Abstract
The inflation and reheating phases of the Universe concern a period where the Universe changes very quickly from a vacuum/constant density domination to an oscillation/matter domination and then a radiation domination. These transitions are not only fast but also violent. We will analyze in detail each of these phases, insisting on the possibility of producing dark matter before reaching the thermal equilibrium. But before, one needs to understand the equation that will lead our expanding Universe: the Hubble law.
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Notes
- 1.
Hubble never had the Nobel prize because cosmology was not recognized as a scientific discipline until his death in 1953.
- 2.
A matter dominated Universe is called a “dust” Universe. It is a Universe composed of matter whose pressure p is negligible compared to its energy density ρ.
- 3.
A more complete statement can be found in Sect. 3.1.2.
- 4.
The redshift is the propensity for a relativistic particle to have its wavelength changed in an expanding Universe or if it is emitted by a moving source. See Sect. 2.1.4.3 for more details.
- 5.
Whereas t e and t 0 are usually very spaced: R(t e) ≠ R(t 0).
- 6.
Or 13.8 Gyrs −380, 000 years to be more precise because the CMB took place 380,000 years after the Big Bang as we will discuss in Sect. 3.3.4.
- 7.
In fact, all the G ii conditions are the same due to the isotropic principle: none of the 3 directions can be distinguished from the others.
- 8.
See Appendix A.2.4 for details.
- 9.
During the phase of inflation, the energy–momentum tensor should depend on the dynamics of the scalar inflaton as we will see in Sect. 2.2.
- 10.
There are subtleties in this argument. One way to see it is to think that in the rest frame of a perfect fluid, a particle with a velocity \(\frac {dx_\theta }{d\tau }\), for instance, will have a counterpart of another particle of velocity \(-\frac {dx_\theta }{d\tau }\), canceling the velocity part of the stress–energy tensor T ii (2.37).
- 11.
We remind the reader that by convention, the radius of the Universe at a time t is written R(t) = a(t)R 0, where R 0 is the present radius of the Universe, of the order of 46 Gpc.
- 12.
A slightly more precise computation commonly used in the literature gives λ = 46.3 × 109 lyrs.
- 13.
To be more precise, one should take into account the physical horizon distance, i.e. the actual distance traveled by the light, which is given by Eq. (2.65), but this approximation is quite valid for the argument.
- 14.
The same result can be obtained by computing the entropy of the Universe in the very early time.
- 15.
That is the phase during which the Universe evolves the more, compared to the matter or dark energy period.
- 16.
Even if the original model of Starobinsky had no scalar, it was shown to be equivalent to a scalar theory.
- 17.
We remind the reader that in our convention of the flat metric, g 00 = g 00 = +1 and g ii = g ii = −1.
- 18.
We will not consider a decaying ϕ at this stage. This is justified because if \(\tilde \Gamma _\phi \) is comparable to the Hubble rate, then the slow-roll regime is not valid anymore: the inflaton decays too fast to obtain the needed 67 e-folds.
- 19.
To be more precise, as we will see in the next section, the inflaton will behave like a dust for a quadratic potential V (ϕ), whereas it will behave like a radiation for a quartic potential.
- 20.
For a detailed study on the subject, see [6].
- 21.
That is almost constant during the whole process.
- 22.
Keeping in mind that we can always decompose S in its fundamental modes, \(S=\sum _k s_ke^{i\vec k. \vec x}\), all contributions add up to the number density n or energy density ρ.
- 23.
This situation corresponds to a narrow resonance if one considers μ ≪ Φ0. The regime μ ≳ Φ0 is a broad resonance regime but exhibits similar features [7].
- 24.
Be careful in the following notations; n s represents the density number of particle S (per unit of space volume), whereas n k represents the occupation number (no units).
- 25.
- 26.
Due to the relative large mass of the inflaton (m ϕ ≃ 1013 GeV), Standard Model particles produced by its decay are obviously ultra-relativistic and can be considered as a form of radiation.
- 27.
ρ M ∝ a −3(1+w) with w = 0 and p = wρ.
- 28.
We make an important hypothesis of a constant number of degrees of freedom in the relativistic bath during all the process. That will not be the case once the Universe cools down as some particles may decouple as we will discuss in the next chapter.
- 29.
Notice that for Γϕ = 0, we recover the classical evolutions (ρ ϕ ∝ a −3, ρ R ∝ a −4) for independent systems with entropy conservation, the a −4 corresponding to the redshifted energy \(E_R (a) = \frac {E_{R} (a_f)}{a}\), and a f being the radius of the Universe just after the inflation.
- 30.
It is possible to find in the scientific literature the parameter v = t × Γϕ to keep the dynamical parameter dimensionless.
- 31.
We will consider in all this section the time t end as being the time just at the end of the inflation, at the beginning of the coherent oscillation.
- 32.
Suppose a coupling to bosons is also possible.
- 33.
Alternatively, one can also draw a loop diagram with two Higgs and a fermion which will give the effective decay \(\phi \rightarrow \overline f f\).
- 34.
It is important to point out that the radiation has thermalized well before t reh.
- 35.
By simplicity, we considered a 2-body decay in this section.
- 36.
When calculating g ρ, one should be careful between the fermionic and bosonic states, as one can see in Eq. (3.26): g ρ = 106.75 (α ≃ 35) for the Standard Model, and g ρ = 213.5 (α ≃ 70) in supersymmetry.
- 37.
To see it easily, one can use uncertainty principle of Heisenberg, for instance, where the number of states N f is \(\frac {d^3x d^3p}{h^3}\), and ħ = 1 ⇒ h = 2π, which implies \(N_f = \frac {V~d^3 p}{(2 \pi )^3}\).
- 38.
Notice that the dark matter particle is never in thermal equilibrium with the plasma during the whole production process.
- 39.
That is the case as the photons and/or Standard Model particles are the 1 and 2 particles, much lighter than the energy at the reheating time, and therefore massless.
- 40.
There are different reasons to consider massless particles: we are in ultra-high energy regimes, the particles are thus ultra-relativistic, and in the meantime, the electroweak phase transition giving masses to the Standard Model sector has not yet occurred.
- 41.
Or more precisely, the branching ratio of the width into Standard Model particles. Decays into dark or hidden sectors are allowed as we will see in the following section.
- 42.
\(H(t)= \frac {\dot a }{ a} \Rightarrow d a = a ~ Hdt\), so during \(dt=\frac {1}{H(t)}, da=a\): the radius of, the Universe has doubled in size.
- 43.
See Sect. 2.3 for the case of non-instantaneous thermalization.
- 44.
We considered T i = 0, in other words, no other thermal sources outside from the inflaton decay.
- 45.
The approximation we need to apply here is that we considered \(H(T_{RH})= \frac {\sqrt {\rho _\phi =\alpha T^4_{RH}}}{\sqrt {3}M_P}\), where we neglected the radiation contribution in ρ because we supposed a Universe still dominated by ρ ϕ. Using \(H(T_{RH})=\frac {\sqrt {\rho \phi +\rho _R}}{\sqrt {3}M_P}= \frac {\sqrt {2 \rho _R}}{\sqrt {3}M_P}\) will also lead to a misleading factor of \(\sqrt {2}\) because the formulae (2.282) could not be applied if the Universe is not anymore dominated by the inflaton field. The exact numerical solution lies between these two approximations.
- 46.
And approximated 1.7 by \(\frac {3}{2}\) to have simplified expressions.
- 47.
The power of T has been chosen so that n corresponds to the dependence of the cross section in the temperatures: \(R(T) \propto n_{SM}^2 \langle \sigma v \rangle \propto T^6 \langle \sigma v \rangle \).
- 48.
In the literature, it is very common to define \(Y = \frac {n}{s}\) and common to define \(Y = \frac {n}{n_\gamma }\). In any case, the dependence in both cases is of the form \(Y = \mathrm {cte} \times \frac {n}{T^3}\). As we will deal with epochs where (locally) the entropy will not be conserved (and thus the definition of s is more difficult), we prefer to define \(Y = \frac {n}{T^n}\) in this section, with n = 3 in a Universe with local entropy conservation.
- 49.
And integrating the coefficient of proportionality in the definition of Λ by simplicity.
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Mambrini, Y. (2021). Inflation and Reheating [M P → T RH]. In: Particles in the Dark Universe. Springer, Cham. https://doi.org/10.1007/978-3-030-78139-2_2
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