# LOG#110. Basic Cosmology (V).

**Posted:**2013/06/23

**Filed under:**Cosmology, Physmatics |

**Tags:**Boltzmann equation, Compton scattering, Coulomb scattering, dark matter, free electron fraction evolution, LSP, photon decoupling, recombination, WIMP, WISP Leave a comment

## 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

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

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

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

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!**