Boltzmann Equation and Its Cosmological Applications

Abstract

We review the derivation of the Boltzmann equation and its cosmological applications in this paper. A novel derivation of the Boltzmann equation, especially the collision term, is discussed in detail in the language of quantum field theory without any assumption of a finite temperature system. We also discuss the integrated Boltzmann equation, incorporating the temperature parameter as an extension of the standard equation. Among a number of its cosmological applications, we mainly target two familiar examples, the dynamics of the dark matter abundance through the freeze-out/in process and a baryogenesis scenario. The formulations in those systems are briefly discussed with techniques in their calculations.

1. Introduction

The early universe, composed of hot plasma, evolves in time while maintaining thermal equilibrium at most epochs, and thus the evolution is described simply by thermodynamics. The cosmologically important events, therefore, are focused on the various turning epochs in which some particle species depart from the thermal bath, as seen in, e.g., Big Bang nucleosynthesis (BBN) and recombination confirmed by both the theory and observations, and the dark matter (DM) relic abundance and the baryogenesis scenario derived from some hypotheses. Qualitatively, the turning epochs can be estimated by comparing the reaction rate and the spatial expansion rate of the universe, but more quantitative treatment to ensure accurate predictions is required by the present cosmological observations.

The Boltzmann equation is a powerful tool for following the out-of-equilibrium dynamics in detail, and can describe the evolution of the particle distribution due to the spatial expansion and the momentum exchanges through interactions. Since the Boltzmann equation provides the abstract relation between the macroscopic and the microscopic evolution of the particle distributions, it is applicable to various situations, and hence, the solving equations differ for each system.

The most successful application of the Boltzmann equation is for the theory of BBN [1,2,3,4,5,6,7,8] (see also, e.g., [9,10,11]), which describes how the initial protons and neutrons form the light elements at the end. As a result, the abundances of the light elements are predicted, and they are fairly consistent with the present observations. The other well-motivated and formulated example is the Kompaneets equation [12,13,14], which is derived from the Boltzmann equation in the photon–electron system. The evolution equation is applied to various analyses in cosmological and astronomical contexts, such as the last scattering surface during the recombination epoch or the Sunyaev–Zel’dovich effect [15,16,17,18], which is the spectral distortion of the Cosmic Microwave Background radiation caused by hot electrons in galaxy clusters.

On the other side, exploring beyond the standard model (BSM), there are many studies on the DM candidate realizing the present density parameter through the relic abundance, which can be predicted using the Boltzmann equation. The most famous and simple scenario for DM candidates is described by weakly interacting massive particles (WIMPs) [19,20], but the current observation requires a more extended scenario or the other mechanism, e.g., feebly interacting massive particles (FIMPs) [21], strongly interacting massive particles (SIMPs) [22], processing into forbidden channels [23,24], co-annihilations [23], co-scattering [25,26], zombie [27,28], and inverse-decays [29,30], etc. (See also [31] for their summarized analysis.) The more complicated models are considered, the more technical treatment in the Boltzmann equation is required. As the other topic for exploring BSM with the Boltzmann equation, there are also many studies for baryogenesis scenarios explaining the origin of (baryonic) matter–antimatter asymmetry in our universe, e.g., GUT baryogenesis [32,33,34,35,36,37], leptogenesis [38], electroweak baryogenesis [39,40,41], and Affleck–Dine baryogenesis [42,43], etc. The analysis tends to be more complicated than the case of DM abundance and requires more technical treatment.

Recent advances have broadened the use of the Boltzmann equation in cosmology. Precision calculations of relic neutrino decoupling have refined predictions [44], underscoring its continued role in linking particle interactions to cosmological observables. The Boltzmann framework also underpins studies of chiral anomalies, magnetic fields, and baryon asymmetry [45], as well as freeze-in production with time-dependent condensates [46], reflecting its expanding relevance in modern cosmology.

The studies using the Boltzmann equation have been more popular and important as approaches to confirm the current physics in detail and explore BSM in the early universe. We aim to provide a clearer understanding of the Boltzmann equation and its techniques with some cosmological examples. This review highlights a novel derivation of the Boltzmann equation directly and accessibly. Especially, the derivation of the collision term is carefully discussed in Section 2.2, starting from quantum field theory under the assumption that the S-matrix is well defined. We emphasize that, once the transition amplitude is known, the Boltzmann equation is derived naturally without any finite temperature system. This approach makes the physical assumptions and approximations transparent, and thus, may serve as a practical entry point for cosmologists and particle physicists interested in applying the Boltzmann framework to out-of-equilibrium phenomena in the early universe. In contrast, the standard derivation of the Boltzmann equation, based on the so-called Kadanoff–Baym equation [47,48], can describe the system more accurately. We do not deal with it in detail to evade the complicated discussion in this paper. For readers who want to apply the Kadanoff–Baym equation, see Section 2.5 and the references there.

This paper is organized as follows. First, we derive the Boltzmann equation by the language of the quantum field theory in Section 2. Next, we demonstrate application examples of how the Boltzmann equation is applied for the DM abundance in Section 3 and the baryogenesis scenario in Section 4 with simple toy models. Finally, we summarize our discussion in Section 5.

2. Boltzmann Equation

The Boltzmann equation is quite a powerful tool for describing the evolution of particles in cosmology. Although the equation is applicable to various situations, the formulae used are also dependent on the circumstances. In this section, we derive the evolution formula of the distribution function $f=f(x^{\mu },p^{\mu })$ from the basic statement

$$\mathbf{L}\left[f\right]=\mathbf{C}\left[f\right]$$

(1)

where $\mathbf{L}$ and $\mathbf{C}$ are called the Liouville operator and the collision operator, respectively. The Liouville operator describes the variation of the distribution of a particle along a dynamical parameter such as a free-streaming evolution under an external geometry. The collision operator describes the source of the variation through the microscopic processes except for the external geometry considered in the Liouville operator.

We note that the structural separation between the Liouville operator and the collision operator reflects a semiclassical perspective: the former encodes classical (gravitational) geometry as the long-distance force, while the latter contains the quantum process as the short-distance force. If one wants to follow the dynamics under the classical electro-magnetic field, that must be included in the Liouville operator such as the Lorentz force (in many cosmological contexts, however, the electromagnetic force effectively acts as the short-distance force because of the shielding by the other charged plasma). If one wants to consider quantum gravity, the collision terms associated with the graviton exchanges appear, although quantum gravity is beyond the scope of this review.

2.1. Liouville Operator

We define the Liouville operator to describe the variation of the distribution along the geodesic line parametrized by the affine parameter $\lambda$. Using the momentum relations

$$\begin{array}{c}\hfill p^{\mu }={\displaystyle \frac{d{x}^{\mu }}{d\lambda }},p^{\mu }{p}_{\mu }=m^2\end{array}$$

(2)

where $p^{\mu }$ is a four-momentum and m is the mass, and the geodesic equation

$$\begin{array}{c}\hfill 0={\displaystyle \frac{d{p}^{\mu }}{d\lambda }}+{\Gamma }_{\nu \rho }^{\mu }{p}^{\nu }{p}^{\rho }\end{array}$$

(3)

where ${\Gamma }_{\nu \rho }^{\mu }$ is the affine connection, the Liouville operator can be written as

$$\begin{array}{ccc}\hfill \mathbf{L}\left[f\right]\equiv {\left.{\displaystyle \frac{df}{d\lambda }}\right|}_{\mathrm{geodesic}\mathrm{line}}& =& {\displaystyle \frac{d{x}^{\mu }}{d\lambda }}{\partial }_{\mu }f+{\displaystyle \frac{d{p}^{\mu }}{d\lambda }}{\displaystyle \frac{\partial f}{\partial p^{\mu }}}\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& =& p^{\mu }{\partial }_{\mu }f-{\Gamma }_{\nu \rho }^{\mu }{p}^{\nu }{p}^{\rho }{\displaystyle \frac{\partial f}{\partial p^{\mu }}}.\hfill \end{array}$$

(5)

In order to derive the concrete representation in the spatial homogeneous and isotropic universe, we apply the FLRW metric $g_{\mu \nu }=\mathrm{diag}(1,-a^2,-a^2,-a^2)$, where $a=a\left(t\right)$ is the scale factor. As a result, the distribution function is also spatially homogeneous and isotropic: $f=f(t,E)$. Here, $E=\sqrt{|\overrightarrow{p}{|}^2+m^2}$ is the single-particle energy defined by the physical momentum ${\left(\overrightarrow{p}\right)}^i\equiv a{p}^i$ and the mass m. Then, the concrete representation of the Liouville operator term can be written as

$$\begin{array}{c}\hfill \mathbf{L}\left[f\right]=E\dot{f}-H{\left|\overrightarrow{p}\right|}^2{\displaystyle \frac{\partial f}{\partial E}}.\end{array}$$

(6)

2.2. Collision Operator

This section highlights our novel points of the derivation of the collision term based on quantum field theory. We define the collision operator $\mathbf{C}\left[f\right]$ as the variation rate of the microscopic processes:

$$\begin{array}{c}\hfill \mathbf{C}\left[f\right]\equiv {\left.{\displaystyle \frac{df}{d\lambda }}\right|}_{\mathrm{microscopic} \mathrm{process}}={\left.E{\displaystyle \frac{df}{dt}}\right|}_{\mathrm{microscopic} \mathrm{process}}.\end{array}$$

(7)

In order to evaluate the variation of the distribution function, we assume the following. First, the process can be described by the quantum field theory. Second, the microscopic process can be evaluated on the Minkowski space-time since the gravitational effect is already evaluated in the Liouville operator part.

With the above assumption, the distribution function f can be regarded as the expectation value of all possible occupation numbers with their probabilities, as we will see later. Also, the probabilities can be evaluated by the quantum field theory on the flat space. To derive the concrete representation of (7), we need to construct the corresponding quantum state and then evaluate the transition probability.

2.2.1. Eigenstate for Occupation Number

At first, we construct a multi-particle state $\left|\right\{n\}\rangle$ in order to include all information about the particle occupations. We impose $\left|\right\{n\}\rangle$ to be the eigenstate satisfying

$$\begin{array}{c}\hfill {\widehat{n}}_a\left(\overrightarrow{k}\right)|\left\{n\right\}\rangle =n_a\left(\overrightarrow{k}\right)|\left\{n\right\}\rangle ,\langle \left\{n\right\}|\left\{n\right\}\rangle =1,\end{array}$$

(8)

where ${\widehat{n}}_a\left(\overrightarrow{k}\right)$ and $n_a\left(\overrightarrow{k}\right)$ are the occupation operator and its corresponding occupation number for species $a\in \left\{n\right\}$ in the unit phase space, respectively. Here, the occupation operator is defined by

$${\widehat{n}}_a\left(\overrightarrow{k}\right)\equiv {\displaystyle \frac{1}{V}}{a}_{\overrightarrow{k}}^{\left(a\right)†}{a}_{\overrightarrow{k}}^{\left(a\right)},$$

(9)

where $V\equiv \int d^{3}x={\left(2\pi \right)}^3{\delta }^3(\overrightarrow{k}=0)$ is a volume of the system, and $a_{\overrightarrow{k}}^{\left(a\right)}$ is an annihilation operator for species a which satisfies the commutation/anti-commutation relations

$$\left.\begin{array}{cc}\mathrm{bosons}:\hfill & [a_{\overrightarrow{k}}^{\left(a\right)},a_{\overrightarrow{p}}^{\left(b\right)†}]\\ \mathrm{fermions}:\hfill & \{a_{\overrightarrow{k}}^{\left(a\right)},a_{\overrightarrow{p}}^{\left(b\right)†}\}\end{array}\right\}={\delta }^{ab}·{\left(2\pi \right)}^3{\delta }^3(\overrightarrow{k}-\overrightarrow{p}),\left(\mathrm{others}\right)=0.$$

(10)

where $[,]$/$\{,\}$ denotes the commutator/anti-commutator. Then, one can obtain the representation of the eigenstate $\left|\right\{n\}\rangle$ by

$$\begin{array}{c}\hfill |\left\{n\right\}\rangle \equiv \prod _{a\in \left\{n\right\}}\left(\prod _{\overrightarrow{p}}{\displaystyle \frac{1}{\sqrt{n_a!}·\sqrt{V^{n_a}}}}{(a_{\overrightarrow{p}}^{\left(a\right)†})}^{n_a}\right)|0\rangle .\end{array}$$

(11)

Note that the occupation number $n_a\left(\overrightarrow{k}\right)$ must be an integer.

Furthermore, it is convenient to define the increased/decreased state from $\left|\right\{n\}\rangle$ for later discussion. We define them by

$$|\left\{n\right\};{\overrightarrow{k}}_a^{(+1)}\rangle ={\displaystyle \frac{1}{\sqrt{1\pm n_a}\sqrt{V}}}{a}_{\overrightarrow{k}}^{\left(a\right)†}|\left\{n\right\}\rangle ,\left(+:\mathrm{bosons},-:\mathrm{fermions}\right),$$

(12)

$$|\left\{n\right\};{\overrightarrow{k}}_a^{(-1)}\rangle ={\displaystyle \frac{1}{\sqrt{n_a}\sqrt{V}}}{a}_{\overrightarrow{k}}^{\left(a\right)}|\left\{n\right\}\rangle .$$

(13)

The coefficients are chosen to be unit vectors:

$$\langle \left\{n\right\};{\overrightarrow{k}}_a^{(\pm 1)}|\left\{n\right\};{\overrightarrow{k}}_a^{(\pm 1)}\rangle =1.$$

(14)

These increased/decreased states also become the eigenstate of the occupation operator:

$${\widehat{n}}_a\left(\overrightarrow{k}\right)|\left\{n\right\};{\overrightarrow{k}}_a^{(+1)}\rangle =\left(1\pm n_a\left(\overrightarrow{k}\right)\right)|\left\{n\right\};{\overrightarrow{k}}_a^{(+1)}\rangle ,\left(+:\mathrm{bosons},-:\mathrm{fermions}\right),$$

(15)

$${\widehat{n}}_a\left(\overrightarrow{k}\right)|\left\{n\right\};{\overrightarrow{k}}_a^{(-1)}\rangle =\pm \left(n_a\left(\overrightarrow{k}\right)-1\right)|\left\{n\right\};{\overrightarrow{k}}_a^{(-1)}\rangle ,\left(+:\mathrm{bosons},-:\mathrm{fermions}\right).$$

(16)

Especially, for the bosonic state of the multi-particle excited, the N-increased/decreased state can be defined by

$$\begin{array}{ccc}\hfill |\left\{n\right\};{\overrightarrow{k}}_a^{(+N)}\rangle & =& \sqrt{{\displaystyle \frac{n_a!}{(n_a+N)!}}{\displaystyle \frac{1}{V^N}}}·{\left(a_{\overrightarrow{k}}^{\left(a\right)†}\right)}^N|\left\{n\right\}\rangle ,\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill |\left\{n\right\};{\overrightarrow{k}}_a^{(-N)}\rangle & =& \sqrt{{\displaystyle \frac{(n_a-1)!}{(n_a+N-1)!}}{\displaystyle \frac{1}{V^N}}}·{\left(a_{\overrightarrow{k}}^{\left(a\right)}\right)}^N|\left\{n\right\}\rangle .\hfill \end{array}$$

(18)

2.2.2. Transition Probability

Using the eigenstates discussed in the previous section, let us consider the transition probability of the process

$$\begin{array}{c}\hfill A,B,\dots \to X,Y,\dots \end{array}$$

(19)

in the background in which other particles ($\left\{n\right\}$) exist. For simplicity, we consider a case in which each species is different. Taking the initial state as $\left|\right\{n\}\rangle$ in order to begin the given occupation numbers, the final state through the process (19) can be represented as $|\left\{n\right\};{\overrightarrow{k}}_A^{(-1)},{\overrightarrow{k}}_B^{(-1)},\dots ,{\overrightarrow{k}}_X^{(+1)},{\overrightarrow{k}}_Y^{(+1)},\dots \rangle$. The probability from the infinite past (in-state) to the infinite future (out-state) on the background particles can be evaluated by

$$\begin{array}{ccc}\hfill \mathcal{P}{(A,B,\dots \to X,Y,\dots )}_{\left\{n\right\}}& =& \prod _{a=A,B,\dots ,X,Y,\dots }\left(V\int {\displaystyle \frac{d^3{\overrightarrow{k}}_a}{{\left(2\pi \right)}^3}}\sum _{g_a}\right)\hfill \\ & & \times {\left|\langle \left\{n\right\};{\overrightarrow{k}}_A^{(-1)},{\overrightarrow{k}}_B^{(-1)},\dots ,{\overrightarrow{k}}_X^{(+1)},{\overrightarrow{k}}_Y^{(+1)},\dots |\widehat{S}|\left\{n\right\}\rangle \right|}^2\hfill \end{array}$$

(20)

where $g_a$ denotes the internal degrees of freedom for species a, and $\widehat{S}$ is the S-matrix operator. If the initial or the final state includes N of the same species, and the extra factor ${\displaystyle \frac{1}{N!}}$ for each duplicated species is needed. The S-matrix element describing the process (19) without the background particles can be represented by the invariant scattering amplitude as

$$\begin{array}{ccc}\hfill {}_{\mathrm{inv}}\langle k_X,k_Y,\dots |\widehat{S}|k_A,k_B,\dots \rangle _{\mathrm{inv}}& =& i\mathcal{M}(k_A,k_B,\dots \to k_X,k_Y,\dots )\hfill \\ & & \times {\left(2\pi \right)}^4{\delta }^4(k_A+k_B+\dots -k_X-k_Y-\dots )\hfill \end{array}$$

(21)

where

$$\begin{array}{c}\hfill |k_a,k_b{,\dots \rangle }_{\mathrm{inv}}\equiv \sqrt{2E_{k_a}2E_{k_b}\dots }{a}_{{\overrightarrow{k}}_a}^{†}{a}_{{\overrightarrow{k}}_b}^{†}\dots |0\rangle \end{array}$$

(22)

is a Lorentz invariant particle state. The representation (21) indicates that the S-matrix operator includes

$$\begin{array}{ccc}\hfill \widehat{S}& \supset & \prod _{a=A,B,\dots ,X,Y,\dots }\left(\int {\displaystyle \frac{d^3{\overrightarrow{k}}_a^{\prime }}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{\sqrt{2E_a^{\prime }}}}\sum _{g_a}\right)\times a_{{\overrightarrow{k}}_X^{\prime }}^{†}{a}_{{\overrightarrow{k}}_Y^{\prime }}^{†}\dots a_{{\overrightarrow{k}}_A^{\prime }}{a}_{{\overrightarrow{k}}_B^{\prime }}\dots \hfill \\ & & \times i\mathcal{M}(k_A^{\prime },k_B^{\prime },\dots \to k_X^{\prime },k_Y^{\prime },\dots )·{\left(2\pi \right)}^4{\delta }^4(k_A^{\prime }+k_B^{\prime }+\dots -k_X^{\prime }-k_Y^{\prime }-\dots ).\hfill \end{array}$$

(23)

Using the above expression, the S-matrix element on the particle background can be written as

$$\begin{array}{ccc}& & \langle \left\{n\right\};{\overrightarrow{k}}_A^{(-1)},{\overrightarrow{k}}_B^{(-1)},\dots {\overrightarrow{k}}_X^{(+1)},{\overrightarrow{k}}_Y^{(+1)},\dots |\widehat{S}|\left\{n\right\}\rangle \hfill \\ & & =\prod _{a=A,B,\dots ,X,Y,\dots }\left(\int {\displaystyle \frac{d^3{\overrightarrow{k}}_a^{\prime }}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{\sqrt{2E_A^{\prime }}}}\sum _{g_a}\right)\hfill \\ & & \times i\mathcal{M}(k_A^{\prime },k_B^{\prime },\dots \to k_X^{\prime },k_Y^{\prime },\dots )·{\left(2\pi \right)}^4{\delta }^4(k_A^{\prime }+k_B^{\prime }+\dots -k_X^{\prime }-k_Y^{\prime }-\dots )\hfill \end{array}$$

(24)

$$\begin{array}{ccc}& & \times \langle \left\{n\right\};{\overrightarrow{k}}_A^{(-1)},{\overrightarrow{k}}_B^{(-1)},\dots {\overrightarrow{k}}_X^{(+1)},{\overrightarrow{k}}_Y^{(+1)},\dots |a_{{\overrightarrow{k}}_X^{\prime }}^{†}{a}_{{\overrightarrow{k}}_Y^{\prime }}^{†}\dots a_{{\overrightarrow{k}}_A^{\prime }}{a}_{{\overrightarrow{k}}_B^{\prime }}\dots |\left\{n\right\}\rangle \hfill \\ & & ={\displaystyle \frac{1}{\sqrt{2E_{A}V·2E_{B}V·\dots 2E_{X}V·2E_{Y}V·\dots }}}\hfill \\ & & \times i\mathcal{M}(k_A,k_B,\dots \to k_X,k_Y,\dots )·{\left(2\pi \right)}^4{\delta }^4(k_A+k_B+\dots -k_X-k_Y-\dots )\hfill \end{array}$$

(25)

$$\begin{array}{l}\times \sqrt{n_A{n}_B\dots (1\pm n_X)(1\pm n_Y)\dots }\end{array}$$

(26)

where $+/-$ is for bosonic/fermionic particles of the produced species $X,Y,\dots$. Substituting the above form into (20), one can obtain

$$\begin{array}{ccc}\hfill \mathcal{P}{(A,B,\dots \to X,Y,\dots )}_{\left\{n\right\}}& =& \prod _{a=A,B\dots ,X,Y,\dots }\left(\int {\displaystyle \frac{d^3{\overrightarrow{k}}_a}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{2E_{k_a}}}\sum _{g_a}\right)\hfill \\ & & \times {\left|\mathcal{M}(k_A,k_B,\dots \to k_X,k_Y,\dots )\right|}^2\hfill \\ & & \times {\left(2\pi \right)}^4{\delta }^4(k_A+k_B+\dots -k_X-k_Y-\dots )·VT\hfill \\ & & \times n_A{n}_B\dots (1\pm n_X)(1\pm n_Y)\dots \hfill \end{array}$$

(27)

where $T=\int dt=2\pi \delta (E=0)$ is the transition time scale.

Although the expression of the probability is derived, the result (27) is constructed by the exact information of quanta represented by the microscopic occupation numbers per unit phase space $n_a$, which is an integer. Since it is impossible to know the exact quantum state, the statistical average should be considered. The probability to realize the state $\left|\right\{n\}\rangle$ can be represented by

$$P_{\left\{n\right\}}\equiv \prod _{a\in \left\{n\right\}}\prod _{\overrightarrow{k}}p\left(n_a\left(\overrightarrow{k}\right)\right),\sum _{n_a=0}^{\infty }p\left(n_a\left(\overrightarrow{k}\right)\right)=1$$

(28)

where $p\left(n_a\left(\overrightarrow{k}\right)\right)$ is a probability to be the occupation $n_a$ on the momentum $\overrightarrow{k}$. Multiplying (28) into (27) and summing over by each occupation number, we can obtain the statistical probability as

$$\begin{array}{ccc}\hfill \langle \mathcal{P}{(A,B,\dots \to X,Y,\dots )}_{\left\{n\right\}}\rangle & \equiv & \sum _{\left\{n\right\}}{P}_{\left\{n\right\}}·\mathcal{P}{(A,B,\dots \to X,Y,\dots )}_{\left\{n\right\}}\hfill \\ & =& \prod _{a=A,B,\dots ,X,Y,\dots }\left(\int {\displaystyle \frac{d^3{\overrightarrow{k}}_a}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{2E_a}}\sum _{g_a}\right)\hfill \\ & & \times {\left|\mathcal{M}(k_A,k_B,\dots \to k_X,k_Y,\dots )\right|}^2\hfill \\ & & \times {\left(2\pi \right)}^4{\delta }^4(k_A+k_B+\dots -k_X-k_Y-\dots )·VT\hfill \\ & & \times f_A{f}_B\dots (1\pm f_X)(1\pm f_Y)\dots \hfill \end{array}$$

(29)

where we denoted

$$f_a\equiv \sum _{n_a=0}^{\infty }p\left(n_a\left(\overrightarrow{k}\right)\right)n_a\left(\overrightarrow{k}\right).$$

(30)

The above procedure is equivalent to the replacement

$$|\left\{n\right\}\rangle \langle \left\{n\right\}|\to \sum _{\left\{n\right\}}{P}_{\left\{n\right\}}|\left\{n\right\}\rangle \langle \left\{n\right\}|$$

(31)

in (20); that is, the initial state is considered by the density operator. The important thing is that the expectation value $f_a$ can be interpreted as the distribution function even though the exact forms of both the probability $p\left(n_a\left(\overrightarrow{k}\right)\right)$ and the related occupation $n_a\left(\overrightarrow{k}\right)$ are unknown.

Using the result of the total probability (29), one can also define the partial probability, as an example, for the species A of the momentum ${\overrightarrow{k}}_A$ by

$$\begin{array}{ccc}\hfill {\mathit{p}}_A(A,B,\dots \to X,Y,\dots )& \equiv & {\displaystyle \frac{d\langle \mathcal{P}{(A,B,\dots \to X,Y,\dots )}_{\left\{n\right\}}\rangle }{{\displaystyle V\frac{d^3{\overrightarrow{k}}_A}{{\left(2\pi \right)}^3}\sum _{g_A}}}}\hfill \\ & =& {\displaystyle \frac{T}{2E_A}}\prod _{a\ne A}\left(\int {\displaystyle \frac{d^3{\overrightarrow{k}}_a}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{2E_a}}\sum _{g_a}\right)\hfill \\ & & \times {\left|\mathcal{M}(k_A,k_B,\dots \to k_X,k_Y,\dots )\right|}^2\hfill \\ & & \times {\left(2\pi \right)}^4{\delta }^4(k_A+k_B+\dots -k_X-k_Y-\dots )\hfill \end{array}$$

(32)

$$\begin{array}{ccc}& & \times f_A{f}_B\dots (1\pm f_X)(1\pm f_Y)\dots .\hfill \end{array}$$

(33)

2.2.3. Expression of Collision Term

The variation of the distribution $\Delta f$ through the microscopic process can be evaluated by

$$\begin{array}{c}\hfill \Delta f_{\varphi }\sim \sum _{\mathrm{all}\mathrm{processes}}\Delta N_{\varphi }·\left[-{\mathit{p}}_{\varphi }(\varphi ,A,B,\dots \to X,Y,\dots )+{\mathit{p}}_{\varphi }(X,Y,\dots \to \varphi ,A,B,\dots )\right]\end{array}$$

(34)

where $\varphi$ is the focusing species, $\Delta N_{\varphi }$ is the change in the number of the quantum $\varphi$ in the process $\varphi ,A,B,\dots \leftrightarrow X,Y,\dots$ ($\Delta N_{\varphi }=1$ in this case), and $p_{\varphi }$ is the partial transition probability for $\varphi$ derived in (33). Finally, the collision term can be evaluated as

$$\begin{array}{ccc}\hfill \mathbf{C}\left[f_{\varphi }\right]& \sim & {\left.E_{\varphi }{\displaystyle \frac{\Delta f_{\varphi }}{\Delta t}}\right|}_{\mathrm{microscopic}\mathrm{process}}\hfill \\ & =& -{\displaystyle \frac{1}{2}}\sum _{\mathrm{all}\mathrm{processes}}\prod _{a\ne \varphi }\left(\int {\displaystyle \frac{d^3{\overrightarrow{k}}_a}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{2E_a}}\sum _{g_a}\right)\hfill \\ & & \times {\left(2\pi \right)}^4{\delta }^4(k_A+k_B+\dots -k_X-k_Y-\dots )\hfill \\ & & \times \Delta N_{\varphi }\left[{\left|\mathcal{M}(k_{\varphi },k_A,k_B,\dots \to k_X,k_Y,\dots )\right|}^2\right.\hfill \\ & & \times f_{\varphi }{f}_A{f}_B\dots (1\pm f_X)(1\pm f_Y)\dots \hfill \\ & & -{\left|\mathcal{M}(k_X,k_Y,\dots \to k_{\varphi },k_A,k_B,\dots )\right|}^2\hfill \end{array}$$

(35)

$$\begin{array}{ccc}& & \left.\times f_X{f}_Y\dots (1\pm f_{\varphi })(1\pm f_A)(1\pm f_B)\dots \right].\hfill \end{array}$$

(36)

In the above derivation, we set $\Delta t=T$. We derived the above result with the single-particle state for all species for simplicity. In the case of including the N-duplicated species in $\varphi ,A,B,\dots$ or $X,Y,\dots$, one needs to multiply an extra factor $1/N!$ for the species.

Equation (36) is the most general result of the collision term. The amplitudes $\mathcal{M}$ for each process are computed using standard Feynman rules in perturbative quantum field theory, typically via the S-matrix formalism. Note that the Boltzmann equation itself does not modify or simplify these calculations, but serves as the framework in which these amplitudes are statistically weighted by the distribution function and integrated over phase space. In this way, it provides a bridge between microscopic quantum processes and macroscopic kinetic evolution.

Note also that all the squared amplitudes in (36) must be regarded as the subtracted state in which the contribution of on-shell particles in the intermediate processes is subtracted in order to avoid the double-counting of the processes. Such a situation will be faced in which the leading contributions of the amplitude consist of the loop diagrams or higher order of couplings, e.g., the baryogenesis scenario, as we discuss later.

2.3. Full and Integrated Boltzmann Equation

Equations (6) and (36) lead the full Boltzmann equation for a species $\varphi$ on the FLRW space-time as

$$\begin{array}{ccc}\hfill {\dot{f}}_{\varphi }-H{\displaystyle \frac{|{\overrightarrow{k}}_{\varphi }{|}^2}{E_{\varphi }}}{\displaystyle \frac{\partial f_{\varphi }}{\partial E_{\varphi }}}& =& -{\displaystyle \frac{1}{2E_{\varphi }}}\sum _{\mathrm{all}\mathrm{processes}}\prod _{a\ne \varphi }\left(\int {\displaystyle \frac{d^3{\overrightarrow{k}}_a}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{2E_a}}\sum _{g_a}\right)\hfill \\ & & \times {\left(2\pi \right)}^4{\delta }^4(k_A+k_B+\dots -k_X-k_Y-\dots )\hfill \\ & & \times \Delta N_{\varphi }\left[{\left|\mathcal{M}(k_{\varphi },k_A,k_B,\dots \to k_X,k_Y,\dots )\right|}^2\right.\hfill \\ & & \times f_{\varphi }{f}_A{f}_B\dots (1\pm f_X)(1\pm f_Y)\dots \hfill \\ & & -{\left|\mathcal{M}(k_X,k_Y,\dots \to k_{\varphi },k_A,k_B,\dots )\right|}^2\hfill \\ & & \left.\times f_X{f}_Y\dots (1\pm f_{\varphi })(1\pm f_A)(1\pm f_B)\dots \right].\hfill \end{array}$$

(37)

Equation (37) for all species describe the detail evolution of the distribution functions, but they are not useful to solve because of a lot of variables. To simplify the equations, the integrated Boltzmann equation is useful and convenient. The momentum integral of the left hand side of (37) leads

$$\begin{array}{ccc}\hfill \int {\displaystyle \frac{d^3{\overrightarrow{k}}_{\varphi }}{{\left(2\pi \right)}^3}}\sum _{g_{\varphi }}\left(\mathrm{LHS}\mathrm{of}\left(37\right)\right)& =& {\dot{n}}_{\varphi }+3H{n}_{\varphi }\hfill \end{array}$$

(38)

where

$$n_{\varphi }\equiv \int {\displaystyle \frac{d^3{\overrightarrow{k}}_{\varphi }}{{\left(2\pi \right)}^3}}\sum _{g_{\varphi }}{f}_{\varphi }$$

(39)

is the number density of $\varphi$. Note in the integration (38) that the variables t and $\left|\overrightarrow{k}\right|=\sqrt{E^2-m^2}$ are independent and a property of the total derivative

$$\int {\displaystyle \frac{d^3\overrightarrow{k}}{{\left(2\pi \right)}^3}}{\displaystyle \frac{|\overrightarrow{k}{|}^2}{E}}{\displaystyle \frac{\partial }{\partial E}}(\dots )=\int {\displaystyle \frac{d^3\overrightarrow{k}}{{\left(2\pi \right)}^3}}\overrightarrow{k}·{\displaystyle \frac{\partial }{\partial \overrightarrow{k}}}(\dots )=-3\int {\displaystyle \frac{d^3\overrightarrow{k}}{{\left(2\pi \right)}^3}}(\dots )$$

(40)

is used. As the result, the numbers of the dynamical variables are reduced from $(\#\mathrm{species})\times (\#t)\times (\#E)$ to (#species)$\times (\#t)$, while the right hand side of the integrated (37) still includes the E-dependent distribution functions. A useful approximation is to apply the Maxwell–Boltzmann similarity distribution

$$\begin{array}{c}\hfill f_a,f_a^{\mathrm{MB}}\ll 1\mathrm{and}{f}_a(t,E_a)\sim {\displaystyle \frac{n_a\left(t\right)}{n_a^{\mathrm{MB}}\left(t\right)}}{f}_a^{\mathrm{MB}}(t,E_a),\end{array}$$

(41)

where

$$\begin{array}{c}\hfill f_a^{\mathrm{MB}}=exp\left[-{\displaystyle \frac{E_a-{\mu }_a}{T_a}}\right],n_a^{\mathrm{MB}}=\int {\displaystyle \frac{d^3{\overrightarrow{k}}_{\varphi }}{{\left(2\pi \right)}^3}}\sum _{g_{\varphi }}{f}_a^{\mathrm{MB}}\end{array}$$

(42)

are the Maxwell–Boltzmann distribution and its number density, respectively. (This well-used approximation can be justified in the case that species a is in the kinetic equilibrium through interacting with the thermal bath. See Appendix A for details. The case in deviating from the kinetic equilibrium is discussed in Section 2.4. Supposing the identical temperature for all species $T_{\varphi }=T_A=\dots \equiv T$, and substituting (41) into the right hand side of (38) and assuming the chemical equilibrium

$${\mu }_{\varphi }+{\mu }_A+{\mu }_B+\dots ={\mu }_X+{\mu }_Y+\dots$$

(43)

where ${\mu }_a$ is the chemical potential of species a, one can obtain the integrated Boltzmann equation as

$$\begin{array}{ccc}\hfill {\dot{n}}_{\varphi }+3H{n}_{\varphi }& =& \int {\displaystyle \frac{d^3{\overrightarrow{k}}_{\varphi }}{{\left(2\pi \right)}^3}}\sum _{g_{\varphi }}\left(\mathrm{RHS}\mathrm{of}\left(37\right)\right)\hfill \\ & =& -\sum _{\mathrm{all}\mathrm{processes}}\Delta N_{\varphi }\left[n_{\varphi }{n}_A{n}_B\dots \times \langle R(\varphi ,A,B,\dots \to X,Y,\dots )\rangle \right.\hfill \\ & & \left.-n_{\varphi }^{\mathrm{MB}}{n}_A^{\mathrm{MB}}{n}_B^{\mathrm{MB}}\dots {\displaystyle \frac{n_X{n}_Y\dots }{n_X^{\mathrm{MB}}{n}_Y^{\mathrm{MB}}\dots }}\times \langle R(\overline{\varphi },\overline{A},\overline{B},\dots \to \overline{X},\overline{Y},\dots )\rangle \right]\hfill \end{array}$$

(44)

where the bar “$¯$” denotes its anti-particle state,

$$\begin{array}{ccc}\hfill R(A,B,\dots \to X,Y,\dots )& \equiv & {\displaystyle \frac{1}{2E_{A}2E_B\dots }}\int {\displaystyle \frac{d^3{k}_X}{{\left(2\pi \right)}^3}}{\displaystyle \frac{d^3{k}_Y}{{\left(2\pi \right)}^3}}\dots {\displaystyle \frac{1}{2E_{X}2E_Y\dots }}\hfill \\ & & \times {\displaystyle \frac{{\sum }_{g_A,g_B,\dots ,g_X,g_Y,\dots }{\left|\mathcal{M}(k_A,k_B\dots \to k_X,k_Y,\dots )\right|}^2}{{\sum }_{g_A,g_B,\dots }}}\hfill \\ & & \times {\left(2\pi \right)}^4{\delta }^4(k_A+k_B+\dots -k_X-k_Y-\dots )\hfill \end{array}$$

(45)

is a reaction rate integrated over the final state, and

$$\begin{array}{ccc}\hfill \langle R(A,B,\dots \to X,Y,\dots )\rangle & \equiv & {\displaystyle \frac{1}{n_A^{\mathrm{MB}}{n}_B^{\mathrm{MB}}\dots }}\int {\displaystyle \frac{d^3{k}_A}{{\left(2\pi \right)}^3}}{\displaystyle \frac{d^3{k}_B}{{\left(2\pi \right)}^3}}\dots f_A^{\mathrm{MB}}{f}_B^{\mathrm{MB}}\dots \hfill \\ & & \times \sum _{g_A,g_B,\dots }R(A,B,\dots \to X,Y,\dots )\hfill \end{array}$$

(46)

is the thermally averaged reaction rate by the Maxwell–Boltzmann distributions of the species appearing in the initial state. We used the property of the CPT-invariance of the amplitude

$$\mathcal{M}(X,Y,\dots \to A,B,\dots )=\mathcal{M}(\overline{A},\overline{B},\dots \to \overline{X},\overline{Y},\dots )$$

(47)

to derive the last term in (44).

Although Equation (44) describing the evolution of number densities is obtained by the integration of (37) directly, in general, the other evolution equations of the statistical quantities

$$Q_a\left(t\right)\equiv \int {\displaystyle \frac{d^3{\overrightarrow{k}}_a}{{\left(2\pi \right)}^3}}\sum _{g_a}{f}_a\left({\overrightarrow{k}}_a\right)q_a(t,E_a)$$

(48)

can also be derived through the same procedure with the corresponding coefficient $q_a(t,E_a)$, e.g., the energy density $Q=\rho$ for $q=E_a$, and the pressure $Q=P$ for $q=|\overrightarrow{k}{|}^2/3E_a$. Such equations help to extract more detailed thermodynamic variables, e.g., to determine the independent temperatures for each species, as we will see in the next subsection.

2.4. Temperature Parameter

The integrated Boltzmann Equation (44) is quite useful and can be applied to many situations. However, it might not be suitable for some situations in which the kinetic equilibrium is highly violated because the formula is based on the approximation by the Maxwell–Boltzmann similarity distribution, which is justified by the kinetic equilibrium of the target particles with the thermal bath, as discussed in Appendix A. Although obtaining the most appropriate solution is to solve the full Boltzmann equation, it takes a lot of cost in the calculation. In this section, we introduce an alternative method based on the integrated Boltzmann equation.

Instead of using the similarity distribution (41), we introduce a more generalized similarity distribution by

$$f_a(t,E_a)\sim {\displaystyle \frac{n_a\left(t\right)}{n_a^{\mathrm{neq}}\left(t\right)}}{f}_a^{\mathrm{neq}}(t,E_a),f_a^{\mathrm{neq}}(t,E_a)\equiv exp\left[-{\displaystyle \frac{E_a-{\mu }_a}{T_a\left(t\right)}}\right].$$

(49)

Here $n_a^{\mathrm{neq}}$ is the number density evaluated by the “non-equilibrium” Maxwell–Boltzmann distribution $f_a^{\mathrm{neq}}$ that is parametrized by the temperature parameter $T_a\left(t\right)$. In general, the temperature parameter is independent of the thermal bath temperature $T\left(t\right)$. While the normalization factor $n_a\left(t\right)/n_a^{\mathrm{neq}}\left(t\right)$ can be regarded as a corresponding quantity to the chemical potential parameter ${\tilde{\mu }}_a\left(t\right)$:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{n_a\left(t\right)}{n_a^{\mathrm{neq}}\left(t\right)}}& =& exp\left[{\displaystyle \frac{{\tilde{\mu }}_a\left(t\right)-{\mu }_a}{T_a\left(t\right)}}\right].\hfill \end{array}$$

(50)

In the case of the Bose–Einstein/Fermi–Dirac type distribution

$$f_a(t,E_a)={\left[e^{(E_a-{\tilde{\mu }}_a\left(t\right))/T_a\left(t\right)}\mp 1\right]}^{-1}(-:\mathrm{boson},+:\mathrm{fermion})$$

(51)

where $T_a\left(t\right)$ and ${\tilde{\mu }}_a\left(t\right)$ are the temperature and the chemical potential parameters, respectively, the temperature parameter can be represented as

$$\begin{array}{ccc}\hfill {\tilde{T}}_a\left(t\right)& =& {\displaystyle \frac{1}{{\rho }_a\left(t\right)+P_a\left(t\right)}}\int {\displaystyle \frac{d^3{\overrightarrow{k}}_a}{{\left(2\pi \right)}^3}}{\displaystyle \frac{{\overrightarrow{k}}_a^2}{3}}{f}_a(t,E_a)\left(1\pm f_a(t,E_a)\right)\hfill \\ & & (+:\mathrm{boson},-:\mathrm{fermion})\hfill \end{array}$$

(52)

with energy density ${\rho }_a$ and pressure $P_a$ evaluated by (51). The above representation is consistent with (55) in the non-relativistic limit: $f_a\ll 1$, $E_a\sim m_a$, and ${\rho }_a\sim m_a{n}_a\gg P_a$. The chemical potential parameter can be obtained by

$${\tilde{\mu }}_a\left(t\right)={\displaystyle \frac{{\rho }_a\left(t\right)+p_a\left(t\right)-T_a\left(t\right)s_a\left(t\right)}{n_a\left(t\right)}}$$

(53)

where $s_a\left(t\right)$ is the entropy density defined by

$$s_a\left(t\right)=\int {\displaystyle \frac{d^3{\overrightarrow{k}}_a}{{\left(2\pi \right)}^3}}\left[\pm (1\pm f_a)ln(1\pm f_a)-f_{a}ln{f}_a\right].(+:\mathrm{boson},-:\mathrm{fermion})$$

(54)

Let us go back to the discussion of the property of the Maxwell–Boltzmann distribution. The temperature parameter can be expressed by the ratio of the pressure and the number density

$$T_a={\displaystyle \frac{P_a^{\mathrm{neq}}}{n_a^{\mathrm{neq}}}}={\displaystyle \frac{P_a}{n_a}}$$

(55)

because of

$$\begin{array}{ccc}\hfill P_a^{\mathrm{neq}}& =& \int {\displaystyle \frac{d^3\overrightarrow{k}}{{\left(2\pi \right)}^3}}\sum _{g_a}{f}_a^{\mathrm{neq}}{\displaystyle \frac{|{\overrightarrow{k}}_a{|}^2}{3E_a}}=\int {\displaystyle \frac{d^3\overrightarrow{k}}{{\left(2\pi \right)}^3}}\sum _{g_a}{\displaystyle \frac{T_a}{3}}\left(-{\overrightarrow{p}}_a·{\displaystyle \frac{\partial }{\partial {\overrightarrow{p}}_a}}\right)f_a^{\mathrm{neq}}\hfill \\ & =& T_a\int {\displaystyle \frac{d^3\overrightarrow{p}}{{\left(2\pi \right)}^3}}\sum _{g_a}{f}_a^{\mathrm{neq}}=n_a^{\mathrm{neq}}{T}_a\hfill \end{array}$$

(56)

and

$$P_a=\int {\displaystyle \frac{d^3\overrightarrow{k}}{{\left(2\pi \right)}^3}}\sum _{g_a}{f}_a{\displaystyle \frac{|{\overrightarrow{k}}_a{|}^2}{3E_a}}={\displaystyle \frac{n_a}{n_a^{\mathrm{neq}}}}\int {\displaystyle \frac{d^3\overrightarrow{p}}{{\left(2\pi \right)}^3}}\sum _{g_a}{f}_a^{\mathrm{neq}}{\displaystyle \frac{|{\overrightarrow{k}}_a{|}^2}{3E_a}}={\displaystyle \frac{n_a}{n_a^{\mathrm{neq}}}}{P}_a^{\mathrm{neq}}.$$

(57)

Therefore, the evolution equation for the temperature parameter can be derived from the pressure one, which can be constructed from the original full Boltzmann Equation (37) multiplied by $|{\overrightarrow{k}}_a{|}^2/3E_a$. After the integration by the momentum, one can obtain the coupled equations for species $\varphi$ as

$$\begin{array}{ccc}& & {\dot{n}}_{\varphi }+3H{n}_{\varphi }\hfill \\ & & =-\sum _{\mathrm{all}\mathrm{processes}}\Delta N_{\varphi }\left[n_{\varphi }{n}_A{n}_B\dots \times {\langle R(\varphi ,A,B,\dots \to X,Y,\dots )\rangle }^{\mathrm{neq}}\right.\hfill \\ & & \left.-n_X{n}_Y\dots \times {\langle R(X,Y,\dots \to \varphi ,A,B,\dots )\rangle }^{\mathrm{neq}}\right],\hfill \end{array}$$

(58)

$$\begin{array}{ccc}& & n_{\varphi }{\dot{T}}_{\varphi }+H{n}_{\varphi }\left(2T_{\varphi }-{〈{\displaystyle \frac{|{\overrightarrow{k}}_{\varphi }{|}^4}{3E_{\varphi }^3}}〉}^{\mathrm{neq}}\right)\hfill \\ & & =-\sum _{\mathrm{all}\mathrm{processes}}\Delta N_{\varphi }\left[n_{\varphi }{n}_A{n}_B\dots \times {〈\left({\displaystyle \frac{|{\overrightarrow{k}}_{\varphi }{|}^2}{3E_{\varphi }}}-T_{\varphi }\right)R(\varphi ,A,B,\dots \to X,Y,\dots )〉}^{\mathrm{neq}}\right.\hfill \\ & & -n_X{n}_Y\dots \times \left({\langle R_{T_{\varphi }}(X,Y,\dots \to \varphi ,A,B,\dots )\rangle }^{\mathrm{neq}}\right.\hfill \\ & & \left.\left.-T_{\varphi }{\langle R(X,Y,\dots \to \varphi ,A,B,\dots )\rangle }^{\mathrm{neq}}\right)\right],\hfill \end{array}$$

(59)

where R is a rate defined in (45) and

$$\begin{array}{ccc}& & R_{T_{\varphi }}(X,Y,\dots \to \varphi ,A,B,\dots )\hfill \\ & & ={\displaystyle \frac{1}{2E_{X}2E_Y\dots }}\int {\displaystyle \frac{d^3{\overrightarrow{k}}_{\varphi }}{{\left(2\pi \right)}^3}}{\displaystyle \frac{d^3{\overrightarrow{k}}_A}{{\left(2\pi \right)}^3}}{\displaystyle \frac{d^3{\overrightarrow{k}}_B}{{\left(2\pi \right)}^3}}\dots \hfill \\ & & \times {\displaystyle \frac{{\sum }_{g_{\varphi },g_A,g_B,\dots ,g_X,g_Y,\dots }{\left|\mathcal{M}(k_X,k_Y,\dots \to k_{\varphi },k_A,k_B,\dots )\right|}^2}{{\sum }_{g_X,g_Y,\dots }}}·{\displaystyle \frac{|{\overrightarrow{k}}_{\varphi }{|}^2}{3E_{\varphi }}}\hfill \\ & & \times {\left(2\pi \right)}^4{\delta }^4(k_{\varphi }+k_A+k_B+\dots -k_X-k_Y-\dots )\hfill \end{array}$$

(60)

is a “temperature weighted” rate, and

$$\begin{array}{ccc}\hfill {〈{\displaystyle \frac{|{\overrightarrow{k}}_{\varphi }{|}^4}{3E_{\varphi }^3}}〉}^{\mathrm{neq}}& =& {\displaystyle \frac{1}{n_{\varphi }^{\mathrm{neq}}}}\int {\displaystyle \frac{d^3{\overrightarrow{k}}_{\varphi }}{{\left(2\pi \right)}^3}}\sum _{g_{\varphi }}{f}_{\varphi }^{\mathrm{neq}}·{\displaystyle \frac{|{\overrightarrow{k}}_{\varphi }{|}^4}{3E_{\varphi }^3}},\hfill \end{array}$$

(61)

$$\begin{array}{ccc}\hfill {\langle R(a,b,\dots \to i,j,\dots )\rangle }^{\mathrm{neq}}& =& {\displaystyle \frac{1}{n_a^{\mathrm{neq}}{n}_b^{\mathrm{neq}}\dots }}\int {\displaystyle \frac{d^3{\overrightarrow{k}}_a}{{\left(2\pi \right)}^3}}{\displaystyle \frac{d^3{\overrightarrow{k}}_b}{{\left(2\pi \right)}^3}}\dots \sum _{g_a,g_b,\dots }\hfill \\ & & \times f_a^{\mathrm{neq}}{f}_b^{\mathrm{neq}}\dots R(a,b,\dots \to i,j,\dots ),\hfill \end{array}$$

(62)

are the thermally averaged quantities by the non-equilibrium distribution $f^{\mathrm{neq}}$, including only the initial species $a,b,\dots$, not the final species $i,j,\dots$. Solving the coupled Equations (58) and (59) for all the species can be expected to obtain more accurate results than the former integrated Boltzmann Equation (44). Following the evolution in practice, the combined quantity

$$y={\displaystyle \frac{m_{\varphi }{T}_{\varphi }}{s^{2/3}}}\propto {\displaystyle \frac{T_{\varphi }}{T^2}}$$

(63)

instead of the solo $T_{\varphi }$, where s is the entropy density, is convenient for the non-relativistic $\varphi$ because of the asymptotic behavior $T_{\varphi }\left(t\right)\propto a{\left(t\right)}^{-2}\propto T{\left(t\right)}^2$ after freezing out.

2.5. Limitation of the Approach by the Boltzmann Equation

Finally, we mention the limitations of the Boltzmann equation. As seen in the derivation of the Boltzmann equation from quantum field theory, the collision term is derived from the transition probability that is described by the S-matrix element. In the usual method of quantum field theory, the evaluated transition probabilities or the corresponding S-matrix elements indicate the process from the infinite past to the infinite future, in both limits of which the distinct one-particle (or multi-particle) state can be well defined. In reality, however, the interaction process happens within a finite time and occurs continuously before the interacting state returns to the actual particle state. If the evolution of the system is described by such a quasi-particle effect (such as involving a significant thermal corrective effect, a resonant effect in the elementary process, a quantum oscillation effect, etc.), one should consider a more accurate treatment of the system.

One of the methods is to consider the Kadanoff–Baym equation [47,48] that describes the evolution of the two-point correlation functions with the self-energy effect through the closed time path formalism (or the Schwinger–Keldysh formalism) [49]. The calculated two-point functions can lead to the number density, the energy density, and so on. Although the formulation is more complicated than the Boltzmann equation, a more accurate treatment including the statistical quantum effects is possible. The applications for the cosmological context can be seen, e.g., refs. [50,51] for DM relic abundance, refs. [52,53,54,55,56,57] for various baryogenesis scenarios. The general derivation of the Boltzmann equation from the Kadanoff–Baym equation applying the WKB approximation can be seen in [58]. See also [59] for a pedagogical review of non-equilibrium quantum field theory and [60] for a recent review of baryogenesis scenarios based on the Kadanoff–Baym equation.

3. Application to DM Abundance

One of the cosmological applications of the Boltzmann equation is for the estimation of the DM abundance. Because DM is stable, the main process changing the particle number is not decay/inverse-decay but the 2-2 annihilation/creation scatterings

$$\chi ,\overline{\chi }\leftrightarrow \psi ,\overline{\psi }$$

(64)

where $\chi$ is a DM and $\psi$ is a standard model particle. This dynamics can be solved once the Lorentz-invariant annihilation cross-section $\sigma$ is given since the rate in the 2-2 scattering can be represented as

$$R(\chi ,\overline{\chi }\to \psi ,\overline{\psi })=\sigma v$$

(65)

where v is the Møller velocity defiend by the pair of 4-momenta $k_1$, $k_2$ and masses $m_1$, $m_2$ of the initial DM particles as

$$v_{12}={\displaystyle \frac{\sqrt{{(k_1·k_2)}^2-m_1^2{m}_2^2}}{k_1^0{k}_2^0}}.$$

(66)

(The Møller velocity can be identical to the relative velocity only in the case of the parallel 3-momenta; ${\overrightarrow{k}}_1·{\overrightarrow{k}}_2=\pm |{\overrightarrow{k}}_1\left|\right|{\overrightarrow{k}}_2|.$ Note that the Møller velocity (66) is not Lorentz-invariant, but $k_1^0{k}_2^0{v}_{12}$ is invariant.) Assuming the symmetric DM $n_{\chi }=n_{\overline{\chi }}$ and the thermal distribution for the standard model particles $n_{\psi }=n_{\overline{\psi }}=n_{\psi }^{\mathrm{MB}}$, the Boltzmann Equation (44) for the DM leads to a simple form:

$$\begin{array}{c}\hfill {\dot{n}}_{\chi }+3H{n}_{\chi }=-\left(n_{\chi }^2-{\left(n_{\chi }^{\mathrm{MB}}\right)}^2\right)\langle \sigma v\rangle .\end{array}$$

(67)

Instead of the particle number to follow its evolution by time, it is convenient to use the yield $Y_{\chi }\equiv n_{\chi }/s$ with a dynamical variable $x\equiv m_{\chi }/T$, where $s={\displaystyle \frac{2{\pi }^2}{45}}{h}_{\mathrm{eff}}\left(T\right)T^3$ is the entropy density and $h_{\mathrm{eff}}\left(T\right)\sim 100$ for $T\gtrsim 100$ GeV is the effective degrees of freedom defined by the entropy density. In the case of no creation/annihilation process, the yield $Y_{\chi }$ becomes a constant since the number and the entropy in the comoving volume are conserved. With these variables, the Boltzmann Equation (67) can be represented as

$$Y_{\chi }^{\prime }=-(1+{\delta }_h){\displaystyle \frac{s\langle \sigma v\rangle }{xH}}\left(Y_{\chi }^2-{\left(Y_{\chi }^{\mathrm{MB}}\right)}^2\right)$$

(68)

where we denote $\prime \equiv d/dx,$ and

$${\delta }_h\equiv {\displaystyle \frac{T}{3h_{\mathrm{eff}}}}{\displaystyle \frac{d{h}_{\mathrm{eff}}}{dT}}.$$

(69)

Since the adiabatic parameter ${\delta }_h$ tends to be negligible in the almost era of the thermal history, we set ${\delta }_h=0$ in the later discussion for simplicity. (If the DM mass scale is around $\mathcal{O}\left(10\right)$ GeV, the freeze-out occurs around the QCD transition scale $T\sim \mathcal{O}\left(100\right)$ MeV, in which $|{\delta }_h|\sim \mathcal{O}\left(1\right)$. Thus, there are few percent level contributions from the adiabatic parameter ${\delta }_h$ even in the WIMP model. See Refs. [61,62,63,64,65,66,67,68] for the determination of that parameter in detail.) Moreover, we denote

$$Y_{\chi }^{\mathrm{MB}}\equiv {\displaystyle \frac{n_{\chi }^{\mathrm{MB}}}{s}}\sim {\displaystyle \frac{g_{\chi }}{h_{\mathrm{eff}}}}{\displaystyle \frac{45}{2^{5/2}{\pi }^{7/2}}}{x}^{3/2}{e}^{-x}(x\gg 1),$$

(70)

where $g_{\chi }$ is the degrees of freedom for the DM particle.

3.1. Relic Abundance in Freeze-Out

As a simple and reasonable setup, we assume that the DM particles $\chi$ are in thermal equilibrium initially. Then, the dynamics described by (68) can be explained as follows. At first, the system is in thermal equilibrium due to the stronger scattering effect than the spatial expansion, but the yield has a small deviation from the thermal value due to the expansion effect as

$$Y_{\chi }\sim Y_{\chi }^{\mathrm{MB}}+\Delta \left(x\right),\Delta \left(x\right)={\displaystyle \frac{xH}{s\langle \sigma v\rangle }}{\displaystyle \frac{-Y_{\chi }^{\prime }}{Y_{\chi }^{\mathrm{MB}}+Y}}\sim {\displaystyle \frac{xH}{2s\langle \sigma v\rangle }}\ll Y_{\chi }^{\mathrm{MB}}$$

(71)

as long as $n^{\mathrm{MB}}\langle \sigma v\rangle \gg xH$. The deviation $\Delta$ continues growing in later time, and finally the evolution of the yield freezes out because the expansion rate exceeds the scattering rate. The freeze-out occurs when $\Delta \left(x_f\right)=c{Y}^{\mathrm{MB}}\left(x_f\right)$, $c\sim \mathcal{O}\left(1\right)$. The freeze-out time $x=x_f$ and the final abundance $Y_{\infty }=Y(x=\infty )$ can be estimated by [9,69]

$$\begin{array}{ccc}\hfill x_f& =& ln\left[c(c+2){\displaystyle \frac{\sqrt{90}}{{\left(2\pi \right)}^3}}{\displaystyle \frac{g_{\chi }}{\sqrt{g_{\mathrm{eff}}\left(T_f\right)}}}{m}_{\chi }{M}_{\mathrm{pl}}{\sigma }_n\right]-\left(n+{\displaystyle \frac{1}{2}}\right)ln{x}_f,\hfill \\ & =& ln\left[c(c+2){\displaystyle \frac{\sqrt{90}}{{\left(2\pi \right)}^3}}{\displaystyle \frac{g_{\chi }}{\sqrt{g_{\mathrm{eff}}\left(T_f\right)}}}{m}_{\chi }{M}_{\mathrm{pl}}{\sigma }_n\right]\hfill \end{array}$$

(72)

$$\begin{array}{ccc}& & -\left(n+{\displaystyle \frac{1}{2}}\right)ln\left(ln\left[c(c+2){\displaystyle \frac{\sqrt{90}}{{\left(2\pi \right)}^3}}{\displaystyle \frac{g_{\chi }}{\sqrt{g_{\mathrm{eff}}\left(T_f\right)}}}{m}_{\chi }{M}_{\mathrm{pl}}{\sigma }_n\right]\right)+\dots ,\hfill \end{array}$$

(73)

$$\begin{array}{ccc}\hfill Y_{\infty }& =& (n+1)\sqrt{{\displaystyle \frac{45}{\pi }}}{\displaystyle \frac{g_{\chi }}{\sqrt{g_{\mathrm{eff}}\left(T_f\right)}}}{\displaystyle \frac{x_f^{n+1}}{M_{\mathrm{pl}}{m}_{\chi }{\sigma }_n}}\hfill \end{array}$$

(74)

where $M_{\mathrm{pl}}=1.22\times {10}^{19}$ GeV is the Planck mass, $g_{\mathrm{eff}}\left(T\right)$ is the effective degrees of freedom defined by the energy density $\rho ={\displaystyle \frac{{\pi }^2}{30}}{g}_{\mathrm{eff}}\left(T\right)T^4$, and $T_f=m_{\chi }/x_f$ is the freeze-out temperature. In the derivation of the analytic results (72) and (74), the temperature dependence of the cross-section is approximated by the most dominant part as

$$\langle \sigma v\rangle ={\sigma }_n{x}^{-n},$$

(75)

where ${\sigma }_n$ is a constant (see also Appendix B for the actual analysis of the thermally averaged cross-section). Especially, $n=0$ and 1 correspond to s-wave and p-wave scattering, respectively. Although a numerical factor c still has uncertainty, choosing $c(c+2)=n+1$ leads to better analysis for the final abundance $Y_{\infty }$ within $5\%$ accuracy for $x_f\gtrsim 3$ [9]. As an example, let us consider a WIMP model. Choosing the parameters as $m_{\chi }=120$ GeV, $n=1$, ${\sigma }_n={\displaystyle \frac{{\alpha }_W^2}{m_{\chi }^2}}$, ${\alpha }_W={\displaystyle \frac{1}{30}}$, $g_{\chi }=2$, $h_{\mathrm{eff}}=g_{\mathrm{eff}}=90$, one can obtain the analytic results

$$x_f=23.2,Y_{\infty }=3.81\times {10}^{-12}.$$

(76)

The actual evolution is depicted in Figure 1.

If the interaction rate becomes much lower than the Hubble rate at the relativistic regime $x\lesssim 1$, the abundance freezes out with the massless abundance (hot relic):

$$Y_{\infty }\sim Y_{\mathrm{hot}}={\displaystyle \frac{45\zeta \left(3\right)}{2{\pi }^4}}{\displaystyle \frac{g_{\chi }}{h_{\mathrm{eff}}\left(T_f\right)}}\times \left\{\begin{array}{cc}1& \left(\mathrm{boson}\right)\\ 3/4& \left(\mathrm{fermion}\right)\end{array}\right.$$

(77)

where $\zeta \left(3\right)=1.202\dots$.

Finally, we need to mention the validity of the approximated results (72) and (74). Their behaviors can deviate easily if the master Equation (67) includes significant extra processes by other species or the singular behavior of the cross-sections. Specifically, some exceptional cases are known: (i) mutual annihilations of multiple species (coannihilations), (ii) annihilations into heaver states (forbidden channels), (iii) annihilations near a pole in the cross-section [23], and (iv) simultaneous chemical and kinetic decoupling (coscattering) [25]. In these cases, the analysis should be performed more carefully. See refs. [23,24,25,26,70,71,72] as their example cases, and also refs. [73,74,75] as examples of the evaluation with the temperature parameter. As comparisons with their results, recent papers [76,77] show results by the full Boltzmann equation without the temperature parameter.

3.2. Constraint on Relic Abundance

The relic abundance for the stable particles through the freeze-out of their annihilation processes, as similar to $\chi$ particles discussed above, is restricted by the cosmological observation results. A useful parameter relating to the relic abundance is the density parameter, defined by

$${\Omega }_{\chi }\equiv {\displaystyle \frac{{\rho }_{\chi }}{\frac{3M_{\mathrm{pl}}^2}{8\pi }{H}^2}}\sim {\displaystyle \frac{16{\pi }^3}{135}}{\displaystyle \frac{h_{\mathrm{eff}}\left(T\right)T^3}{M_{\mathrm{pl}}^2{H}^2}}·m_{\chi }{Y}_{\chi }.$$

(78)

Since the yield maintains a constant after the freeze-out unless the additional entropy production occurs in the later era, one can estimate the present density parameter of $\chi$ with the present values. The current observation through the the Cosmic Microwave Background [78] provides $T_0=2.726$ K, $h_{\mathrm{eff}}\left(T_0\right)=3.91$, $H_0=100h^2$ km/s/Mpc, $h=0.677$; therefore, one can estimate

$${\Omega }_{\chi ,\mathrm{now}}\sim {\displaystyle \frac{m_{\chi }{Y}_{\chi }}{3.64h^2\times {10}^{-9}\mathrm{GeV}}}.$$

(79)

Because the present density parameter for the cold matter component is observed as ${\Omega }_c{h}^2=0.119$ and it must be larger than the $\chi$ component, one can obtain a bound as

$$m_{\chi }{Y}_{\chi }<4.36\times {10}^{-10}\mathrm{GeV}.$$

(80)

The set of parameters shown in (76) is seemingly suitable for the above constraint with a bit of modification. However, that would fail by taking into account the direct detections of the DM that focuses on the process of $\chi ,\psi \leftrightarrow \chi ,\psi$, where $\psi$ is a standard model particle. If the annihilation process occurs through a similar interaction to the electroweak gauge interaction, the cross-section for $\chi$-$\psi$ elastic scattering also relates to the same gauge interaction. One can estimate ${\sigma }_{\chi \psi \to \chi \psi }\sim G_F^2{m}_{\chi }^2\sim {10}^{-36}{\mathrm{cm}}^2$, but it is already excluded by the direct detection [79].

3.3. Relic Abundance in Freeze-In

The discussion and the result in the previous subsections are based on the freeze-out scenario in which the DM particles are in thermal equilibrium initially. However, it is not satisfied if the interaction between the DM particles and the thermal bath is too small—the so-called FIMP (feebly interacting massive particle) scenario [21,80]. In this situation, the yield of DM evolves from zero through the thermal production from the thermal bath. Although the DM never reaches the thermal equilibrium, the yield freezes in with a non-thermal yield at last.

We discuss here the relic abundance by the freeze-in scenario in two cases of the pair-creation of DM by (1) scattering from thermal scattering and (2) decay from a heavier particle.

3.3.1. Pair-Creation by Scattering

In the case that DM-pair ($\chi \overline{\chi }$) is produced by thermal pair particles ($\psi \overline{\psi }$), the Boltzmann equation is given by (68) as

$$\begin{array}{ccc}\hfill Y_{\chi }^{\prime }& \sim & {\displaystyle \frac{s\langle \sigma v\rangle }{xH}}{\left(Y_{\chi }^{\mathrm{MB}}\right)}^2,\hfill \end{array}$$

(81)

where we approximated $Y_{\chi }\ll Y_{\chi }^{\mathrm{MB}}$ and the adiabatic degrees ${\delta }_h\sim 0$ until the freezing-in. For simplicity, we consider the simple interaction described by

$${\mathcal{L}}_{\mathrm{int}}=\lambda \left({\chi }^{†}\chi \right)\left({\psi }^{†}\psi \right).$$

(82)

where $\chi$ is the bosonic DM, ${\psi }_i$ labeled i are the massless bosons in the thermal bath, and y is a coupling constant. The thermally averaged cross-section is given by

$$\begin{array}{ccc}\hfill \langle \sigma v\rangle & =& {\displaystyle \frac{g_{\chi }^2{g}_{\psi }^2}{{\left(n_{\chi }^{\mathrm{MB}}\right)}^2}}{\displaystyle \frac{{\lambda }^2}{{\left(2\pi \right)}^5}}T{\int }_{4m_{\chi }^2}^{\infty }ds\sqrt{s-4m_{\chi }^2}{K}_1(\sqrt{s}/T).\hfill \end{array}$$

(83)

where $g_{\chi }$ and $g_{\psi }$ are the degrees of freedom for each species. Therefore, one can estimate the final yield at $x=m_{\chi }/T=\infty$ as

$$\begin{array}{ccc}\hfill Y_{\chi }(\infty )& \sim & {\int }_0^{\infty }dx{\displaystyle \frac{s\langle \sigma v\rangle }{xH}}{\left(Y_{\chi }^{\mathrm{MB}}\right)}^2\hfill \end{array}$$

(84)

$$\begin{array}{ccc}& =& {\displaystyle \frac{3{\pi }^2}{128}}·g_{\chi }^2{g}_{\psi }^2{\displaystyle \frac{{\lambda }^2}{{\left(2\pi \right)}^5}}·{\displaystyle \frac{m_{\chi }^4}{H(T=m_{\chi })s(T=m_{\chi })}}.\hfill \end{array}$$

(85)

This result implies that the yield freezing occurs around the earlier stage $x\sim \mathcal{O}\left(1\right)$ because (85) can be regarded as $Y_{\chi }(\infty )\sim {\left.{\displaystyle \frac{n_{\chi }^{\mathrm{MB}}\langle \sigma v\rangle }{H}}{Y}_{\chi }^{\mathrm{MB}}\right|}_{x\sim 1}$.

Applying the obtained relic abundance (85) to the relation of the present density parameter (79), one can obtain the required strength of the coupling as

$$\lambda =1.0\times {10}^{-12}·{\displaystyle \frac{1}{g_{\chi }{g}_{\psi }}}·{\left({\displaystyle \frac{g_{\mathrm{eff}}(T=m_{\chi })}{100}}\right)}^{1/4}{\left({\displaystyle \frac{h_{\mathrm{eff}}(T=m_{\chi })}{100}}\right)}^{1/2}·{\left({\displaystyle \frac{{\Omega }_{\chi ,\mathrm{now}}{h}^2}{0.119}}\right)}^{1/2}.$$

(86)

3.3.2. Pair-Creation by Decay

The other possible freeze-in scenario is due to the pair production from a heavier particle: $\sigma \to \chi \overline{\chi }$ [21,81]. The original Boltzmann equation for DM is given by

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d{Y}_{\chi }}{d{x}_{\sigma }}}& =& (1+{\delta }_h){\displaystyle \frac{{\Gamma }_{\sigma \to \chi \overline{\chi }}}{x_{\sigma }H}}{\displaystyle \frac{K_1\left(x_{\sigma }\right)}{K_2\left(x_{\sigma }\right)}}\left(Y_{\sigma }-{\left({\displaystyle \frac{Y_{\chi }}{Y_{\chi }^{\mathrm{MB}}}}\right)}^2{Y}_{\sigma }^{\mathrm{MB}}\right)\hfill \end{array}$$

(87)

$$\begin{array}{ccc}& \sim & {\displaystyle \frac{{\Gamma }_{\sigma \to \chi \overline{\chi }}}{x_{\sigma }H}}{\displaystyle \frac{K_1\left(x_{\sigma }\right)}{K_2\left(x_{\sigma }\right)}}{Y}_{\sigma }^{\mathrm{MB}}\hfill \end{array}$$

(88)

where ${\Gamma }_{\sigma \to \chi \overline{\chi }}$ is a decay constant and $x_{\sigma }\equiv m_{\sigma }/T$. We also approximated $Y_{\sigma }\sim Y_{\sigma }^{\mathrm{MB}}$, $Y_{\chi }\ll Y_{\chi }^{\mathrm{MB}}$, and ${\delta }_h\sim 0$ in the second line. Therefore, the final yield can be estimated as

$$\begin{array}{ccc}\hfill Y_{\chi }(\infty )& \sim & {\int }_0^{\infty }d{x}_{\sigma }{\displaystyle \frac{{\Gamma }_{\sigma \to \chi \overline{\chi }}}{x_{\sigma }H}}{\displaystyle \frac{K_1\left(x_{\sigma }\right)}{K_2\left(x_{\sigma }\right)}}{Y}_{\sigma }^{\mathrm{MB}}\hfill \end{array}$$

(89)

$$\begin{array}{ccc}& =& {\displaystyle \frac{3g_{\sigma }}{4\pi }}·{\displaystyle \frac{{\Gamma }_{\sigma \to \chi \overline{\chi }}}{H(T=m_{\sigma })}}{\displaystyle \frac{m_{\sigma }^3}{s(T=m_{\sigma })}}.\hfill \end{array}$$

(90)

where $g_{\sigma }$ is the degrees of freedom for $\sigma$. If the decay constant can be represented by the coupling constant y as

$${\Gamma }_{\sigma \to \chi \overline{\chi }}=g_{\chi }·{\displaystyle \frac{y^2}{8\pi }}{m}_{\sigma },$$

(91)

the required magnitude of the coupling with the relation formula to the density parameter (79) can be estimated as

$$y^2\sim 2.7\times {10}^{-24}·{\displaystyle \frac{m_{\sigma }}{m_{\chi }}}·{\displaystyle \frac{1}{g_{\sigma }{g}_{\chi }}}·{\left({\displaystyle \frac{g_{\mathrm{eff}}(T=m_{\sigma })}{100}}\right)}^{1/2}{\displaystyle \frac{h_{\mathrm{eff}}(T=m_{\sigma })}{100}}·{\displaystyle \frac{{\Omega }_{\chi ,\mathrm{now}}{h}^2}{0.119}}.$$

(92)

4. Application to Baryogenesis

The other popular application of the Boltzmann equation in cosmology is the baryogenesis scenario that describes the dynamical evolution of the baryon number in the universe from zero at the beginning to the non-zero at present. The present abundance of the baryons can be estimated from (79). Replacing $\chi$’s mass $m_{\chi }$ into the nucleon mass $m_N=939$ MeV and using the present density parameter for the baryon ${\Omega }_b{h}^2=0.0224$ [78], one can obtain

$$Y_{B,\mathrm{now}}=8.69\times {10}^{-11}.$$

(93)

There are three conditions suggested by A. D. Sakharov [82] in order to develop the baryon abundance from $Y_B=0$ to non-zero: (1) baryon number (B) violation, (2) C and $CP$ violation, (3) non-equilibrium condition. Their brief reasons are as follows. The B violation is trivial by definition. If the baryon number violating processes conserve C or $CP$, their anti-particle processes happen with the same rate. As a result, the net baryon number is always zero. Even if the processes violate the baryon number, C and $CP$, the thermal equilibrium reduces the baryon asymmetry due to their inverse processes. The Boltzmann equation, especially, provides a powerful tool to quantify the third condition.

To see how to construct the Boltzmann equations for the baryogenesis, let us consider with a toy model, as shown in

Table 1. Matter contents and their baryon number. All the anti-particles have the opposite sign of the baryon number.

Species	Particle Statistic	#B
$X_a$	Chiral fermion (Majorana)	−
${\psi }_i$	Chiral fermion	b
$\varphi$	Complex scalar	0

and

Table 2. Possible processes up to the 4-body B-violating interactions and their variation of the baryon number. The bar ($¯$) on each species denotes the anti-particle. In addition to the shown processes here, their inverse processes are also possible. Although there are elastic scatterings $X_a,{\psi }_i,\to X_b,{\psi }_j$, $X_a,\varphi \to X_b,\varphi$, and ${\psi }_i,\varphi \to {\psi }_j,\varphi$, we omitted them because they do not change the baryon number.

Process	$\Delta \mathit{B}$
$X_a\to {\psi }_i,\varphi$	b
$X_a\to {\overline{\psi }}_i,\overline{\varphi }$	$-b$
${\psi }_i,{\psi }_j\to \overline{\varphi },\overline{\varphi }$	$-2b$
${\psi }_i,\varphi \to {\overline{\psi }}_j,\overline{\varphi }$	$-2b$

. The model includes Majorana-type of chiral fermions $X_a$ for $a=1,\dots ,N_X$, baryonic chiral fermions ${\psi }_i$ for $i=1,\dots ,N_{\psi }$ with the common baryon number b, and a non-baryonic complex scalar $\varphi$. For simplicity, the baryonic fermions ${\psi }_i$ and the scalar $\varphi$ are massless and they are always in the thermal equilibrium. Because $X_a$ are the Majorana fermion, $X_a$ and ${\overline{X}}_a$ can be identified. Thus, once we set the fundamental interaction to provide a decay/inverse-decay processes $X_a\leftrightarrow {\psi }_i,\varphi$, their anti-particle processes $X_a\leftrightarrow {\overline{\psi }}_i,\overline{\varphi }$ also exist. These 3-body interactions also induce the B-violating 2-2 scatterings exchanging $X_a$ fermions.

We comment on the toy model introduced above. Replacing X, $\psi$, $\varphi$ into the right-handed neutrino, left-handed neutrino, Higgs doublet in the standard model, respectively, one can obtain the type-I seesaw model that can realize the well-known leptogenesis scenario [38]. However, the correspondence is incomplete: the type-I seesaw model includes the gauge interactions that induce the $\Delta L=1$ scattering process. See, e.g., ref. [83] for a review of the leptogenesis scenario and [60,84] for the treatment by the Kadanoff–Baym equation.

4.1. Mean Net Baryon Number

At first, we consider only the decay processes for simplicity. This situation is realized when $X_a$ particles start to decay after the scattering processes freeze out. Here, we define the mean net baryon number by

$$\begin{array}{ccc}\hfill {ϵ}_a& =& \sum _f\Delta B_f\left(r_{X_a\to f}-r_{{\overline{X}}_a\to \overline{f}}\right)\hfill \end{array}$$

(94)

where the summation runs for all decay processes, $\Delta B_f$ and $r_{X_a\to f}$ are the generated baryon number through the process of $X_a\to f$ and its branching ratio, respectively. The physical meaning of the mean net baryon number ${ϵ}_a$ is an average of the produced baryon number by a single quantum of $X_a$. In the case of our toy model, this quantity can be represented as

$$\begin{array}{ccc}\hfill {ϵ}_a& =& b\sum _i\left(r_{X_a\to {\psi }_i,\varphi }-r_{X_a\to {\overline{\psi }}_i,\overline{\varphi }}\right)\hfill \end{array}$$

(95)

This result reflects the requirements of B-violation and C, $CP$ violation. If the decay processes are B-conserving $b=0$ or C, $CP$ conserving processes $r_{X_a\to {\psi }_i,\varphi }=r_{X_a\to {\overline{\psi }}_i,\overline{\varphi }}$, the mean net baryon number is vanished.

Supposing that only a single flavour $X_1$ survives and all of $X_1$ particles decay into the baryonic fermions ${\psi }_i$, the generated baryon number can be estimated by $Y_B\sim {ϵ}_1{Y}_{X_1}$. Especially, the baryon abundance can be maximized if $X_1$ particles are the hot relic $Y_{X_1}\sim Y_{\mathrm{hot}}\sim {\displaystyle \frac{45}{2{\pi }^4}}{\displaystyle \frac{g_{X_1}}{h_{\mathrm{eff}}\left(T_f\right)}}$:

$$Y_B\sim {\displaystyle \frac{45}{2{\pi }^4}}·{\displaystyle \frac{{ϵ}_1{g}_{X_1}}{h_{\mathrm{eff}}\left(T_f\right)}},$$

(96)

where $g_{X_1}=2$ is the degrees of freedom of the Majorana-type fermion $X_1$.

4.2. Boltzmann Equations in Baryogenesis Scenario

Although we considered quite a simplified situation in the previous subsection, in reality, the situation is more complicated since the system includes the dynamical decay/inverse-decay and scattering processes. In order to quantify the actual evolution of the baryon abundance including the scattering effects, we need to construct the Boltzmann equations in this system and solve them.

For simplicity, we suppose again that only a single flavor $X_1$ affects to the evolution of the net baryon number. Using the definition of the net baryon density $n_B=b{\sum }_i\left(n_{{\psi }_i}-n_{{\overline{\psi }}_i}\right)$, the evolution of the system including the processes in Table 2 are described by

$$\begin{array}{ccc}\hfill {\dot{n}}_{X_1}+3H{n}_{X_1}& =& -〈{\displaystyle \frac{M_{X_1}}{E_{X_1}}}〉{\Gamma }_{X_1}(n_{X_1}-n_{X_1}^{\mathrm{MB}})+\dots ,\hfill \\ \hfill \end{array}$$

(97)

$$\begin{array}{ccc}\hfill {\dot{n}}_B+3H{n}_B& =& {ϵ}_1〈{\displaystyle \frac{M_{X_1}}{E_{X_1}}}〉{\Gamma }_{X_1}\left(n_{X_1}-n_{X_1}^{\mathrm{MB}}\right)-2{\Gamma }_S{n}_B+\dots ,\hfill \end{array}$$

(98)

where $M_{X_1}$ ($E_{X_1}$) is the mass (energy) of $X_1$, ${\Gamma }_{X_1}\sim {\sum }_i\left({\Gamma }_{X_1\to {\psi }_i\varphi }+{\Gamma }_{X_1\to {\overline{\psi }}_i,\overline{\varphi }}\right)$ is the total width of $X_1$ and

$${ϵ}_1\equiv b·{\displaystyle \frac{{\sum }_i\left({\Gamma }_{X_1\to {\psi }_i\varphi }-{\Gamma }_{X_1\to {\overline{\psi }}_i,\overline{\varphi }}\right)}{{\Gamma }_{X_1}}}$$

(99)

is the mean net baryon number corresponding to (95), and

$$\begin{array}{ccc}\hfill {\Gamma }_S& =& n_{\varphi }^{\mathrm{MB}}\langle {\sigma }_{\psi \varphi \to \overline{\psi }\overline{\varphi }}v\rangle +n_{\psi }^{\mathrm{MB}}\langle {\sigma }_{\psi \psi \to \overline{\varphi }\overline{\varphi }}v\rangle \hfill \end{array}$$

(100)

is the reaction rates through the B-violating scatterings up to the tree level. (See Appendix B for the treatment of the thermally averaged quantities and, also, see Appendix C for the detail of the derivation of the equations, especially the treatment of the real intermediate state (RIS) to avoid the double-counting.) The omitted parts “…” denote the sub-leading processes in terms of the order of couplings. Equation (97) describes the dissipation of $X_a$, and it converts to the baryon with the rate ${ϵ}_a$ and flows into the baryon sector. However, the produced baryons also wash themselves out through the B-violating scattering processes due to the last term in (98). Therefore, the smaller B-violating scattering effect is favored for remaining the more net baryons as long as $X_a$ can be thermalized enough at the initial.

To solve the equations of motion (97) and (98), it is convenient to use the yields $Y_{X_1}=n_{X_1}/s,Y_B=n_B/s$ and the variable $x=M_{X_1}/T$ as

$$\begin{array}{ccc}\hfill Y_{X_1}^{\prime }& =& -{\gamma }_D(Y_{X_1}-Y_{X_1}^{\mathrm{MB}}),\hfill \end{array}$$

(101)

$$\begin{array}{ccc}\hfill Y_B^{\prime }& =& {ϵ}_1{\gamma }_D(Y_{X_1}-Y_{X_1}^{\mathrm{MB}})-2{\gamma }_S{Y}_B,\hfill \end{array}$$

(102)

where

$$\begin{array}{ccc}\hfill Y_{X_1}^{\mathrm{MB}}& =& {\displaystyle \frac{n_{X_1}^{\mathrm{MB}}}{s}}={\displaystyle \frac{45}{4{\pi }^4}}{\displaystyle \frac{g_{X_1}}{h_{\mathrm{eff}}}}{x}^2{K}_2\left(x\right),\hfill \end{array}$$

(103)

$$\begin{array}{ccc}\hfill {\gamma }_D& =& {\displaystyle \frac{1}{x}}·{\displaystyle \frac{{\Gamma }_{X_1}}{H\left(T\right)}}〈{\displaystyle \frac{m_{X_1}}{E_{X_1}}}〉=\sqrt{{\displaystyle \frac{45}{4{\pi }^3{g}_{\mathrm{eff}}}}}{\displaystyle \frac{M_{\mathrm{pl}}}{M_{X_1}}}·{\displaystyle \frac{K_1\left(x\right)}{K_2\left(x\right)}}{\displaystyle \frac{{\Gamma }_{X_1}}{T}},\hfill \end{array}$$

(104)

$$\begin{array}{ccc}\hfill {\gamma }_S& =& {\displaystyle \frac{1}{x}}·{\displaystyle \frac{{\Gamma }_S}{H\left(T\right)}}=\sqrt{{\displaystyle \frac{45}{4{\pi }^3{g}_{\mathrm{eff}}}}}{\displaystyle \frac{M_{\mathrm{pl}}}{M_{X_1}}}·{\displaystyle \frac{{\Gamma }_S\left(T\right)}{T}},\hfill \end{array}$$

(105)

with the n-th order of the modified Bessel function $K_n\left(x\right)$. We assumed the adiabatic evolution of the relativistic degrees $h_{\mathrm{eff}}^{\prime }/h_{\mathrm{eff}}\sim 0$ to obtain (101) and (102). The dimensionless parameters ${\gamma }_{D,S}$ are the reaction rates normalized by the Hubble parameter. In general, ${\gamma }_D$ is proportional to $x^2$ ($x^1$) at the limit of $x\ll 1$ ($x\gg 1$), whereas the behavior of ${\gamma }_S$ depends on the detail of the interaction, as we will see its concrete form with an example model later.

Equations (101) and (102) can provide the analytic form of $Y_B$ as

$$\begin{array}{ccc}\hfill Y_B(\infty )& =& -{ϵ}_1{\int }_0^{\infty }dx{Y}_{X_1}^{\prime }\left(x\right)exp\left[-2{\int }_x^{\infty }d{x}^{\prime }{\gamma }_S\left(x^{\prime }\right)\right].\hfill \end{array}$$

(106)

Especially in the weakly scattering case, ${\int }_0^{\infty }dx{\gamma }_S\lesssim 1$, one can approximate the above result as

$$\begin{array}{ccc}\hfill Y_B(\infty )& \sim & -{ϵ}_1{\int }_0^{\infty }dx{Y}_{X_1}^{\prime }\left(x\right)={ϵ}_1{Y}_{\mathrm{hot}}.\hfill \end{array}$$

(107)

The physical interpretation is that the whole $X_1$ particles existing from the beginning can convert to the net baryons without any wash-out process in this case. Hence, the approximated result does not depend on the details of the decay process ${\gamma }_D$. The result (107) is consistent with the former estimation in (96). On the other hand, the strongly scattering case causes a significant wash-out process, and thus, the final net baryon abundance is strongly suppressed from the result of (107).

To see the concrete evolution dynamics, we consider the following interaction

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\mathrm{int}}& =& -\sum _{a,i}{y}_{ai}\varphi X_a{\psi }_i+\left(\mathrm{h}.\mathrm{c}.\right)\hfill \end{array}$$

(108)

with the Yukawa coupling $y_{ai}$ and the two-component spinors $X_a$ and ${\psi }_i$. This interaction leads to the concrete representation of the decay width and the scattering rate as

$$\begin{array}{ccc}\hfill {\Gamma }_{X_1}& =& \tilde{\alpha }{M}_{X_1},\hfill \end{array}$$

(109)

$$\begin{array}{ccc}\hfill {\Gamma }_S& =& T·{\displaystyle \frac{8{\tilde{\alpha }}^2}{\pi g_{\psi }}}{\tilde{\gamma }}_S\left(x\right)\hfill \end{array}$$

(110)

where we denoted

$$\begin{array}{ccc}\hfill \tilde{\alpha }& =& \sum _i{g}_{{\psi }_i}{g}_{\varphi }{\displaystyle \frac{|y_{1i}{|}^2}{32\pi }},\hfill \\ \hfill {\tilde{\gamma }}_S& =& {\displaystyle \frac{1}{8}}{\int }_0^{\infty }dz{K}_1\left(z\right)\left[2\left({\displaystyle \frac{z^4}{z^2+x^2}}+{\displaystyle \frac{x^2{z}^2}{z^2+2x^2}}ln\left(1+{\displaystyle \frac{z^2}{x^2}}\right)\right)\right.\hfill \\ & & +{\displaystyle \frac{x^2{z}^4}{{(z^2-x^2)}^2+{\tilde{\alpha }}^2{x}^4}}+2\left(z^2-x^{2}ln\left(1+{\displaystyle \frac{z^2}{x^2}}\right)\right)\hfill \end{array}$$

(111)

$$\begin{array}{ccc}& & \left.+{\displaystyle \frac{4x^2(z^2-x^2)}{{(z^2-x^2)}^2+{\tilde{\alpha }}^2{x}^4}}\left(z^2-\left(z^2+x^2\right)ln\left(1+{\displaystyle \frac{z^2}{x^2}}\right)\right)\right]\hfill \end{array}$$

(112)

$$\begin{array}{ccc}& \sim & \left\{\begin{array}{cc}1& (x\ll 1)\hfill \\ 8/x^2& (x\gg 1)\hfill \end{array}\right.,\hfill \end{array}$$

(113)

and $g_{\psi }\equiv {\sum }_i{g}_{{\psi }_i}=N_{\psi }{g}_{{\psi }_i}$. Here, $g_{{\psi }_i}$ and $g_{\varphi }$ are the degrees of freedom of the chiral fermion ${\psi }_i$ and the scalar $\varphi$, not including their anti-particle state. The asymptotic behaviors for each reaction rate are governed by

$${\gamma }_D\left(x\right)\propto \left\{\begin{array}{cc}{x}^2& (x\ll 1)\\ x^1& (x\gg 1)\end{array}\right.,{\gamma }_S\left(x\right)\propto \left\{\begin{array}{cc}\left(\mathrm{constant}\right)& (x\ll 1)\\ x^{-2}& (x\gg 1)\end{array}.\right.$$

(114)

The actual behaviors of ${\gamma }_D$ and ${\gamma }_S$ with concrete parameters are shown in the upper side of Figure 2. The asymptotic behaviors at $x\ll 1$ and $x\gg 1$ are consistent with (114). The enhancement structures for each ${\gamma }_S$ seen around $x\sim \mathcal{O}\left(1\right)$ are induced by the resonant process through the on-shell s-channel shown in (112).

The lower side in Figure 2 shows the evolution of $Y_B/{ϵ}_1$, which is the numerical result from the coupled Equations (101) and (102). The result shows that the heavier mass of X can generate the net baryon number more because the reaction rates are reduced for the heavier case, and hence, the generated baryons can avoid the wash-out process. Specifically, the plot for $M_{X_1}={10}^{16}$ GeV leads the close result to the hot relic approximation (107), whereas the plot for $M_{X_1}={10}^{13}$ GeV shows the dumping by the wash-out effect at the late stage. The milder decrease at the middle stage is caused by the decay of X particles that supply the net baryons to compensate for the wash-out effect.

Finally, the obtained yield of the net baryon number $Y_B$ should be compared with the current bound (93), $Y_{B,\mathrm{now}}\sim {10}^{-10}$. Since the mean net baryon number can roughly be estimated by ${ϵ}_1\sim {\tilde{\alpha }}^2{sin}^2{\theta }_{\mathrm{CP}}$, where ${\theta }_{\mathrm{CP}}$ is a CP phase in the considered model, one can obtain the constraint from the current observation as $Y_{B,\mathrm{now}}/{ϵ}_1\gtrsim {10}^{-10}/{\tilde{\alpha }}^2\sim {10}^{-6}$, where we used $\tilde{\alpha }=0.01$. Therefore, one can find that $M_{X_1}\gtrsim {10}^{14}$ GeV is allowed by comparison with the lower plot in Figure 2.

5. Summary

In this paper, we have demonstrated the derivation of the Boltzmann equation from the microscopic point of view with the quantum field theory, in which the transition probability has been constructed with the statistically averaged quantum states. Although both results of the full and the integrated Boltzmann Equations (37) and (44) are consistent with the well-known results, our derivation ensures that especially the full Boltzmann equation is widely applicable even in the non-equilibrium state, since the derivation does not assume any distribution type nor the temperature of the system. Specifically, the integrated Boltzmann Equation (44) is quite convenient and applicable to a wide range of situations. In the particular case that the kinetic equilibrium cannot be ensured, the coupled equations with the temperature parameter (58) and (59) are better for following the dynamics.

As the application examples of the (integrated) Boltzmann equation in cosmology, we have reviewed two cases: the relic abundance of the DM and the baryogenesis scenario. For the former case, we have shown the Boltzmann equation and its analysis. The analytic results (72) and (74) are quite helpful for estimating the final relic abundance of the DM and its freeze-out epoch. For the latter case, we have derived the Boltzmann equation with a specific model and shown the numerical analysis. The final net baryon number can be estimated by the analytic result (107) in the case of the weakly interacting system, whereas that is strongly suppressed by the wash-out effect in the case of the strongly interacting system.

The Boltzmann equation is a powerful tool for following the evolution of the particle number or other thermal quantities, and thus, it will be applied to many more situations in the future and will open a new frontier in current physics. We hope this paper will be helpful in using the Boltzmann equation and its techniques thoughtfully.