Thermal and Non-Thermal dark matter I

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====> 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 time in kinetic decoupling


Using the Boltzmann equation, it is then trivial to compute the abundance of cold dark matter today


https://cds.cern.ch/record/2702846/files/938-4114-1-PB.pdf

https://en.wikipedia.org/wiki/Matter_power_spectrum

https://en.wikipedia.org/wiki/Boltzmann_equation

https://www.ippp.dur.ac.uk/~dcerdeno/Dark_Matter_Lab_files/DM.pdf


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