Did cosmological inflation occur at speeds greater than $c$? If cosmological inflation occurred at speeds less than $c$, wouldn't we see the CMB right in front of our noses (or not at all) instead of at 13.7 billion light years?
And if it happened faster than $c$, does that mean inflation occurred before the current laws of physics (read the four basic forces) precipitated, was $c$ greater then than it is now?
The simple answer to your question is "yes" - the universe expanded at much greater speeds than $c$ during the inflationary epoch. This period of time was very quick but very dramatic, lasting from about $10^{-36}$ to $10^{-32}$ seconds. The universe expanded, in this very short period, by a factor of $10^{26}$. That's pretty incredible, when you think about it.

Inflation was originally proposed to solve, among other things, the horizon problem - that is, why the universe is isotropic and homogenous (on a large scale). This would mean that all the parts of the universe were in "causal contact" at one point in time. Inflation is the explanation for this.

Now, what does this all have to do with the CMB? Well, the temperature of the CMB is the same throughout the universe - a toasty 2.7 Kelvin. For the temperature to be uniform, all the regions of the universe would have had to be in causal contact in some point in time; hence, inflation explains the uniform temperature of the CMB.

However, the CMB was not around during the inflationary epoch. Far from it. It formed a lot later, when the universe was at the ripe old age of $379,000$ years. But the reason that it was formed (photon decoupling) equally throughout the universe is because the conditions were roughly equal, because of inflation. The CMB was, and still is, everywhere. It was never around during inflation, and as such was not otherwise affected by it.

I hope this helps.



My sources for the times:

Inflationary epoch

NASA