ideasrule said:
The rate of increase in scale factor is the rate of percentage change. If the scale factor increases at 0.01/million years, that means the distance between 2 objects increases 1% per million years. I don't see how this leads to inflationary expansion, in which the time derivative of the scale factor, and the 2rd time derivative, and the third, and so on all increase exponentially with time.
No, it isn't. Really.
If you have a function that increases by a fixed percentage, or factor, per unit time, then you have an exponential function. Think about compound interest.
Here are more technical details specifically on cosmology. The co-moving distance co-ordinate is unaltered by expansion. The scale factor is the ratio between proper distance and co-moving distance.
The scale factor is often written as
"a", and it is a function of proper time. When you have constant expansion, with no acceleration or deceleration, the scale factor is a linear function of time. In this case, if you pick any two co-moving galaxies, they continue to separate at a constant velocity.
If you have the scale factor increasing by a fixed factor, or percentage, per unit time, then you have an exponential relation, corresponding to inflation.
Another well known simple model is the matter-critical flat model, with no dark energy. In this case the expansion is slowed by the effects of gravity, but not quite enough to stop expansion continuing to indefinitely.
The three solutions can be given as follows:
\begin{align*}<br />
a &= H_0 t & \mbox{Linear expansion; empty universe} \\<br />
a &= e^{H_0 t} & \mbox{Exponential expansion; inflation} \\<br />
a &= \left(1.5 H_0 t \right) ^\frac{2}{3} & \mbox{Matter critical expansion}<br />
\end{align*}
The first case has no acceleration, and the scale factor increases by a fixed amount per unit time. The second case is accelerating, and the scale factor increases by a fixed percentage per unit time. The third case is decelerating.
The constant H
0 has units of inverse time, and it is actually the Hubble constant in more sensible units. It is equal to the first derivative of a at the time when a itself is equal to 1, typically chosen as the present.
The above should serve to address the matter of what acceleration means... here is a bit more technical stuff closer to what our universe is doing. Our own universe, as far as we can tell, behaves on large scales pretty much as given by the following differential equation:
\begin{align*}<br />
\frac{da}{dt} & = a H_0 \sqrt{ \Omega_M a^{-3} + \Omega_V }<br />
\end{align*}
This is the FRW solution for a flat universe, with matter as a fraction Ω
M of critical and dark energy as a faction Ω
V of critical. There's a bit of work nailing down those parameters, but they seem to be about 0.27 and 0.73 respectively. You'll often see these numbers used in recent descriptions of modern cosmology; they are based on the WMAP experiment; and there may be slightly updated estimates used with recent more accurate work nailing down the Hubble "constant".
The Ω
V drives acceleration, and the Ω
M factor retards it. If you solve for the next derivative you get
<br />
\frac{d^2a}{dt^2} = H_0^2 a (\Omega_V - 0.5 \Omega_M a^{-3})<br />
In this model, the acceleration took over from deceleration when the scale factor was
a = \left(\frac{\Omega_M}{2 \Omega_V} \right)^\frac{1}{3} \approx 0.57