AcademyParticles and Cosmology
Academy
Early Universe
Level 1 - Physics topic page in Particles and Cosmology.
Principle
The early universe was hotter, denser, and more radiation dominated. As it expanded, it cooled, allowing particles, nuclei, atoms, and the cosmic microwave background to form.
Notation
\(T\)
temperature
\(\mathrm{K}\)
\(a\)
scale factor
1
\(z\)
redshift
1
\(k_B\)
Boltzmann constant
\(\mathrm{J\,K^{-1}}\)
\(E\)
typical particle energy
\(\mathrm{J,\;eV}\)
\(\rho\)
energy density
\(\mathrm{J\,m^{-3}}\)
Method
Derivation 1: Expansion cools radiation
Photon wavelengths stretch with the scale factor, so photon energies and radiation temperature fall as the universe expands.
Photon energy
\[E=\frac{hc}{\lambda}\]
Temperature scaling
\[T\propto\frac1a\]
Derivation 2: Redshift gives past temperature
Because \(1+z=a_0/a\), the temperature at redshift \(z\) was higher by the same factor.
Scale factor
\[1+z=\frac{a_0}{a}\]
Temperature-redshift relation
\[T(z)=T_0(1+z)\]
Derivation 3: Matter and radiation dilute differently
Matter density falls with volume, while radiation also loses photon energy through redshift.
Matter density
\[\rho_m\propto a^{-3}\]
Radiation density
\[\rho_r\propto a^{-4}\]
Rules
Temperature scaling
\[T\propto\frac1a\]
Redshift temperature
\[T(z)=T_0(1+z)\]
Thermal energy scale
\[E\sim k_BT\]
Examples
Question
If today's CMB temperature is
\[2.7\,\mathrm K\]
estimate the temperature at \[z=999\]
Answer
\[T=T_0(1+z)=2.7(1000)=2700\,\mathrm K\]
Checks
- Earlier times correspond to smaller \(a\), larger \(z\), and higher \(T\).
- Nucleosynthesis required hot dense conditions but not temperatures so high that nuclei were instantly broken apart.
- The CMB comes from the epoch when photons decoupled from matter.
- Radiation density falls faster than matter density during expansion.