galaxies each. And there are now slice surveys that are going out quite a bit further than this, as in this one called the 2dF survey - The numbers of galaxies for a given apparent brightness falls off as we go further out due to the inverse square law!
So our current view has galaxies clustered together on filaments with big voids in between where there are fewer galaxies. Yet we said that the very early universe was a very smooth place - the cosmic microwave background is amazingly uniform in temperature. So how do we start out with something so smooth and get to something so lumpy ? Well, gravity is inherently very very non--linear. If we had regions in the very early universe that were ever so slightly more dense than the rest then these would have ever so slightly more gravity, so would attract the surrounding material and grow more dense, so they'd have even more gravity and attract even more material and get even more dense...... But if these regions were ever so slightly more dense then their cosmic microwave background radiation would have to come out through more gravity so it would be redshifted - slightly cooler. These temperature fluctuations in the cosmic microwave background measure on average 1 part in 100,000!! thats pretty small! But its enough! starting from these tiny fluctations, and with Omega=0.3 in dark matter we can make numerical models of the growth of structure in the Universe from these tiny beginnings to the present day. And these numerical simulations with Omega=0.3 come out giving the same sort of lumpy pattern as is seen in the current Universe.
Neat! so can we use this to figure out what Omega is ? The sort of clustering we get looks slightly different if we have Omega=1 in dark matter than if we have Omega=0.3. And Omega=0.3 gives something rather closer to what we see. Not exactly proof, but consistent with the measure of Omega from the local Universe.
And there is a considerable amount of evolution - as we look outwards then we are looking back in time becasue of the finite (though very fast) speed of light. So big telescopes look into the past. Given that the early universe was a rather different place - very hot and dense, then we might expect to see considerable evolution as we look back in time (ie further away in space). And indeed, we do see changes e.g. the density of active galaxies and quasars increases rapidly out to a redshift of around 2 - they were MUCH more abundant in the past than now. And galaxies were somewhat different when they were young.
This also gives an answer to a very old question - why is the sky dark at night ? This is called Olbers paradox. Firstly the expansion of the universe means that the light from distant objects is redshifted. But secondly and more importantly, because the universe is not infinitely old, light from very distant objects hasn't had time to reach us yet! We cannot see back beyond 10-20 billion light years because thats before the time at which the Universe was born.