Abstract
We discuss the thermodynamics of elementary particles in the early Universe. We formulate the conditions for the establishment of thermal equilibrium and we discuss the properties of the canonical equilibrium distribution functions, including the conditions for the formation of Bose-Einstein condensates in cosmology. We compute frozen densities of massless and massive particles. It is proven that the equilibrium distribution of massless particles is not distorted by the cosmological expansion. On the other hand, it is shown that massless neutrinos in the early Universe deviate from equilibrium.
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- 1.
We can still use Eq. (5.11) in the case of one or more decoupled relativistic components with possible different temperatures, but now \(g_*\) is
where the first summation is over all the boson species and the second one over all the fermion species, \(g_i\) is the number of spin states of the species i, and \(T_i\) is the temperature (or effective temperature in the case of decoupled particles) of the species i.
- 2.
If all the components of the plasma have the same temperature, \(g_* = g_*^s\). If this is not the case, we have the counterpart of Eq. (5.12) and \(g_*^s\) is
which may be different from \(g_*\).
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Problems
Problems
5.1
Derive Eq. (5.9).
5.2
Derive Eq. (5.10).
5.3
Derive Eq. (5.16). [Hint: the calculations of integrals with the equilibrium distribution functions are discussed in detail in Landau and Lifshitz (1980).]
5.4
Check that the equilibrium Bose and Fermi distribution functions (5.4) annihilate the collision integral (5.19) if the T-invariance is not broken.
5.5
Show that the Bose condensed distribution functions (5.25) annihilate the collision integral (5.19) if and only if \(\mu =m\).
5.6
Why in the distribution function p and t are taken as independent variables, while in Eq. (5.30) we treated the momentum as a function of the time, namely \({p = p(t)}\)?
5.7
Find the frozen number densities of protons and electrons in a charge-symmetric universe. [Answer: \({n_p/n_\gamma \approx 10^{-19}}\), \({n_e/n_\gamma \approx 10^{-16}}\).]
5.8
What number density would have antiprotons if \((n_p-n_{\bar{p}})/n_\gamma = 10^{-9}\)?
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Bambi, C., Dolgov, A.D. (2016). Kinetics and Thermodynamics in Cosmology. In: Introduction to Particle Cosmology. UNITEXT for Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48078-6_5
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