Niles said:
Yes. Exactly right. It is how it is defined.
t_0 is the symbol used to stand for the present moment.
We can know the scalefactor at other times only relative to the present. We don't have it in absolute units like inches or kilometers. We can only say that at such and such epoch it was 1/1000 of what it is today, or 0.3 of what it is today.
So the simplest way to define it is to define a(present) = 1
and that makes it possible to specify it for other times.
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For example if we see a quasar with redshift z = 6
then we know the univese has expanded by a factor of 7 while the light has been traveling to us
so we can immediately say that the scalefactor was equal to 1/8 at the moment when the light left the quasar and began its journey
a(then) = 1/8.
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in the case of scalefactor, saying a(present) = 1 is just a matter of simple convenient bookkeeping
it is the conventional definition
but in the case of Omega there is more to it. We can OBSERVE that the universe is approximately spatially flat. There is discussion and uncertainty about what Omega is exactly. Some say it is exactly 1 and some just say it is like 1.01 plus or minus some percentage uncertainty---they give an ERROR BAR for it. But either way everybody agrees that it is NEARLY one.
So a teacher will be tempted to just tell the students to take it equal to one, and not get into the messy business of different studies and data and uncertainty and errorbars.
But it isn't one by definition. Omega is a RATIO of the actual largescale density to the critical density which the universe would have in order to be perfectly flat (largescale average). Since universe is approx flat, the two densities are approx equal, therefore their ratio (Omega) is approx equal to one.
It isn't by definition, it is because of observations.
You should ask Wallace or one of the other professional astronomers here to explain to you how they actually figured out that the universe spatially is nearly flat. heh heh. it is rather neat.
one way is by galaxy redshift surveys
after adjusting for the expansion history you can plot how many galaxies are now (at present) in a ball of radius R
assuming uniform distrib'n the number of galaxies you count tells you the VOLUME of the ball of radius R.
If that volume increases as the cube of R, as R increases, then we have spatial flatness.
if it doesn't increase as the cube. if it increases slower than the cube, then we have some largescale positive curvature.
it is high school geometry reasoning at work, but still it is kind of elegant. they also, as a double check, use the CMB map, the bumpiness of the temperature distribution. I like it. ask your teacher or start a thread here, if you are curious.
IOW how do we know Omega is approximately 1, or that we have spatial near-flatness.