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I have read that the cosmic background radiation was formed 380,000 years after the big bang, when stuff changed from being opaque to light, because of free electrons, to becoming transparent.

However, I don't understand how we got to where we are in space faster than the CMB? I'm imagining in my mind that the big bang started at a small infinitely small point, and then stuff spread outwards from that. I'm imagining that we moved away from that point at a speed slower than the speed of light? But the CMB was presumably travelling at the speed of light? So, how did we get to where we are before the CMB got to us?

Qmechanic
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Welcome to PSE! The main point of confusion seems to be that you think the Big Bang happened at a single location in space. This is a very common misconception, and the name itself probably contributes.

The generation of matter in the Big Bang, and thus the transition from opacity that would allow light to begin traveling freely thousands of years later giving us the CMB, did not happen at any one place - it filled all of space, with space itself being the thing that was infinitely dense at the moment of the BB according to current theory.

So the Big Bang didn’t occur at some far-off location - it occurred right under your feet. The reason the CMB is a microwave background and not light of shorter wavelengths is attributed to the expansion of the universe since that time - some day it’ll be the Cosmic Radio Background.

JPattarini
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It's a very good question and has been asked before maybe, but it highlightens how one thinks of a big bang. Namely a bang in an infinite space. If the matter expands as such and electrons and protons recombine then the photons would finally escape the expanding ball, like photons escape the Sun, although the Sun ain't expanding. You would see a glowing expanding sphere, from the outside. That eventually would be devoid of photons, which would not decrease in energy to get to the level of the CMBR energy.

Space starts out small. Like a spatially 3D ballon that gets inflated. So where ever in the balloon you are, you are not near and edge or boundary of an expanding 3D mass in 3D space. The photons get stretched by the expansion and their energy is lowered. You can't visualize this, so a 2D balloon picture is offered often. But in reality, the balloon is 3D.