Garth said:
The question with the scale factor and the 'size of the universe' is,
"How do you measure it?"
...
Garth, you probably have read up on this and have some ideas about
how cosmologists establish the various parameters. It is not a one-post thing. IMO it would take one of us several attempts to discuss it.
I've found Lineweaver's "inflation and the CMB" article helpful
and there is a great picture there of how the scale factor has evolved over time. Figure 14
In his notation, which is fairly common, he writes R(t) for the scale factor.
But many people write a(t)
in many treatment the scale factor is dimensionless---so no units to worry about---and
it is normalized so that R(present epoch) = 1
IIRC this is how it is in Lineweaver's figure 14.
You will see in Figure 14 that the past history of R(t) is calculated for several different assumptions----like assuming dark energy or not assuming dark energy. The curves you get for past R(t) are not very different for various reasonable assumptions. the idea is that the estimate
is fairly robust, not terribly sensitive to whatever assumptions.
In this post, for simplicity i will assume that dark energy exists and has constant density (leading to equation of state w = -1)
Since the problem of how to measure R(t) for the present moment is solved by convention (just set R=1), the problem is to infer it for times in the past.
One useful fact is that the official definition of the Hubble parameter H(t) is this:
H(t) = R'(t)/R(t)
Since one can measure the present Hubble H(now) and since R(now) = 1, one has a figure for R'(now). It will serve as "initial condition" for starting to solve the differential equation for R(t) later when we have the rest of the data.
And the formula for redshift of light received at the present is this:
z + 1 = R(present)/R(time emitted) = 1/R(time emitted)
Here I am using the fact that R(time received) = R(present) = 1
Another useful fact is that one can measure the density of stuff in our immediate neighborhood, where the effect of expansion can be neglected because it is so slow.
And one ASSUMES that the density of stuff is everywhere the same on average over large distances. this uniformity is a very useful assumption
to say the least!
For today's density of matter (dark and light) I will use the nightmare notation rhoM(now)
rho means density, M means matter, and now means now.
Then one knows the density of matter in the past is just
rhoM(t) = rhoM(now)/R(t)^3
and total density is matter density plus constant DE density
rho(t) = rhoM(t) + rhoDE
This is the most dreadful notation I can ever remember using. But then my memory is not too good, so it might not be.
Further, people seem able to infer that the universe is spatially approximately flat by looking at the bumps in the CMB. they also have other evidence for this. I find that this is a bit arcane, but i take their word for it.
Incidentally the equation I just wrote is how we know rhoDE.
Because we can specialize it for t = now
rho(now) = rhoM(now) + rhoDE
And then rho(now) we know from observed flatness plus observed Hubble. because H(now) tells us the critical density and flatness tells us that the actual density equals critical density. So rhoDE is just the difference of two known things.
It certainly makes things mathematically simpler to assume that the universe is spatially flat! A special argument (see Lineweaver) shows that flatness now implies flatness way back in time---all the way back to the very early inflationary period, which may have been the cause of spatial flatness in the first place.
Flatness at time t means that the actual density rho(t) is equal to the critical density
3H(t)^2/(8piG)
If we write that out, we will just get the Friedmann equation:
rhoM(t) + rhoDE = 3H(t)^2/(8piG) = (3/8piG) (R'(t)/R(t))^2
Then switching around:
(R'(t)/R(t))^2 = (8piG/3) [rhoM(now)/R(t)^3 + rhoDE]
In the horrible notation I find myself using, this is just the Friedmann equation. And most of the terms we know. 8piG we know. rhoDE we know. rhoM(now) we know.
So we just have to solve the differential equation for R(t)!
I guess if you multiply thru by R(t)^2 it looks even simpler
R'(t)^2 = (8piG/3) [rhoM(now)/R(t) + R(t)^3 rhoDE]
this would be easy to do by machine. one would just start at present, with
R(now) = 1 and one would plug in known rhoDE and rhoM.
Remember that R'(now) is known from measuring the Hubble parameter.
So you can start out taking small steps, get a new R(now - delta),
get a new R'(now - delta), and then use that to get a new R(...) and so on.
There is probably some analytical way to solve it too, the point is that it is solvable.
This is ONE OF THE WAYS that can be used to infer what the scale factor R(t) has been in the past.
But there are several ways to infer the past history of R(t) and any avenue of inference involves making some assumptions. The game is to make several independent inferences of R(t) in the past and have them CHECK so the consistency gives confidence that the inferences are well founded.
Here is something I didnt use yet:
z + 1 = R(now)/R(time emitted) = 1/R(time emitted)
this means that the past history of the scale factor is precisely what we need to relate redshift to time-----to relate the redshift to the length of time the light has been in flight towards us
or to relate the redshift to the age of the universe when the light was emitted.
So there are going to be various ways to check our history R(t) of the scale factor by looking back at historical events like galaxy formation and star formation and reasoning about conditions and mapping out the history and comparing with the redshift and seeing if it all matches well or not.
And then there is the supernova IA business where you actually have a way of estimating the distance (by standard candle) independent of redshift. So again you can check if there is consistency with the estimated history R(t) of the scalefactor.
Hope this is not too muddled and is free of major bungles. One attempt to answer the question anyway. Room for more. how R(t) is determined is a complex issue.