Some of the most convincing evidence to this effect involves the
"geometrical" measurement of distance using proper motions. There are
some water masers (that is, sources of radio emission) at the center of
the galaxy NGC 4258 that have had their proper motions measured (that is,
shifts in position on the sky). These proper motions can be compared with
the line-of-sight velocities (as determined with Doppler shifts) to
directly determine the distance -- in this case, no less than 6.7
megaparsecs (22 million light years) away. (Herrnstein et al. 1999,
Nature, vol 400, p 539; see also http://xxx.lanl.gov/abs/astro-ph/9907013)
And thus, the universe is at least 22 million years old.
To refute this conclusion, one would have to either claim that A)
Doppler shifts are not a good measure of velocity, B) the galaxy's light
was created "in transit", or the speed of light is not constant, or C) the
wrong model has been assumed for the maser motions (in which they're
orbiting around the galactic center). Claims A and B are ad-hoc, and make
no sense in the light of other knowledge about the world. As for claim C,
the current model is strongly self-consistent, and some kind of extremely
special configuration would be needed to imply a distance of much less.
This distance is of course quite consistent (to within 10-20%) with all
sorts of other (indirect) ways of estimating the distance. All of
astronomy hangs together pretty well in this way. For example, some
galaxies can be easily deemed to be farther away than others because they
show the same morphology, colors, etc., but appear smaller and fainter,
as though seen at a distance. The same goes for stars, planetary nebulae,
globular clusters, supernovae, etc...
- Aaron Romanowsky