That is exactly what I am asking. Is the universe only "thousands" times smaller at 12, or even say, 8 billion years ago (when did the inflation end?)? It does not sound right.
Can the equation of calculation be shown?
Well, that's because the expansion has slowed with time. The expansion rate H (usually provided in units of km/sec/Megaparsec) is given by the following equation:
H^2 = H_0^2 * (Om / (a^3) + Or / (a^4) + Olambda)
Here are the other variables in the above equation:
H_0 = expansion rate today (about 70km/sec/Mpc).
Om = proportion of energy density today made up by matter (normal + dark, about 0.25).
Or = proportion of energy density today made up by radiation (about 10^-5)
Olambda = proportion of energy density today made up by dark energy (about 0.75)
a = scale factor of the universe. a = 1 today. a = 0.5 when the universe was half its current size, and so on.
For a bit of understanding why the above factors are the way they are, the above equation has to do with how things are diluted out. In general, the expansion rate goes as the energy density:
H^2 = 8*pi*G/3*rho
...where rho is the total energy density of the universe. So how the expansion rate changes with time depends upon how the energy density of the universe changes with time. So how do the various things dilute?
Well, normal matter is easy. Normal matter just goes by the volume. Increase the volume by a factor of a^3, and the density of the matter drops by a factor of a^3. Nice and easy: just divide by a^3.
Radiation is a tiny bit more difficult, because it doesn't just dilute. It also redshifts. So not only do the number of photons per volume drop by a^3, but the energy of each individual photon also drops by a factor of a. So the total drop in energy density is a^4.
And, if we assume that the dark energy is a cosmological constant (not yet known, but good enough for purposes of this calculation), then it doesn't dilute at all. Its energy density stays the same.
So here's the picture: in the very early universe, it was dominated by radiation. This stuff dilutes very rapidly, however, as a^4, and so it goes to small values pretty quickly. But during this epoch, the expansion rate of the universe would have been at its fastest.
Later, matter dominates. Again, the expansion rate is fast, but it's slowing. It doesn't slow quite as rapidly as radiation, but the matter still dilutes, and it still slows down.
Still later, dark energy comes to dominate. Here the dilution stops. Once all the matter and the radiation are diluted away to essentially nothing, we're left with a constant energy density, which is a constant expansion rate.
You may be a bit confused now about a particular point: why do we call it acceleration? It's because of what the scale factor (a) does in such a situation. The definition of the expansion rate is as follows:
H = 1/a * (da/dt).
So, if H = constant, then we have the differential equation:
da/dt = a*H
...in which case, the solution is:
a(t) = e^Ht
...which is exponential expansion! A constant expansion
rate, then, leads to a scale factor which increases in scale at an accelerated pace. Looking carefully at the above equations shows that in the very early universe, the scale factor 'a' was increasing at an extremely rapid pace in the early universe, but under the influence of the dilution of radiation and later matter, it slowed down dramatically. Until, that is, the recent accelerating phase, where we're dominated by dark energy.
So, if you want to know the time to any given scale factor, all you need to do is integrate time:
t = integral from 0 to t (dt)
We can then use a simple change of variables knowing that H = 1/a * da/dt, to get:
dt = a*da/H
...to get the time to any point a is:
t = integral from 0 to a (a*da/H)
It then just becomes an exercise in performing a numerical integral, using the above definition of H given the known energy densities of the universe at the current time. For example, integrating the above equation (with the most accurate values for these parameters that we have) from 0 to 1 gives t = 13.7 billion years (since a = 1 now). Integrating from 0 to 0.001 gives t = 380,000 years (when a = 0.001, the universe was 1/1000th the size in each direction).