Boltzmann equation plays important roles in particle cosmology in studying the evolution of distribution functions (also called as occupation numbers) of various particles. The success of the standard cosmology or cold dark matter (ΛCDM) essentially relies on the use of the Boltzmann equation in an expanding Universe. In cosmology, however, the stimulated emission or Pauli blocking effect is usually omitted.

Scalar fields may form condensation; examples of such scalar fields include inflaton, curvaton, axion, Affleck-Dine field for the baryogenesis, and so on. (Hereafter, such a scalar condensation is called φ.) If φ can decay into a pair of bosonic particles (called χ) as φ → χχ, the effects of the stimulated emission can be important since the daughter particles may be enormously populated. In particular, today itself, I wrote an article on a mechanism of producing bosonic dark matter from the inflaton decay, in which a stimulated emission of the bosonic dark matter plays an important role.

Such systems have been studied by employing the Boltzmann equation or in the context of the parametric resonance. In particular, particle production from an oscillating scalar field has been intensively investigated, particularly using the Mathieu equation. In the previous studies, the evolution equation of the expectation value of the field operator (which we call wave function) for the final-state particle is converted to the Mathieu equation, based on which the occupation number of the final state particle has been estimated. Then, it has been shown that there exist resonance bands and that the occupation numbers in the resonance bands grow exponentially. In the broad resonance region, it has been known that the particle production is non-perturbative and that the process cannot be described by the Boltzmann equation. On the contrary, sometimes it has been argued that the Boltzmann equation can provide a proper description in the narrow resonance regime.

Now, Moroi and Yin, in this paper, have studied the particle production from an oscillating scalar field (φ) in the narrow resonance regime (or assuming that the final state particle (χ) is very weakly interacting). They have paid particular attention to the consistency of the results from the Boltzmann equation and those from the QFT calculation. They have concentrated on the case that the production of χ is via the process φ → χχ.

“We study the particle production including the possible enhancement due to a large occupation number of the final state particle, known as the stimulated emission or the parametric resonance.”— told Moroi, first author of the study.

First, they have considered particle production in the flat spacetime. In such a case, they have discussed the evolution of the occupation number of each mode (i.e., mode with a fixed momentum *k*) separately in the narrow resonance regime. They have derived the evolution equations for the occupation number of each mode based on the QFT. A resonance band shows up at ω_k close to 1/2 ×m_φ, which corresponds to the lowest resonance band in the context of the parametric resonance. The modes within the resonance band can be effectively produced. For the timescale much longer than (qmφ)¯^{1}, the occupation numbers of the modes in the resonance band exponentially grow; the growth rate obtained by them in analysis is consistent with that given by the study of the parametric resonance using the Mathieu equation. Then, comparing the occupation number obtained from the QFT calculation with that from the Boltzmann equation, they have found that they do not agree well when the occupation number is larger than ∼ 1. On the contrary, when f_k << 1, they have found a good agreement of two results. They have also argued how their evolution equation based on the QFT could be related to the ordinary Boltzmann equation. When the occupation number is larger than ∼ 1, some of the approximation and assumption necessary for such an argument cannot be justified, which, they expect, causes the disagreement.

Then, they have studied particle production taking into account the effects of cosmic expansion. With the cosmic expansion, the physical momentum redshifts. The momentum of each mode stays in the resonance band for a finite amount of time and then exits the resonance band. The exponential growth of the occupation number occurs only in the resonance band. The growth factor has been studied numerically and analytically, adopting the evolution equations based on the QFT. The agreement between numerical and analytical results is excellent. They have also analyzed the system by using the conventional Boltzmann equation and found that the growth rate obtained by solving the Boltzmann equation is a factor of 2 larger than that based on the QFT. Thus, the occupation number from the Boltzmann equation may become exponentially larger than that from the QFT, and a naive use of the conventional Boltzmann equation may result in a significant overestimation of the number density of χ.

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In this paper, we have considered the production of a bosonic particle, concentrating on the lowest resonance band of the parametric resonance. Consideration of the production of fermionic particles and the study of the higher resonance bands, as well as the use of the evolution equations based on the QFT to other phenomena, are left as future works.“— concluded authors of the study.

**Reference***: Takeo Moroi, Wen Yin, “Particle Production from Oscillating Scalar Field and Consistency of Boltzmann Equation”, ArXiv, pp. 1-23, 2021. **https://arxiv.org/abs/2011.12285*

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