Chapter 9

When the universe was much hotter than it was at the time of last scattering, it had a temperature and density not unlike the core of the Sun, where hydrogen is fused into helium. So, it makes sense to expect that fusion occurred for some period of time in the first few minutes of the universe. But how much? If fusion lasted a long enough, it could convert all the hydrogen in the early universe into helium. However (spoiler alert), conditions were only suitable to convert about 1/4th of the hydrogen into helium. A fun (or, perhaps more accurately, dumb) way that astronomers characterize the composition of things is by the fraction of atoms that are hydrogen, helium, and everything else (generally called metals), denoted by $X$, $Y$, and $Z$, respectively. Therefore, the primordial fraction (i.e., resulting from the Big Bang) of baryons in helium atoms is given by $\boxed{Y_p \equiv \rho(^4{\rm He}) / \rho_{\rm bary}}$.

Neutrons & Protons

First, let's figure out what happens with the hadrons (protons and neutrons being the stable-ish ones), since that will affect the progression of nucleosynthesis, or creation of nuclei. When the universe is less than a second old, it's so hot that electrons and positrons (anti-electrons) are freely created from photon-photon collisions and vice versa (possible since the temperature $kT \sim 10~{\rm MeV}$ is larger than the combined rest masses of the particles, $\sim1~{\rm MeV}$. At this point neutrons and protons have condensed out of the quark soup, but since the difference in their mass $\Delta E = 1.29~{\rm MeV} < kT$, they can collide with neutrinos or electrons/positrons to convert between each other.

Just as in the last chapter, when we wanted to understand how the balance of electrons, protons, and hydrogen atoms depended on temperature, the same can be done between neutrons and protons at higher temperatures (using the same equation as before):

where $Q_n = 1.29~{\rm MeV}$, the mass difference. At much higher temperatures than this, neutrons and protons are present in nearly equal numbers, and as the temperature falls, the relative number of neutrons will fall exponentially as the left-to-right direction of the reactions

are energetically favored. However, the cross-section for neutron-neutrino interactions $\sigma_w \propto T^2$, so as the temperature falls, the likelihood of these collisions becomes increasingly rare. As with decoupling between photons and electrons as electrons get captured into atoms, neutrinos start to "disappear" and the scattering rate $\Gamma_\nu$ falls, and once $\Gamma_\nu <H$, the reaction effectively stops — in other words, the above equation will no longer hold and the neutron-proton ratio will become fixed or "frozen." This occurs at $kT_{\rm freeze} = 0.8~{\rm MeV}$, around when the universe is 1 second old, and plugging values into the above equation gives $n_n/n_p \approx 0.2$, one neutron for every 5 protons.

Fusion between 2 protons directly is very slow, and during the short period of time when the energy is high enough for fusion reactions to occur (less than an hour after the Big Bang), it can't work. However, neutron plus proton reactions are easy, so we can assume that the maximum amount of helium that can be possibly be made (but less will be remain at the end of nucleosynthesis if other elements are also made) uses all the leftover neutrons. Since helium consists of 2 neutrons and 2 protons, and there are 10 protons for every 2 neutrons,

Deuterium & Beyond

Big Bang nucleosynthesis (BBN) begins with neutrons and protons combining to form deuterium via $p+n \rightleftharpoons {\rm D} +\gamma$, where ${\rm D} =~^2{\rm H}$. The equivalent expression for the Saha equation for deuterium production is

where $B_{\rm D} = 2.22~{\rm MeV}$ is the difference in masses. Following the same procedure as with decoupling, the temperature of deuterium nucleosynthesis $kT_{\rm nuc} \approx 0.066~{\rm MeV} \approx B_{\rm D}/34$ can be calculated, which corresponds to an age of the universe $t_{\rm nuc} \approx 200~{\rm s}$. This is when deuterium nucleosynthesis can begin.

Neutron Decay

There is another effect that matters on this timescale; free neutrons decay via $n \rightarrow p + e^- + \bar{\nu}_e$ with a decay time of $\tau_n \approx 880~{\rm s}$, where the number of neutrons remaining over time is given by $N_n(t) = N_n(t=0) \exp(-t/\tau_n)$. This is not negligible relative to the age of deuterium creation, which will change our estimate of $Y_{\rm max}$. While $N_n(t)$ falls, $N_p(t)$ grows by the same amount since each decayed neutron produces one proton, and by this time $Y_{\rm max}$ will consequently fall from 0.33 to 0.27.

Elements with higher atomic number

Once you have deuterium, other elements can be made via reactions between deuterium and protons and their products, which include helium-3 ($^3{\rm He}$), tritium ($^3{\rm H}$), and helium-4 ($^4{\rm He}$). These reactions don't require neutrinos, and so they happen quickly once there's enough deuterium around.

Helium-4 is more tightly bound than elements with higher atomic number, such as lithium and beryllium, so most of the neutrons end up in $^4{\rm He}$, which explains why the observed primordial value of the helium fraction, $Y_p \approx 0.24$ is so close to the updated maximum value. However, trace amounts of higher elements, plus leftover amounts of tritium and helium-3, are produced, the exact amounts of which are tricky to accurately compute. Modern theoretical estimates of the abundances of these elements and observed cosmic abundances are generally in agreement, although some discrepancies remain.

Matter-antimatter Asymmetry

Going back to very early times, when the universe was a quark soup, photons and quarks/anti-quarks collided and annihilated to produce each other:

According to our understanding of physics at these energies, this reaction should have no asymmetries to it, and all of the quark-antiquark pairs should have converted into photons once the temperature dropped below the quark mass scale (quark annihilations could still occur, but photons would no longer be able to create quark particle pairs). However, we exist, and the ratio of baryons to photons, $\eta \approx 6 \times 10^{-10}$, implies that for every 800,000,000 antiquarks in the early universe, there were 800,000,003 quarks, and all the matter we see around us today is made up of those leftover 3 quarks, equivalent to one proton or neutron, with 800,000,000 CMB photons for each baryon, which originated from those annihilations. The origin of this imbalance is unknown and there is surely a Nobel prize waiting for whoever discovers it.