Unit 3: How Much Helium can we Make?

How Much Helium can we Make?

In order to figure out how much Helium we can make at the end of Big Bang Nucleosynthesis, we first need to know how many of the composite particles we have: mainly neutrons and Deuterium. The remaining total of Helium, Hydrogen, etc. we have at the end of BBN is called the freeze-out abundance, called this because at the end of BBN the universe expanded and cooled to the point where particle interactions are no longer happening and freezes the final abundances to this value. During this time, the universe was hot, dense, and expanding. To consider how many interactions we can get before the universe expands too much, we need to look at the number density of particles at a given time or temperature. The higher the number of particles squished into a box, the more interactions and collisions we should get. However, as the universe expands the box gets larger and limits the number of interactions. We can relate the expansion of the universe, particle interaction rates, and the number density of particles through the Boltzmann Equation, given as follows

We see from this equation that the number density of particles is affected both by the expansion of the universe and the interaction rate between other particles. As the universe expands, the number density goes down; even if the number of particles remains the same, the number density decreases because the universe's expansion is increasing the volume. This also means that the interaction rate also goes down as the universe expands, because the particles will have a smaller and smaller chance of interacting with one another when the volume they're in gets larger and larger. 

Particle decoupling occurs when the Hubble Constant (the rate of expansion of the universe) becomes greater than the interaction rate between other particles, indicating that these particles now evolve/interact independently of others. Particle freeze-out occurs when the interaction rate becomes zero, indicating that no more interactions will take place and whatever abundance of particles you have cannot be changed. To continue our conversation on the final freeze-out abundances of elements, we start by looking at when particles in the universe were in simple, harmonious equilibrium with one another.

Thermal Equilibrium in the Early Universe

For simplicity, we will assume all of the particles that are interacting with one another in the Early Universe were in thermal equilibrium, or that they all have the same temperature, T.  

There's also one other variable that we need need to introduce, μi, called the chemical potential. This variable tells us how the energy of an interaction changes when particles form or are annihilated. We will go into more detail in the next subsection.

Chemical Equilibrium in the Early Universe

This equation is special for the case of chemical equilibrium. The chemical potential does, indeed, depend on the temperature of the universe; at some point in the universe's history, the annihilation/creation processes will only go one way instead of back and forth. This will break that lovely equation we just found. 

There are a few things to note about chemical potentials that will help us in figuring out the final distribution of Helium in the universe. Firstly, that the chemical potential of photons, μˠ, is zero. Secondly, that antiparticles have the negative chemical potential of their particle counter-parts. 

Finding the ratio of Neutrons to Protons at Chemical Equilibrium: A Derivation

We now possess all of the information we need to figure out how many neutrons there were in the early universe compared to protons. We know what's involved in the process of neutron creation/annihilation, but let's assume that the contributions from electrons and neutrinos are too small to care about. Let's take a ratio of the number densities on each side of the process at equilibrium using the Boltzmann distribution, simplifying a little to cancel out some like-terms, to investigate the processes involved.

Since this process is at chemical equilibrium, we know that we can enforce μp= μn, which cancels out an annoying term in the exponent of what we found. Additionally, it turns out that gn= gp, allowing us to simplify this picture even more. The final ratio of neutrons to protons at equilibrium will then only depend on the masses of the particles and the temperature in the following way:

Well this expression is much simpler than the original one! Looking at some of these terms, we see a ratio of the neutron and proton masses. These masses are very similar, so we can take yet another approximation to say that this ratio is 1! Looking at the terms in the exponent, we see that there is a term that depends on the difference between the neutron and proton masses. We can recognize this as the binding energy for a neutron. Because this term is in an exponent, this term will have a large contribution. We can't take the same approximation of the masses as before and need to leave it as-is. Defining Q to be the neutron binding energy, 1.30 MeV, let's re-write our equation to its final form.

A Rough Estimate of the Helium Freeze-Out Abundance

More Detail on Calculating the Helium Freeze-Out Abundance

As mentioned in the previous section, we're in a race against time before all of the neutrons in the universe decay away and the formation of Helium relies on the production of Deuterium. Not all neutrons will fuse with a proton to become a Deuterium atom. Cosmologists call having a limited number of Deuterium atoms to fuse into heavier elements the Deuterium Bottleneck. 

If we followed the same procedures to find the abundances of Deuterium, protons, and neutrons at chemical equilibrium, we would arrive at the following expression:

This becomes trickier because we need to figure out the number densities of all of these particles at this new equilibrium: we can't use our previous results to help us! We would need to solve for each Boltzmann equation, know how to characterize the interaction rate between particles in the early universe, and take into account the decay of neutrons in the formation of Deuterium. Calculations like these are beyond the scope of this course. But know that this can get pretty challenging! Cosmologists can only calculate these quantities numerically using computers. 

And just like that, we've formed the atomic building blocks of the universe! The universe is mostly composed of Hydrogen and Helium. There are other processes, though, that make heavier elements. If the universe was made up only of these two elements, we wouldn't even exist! Let's investigate some of the processes that make the other elements in the next page: How to make elements heavier than Helium.