10-35 seconds after the Big Bang
Current theories involve a very brief, but very rapid period of expansion in the early Universe. This period was very brief - in fact only around 10-35 seconds (that's 0.00000000000000000000000000000000001 seconds in "long-hand") but in that time it is thought that the Universe expanded by a factor of around 1060 (1 million million million million million million million million million million times!). Any tiny "quantum fluctuations" left by the Big Bang are inflated to be much larger, than they were before. To anyone inhabiting the Universe at this time, it would have seemed to get a lot smoother all of a sudden.
Imagine a golf ball, around 5cm in diameter with dimples around 4mm wide and 0.2mm deep. Most people would agree that a golf ball is quite bumpy. The golf ball can be thought of as the early Universe and the dimples are the tiny fluctuations in it. Now let's blow up the golf ball to the size of the Earth, an increase of around 250,000,000 times. Now we have a ball the size of the Earth, with dimples 1000km wide and 50km deep, making it much smoother than the actual Earth. Since the Universe expanded so quickly during Inflation, faster than the speed of light in fact, we can only see a tiny fraction. In fact the region we can see is, in our analogy, still only about the size of a golf ball, and is now much smoother that the golf ball was to start with.
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