# What paper can I cite for the estimated number of atoms in the observable universe to be lower than $10^{82}$?

The title contains the whole question. I am a logician in theoretical computer science and want to use the estimated number of atoms in the observable universe to show what a ridiculous amount of space a certain algorithm would require in the implementation. But whenever I am looking for a paper to cite, I only find websites, forum discussions and documentaries linked (such as this answer). I need a paper published by scientists which estimates the number of atoms in the observable universe.

I searched on Google and Wikipedia, but couldn't find anything.

Update: Commentators guess that there is no such paper, they don't think it is 'fundamental' knowledge. But I also found a paper to cite for the estimated age of the universe even though the age changes and might not appear 'fundamental' to you. Since I don't want to pollute my references with papers of natural sciences, which paper would you then suggest to cite as a basis for the estimation?

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• I did that already. – xamid Apr 1 '17 at 14:41
• What you did and what you didn't do is not obvious at all, and your tone is not helping. – AccidentalFourierTransform Apr 1 '17 at 14:44
• You could have seen that there are only papers to different subjects (positioning of galaxies ; alien life ?!), websites, and a documentary linked. – xamid Apr 1 '17 at 14:51
• I doubt such a paper exists. It probably exists in something as a passing reference, but definitely not the focus of the paper... it's a trivial estimate to make, so from an astrophysics standpoint it shouldn't really need a citation. – DilithiumMatrix Apr 1 '17 at 14:57
• I doubt anyone has written a "paper" on this, since it isn't "fundamental". It is a completely arbitrary number that changes from day to day as the universe expands. – Rob Jeffries Apr 1 '17 at 14:57

You can cite Planck Collaboration et al. (2016).

This paper describes the latest result from the Planck satellite, which measures the anisotropies of the CMB and infers, among other things, the densities of the various components of the Universe. The density of baryons is measured to be $\Omega_\mathrm{b} = 0.0484$ and the helium fraction $Y = 0.2453$. Ignoring the small correction from metals$^\dagger$, the hydrogen fraction is then $X=1-Y=0.7547$, and the mean atomic mass is $$\bar{m} = (X + 4Y) \, m_\mathrm{H} = 1.718\, m_\mathrm{H}.$$

The Hubble parameter is $H_0 = 67.81\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$, yielding a critical density of $\rho_\mathrm{c} \equiv 3H^2/8\pi G = 8.638\times10^{-30}\,\mathrm{g}\,\mathrm{cm}^{-3}$.

These numbers together give you the number density of atoms of $$n = \Omega_\mathrm{b}\,\frac{\rho_\mathrm{c}}{\bar{m}} = 1.44\times10^{-7}\,\mathrm{cm}^{-3}.$$

Using the rest of the cosmological parameters and integrating the Friedmann equation gives the distance to the particle horizon, $R = 14.45\,\mathrm{Gpc}$, corresponding to a volume of $V = 4\pi R^3/3 = 3.71\times10^{86}\,\mathrm{cm}^{3}$.

Thus, the total number of atoms in the observable Universe is $$N = n\,V \simeq 5.3\times10^{79}.$$

$^\dagger$"Metals" means anything heavier than helium. The Planck observations do not deal with metals, which increase the mean atomic mass a little, and hence decrease the total number of atoms. However, firstly the metallicity $Z=1-X-Y$ is small ($Z\simeq0.02$) in galaxies, secondly most of the metals are in galaxies, and thirdly most of the atomic mass is outside the galaxies in the intergalactic medium (see Calura & Matteucci 2004), so the average cosmic metallicity is more like $Z\sim0.01$, increasing $\bar{m}$ to perhaps $1.8\,m_\mathrm{H}$, which in turn decreases $N$ to $5.0\times10^{79}$. These numbers are a bit uncertain because measurements of the metallicity in the IGM are difficult due to the high ionization state of the atoms.

• I cited exactly that paper for the estimation of the age of the universe already. But I couldn't find something about the total number of atoms in the observable universe. Do I correctly understand your answer, that it isn't really mentioned in the paper, but can easily be calculated based on the information provided? – xamid Apr 1 '17 at 15:20
• @xamid: Yes. Usually the densities are the interesting parts. The total number of atoms, or anything, in the observable Universe is rarely interesting in a scientific context. But it is easily inferred from the numbers found in that paper. – pela Apr 1 '17 at 16:34
• Note (to the OP) that this number is the number of nuclei. The number of atoms is smaller since a large fraction are completely ionised. – Rob Jeffries Apr 2 '17 at 8:33
• Unless that paper actually has that calculation and specific number in it, I would think that the citation is pretty misleading, to be honest. – Emilio Pisanty Apr 8 '17 at 18:51
• @EmilioPisanty: Yes, you're right that simply stating the number and referring to the paper is a bit far-fetched. But you could write something like "The number of atoms in the observable Universe is 1e80, which can be inferred from the cosmological parameters determined by the Planck Collaboration et al. (2016)". – pela Apr 8 '17 at 22:47