Solving the Boltzmann Equation
This page solves the Boltzmann equation for the dark matter comoving
number density and relic density. $$ \frac{dY}{dx} =
-\sqrt{\frac{\pi}{45}}\frac{m_{\chi}M_{pl}}{x^2}g_{*}^{1/2}\langle\sigma
v\rangle \left(Y^2-Y_{\mathrm{eq}}^{2}\right) $$ where: $$
\begin{align} Y &= n_{\chi} / s, & x &= m_{\chi} / T, &
\langle\sigma v\rangle &\sim \langle\sigma v\rangle_{0} x^{-n}
\end{align} $$
Glossary
- \(m_{\chi}\): Dark Matter mass
-
\(\langle\sigma v\rangle_{0}\): Leading-order coefficient of the
thermally averaged annihilation cross section
-
\(n\): Leading-order power in \(x\) of the thermally averaged
annihilation cross section
- \(T\): Temperature of the Standard Model bath
-
\(x\): Scaleless temperature (Dark Matter mass divided by Standard
Model temperature)
-
\(Y\): Comoving number density (number density \(n_{\chi}\)
divided by Standard Model entropy density \(s\))
- \(Y_{\mathrm{eq}}\): Equilibrium value of \(Y\)
-
\(M_{\mathrm{pl}}\): Plank mass (taken to be \(1.220910 \times
10^{19}\))
Use the input boxes below to changes the parameter values.