P_Ravensorow said:
But why those specific numbers? Are there any theories that suggest these decimaly accurate numbers?
You want to know the theoretical basis for how these numbers hang together and fit the data. It's a reasonable thing to ask about, for sure. The basis is the GR equation. Don't get put off by the fact that it is mathematics. I will explain in words. Touch base for a moment with the equations if you want, in Wikipedia, without getting bogged down. If you want, look up General Relativity in Wikipedia just to glance at the equation and then look up FRIEDMANN EQUATION. That's the important one. It is a simplification derived from the GR equation by assuming an approximately even distribution of matter and roughly constant spatial curvature (which could be zero, the measurements always come out very close to zero, so the flat case of the Friedmann equation is assumed a lot, for fitting data)
I'm not trying to confuse or snow anybody, it is actually very simple. The Freidmann describes
the growth of the size factor a(t) over time. a(t) is normalized to be a = 1 at present. and it only changes by about 1/140 of one percent per million years. So we don't see it change from year to year.
That means if you have a figure for the dark matter density NOW, you can say what it was when a(t) was 0.5.* or some other time in the past when the size factor was something else.
this is the backbone of cosmology, it is what let's you fit all the observational data about different stages of the universe together and make sure they are consistent with the estimated numbers for the present. a(t) is really really important, and the Friedmann equation is what controls it and tells how it must grow.
Why do we believe the Friedmann? Because the GR equation has been tested and tested and tested ever since around 1919 almost for a hundred years, and it always passes the observational tests with flying colors and now gets verified out to 6 decimal places in some experiments etc etc. And the Friedmann is just a simplified form of it.
What else to we need to know in order to get those numbers you asked about? the most important thing is the observation of approximate spatial
flatness . The Friedman equation is especially simple in the spatial flat case. It relates the percentage growth rate of a(t) to the density of matter and energy in a simple way.
Measuring the current percentage growth rate tells us the present-day density!
Now we can check flatness by measuring angles, and comparing radius and areas, the usual geometry and trig formulas work if space is flat and don't if it is not flat. Fortunately checking the geometry keeps assuring people that spatial curvature is either zero or very nearly zero.
This is a bit of luck. We are lucky the Friedmann equation is so simple. We are lucky the geometry measurements say near flatness--making the equation even simpler. We then have a straightforward way to find out the current overall average density of matter and energy. Just by observing the current percentage growth rate of a(t)
*If a(t) = 1/2 that means distances were half a big, so volumes were 1/8 what they are today, so densities of matter were 8 times what they are today.
Some light comes to us from a time when a(t) was 1/1000, so distances were 1/1000 today's size and matter density was a billion times what it is today. A lot of information is gotten by observing that ancient light. Knowing the history of a(t) is what let's people extrapolate back and understand what they see and keep checking that it is consistent with those today numbers that you asked about.