How much has the universe expanded?

Expansion speed of universe is 68 km/sec, which is 1/4400 times the speed of light. From the big bang 13.7 billion years ago, the universe would have expanded a few hundred million light years at these speeds, even if initially the universe expanded with much higher speeds during creation. Could it have expanded so much that the distances between galaxies are being reported to be in billions of light years? I think one galaxy has been reported at a distance of 12 billion light years, picked up by telescopes? Secondly, 12 billion years ago, it would have been just 1 billion or so years after creation of universe, where it is difficult to believe that the light is being picked up now (in present time), where as light of that era should have traveled much beyond by now, since expansion speed of universe is only 68 km/sec. What is it that I am missing out in understanding the matter?

• When you say 68 km/s I think you mean 68 km/MPc/s, which is much more. – J.G. May 7 '18 at 8:33

The Big Bang didn't happen at a point, it happened everywhere at once. If the universe is infinite, it has always been infinite. The distance between any two random points now may tend towards zero as you look back towards the Big Bang, but even $.000000000000001~\rm s$ after the Big Bang the universe as a whole was infinite. You could say that the universe right at the moment of the Big Bang was of size $0*\infty$, but that doesn't mean it's $0$!
You also miss that the universe could have been expanding much, much faster at one point. Inflationary theories have the universe expanding by about 26 orders of magnitude in around $10^{-32}~\rm s$.
• @AtulPant Yes, but a "singularity" and a "point" are different things. A singularity is just where the math gets singular- in this case, the density of the universe goes to $\infty$. That doesn't mean the size can't also be $\infty$. – Chris May 10 '18 at 17:56