# Time-density and time-temperature functions of the Universe since the Big Bang

I am looking for these... functions. I want to be able to say, "at a 1873. second there were around 54230000000 K temperature and 0.00435 kg/m^3 density".

On the net I could find only tables or diagrams, as this: .

(Here is a link to the page it accompanies: http://www.astro.ucla.edu/~wright/BBhistory)

I am practically interested in the interval between the first seconds and the first some ten millions of years.

Is it possible? Some simple formula were also enough (although I think, there are different formulas for the different eras after the Big Bang).

• hyperphysics.phy-astr.gsu.edu/hbase/astro/timlin.html Sep 12 '14 at 17:12
• Why is the graphic pointed downwards? Most Big Bang diagrams I have seen (actually, all of them) use width to indicate universal expansion. Does the downwards point have any relevance? Sep 13 '14 at 16:52
• @Aaron There seems to be an issue with the link. It's not working properly (although I think that could be thanks to hyperphysics). Sep 13 '14 at 21:59
• @HDE226868 "Why is the graphic pointed downwards?" This diagram does not represent expansion of the universe. It represents the visible part (space-time) of the universe considering the finite speed of propagation of the light. It is a depiction of a light cone. ------ Here is the same image with more explicit depiction of the "last scattering": Tests of Big Bang: The cosmic microwave background. Sep 15 '14 at 11:56
• @HDE226868 the site is back up now. Sep 16 '14 at 0:31

## 1 Answer

Both the temperature and the density of a toy universe can, in a way, be linked to the age of the universe ($t$). Certain properties of the universe are dependent upon a scale factor $a(t)$ that is a function of time. The expansion of such a universe can be written in the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, generally written as $$ds^2 = dt^2 - a(t)^2(d\Sigma^2)$$ where, in spherical coordinates, $d\Sigma$ is given by $$d\Sigma^2 = \frac{dr^2}{1-kr^2} + r^2d\theta^2 + r^2\sin^2\theta d\phi^2$$ where $k$ is a constant. This shows that, in a toy universe, $r$ and $dr$ is proportional to $a(t)$, and so in a perfectly spherical toy universe, you could relate the volume to $t$. We know that, for a sphere, $$V=\frac{4}{3}\pi r^3$$ and because $d\Sigma^2$ (and by extension $r^2$) is related to $a(t)^2$, we can relate volume to time. If you knew the mass-energy of the universe, you could then relate the change in density to $t$. Unfortunately, our universe does not appear to be perfectly spherical, and so this may not apply.

You can relate something else to the age of the universe: the temperature of the cosmic microwave background. It has cooled over time, and is presently at about 2.7 Kelvin. While the temperature of the CMB may not be governed by the scale factor $a(t)$, it is certainly governed by time, and thus one you could in fact write the temperature of the universe as function of time. Unfortunately, I haven't been able to find data on the evolution of the CMB, but I can try, and perhaps this will lead you to an answer.

To summarize: In a perfectly spherical universe, assuming you know the matter-energy content and the scale factor $a(t)$, you could write an equation for the density of the universe. In any universe, you can also write an equation for the temperature of the CMB - the "temperature of the universe".

I hope this helps.

Source for FLRW metric: http://www.universe-galaxies-stars.com/Robertson-Walker_coordinates.html