Can space expand with unlimited speed? According to this article on the European Space Agency web site just after the Big Bang and before inflation the currently observable universe was the size of a coin. One millionth of a second later the universe was the size of the Solar System, which is an expansion much much faster than speed of light. Can space expand with unlimited speed?
 A: I'm going to go with "yes, but that's less interesting than you may think."
1. The laws of physics are local
Every law of physics that we know of only "sees" a tiny portion of the universe. The universe seems to consist of the same physical laws being applied identically and independently to every little part of itself.
If you look at any tiny part of an expanding universe, nothing untoward is going on. Everything is following the same laws as in any other situation, and nothing is exceeding the speed of light. When you stitch all of these pieces together, you get a global spacetime where the total volume of space seems to increase very quickly, but this "total volume of space" doesn't appear in any physical law, and in some sense you could think of it as a human invention.
2. Even globally, it's not clear that anything untoward is going on
The Milne model is the zero-density limit of the standard (FLRW) expanding cosmological model. It's a useful source of counterexamples to misconceptions about cosmology because it's actually just a portion of Minkowski space (the flat spacetime of special relativity) in different coordinates, so you can apply your special-relativistic intuition and calculational techniques to problems in cosmology, often getting results that contradict what might appear to be true in the FLRW coordinates.
In the Milne model, recessional velocities between objects can be arbitrarily high (exceeding $c$ or any particular multiple of $c$). This doesn't contradict special relativity because the definition of "recessional velocity" doesn't match the usual definition of "velocity" in special relativity. The recessional velocity is, in SR terms, the rapidity (times $c$).
In the Milne model, you and your friend (both at rest relative to the Hubble flow) can be 1 meter apart, wait for 1 second (measured by your respective watches), and at the end of that second be 10100 meters apart – or any other time interval and two distances you like, as long as the later distance is larger than the earlier one.
How can there possibly be "room" for this in Minkowski space? It's pretty easy to see what's going on. Since any inertial frame is as valid as any other, I'll pick one where you and your friend have equal and opposite velocities $\pm \mathbf v$. After a time $\tau$ has elapsed on your watches, your $t$ coordinates will have increased by $\gamma\tau$ and your x coordinate by $\pm\mathbf v\gamma\tau$. Since $\gamma\to\infty$ as $|\mathbf v|\to c$, these coordinate changes can be arbitrarily large, so there can be plenty of "room" at the end even if $\tau$ is small.
Another way of looking at this is that the triangle inequality doesn't work in spacetime. You might expect that if you and your friend start at the same point and you each travel in a straight line (inertial motion) for 1 second (elapsed proper time = length of worldline), that the distance between you should be at most 2 light seconds. In fact, though, the distance can be anything. If we classify that as "superluminal expansion of space" (and I think we should, since we are literally doing FLRW cosmology here), then superluminal expansion of space is allowed even in special relativity.
When you move from this special case to general FLRW cosmology, you lose the special-relativistic correspondence, but I don't think that makes the possibility of "superluminal" expansion any more surprising. On the contrary: if it can happen in special relativity, then of course it can happen in general relativity.
A: I should add, in order to avoid inconsistencies in the text and in the derived formulas, that the expression for $D(z_{ob},t_{ob})$ as it written down here already IS a comoving distance, which forces you to set $a_{ob} = a(t_{0}) = 1$ and $t_{ob} = t_0$, $z_{ob} = 0$.  This is a direct consequence of setting $ds^2 = 0$ in the FRLW line element, yielding the light-cone equation.
This distance is also the comoving distance given on the horizontal axis of the diagram.
The text and the derived formulas should be adapted to these notions.
For a correct treatment, refer to the Davis and Lineweaver papers.
rhkail
A: Yes, the expansion of space itself is allowed to exceed the speed-of-light limit because the speed-of-light limit only applies to regions where special relativity – a description of the spacetime as a flat geometry – applies. In the context of cosmology, especially a very fast expansion, special relativity doesn't apply because the curvature of the spacetime is large and essential.
The expansion of space makes the relative speed between two places/galaxies scale like $v=Hd$ where $H$ is the Hubble constant and $d$ is the distance. When this $v$ exceeds $c$, it means that the two places/galaxies are "behind the horizons of one another" so they can't observer each other anytime soon. But they're still allowed to exist.
In quantum gravity i.e. string theory, there may exist limits on the acceleration of the expansion but the relevant maximum acceleration is extreme – Planckian – and doesn't invalidate any process we know, not even those in cosmic inflation.
A: Your question is based on a fundamental misconception. You say:

At the beginning, right after the Big Bang, the universe was the size of a coin

but it's more accurate to say "the observable universe was the size of a coin" i.e. the 13.7 billion light year bit that we can currently see was at one time the same radius as a coin. The universe may well be infinite in size, and if so it has always been infinite in size right back to the moment of the Big Bang.
There is no point in the observable universe that is moving away from us at faster than the speed of light, but assuming the universe is infinite, or at least much bigger than the bit we can see, everything farther away from us than the edge of the observable universe is moving away from us faster than the speed of light. As Luboš says this doesn't violate relativity since it's space that's expanding not the objects themselves moving, and there is no limit to the expansion rate of space. In fact if there was a period of inflation immediately after the Big Bang, during this period space expanded at a rate that makes the speed of light look positively glacial.
If you're interested in a bit more detail about how we model the expansion of the universe search this site for "FLRW metric", or Google for it.
