Thermal and Non-Thermal dark matter I
====> Thermal Dark Matter Equilibrium reactions of dark matter particles with Forward-rate $=$ Backward-rate -- Probability of interacting in time t is: Number-density$\times$ cross-section $\times$ velocity $\times$ time -- reaction rate: Number-density$\times$ cross-section $\times$ velocity -- So $\Gamma =n <\sigma v>$ distribution function = $f_A = \frac{1}{e^{(E - u)/T} \pm 1}$ --non-relativistic:: $n^{eq} = \frac{g}{2 \pi^3}\int f(p) d^3p$ --relativistic:: $n^{eq} = \frac{\xi(3)}{\pi^2} g T^4$ multiplied by 1 if bosons or 3/4 if fermions For non-relativistic $\Gamma_{inelastic}$ ~$T^{\frac{3}{2}}e^{\frac{m}{T}} <\sigma v>$ Forrelativistic $\Gamma_{elastic}$ ~$T^3 <\sigma v> $ $\Gamma_{inelastic} \neq \Gamma_{elastic}$ -- Can identify with Hubble rate to get "freeze-out " values! ----- Kinetic decoupling ----- One process dominates and continues after the other shuts off --- Consider delta in tim