In the mid-forties G.A. Gamov proposed the idea
of the "hot" origin of the World. Therefore thermodynamics
was introduced into cosmology, and nuclear physics too.
Before him the science of the evolution of Universe
contained only dynamics and geometry of the World.
A.D.Chernin.
The Standard Model particles and their properties.
| Particles
|
Mass
|
Number of states
|
$g$ (particles and anti-particles)
|
| spin
|
color
|
| Photon ($\gamma$) |
0 |
2 |
1 |
2
|
| $W^+,W^-$ |
$80.4\, GeV$ |
3 |
1 |
6
|
| $Z$ |
$91.2\,GeV$ |
3 |
1 |
3
|
| Gluon ($g$) |
0 |
2 |
8 |
16
|
| Higgs boson |
$>114\, GeV$ |
1 |
1 |
1
|
| Bosons
|
|
28
|
| $u,/,\bar{u}$ |
$3\, MeV$ |
2 |
3 |
12
|
| $d,/,\bar{d}$ |
$6\, MeV$ |
2 |
3 |
12
|
| $s,/,\bar{s}$ |
$100\,MeV$ |
2 |
3 |
12
|
| $c,/,\bar{c}$ |
$1.2\,GeV$ |
2 |
3 |
12
|
| $b,/,\bar{b}$ |
$4.2\, GeV$ |
2 |
3 |
12
|
| $t,/,\bar{t}$ |
$175\,GeV$ |
2 |
3 |
12
|
| $e^+,\,e^-$ |
$0.511\, MeV $ |
2 |
1 |
4
|
| $\mu^+,\,\mu^-$ |
$105.7\,MeV$ |
2 |
1 |
4
|
| $\tau^+,\,\tau^-$ |
$1.777\, GeV$ |
2 |
1 |
4
|
| $\nu_e,\,\bar{\nu_e}$ |
$<3\,eV$ |
1 |
1 |
2
|
| $\nu_\mu,\,\bar{\nu_\mu}$ |
$<0.19\,MeV $ |
1 |
1 |
2
|
| $\nu_\tau,\,\bar{\nu_\tau}$ |
$<18.2\, MeV $ |
1 |
1 |
2
|
| Fermions
|
|
90
|
6.1.1
Problem 6.1.1
Find the energy and number densities for bosons and fermions in the relativistic limit.
solution
The energy density equals to
$$
\rho = \left\{ \begin{array}{c}
\frac{\pi^2}{30}gT^4 \quad \; for\;bosons\\
\frac{7}{8}\frac{\pi^2}{30}gT^4 \quad for \;fermions
\end{array}
\right.
$$
Here $g$ equals to the number of internal degrees of freedom for the particle. For photon (with spin $S=1$) $g=2$ due to absence of longitudinal polarization for the zero-mass particles.
The number density is the following:
\[
n = \left\{ \begin{array}{c}
\frac{\zeta (3)}{\pi ^2 }gT^3 \quad \;for\;bosons \\
\frac{3}
{4}\frac{\zeta (3)}{\pi ^2 }gT^3 \quad \; for\;fermions
\end{array} \right.
\]
Here $\zeta (x)$ is the Riemann $\zeta$-function; $\zeta (3) \approx
1.202$.
6.1.2
Problem 6.1.2
Find the number densities and energy densities, normalized to the photon energy density at the same temperature, for neutrinos, electrons, positrons, pions and nucleons in the relativistic limit.
solution
\[
\frac{n_i}{n_\gamma} = \left\{
\begin{gathered}
\frac{1}{2}g_i \quad for\; bosons\\
\frac{3}{8}g_i \quad for\; fermions\\
\end{gathered}
\right.;\quad
\frac{\rho_i}{\rho_\gamma} = \left\{
\begin{gathered}
\frac{1}{2}g_i \quad for\; bosons\\
\frac{7}{16}g_i \quad for\; fermions\\
\end{gathered} \right.
\]
\[
\begin{gathered}
\left(\frac{n_i }{n_\gamma},\frac{\rho_i}{\rho_\gamma} \right)_{\nu_{\mu ,e}} = \left( \frac{3}{8},\frac{7}{16} \right);\;\;\left( \frac{n_i}{n_\gamma},
\frac{\rho_i}{\rho_\gamma} \right)_{e^\pm} = \left(
\frac{3}{4},\frac{7}{8} \right);\\
\left( \frac{n_i}{n_\gamma},\frac{\rho_i}{\rho_\gamma} \right)_\pi = \left( \frac{1}{2},\frac{1}{2} \right);\;\;\left( \frac{n_i}{n_\gamma},\frac{\rho_i}{\rho_\gamma} \right)_{n,p} = \left( \frac{3}{4},\frac{7}{8}
\right) \\
\end{gathered}
\]
6.1.3
Problem 6.1.3
Calculate the average energy per particle in the relativistic and non-relativistic limits.
solution
Average energy per particle reads$\left\langle \varepsilon \right\rangle = \rho /n$. For $T \gg m$
\[
\left\langle \varepsilon \right\rangle = \left\{
\begin{gathered}
\frac{\pi ^2}{30\zeta (3)}T\quad \; for\; bosons\\
\frac{7\pi ^4}{180\zeta(3)}T\quad for\; fermions\\
\end{gathered} \right.
\]
In the non-relativistic limit $(T \ll m)$ one obtains the following:
\[
\left\langle \varepsilon \right\rangle = m + \frac{3}{2}T \approx m
\]
6.1.4
Problem 6.1.4
Find the number of internal degrees of freedom for a quark.
solution
For quark one obtains the following:
\[
g = 2(\, spin) \times 3(\, particle) \times 2(\, particles + \, antiparticle) = 12
\]
6.1.5
Problem 6.1.5
Find the entropy density for bosons end fermions with zero chemical potential.
solution
In the considered case one gets:
\[
s = \frac{\rho + p}{T};\quad p = \frac{1}{3}\rho
\]
Using the results of problem of the present Chapter one obtains the following:
\[
s = \frac{2\pi ^2}{45}g^* T^3,
\]
where $g^* = 1$ for bosons and $g^* = 7/8$ for fermions.
6.1.6
Problem 6.1.6
Find the effective number of internal degrees of freedom for a mixture of relativistic bosons and fermions.
solution
\[
\rho = \sum\limits_{i = bosons} g_i \frac{\pi^2}{30} T^4 +
\sum\limits_{j = fermions} \frac{7}{8}g_j\frac{\pi^2}{30}
T^4 \frac{\pi^2}{30}g^* T^4,
\]
where $g^* $ is the effective number of degrees of freedom:
\[
g^* = \sum\limits_{i = bosons} g_i + \frac{7}{8}\sum\limits_{j
= fermions}g_j.
\]
6.1.7
Problem 6.1.7
Generalize the results of the previous problem for the case when some $i$-components have temperature $T_i$ different from the CMB temperature $T$.
solution
\[
g^* = \sum\limits_{i = bosons}g_i \left(\frac{T_i}{T} \right)^4 + \frac{7}{8}\sum\limits_{j = fermions}g_i \left( \frac{T_j}{T}\right)^4
\]
6.1.8
Problem 6.1.8
Find the effective number of internal degrees of freedom for the Standard Model particles at temperature $T>1TeV$.
solution
At temperature $1\, TeV$ all the Standard Model
particles are relativistic (see table). Therefore
\[
g^* = 28 + \frac{7}{8} \times 90 = 106.75.
\]
6.1.9
Problem 6.1.9
Find the change of the number of internal degrees of freedom due to the quark
hadronization process.
solution
At temperature $T \approx 0.2\, GeV$ quarks and gluons transform into colorless hadrons (protons and neutrons). Therefore
\[
\Delta g = 16\,(gluons) + 12 \times 6\,(quarks) -
8\,(nucleons) = 80.
\]
6.1.10
Problem 6.1.10
Find the relation between the energy density and temperature at
$10^{10}\, K$.
solution
Represent the expression for energy density of relativistic particles in the following form:
\[
\rho = \left( \sum\limits_{i = bosons} \frac{g_i}{2} +
\sum\limits_{j = fermions} \frac{7}{16}g_j \right)\rho
_\gamma.
\]
At the temperature under consideration the relativistic particles are presented by photons, electron, positrons and neutrinos. Therefore
\[
\rho = \left( 1 + \frac{7} {4} + \frac{7}{4} \right)\rho _\gamma
= \frac{9}{2}\rho _\gamma = \frac{9}{2}\alpha T^4,
\]
where the radiation constant \[\alpha =
\frac{\pi^2}{15}\frac{k^4}{(\hbar c)^3} \approx 7.56 \cdot
10^{-16}\frac{ J}{ m^3K^4}.\]
6.1.11
Problem 6.1.11
Find the ratio of the energy density at temperature $10^{10}\, K$ to that at $10^8\, K$.
solution
Unlike the case considered in the previous problem, now only photons and neutrinos remain relativistic, because $10^8 K \sim 10^{ - 2} \, MeV$, and electron mass is $0.511 \, MeV$ is considerably greater. Therefore in the case under consideration one obtains the following:
\[
\rho = \left( 1 + \frac{7}{4} \right)\rho _\gamma =
\frac{11}{4}\rho _\gamma \Rightarrow
\frac{\rho\left(T_2 = 10^{10} K \right)}{\rho \left( T_1 = 10^8 K \right)} = \frac{18}
{11}\left(\frac{T_2}{T_1}\right)^4\simeq1.6\cdot10^8.\]
6.1.12
Problem 6.1.12
Determine Fermi energy of cosmological neutrino background (CNB) (a gas).
solution
For a relativistic Fermi-gas the Fermi energy reads the following
$E_F = \hbar c\left( 3\pi ^2 n \right)^{1/3} ;$ $\hbar c \simeq 197.3 \, MeV\cdot \, fm$, $n \simeq 300 \, cm^{-3}$, so
$$E_F \simeq 0.4 \times 10^{-3} \, eV.$$