# Chapter 7 Thermal History of the Universe

## Intro

### Average density of photons

* The total energy density of a black body is

  $$
  u=aT^4
  $$

  where $a={4\sigma}/{c}=7.57\times10^{-15}\text{ erg}\text{ cm}^{-3}\text{ K}^{-4}$ is the radiation constant
* Average energy of a photon (black body)

  $$
  \langle u\rangle=\frac{\int h\nu B\_\nu(T)\text d\nu}{\int B\_\nu(T)\text d\nu}=2.70 k T
  $$

where $k=1.38 \times 10^{-16} \text { erg } \mathrm{K}^{-1}$ is Boltzmann constant

* Number density of CMB photons (which make up most of the photons at present) today

  $$
  n\_{\gamma, 0}=\frac{u}{\langle u\rangle}=\frac{a}{2.70 k} T^{3}=4.1 \times 10^{2} \mathrm{cm}^{-3}
  $$

### Average density of baryons

$$
n\_{\mathrm{b}, 0}=\frac{\Omega\_{\mathrm{b}, 0} \cdot \rho\_{\mathrm{crit}}}{\left\langle m\_{\mathrm{b}}\right\rangle}=\frac{0.0488 \cdot 8.5 \times 10^{-30} \mathrm{g}\cdot \mathrm{cm}^{-3}}{1.67 \times 10^{-24} \mathrm{g}}=2.5 \times 10^{-7} \mathrm{cm}^{-3}
$$

* $\Omega\_{\text b,0}=0.0488$, which is a small fraction compared to the dark matter

### Density ratio of photons and baryons

* Number density

  $$
  \frac{n\_{\mathrm{b}, 0}}{n\_{\gamma, 0}}=6.1 \times 10^{-10}
  $$
* Energy density

  $$
  \frac{u\_{\mathrm{b}, 0}}{u\_{\gamma, 0}}=\frac{n\_{\mathrm{b}}}{n\_{\gamma}} \frac{\left\langle m\_{\mathrm{b}}\right\rangle c^{2}}{\langle u\rangle}=9.0 \times 10^{2}
  $$

  If we include non-relativistic cold dark matter (matter) and relativistic neutrinos (radiation)

  $$
  \frac{u\_{\mathrm{m}, 0}}{u\_{\mathrm{rad}, 0}} \simeq 3.4 \times 10^{3}
  $$

  So the present universe is **matter dominated**
* However, in the past, since

  $$
  \rho\_{\mathrm{m}}=\rho\_{0}(1+z)^{3}\\
  \rho\_{\mathrm{rad}}=\rho\_{0}(1+z)^{4}
  $$

  we find the redshift that these two density equal

  $$
  z\_{eq}\approx3370
  $$

  before which the **radiation dominated**

### The evolution of temperature

* $\lambda\_{max}T$ is a constant

  $$
  T\_\text{CMB}=2.73(1+z)\text{ K}=2.73a^{-1}\text{ K}
  $$
* At early universe, the typical photons were sufficiently energetic that they interacted strongly with matter - the temperature was dominated by the radiation

  The evolution of scale factor

  $$
  a(t)=\sqrt{\frac{t}{t\_0}}\Rightarrow t(\mathrm{s})=\left(\frac{T}{1.5 \times 10^{10}\ \mathrm{K}}\right)^{-2}=\left(\frac{T}{1.3\ \mathrm{MeV}}\right)^{-2}
  $$

## The Universe at $t<1\text{ s}$

### Planck time

* According to the formula

  $$
  T=2.73/a\ (\text{K})
  $$

  $T$ diverges as $a\to\infty$
* This extrapolation fails when the wavelength associated with the particle approaches its Schiwarzschild radius

  $$
  \lambda=\frac{2\pi\hbar}{mc}=\pi r\_s=\frac{2\pi Gm}{c^2}
  $$

  * **Planck mass**

    $$
    m\_{\mathrm{P}}=\left(\frac{\hbar c}{G}\right)^{1 / 2} \simeq 10^{19}\ \mathrm{GeV}
    $$
  * **Planck length**

    $$
    l\_{\mathrm{P}}=\frac{\hbar}{m\_{\mathrm{P}} c}=\left(\frac{\hbar G}{c^{3}}\right)^{1 / 2} \simeq 10^{-33}\ \mathrm{cm}
    $$
  * **Planck time**

    $$
    t\_{\mathrm{P}}=\frac{l\_{\mathrm{P}}}{c}=\left(\frac{\hbar G}{c^{5}}\right)^{1 / 2} \simeq 10^{-43}\ \mathrm{s}
    $$

    At $t\approx t\_P$, classical spacetime dissolves into a **foam** of quantum black holes - Quantum gravity limit

### Freeze-out

* Interaction rate (1/s): $\Gamma$
* Expansion rate of the universe: $H$
  * If $\Gamma\gg H$, then the timescale of particle interactions is much smaller than the characteristic expansion timescale

    $$
    t\_{\mathrm{c}}=\frac{1}{\Gamma} \ll t\_{\mathrm{H}}=\frac{1}{H}
    $$

    and **local thermal equilibrium** is built
  * If $t*c\sim t*\ce{H}$

    The particles in question decouple from the thermal plasma

### Major milestones

#### Planck time $t\sim10^{-43}\text{ s}$

* Four fundamental interactions were united before Planck time
* Gravity became separate from the other three forces
* Beginning of GUT era

#### GUT transition $t \sim 10^{-35} \mathrm{s}$

$$
T \sim 10^{27}\ \mathrm{K} \simeq 10^{14}\ \mathrm{GeV}
$$

* The Electroweak and Strong forces emerged
* Quarks (that interact mostly through the strong force) and leptons (which interact mostly through the weak force) and their anti-particles acquired individual identities

#### End of GUT era

**Inflation $t\sim10^{-36}-10^{-34}\text{ s}$**

* Horizon problem (causal link)
* Flatness problem
* Monopole problem

**Baryogenesis**

* Absence of anti-matter

#### Electro-Weak transition $t \sim 10^{-12} \mathrm{s}$

$$
T \sim 10^{15}\ \mathrm{K} \simeq 100\ \mathrm{GeV}
$$

* The electromagnetic and weak forces become separate
* Leptons acquired mass
* Corresponding bosons appeared (and decayed at the temperature corresponding to their mass)
* Baryongensis continued

#### QCD (Quark-Hadron) transition $t \sim 10^{-6} \mathrm{s}$

$$
T \sim 10^{12-13}\ \mathrm{K} \simeq 200\ \mathrm{MeV}-1\ \mathrm{GeV}
$$

* Quarks can no longer exist on their own. They combine into hadrons (baryons and mesons), glued together by gluons (strong force bosons)
* Quark confinement commences

## The Universe at $t>1\text{ s}$

* The Lepton era $t \sim 10^{-6} \text { to } t \sim \text { a few } \mathrm{s}$
* $T\le1\text{ GeV}$, the physics is well understood and accessible to experimental verification with particle accelerators (CERN)

### Decoupling of Neutrinos $t \simeq 1 \mathrm{s}$

$$
T \simeq 1\ \mathrm{MeV} \simeq 10^{10}\ \mathrm{K}
$$

* The energy is much less then the rest-mass of protons and neutrons
* All the baryons that exist today must have already been present when the Universe was one millionth of a second old
* So long as the reaction rate is faster than the expansion rate, electrons, neutrinos and their anti-particles and photons are kept in equilibrium by the following reactions

  * Compton scattering

    $$
    \ce{e^{\pm} + \gamma <-> e^{\pm} + \gamma }
    $$
  * Pair production and annihilation

    $$
    \ce{e+ + e- <-> {\gamma} + \gamma}
    $$
  * Neutrino-antineutrino scattering

    $$
    \ce{\nu + \bar{\nu} <-> e+ + e-}
    $$
  * Neutrino-electron scattering

    $$
    \ce{\nu + e^{\pm} <-> \nu + e^{\pm} }
    $$

  The reactions involving neutrinos are mediated by the weak force

  $$
  \frac{\Gamma}{H} \simeq\left(\frac{T}{1.6 \times 10^{10} \mathrm{K}}\right)^{3}
  $$

  When $T$ falls below $10^{10}\text{ K}$, the neutrinos are no longer in equilibrium and **decouple** from the rest of the plasma
* At freeze-out the neutrinos are still relativistic with a thermal distribution at the same temperature as the electrons and photons that remained in mutual equilibrium
* The neutrinos have kept their thermal distribution to the present day, with $T\propto 1/a$ - difficult to detect

### Electron-Positron Annibilation $t\simeq 5\text{ s}$

$$
T\sim500\text{ keV}\sim 5\times10^9\text{ K}
$$

* The number density of photons with energies above the pair production threshold is insufficient to maintain the reaction $\ce{e+ + e- <-> {\gamma} + \gamma}$, and that the reaction proceeds preferentially in the right direction
* Only a small number of electrons are left to balance the protons produced by baryogenesis - electrically neutral universe
* Photon gas is re-heated (not neutrinos, since they are no longer in thermal equilibrium with photons, except for some high energy neutrinos)
* After annihilation

  $$
  T\_{\nu}=\left(\frac{4}{11}\right)^{1 / 3} T\_{\gamma}
  $$

  which has been maintained ever since, $T\_{\nu,0}=0.71 \times 2.73=1.95 \mathrm{K}$

## Equilibrium Thermodynamics

* See in any textbook of statistical mechanics
* Energy density for (relativistic)

  * Bosons

    $$
    u=\frac{g}{2} a T^{4}
    $$
  * Fermions

    $$
    u=\frac{7}{8} \frac{g}{2} a T^{4}
    $$

  The total energy density of the mixture of photons (2 polarization states), electrons, positrons (two spin freedoms), neutrinos and antineutrinos (only one helicity state) at time $t\sim1 \text{ s}$ is thus

  $$
  u=\rho(T) c^{2}=c^{2} \sum \rho\_{\mathrm{i}}(T)=\frac{1}{2} g\_{\*} a T^{4}
  $$

  where

  $$
  g\_{\*}=\sum\_{\text {bosons}} g\_{i}+\frac{7}{8} \sum\_{\text {fermions}} g\_{j}=2+\left(2\cdot\frac78+2\cdot\frac78\right)+2\cdot\frac78\mathcal N\_\nu
  $$

  $\mathcal N\_\nu$ is the number of neutrino families