Recombination
When the Universe cooled up to 
, the neutrinos decoupled from the primordial plasma (soup). Protons, electrons and photons remained tighly coupled by 2 main types of scattering processes:
1) Compton scattering: 
2) Coulomb scattering: 
Then, there were little hydrogen (H) and though 
due to small baryon fraction 
.
The evolution of the free electron fraction provided the ratio
where 
and the second equality is due to the neutrality of our universe, i.e., to the fact that 
(by charge conservation). If 
remains in the thermal equilibrium, then
![\dfrac{n_en_p}{n_H}=\dfrac{n_e^0n_p^0}{n_H^0}\longrightarrow \dfrac{X_e^2}{1-X_e}=\dfrac{1}{n_e+n_H}\left[\left(\dfrac{m_eT}{2\pi}\right)^{3/2}e^{-\left[m_e+m_p-m_H\right]/T}\right]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-ecae159d2fa5771052dd11181d3636ea_l3.png "Rendered by QuickLaTeX.com")
where we have
It gives 
and the last equality is due to the fact we take 
. It means that 
at . As we have 
, we are out of the thermal equilibrium.
From the Boltzmann equation, we also get
or equivalently
i.e.
Using that 
and 
, we obtain
![\dfrac{dX_e}{dt}=\left[(1-X_e)\beta -X_e^2n_b\alpha^{(2)}\right]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-f6a516834c06dd447af6a75e918a4918_l3.png "Rendered by QuickLaTeX.com")
with
, the ionization rate, and
the so-called recombination rate. It is taken the recombination to the n=2 state of the neutral hydrogen. Note that the ground state recombination is NOT relevant here since it produces an ionizing photon, which ionizes a neutral atom, and thus the net effect is zero. In fact, the above equations provide
The numerical integration produces the following qualitative figure
The decoupling of photons from the primordial plasma is explained as
Mathematicaly speaking, this fact implies that
where 
is the Thomson cross section. For the processes we are interesting in, it gives
and then
Thus, we deduce that
![\dfrac{n_e\sigma_T}{H}=113X_e\left(\dfrac{\Omega_bh^2}{0\mbox{.}02}\right)\left(\dfrac{0\mbox{.}15}{\Omega_mh^2}\right)^{1/2}\left(\dfrac{1+z}{1000}\right)^{3/2}\left[1+\dfrac{1+z}{3600}\dfrac{0\mbox{.15}}{\Omega_mh^2}\right]^{-1/2}](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-48d0f93f4897337d24a1519fb0f7fa6e_l3.png "Rendered by QuickLaTeX.com")
and where 
implies that the decoupling of photons occurs during the time of recombination! In fact, the decoupling of photons at time of recombination is what we observe when we look at the Cosmic Microwave Background (CMB). Fascinating, isn’t it?
Dark Matter (DM)
Today, we have strong evidences and hints that non-baryonic dark matter (DM) exists (otherwise, we should modify newtonian dynamics and or the gravitational law at large scales, but it seems that even if we do that, we require this dark matter stuff).
In fact, from cosmological observations (and some astrotronomical and astrophysical measurements) we get the value of the DM energy density
The most plausible candidate for DM are the Weakly Interacting Massive Particles (WIMPs, for short). Generic WIMP scenarios provide annihilations
where 
is some “heavy” DM particle and the (ultra)weak interaction above produces light particles in form of leptons and antileptons, tighly couple to the cosmic plasma. The Boltzmann equation gives
Define the yield (or ratio) 
. It is produced since generally we have
and since 
, then 
. Thus,
and
Now, we can introduce a new time variable, say
Then, we calculate
For a radiation dominated (RD) Universe, 
implies that 
and 
In this case, we obtain
with 
The final freeze out abundance is got in the limit 
. Typically, 
, and when 
and 
, for 
, and there, the yield drops exponentially
or
Integrating this equation,
and then
Generally, 
and the freeze out temperature for WIMPs is got with the aid of the following equation
Indeed, 
A qualitative numerical solution of the “WIMP” miracle (and its freeze out) is given by the following sketch
The present abundance of heavy particle relics gives
and where the effect of entrpy dumping after the freeze-out is encoded into the factor
with 
Moreover, the DM energy density can also be estimated:
so
![\Omega_X=\left[\dfrac{4\pi^3Gg_\star (m)}{45}\right]^{1/2}\dfrac{x_fT_0^3}{30\langle \sigma v\rangle \rho_c}=0\mbox{.}3h^{-2}\left(\dfrac{x_f}{10}\right)\left(\dfrac{g_\star (m)}{100}\right)^{1/2}\dfrac{10^{-39}cm^2}{\langle \sigma v\rangle}](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-ad407007e8dd63b69691ce7c0ed06277_l3.png "Rendered by QuickLaTeX.com")
The main (current favourite) candidate for WIMP particles are the so called lightest supersymmetric particles (LSP). However, there are other possible elections. For instance, Majorana neutrinos (or other sterile neutrino species), Z prime bosons, and other exotic particles. We observe that here there is a deep connection between particle physics, astrophysics and cosmology when we talk about the energy density and its total composition, from a fundamental viewpoint.
Remark: there are also WISP particles (Weakly Interacting Slim Particles), like (superlight) axions and other exotics that could contribute to the DM energy density and/or the “dark energy”/vacum energy that we observe today. There are many experiments searching for these particles in laboratories, colliders, DM detection experiments and astrophysical/cosmological observations (cosmic rays and other HEP phenomena are also investigated to that goal).
See you in a next cosmological post!
