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 $tc\sim t\ce{H}$

    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