# Calculating the % hydrogen and % helium

1. May 4, 2014

### manav95

Hello, I have been studying the early universe where everything was a bunch of hydrogen atoms and helium atoms. Apparently they calculated the %composition of the universe at the time. How did they do that? What sort of equations can be used to calculate these quantities?

2. May 4, 2014

3. May 4, 2014

### Mordred

Good question, the answer is rather involved. Essentially its a correlation of our knowledge of particle physics and thermodynamics. In a process described as nucleosynthesis. Rather than try to explain such a lengthy step by step process, QCD plasma to photons to neutrinos to protons etc. Its best to follow each step your self. Keep in mind the process also heavily involves the ideal gas laws.

Particle physics of the early universe by Uwe-Jens Wiese

http://www.wiese.itp.unibe.ch/lectures/universe.pdf abundances of hydrogen and helium are covered section 5.2 however its best to start at the beginning and work up from there. However if you already understand SR and the EFE the first couple of chapters you can skip. This article is particularly well done as it also correlates to Scott Dodelson's Modern Cosmology textbook in the same metrics. The same metrics I've seen in other articles, however this one is easier to follow

a couple of key formulas used in the earlier steps is Bose-Einstein distributions (bosons) and Fermi-Dirac distributions (fermions), however other formulas are also necessary, The Saha equations in particular apply to %of hydrogen and helium. The use starts at chapter 3 or 4 I'd have to look again lol

http://www.phy.ohiou.edu/~mboett/astro401_fall12/saha.pdf

Last edited: May 4, 2014
4. May 4, 2014

### Mordred

found this article as well, its done in the same manner as the previous articles I mentioned, however has some added details. As well as being shorter with some good visual aids

Lecture Notes on CMB Theory:
From Nucleosynthesis to Recombination by Wayne HU

http://arxiv.org/pdf/0802.3688.pdf

note: its not specific to nuceosynthesis, however has the related equations

Last edited: May 4, 2014
5. May 5, 2014

### phsopher

If you want to have some very basic qualitative understanding of the story consider the following. The number density of non-relativistic particles is determined by temperature and the mass of the particles. Since protons and neutrons have very similar masses their abundances in the early universe were pretty much equal. Additionally, they can turn into each other via the following reaction mediated by weak interaction

$$n \leftrightarrow p + e^- + \bar \nu_e$$

and other permutations of it. This means that a neutron may decay into a proton, an electron and an anti-neutrino. Alternatively tha latter three can collide to form a neutron. In the very early universe these two reactions happened at equal rates and so the numbers of particles stayed the same, but at some point the universe will have expanded so much that the reaction from right to left will have become inefficient because the particles are too rare to find each other and collide. So before this point there were approximately equal numbers of protons and neutrons. After this point neutrons started to dacay into protons and so the ratio of neutrons to protons went down.

At some point after that the universe cooled down so much that it became energetically favourable for protons and neutrons to form nuclei (nucleosynthesis). As a very crude approximation assume that only helium and hydrogen nuclei are produced (and ingnore different isotopes). Then all of the neutrons go into helium nuclei (which also claim two protons for each two neutrons) and the leftover protons are the hydrogen nuclei. Thus you arrive at the ratio of helium to hydrogen.

In this very rudimentary picture you only need the basic nuclear decay equation into which you need to plug the life-time of a neutron and the time period between the moment when the reaction above falls out of equilibrium and the moment of nucleosynthesis.

Of course this is a very crude approximation that doesn't take into accound many of the issues such as different isotopes, other light elements being produced, protons and neutrons having different masses, etc. In order to make an accurate calculation one needs to understand all of the different nuclear reactions involved which is quite complicated.

6. May 5, 2014

### Mordred

One key problem with this VERY rough estimate is that it does not take into consideration the half life of the free neutrons compared to the proton. the decay of a neutron emits electrons and electron anti-neutrinos to become protons. So the ratio of neutrons will depend on the temperature and the number of stable reactions it can find before the neutrons can decay.

However as Phsopher mentioned his visualization is rudimentary