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Big Bang Cosmology

Friday, April 22, 2022

Running Time Backwards

Right now, the universe is relatively cold and has a relatively low density. However, the universe has been expanding and cooling. This means it used to be hot and dense. It used to be so hot it could ionize atoms. Before then, free nucleons roamed, when the strong force dominated. Even earlier, the weak force played a significant role. However, before this, the universe was just quarks and leptons. Since we have never seen a free quark, we do not know how they interact and therefore cannot make predictions about the universe before $10^{-43}~\text{s}$ (called the Planck time). At that time, quantum effects and gravity were intertwined and none of our current theories work in that realm.

After the Planck time but before the condensation of bulk matter, the universe was in approximate thermal equilibrium at some temperature $T$. It was dominated by radiation, so using the equation for radiation density, we can find

$$T=\frac{1.5\times 10^{10}~\text{s}^{1/2}\cdot\text{K}}{t^{1/2}}$$

where $T$ is the temperature and $t$ is the time in seconds. High energy photons, of energy $kT$ interacted with matter in two ways:

$$\text{photons}\rightarrow\text{particle}+\text{antiparticle}\newline\text{particle}+\text{antiparticle}\rightarrow\text{photons}$$

$t=10^{-6}~\text{s}$

At one microsecond, we find that $T=1.5\times 10^{13}~\text{K}$. The universe at this time was the size of the solar system and consisted of only $\text{p}$, $\overline{\text{p}}$, $\text{n}$, $\overline{\text{n}}$, $\text{e}^-$, $\text{e}^+$, $\mu^-$, $\mu^+$, $\pi^0$, $\pi^-$, $\pi^+$, some other particles, and photons, neutrinos, and antineutrinos. Because of pair production, the number of particles roughly equals the number of antiparticles.

Additionally, the relative number of neutrons and protons is determined by

1. The Boltzmann factor $e^{-\Delta ElkT}$: since protons have less rest energy than neutrons, there are more of them at any given time.
2. Nuclear reactions: Protons and neutrons can turn into one another so long as there are enough electrons, positrons, neutrinos, and antineutrinos around.
3. Neutron decay: Neutrons have not had enough time to decay yet.

$t=10^{-2}~\text{s}$

At this time, the temperature is only about $T=1.5\times 10^{11}~\text{K}$. Photons now have too little energy to produce pions and muons, which have by this time decayed into electrons, positrons, and neutrinos. Pair production of nucleons no longer occurs. There is a very slight imbalance of matter to antimatter, making all the antimatter and most (99.9999999%) of matter annihilate.

$t=1~\text{s}$

At this time, the temperature drops to about $T=1.5\times 10^{10}~\text{K}$. Also at this time, the Boltzmann factor becomes different than 1, resulting in nucleons being about 73% protons and 27% neutrons. Neutrinos also lose much of their influence at this time since they no longer have enough energy to convert protons to neutrons. This is the start of "neutrino decoupling."

$t=6~\text{s}$

By six seconds, the temperature of the universe is about $T=6\times 10^9~\text{K}$. Photons do not have enough energy for pair production anymore and electrons do not have enough energy to convert protons to neutrons. Therefore, the only processes to switch nucleons are decay processes. Nucleons are now about 84% protons and 16% neutrons.

After 6 Seconds

The universe after six seconds consists of about $N$ protons, $N$ electrons, and $0.2N$ neutrons. There are about $10^9N$ photons and about that many neutrinos as well.