Abstract
Once the Universe has terminated its inflationary phase, and produced a thermal bath through the Inflaton decay, its evolution follows the evolution of a classical plasma in an expanding Universe. The law undergoes (relativistic) statistical laws and one can apply our knowledge of this field to the production and decoupling of elements, from neutrino to dark matter. We propose in this chapter to review in details the thermal evolution of the primordial plasma, and the possibility to produce weakly interacting massive particles (WIMP) from it.
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Notes
- 1.
By default when one talks about temperature, T represents the photon temperature T γ, or more generically, the temperature of the thermal bath.
- 2.
The computation of the degrees of freedom will be detailed in Sect. 3.1.5.
- 3.
For a more technical explanation of the factor \(\frac {1}{3}\) originated from an integration of \(\cos ^2 \theta \) (θ being the scattering angle), see Eq. (2.203).
- 4.
This assumption is obviously not valid in the case of the Standard Model, since all its particles have internal degrees of freedom.
- 5.
See Sect. A.5 for the demonstration of this law called Laplace’s law.
- 6.
Another way to understand it is to imagine a square box made of sides corresponding to n i = 0 and n i = 1. Each side of the box has a size of h∕L, so the box has a volume of h 3∕L 3. However, each of the eight corners of the box corresponds to a state, pertaining to 8 boxes surrounding it, so counting as 1∕8 of states par box. Then, a box of 8 corners contains 8 × 1∕8 = 1 state in its h 3∕L 3 volume. In other words, Δp x Δp y Δp z∕(h 3∕L 3) represents the number of states [number of boxes] having momentum \(\vec p\) in the range (Δp x, Δp y, Δp z).
- 7.
It is important to note that this hypothesis will no longer be valid when it comes to the quark-hadrons phase transition, where the strong interactions confine quarks inside protons, neutrons, or pions.
- 8.
This is true in the early Universe, but not at a later time when eventually the rest masses of the particles left over from annihilation begin to dominate and we enter a matter dominated era, see Sect. 3.1.6.
- 9.
Notice that the 2 degrees of freedom of the gravitino h μν and the 2 degrees of freedom of its partner the gravitino are not counted here because they do not participate to the thermal bath.
- 10.
At least two neutrino flavors being nowadays non-relativistic from the oscillations measurements, see the text in the box below Eq. (3.90) for a discussion on the subject.
- 11.
The assumption of a weakly interacting gas of particles still holds for the baryons and the mesons, but not for the individual quarks and gluons.
- 12.
Keeping in mind the possibility that all the three families of neutrino are massive and thus do not contribute at all to the radiation density.
- 13.
Be careful to not be confused with the ratio η b = n b∕n γ ≃ 6 × 10−10. Indeed, a photon energy is around 10−4 eV at present time, whereas a baryon mass is 1 GeV. ρ b∕ρ γ ≃ 1000 gives you n b∕n γ ≃ 10−10. The number density of the photon is much larger than the number density of the baryons nowadays, this is the baryogenesis concept.
- 14.
The fact of taking into account three species of neutrino, even if it is known from the measurement of Δm atm and Δm sol that some are non-relativistic nowadays, comes from the fact that, once decoupled, the massive neutrinos still follow a classical relativistic distribution function (see Sect. 3.2 for more details).
- 15.
The analysis does not depend on the nature –fermionic/bosonic– of the particles we consider.
- 16.
Notice that σ scat v SM = σ scat as the SM particles are still largely relativistic in the thermal bath : v SM = c.
- 17.
Notice that we made the supposition in this section that the dark matter bath is in thermal equilibrium with itself (with a temperature T′) or from the self-scattering of the dark matter on itself or from the scattering of other particles of the dark sector on themselves, i.e. Γscatter(T) > H(T). For a more detailed description of the mechanism of thermalization, see the next section.
- 18.
We took Dirac fermions for the dark matter and particles 1 and 2: g 1 = g 2 = g χ = 4.
- 19.
Two degrees of freedom for the photon, 2 fermionic states for the electron, plus 2 for the positrons.
- 20.
Notice that the mass density of a massive neutrino does not follow the Boltzmann suppressed evolution of classical massive particles as neutrino have already decoupled from the thermal bath, and there density thus always follows the T 3 evolution by number of particle conservation (n ν(t) × a(t) ∝ n(t) × T −3 = cst), a(t) being the scale factor of the Universe.
- 21.
We keep in this paragraph the notation m ν for the dark matter mass because it was the first candidate of that sort proposed in the literature and still can be under the form of a sterile neutrino, for instance.
- 22.
- 23.
Which corresponds to 3 active neutrinos, e ± and photons.
- 24.
with g e = g p = 2 (spin 1/2) and g H = 4 (combination of two spins 1/2).
- 25.
The Thomson scattering is the low energy limit of the Compton scattering. Both effects are a diffusion on the electromagnetic field by a charged particle. In the CMB, the electron being non-relativistic, the Thompson limit is a valid one.
- 26.
\(\frac {d[n_A \times a^3]}{dt}=0\) when \(n_A=n_A^{eq}\).
- 27.
More precisely, the particles A talk to the photons but cannot hear them.
- 28.
〈σv〉 has been normalized to a typical electroweak cross section for a 100 GeV particle: 10−9 GeV−2 = 1.2 × 10−26cm3 s−1, Eq. (3.154).
- 29.
We define the relative velocity between two particles i and j by \(v_{ij} = \frac {\sqrt {(p_i.p_j)^2-m_i^2 m_j^2}}{E_i E_j}\), with p i and E i being four-momentum and energy of particle i.
- 30.
See the Sect. 3.5.5.1 for another way to lead the integration for the mean 〈σv〉.
- 31.
We must be careful that the solid angle d Ω is the one between the outgoing particles m 3 and m 4 in the center of mass of the colliding particles, to not confuse with the solid angle of the colliding particles \(\cos \theta \) in which we perform the statistical average.
- 32.
For instance, by Pete Hut [9].
- 33.
- 34.
As an indication, \(Y_{\infty } \simeq 3.3 \times 10^{-12} \left ( \frac {100~\mathrm {GeV}}{m}\right ) \left (\frac {\Omega h^2}{0.1} \right )\).
- 35.
See the Sect. 3.1.8 for a deeper understanding of this decoupling condition.
- 36.
To be more precise, \(\Lambda \sim \sqrt {M_Pm_{3/2}}\), m 3∕2 being the gravitino mass.
- 37.
Supersymmetric partner of the gluon.
- 38.
After using \(Y_\infty \simeq 3.3 \times 10^{-12} \left ( \frac {100~\mathrm {GeV}}{m} \right ) \left ( \frac {\Omega h^2}{0.1} \right )\) from Eq. (3.198).
- 39.
This remark is also valid in the case where the dominant process is the annihilation one.
- 40.
We have incorporated in g A the multiplicity of the decay rate as A decays into 2 dark matter particles.
- 41.
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Mambrini, Y. (2021). A Thermal Universe [T RH → T CMB]. In: Particles in the Dark Universe. Springer, Cham. https://doi.org/10.1007/978-3-030-78139-2_3
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