# LOG#110. Basic Cosmology (V).

## Recombination

When the Universe cooled up to $T\sim eV$, 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: $e+\gamma \leftrightarrow e+\gamma$

2) Coulomb scattering: $e^-+p\leftrightarrow H+\gamma$

Then, there were little hydrogen (H) and though $B_H>T$ due to small baryon fraction $\eta_b$.

The evolution of the free electron fraction provided the ratio

$X_e\equiv =\dfrac{n_e}{n_e+n_H}=\dfrac{n_p}{n_p+n_H}$

where $n_p+n_H\approx n_b$ and the second equality is due to the neutrality of our universe, i.e., to the fact that $n_e=n_p$ (by charge conservation). If $e^-+p\longrightarrow H+\gamma$ remains in the thermal equilibrium, then

$\dfrac{n_en_p}{n_H}=\dfrac{n_e^0n_p^0}{n_H^0}\longrightarrow \dfrac{X_e^2}{1-X_e}=\dfrac{1}{n_e+n_H}\left[\left(\dfrac{m_eT}{2\pi}\right)^{3/2}e^{-\left[m_e+m_p-m_H\right]/T}\right]$

where we have

$\dfrac{1}{n_e+n_H}\left(\dfrac{m_eT}{2\pi}\right)^{3/2}=n_p+n_H=n_b-4n(He)\approx n_p+n_H=n_b=\eta_b\eta_\gamma$

It gives $\eta_b\eta_\gamma\sim 10^{-9}T^3\approx 10^{15}$

and the last equality is due to the fact we take $T\sim E_0$. It means that $X_e\approx 1$ at $T\sim E_0$. As we have $X_e\longrightarrow 0$, we are out of the thermal equilibrium.

From the Boltzmann equation, we also get

$a^{-3}\dfrac{d(n_ea^3)}{dt}=n_e^0n_p^0\langle \sigma v\rangle \left( \dfrac{n_Hn_\gamma}{n_H^0n_\gamma^0}-\dfrac{n_e^2}{n_e^0n_p^0}\right)$

or equivalently

$a^{-3}\dfrac{d(n_ea^3)}{dt}=n_b\langle \sigma v\rangle \left(\dfrac{n_H}{n_b}\dfrac{n_e^0n_p^0}{n_H^0}-\dfrac{n_e^2}{n_b}\right)$

i.e.

$a^{-3}\dfrac{d(n_ea^3)}{dt}=n_b\langle \sigma v\rangle \left( (1-X_e)\left(\dfrac{m_eT}{2\pi}\right)^{3/2}e^{-E_0/T}-X_e^2n_b\right)$

Using that $n_e=n_bX_e$ and $\dfrac{d}{dt}(n_ba^3)=0$, we obtain

$\dfrac{dX_e}{dt}=\left[(1-X_e)\beta -X_e^2n_b\alpha^{(2)}\right]$

with

$\beta\equiv \langle \sigma v\rangle \left(\dfrac{m_eT}{2\pi}\right)^{3/2}e^{-E_0/T}$, the ionization rate, and

$\alpha^{(2)}\equiv \langle \sigma v\rangle$ 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

$\alpha^{(2)}=9\mbox{.}78\dfrac{\alpha^2}{m_e^2}\left(\dfrac{E_0}{T}\right)^{1/2}\ln \left(\dfrac{E_0}{T}\right)$

The numerical integration produces the following qualitative figure

The decoupling of photons from the primordial plasma is explained as

$\mbox{Compton scattering rate}\sim\mbox{Expansion rate}$

Mathematicaly speaking, this fact implies that

$n_e\sigma_T=X_en_b\sigma_T$

where $\sigma_T$ is the Thomson cross section. For the processes we are interesting in, it gives

$\sigma_T=0\mbox{.}665\cdot 10^{-24}cm^2$

and then

$n_e\sigma_T=7\mbox{.}477\cdot 10^{-30}cm^{-1}X_e\Omega_bh^2a^{-3}$

Thus, we deduce that

$\dfrac{n_e\sigma_T}{H}=0\mbox{.}0692a^{-3}X_e\Omega_bh\dfrac{H_0}{H}$

$\dfrac{n_e\sigma_T}{H}=113X_e\left(\dfrac{\Omega_bh^2}{0\mbox{.}02}\right)\left(\dfrac{0\mbox{.}15}{\Omega_mh^2}\right)^{1/2}\left(\dfrac{1+z}{1000}\right)^{3/2}\left[1+\dfrac{1+z}{3600}\dfrac{0\mbox{.15}}{\Omega_mh^2}\right]^{-1/2}$

and where $X_e\leq 10^{-2}$ 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

$\Omega_{DM}\sim 0\mbox{.}2-0\mbox{.}3$

The most plausible candidate for DM are the Weakly Interacting Massive Particles (WIMPs, for short). Generic WIMP scenarios provide annihilations

$X_{DM}+\bar{X}_{DM}\leftrightarrow l+\bar{l}$

where $X_{DM}$ 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

$a^{-3}\dfrac{d(n_Xa^3)}{dt}=\langle \sigma_X v\rangle \left( n_X^{(0)2}-n_X^2\right)$

Define the yield (or ratio) $Y_X=\dfrac{n_X}{T^3}$. It is produced since generally we have

$Y=\dfrac{n_X}{s}$

and since $sa^3=constant$, then $s\propto T^3$. Thus,

$\dfrac{dY}{dt}=T^3\langle \sigma v\rangle \left( Y_{EQ}^2-Y^2\right)$

and

$Y_{EQ}=\dfrac{n_X^0}{T^3}$

Now, we can introduce a new time variable, say

$x=\dfrac{m}{T}$

Then, we calculate

$\dfrac{dx}{dt}=-\dfrac{m}{T^2}\dfrac{dT}{dt}=-\dfrac{m}{T^2}\left(-\dfrac{\dot{a}}{a}T\right)=xH$

For a radiation dominated (RD) Universe, $\rho\propto T^4$ implies that $H\propto T^2$ and $H(x)=-\dfrac{H(m)}{x^2}$

In this case, we obtain

$\dfrac{dY}{dx}=\dfrac{\lambda}{x^2}\left(Y^2-Y_{EQ}^2\right)$

with $\lambda=\dfrac{m^3\langle \sigma v\rangle}{H(m)}$

The final freeze out abundance is got in the limit $Y_\infty=Y(x\longrightarrow \infty)$. Typically, $\lambda >>1$, and when $Y_{EQ}\sim 1$ and $Y\approx Y_{EQ}$, for $x>>1$, and there, the yield drops exponentially

$\dfrac{dY}{dx}\approx \dfrac{\lambda Y^2}{x^2}$

or

$\dfrac{dY}{Y^2}\approx \dfrac{\lambda dx}{x^2}$

Integrating this equation,

$\displaystyle{\int_{Y_f}^{Y_\infty}\dfrac{dY}{Y^2}=\int_{x_f}^\infty \dfrac{\lambda}{dx}{x^2}}$

and then

$\dfrac{1}{Y_\infty}-\dfrac{1}{Y_f}=\dfrac{\lambda}{x_f}$

Generally, $Y_f>>Y_\infty$ and the freeze out temperature for WIMPs is got with the aid of the following equation

$Y_\infty=\dfrac{x_f}{\lambda}$

Indeed, $n\langle \sigma v\rangle= H\longrightarrow x_f\sim 10$

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

$\rho_X=mY_\infty T_0^3\left(\dfrac{a_1T_1}{a_0T_0}\right)^3\approx \dfrac{mY_\infty T_0^3}{30}$

and where the effect of entrpy dumping after the freeze-out is encoded into the factor

$\left(\dfrac{a_1T_1}{a_0T_0}\right)^3$ with $\left(\dfrac{g_\star (0)}{g_\star (f)}\right)^3\approx \dfrac{1}{30}$

Moreover, the DM energy density can also be estimated:

$\Omega_X=\Omega_{DM}=\dfrac{x_f}{\lambda}\dfrac{mT_0^3}{30\rho_c}=\dfrac{H (m) x_fT_0^3}{30m^2\langle \sigma v\rangle\rho_c}$

so

$\Omega_X=\left[\dfrac{4\pi^3Gg_\star (m)}{45}\right]^{1/2}\dfrac{x_fT_0^3}{30\langle \sigma v\rangle \rho_c}=0\mbox{.}3h^{-2}\left(\dfrac{x_f}{10}\right)\left(\dfrac{g_\star (m)}{100}\right)^{1/2}\dfrac{10^{-39}cm^2}{\langle \sigma v\rangle}$

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!

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