# What was the size of the universe around 400 million years after the Big Bang?

I was trying to find out how large the universe was around 400 million years after the Big Bang as this is when some of the first galaxies were formed but I am having trouble finding out the answer or working it out.

I found some information on the size of the universe after 380,000 years when the CMB was created and by calculating the red shift from the time of creation a size of roughly 80 million light years is given by reducing the observable universes size by a factor of 1100 but I am not sure how to use this to calculate the universes size after 400 million years and I am not sure if as the above calculation is for the observable universe how that related to the actual size.

Is the size of the universe in terms of its length or diameter known or can be calculated after 400 million years from the Big Bang event?

• The issue: By "universe" do you mean "Universe", or "the observable universe". As far as we can tell, the universe may be infinitely large and always has been. Or it might not be. Observations limited by the speed of light can't tell. – James K Sep 10 '20 at 20:03
• @JamesK When you say the Universe may have always been infinite is that based on the Big Bang occurring in an existing space? I do mean the whole universe not just the observable but i may be misinterpreting theories and getting confused but i mean the distance that all the matter and energy that came out of what we call the Big Bang had spread to. – user34615 Sep 10 '20 at 20:14
• No, ...... The whole universe was hot and dense and infinite. Now it is cold, rare and infinite. The big bang didn't occur in the universe. The big bang is the universe – James K Sep 10 '20 at 20:17
• @JamesK Where does the singularity and inflation come in at the early stages if it was already infinite? – user34615 Sep 10 '20 at 20:37

There are two issues: what counts as "the size of the universe", and how to rescale this. The second issue is easier.

If you have something at distance $$d_1$$ at time $$t_1$$ it will be at distance $$d_2 = d_1 a(t_2)/a(t_1)$$ at time $$t_2$$, where $$a(t)$$ is the scale factor at the relevant time. To further make things nice, $$a(t)=1/(1+z)$$ where $$z$$ is the redshift at the time.

For $$\approx 400$$ myr and a flat universe, $$z=11.35$$ and $$a=0.081$$ (from this calculator).

Note the word "flat" above: flat and open universes are by default unbounded. So in a literal sense they are always infinitely large. However, it is entirely possible that they have a nontrivial topology (imagine a universe that repeats like a hall of mirrors; various cases are possible), except that there is at present no evidence for this possibility - if there is something like going on it is likely larger than our current cosmological horizon.

The observable universe on the other hand is finite, and grows both because light from more and more distant objects arrive and because of the expansion makes their distance larger. This is why the size is not just $$ct$$, it depends on the expansion. The calculator gives 367.92 Mpc as the distance to the horizon at $$z=11.35$$, or 1.2 giga-lightyear.

• Thanks, So if I am understanding this correctly all the mass of the observed universe would crammed into a space of 1.2 billion light years? – user34615 Sep 10 '20 at 22:30
• Not quite: the 1.2 billion light years is the observable universe back then. However, you can divide the radius of our observable universe by 11.35 to get the radius back then of the region that would expand to become the current observable universe; since matter is not moving fast most matter in these two spheres is the same. – Anders Sandberg Sep 11 '20 at 13:00
• Ok thanks so that puts 90 billion light years diameter of superclusters and filaments into around 8 billion light years of space. – user34615 Sep 11 '20 at 13:46