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## 物理代写|宇宙学代写cosmology代考|The collisionless Boltzmann equation for radiation

The Boltzmann equation for radiation, i.e. ultra-relativistic particles, in the perturbed universe is a straightforward generalization of the treatment in Sect. 3.2.2 which led us to Eq. (3.39). Moreover, we have done the hard part already by computing the expressions for $d x^{i} / d t$ [Eq. (3.62)] and $d p^{i} / d t$ [Eq. (3.69)]. We simply specialize them to the case $m=0$, i.e. $E=p$. We can then write Eq. (3.33) as
\begin{aligned} \frac{d f}{d t}=& \frac{\partial f}{\partial t}+\frac{\partial f}{\partial x^{i}} \frac{\hat{p}^{i}}{a}(1-\Phi+\Psi)-\frac{\partial f}{\partial p}\left{[H+\dot{\Phi}] p+\frac{1}{a} p^{i} \Psi_{, i}\right} \ &+\frac{\partial f}{\partial \hat{p}^{i}} \frac{1}{a}\left[(\Phi-\Psi){, i}-\hat{p}^{i} \hat{p}^{k}(\Phi-\Psi){, k}\right] \end{aligned}
This is the complete, linear-order left-hand side of the Boltzmann equation for radiation. However, we can simplify it further by making use of our knowledge of the zeroth-order distribution function $f(\boldsymbol{x}, \boldsymbol{p}, t)$. In the homogeneous universe, this distribution is of the Bose-Einstein form Eq. (2.65). This equilibrium distribution obviously does not depend on position $\boldsymbol{x}$, but it also does not depend on the direction of the momentum vector $\hat{\boldsymbol{p}}$ since it is isotropic. We now make the ansatz that the deviations from the equilibrium distribution of radiation in the inhomogeneous universe are of the same order as the spacetime perturbations $\Phi, \Psi$. We will see in subsequent chapters that this ansatz not only makes our life much easier, but is indeed valid.

With this working assumption, we can immediately drop the last term, $\propto \partial f / \partial \hat{p}^{i}$, in Eq. (3.73). Recall that $\partial f / \partial \hat{p}^{i}$ is nonzero only if we consider a perturbation to the zeroth order $f$; i.e., it is a first-order term. But so is the term which multiplies it. So we can neglect it.

Further, it is easy to see that the potentials in the second term $\propto \partial f / \partial x^{i}$ in Eq. (3.73) are higher order as well, because they multiply $\partial f / \partial x^{i}$ which is a first-order term (again, the zeroth-order distribution function does not depend on position). We finally obtain the Boltzmann equation for radiation consistently expanded to linear order:
$$\frac{d f}{d t}=\frac{\partial f}{\partial t}+\frac{\hat{p}^{i}}{a} \frac{\partial f}{\partial x^{i}}-\left[H+\dot{\Phi}+\frac{1}{a} \hat{p}^{i} \frac{\partial \Psi}{\partial x^{i}}\right] p \frac{\partial f}{\partial p}$$
Eq. (3.74) will lead us directly to the equations governing CMB anisotropies.

## 物理代写|宇宙学代写cosmology代考|The origin of species

The very early universe was hot and dense. As a result, interactions among particles occurred much more frequently than they do today. As an example, a photon in the visible band today can typically travel across much of the observable universe without deflection or capture, so it has a mean free path greater than $10^{28} \mathrm{~cm}$. When the age of the universe was equal to $1 \mathrm{sec}$, though, the mean free path of a photon was about the size of an atom. Thus, in the time it took the universe to expand by a factor of 2, a given photon interacted many, many times. These multiple interactions kept many of the constituents in the universe in equilibrium. Nonetheless, there were times when reactions could not proceed rapidly enough to maintain equilibrium conditions. Not coincidentally, these times are of the utmost interest to cosmologists.

Indeed, we will see in this chapter that out-of-equilibrium phenomena played a role in (i) the formation of the light elements during Big Bang Nucleosynthesis; (ii) recombination of electrons and protons into neutral hydrogen; and possibly in (iii) the production of dark matter in the early universe. It is important to understand that all three phenomena are the result of nonequilibrium physics and that all three can be studied with the same formalism: the Boltzmann equation in the homogeneous universe, as introduced in Sect. 3.2. Sects. $4.2-4.4$ of this chapter are simply applications of this general formula.

To summarize, in this chapter we will go beyond our treatment in Ch. 2 by considering out-of-equilibrium processes in the universe, but we still work within the framework of a homogeneous universe. In succeeding chapters, we will then move beyond uniformity and explore distribution functions for matter and radiation that depend on both position and direction of propagation.

## 物理代写|宇宙学代写cosmology代考|The homogeneous Boltzmann equation revisited

Suppose that we are interested in the number density $n_{1}$ of species 1 . For simplicity, we will assume that the only process affecting the abundance of this species is a reaction with species 2 producing two particles, imaginatively called 3 and 4 . Schematically, $1+2 \leftrightarrow$ $3+4$; i.e., particle 1 and particle 2 can annihilate producing particles 3 and 4 , or the inverse process can produce 1 and 2 . The Boltzmann equation for this system in an expanding universe was derived in Sect. 3.2.2, and the corresponding collision term in Sect. 3.2.3. Combining the general results Eq. (3.43) and Eq. (3.48), we obtain the following evolution equation for $n_{1}$ :\begin{aligned} a^{-3} \frac{d\left(n_{1} a^{3}\right)}{d t}=& \int \frac{d^{3} p_{1}}{(2 \pi)^{3} 2 E_{1}} \int \frac{d^{3} p_{2}}{(2 \pi)^{3} 2 E_{2}} \int \frac{d^{3} p_{3}}{(2 \pi)^{3} 2 E_{3}} \int \frac{d^{3} p_{4}}{(2 \pi)^{3} 2 E_{4}} \ & \times(2 \pi)^{4} \delta_{\mathrm{D}}^{(3)}\left(\boldsymbol{p}{1}+\boldsymbol{p}{2}-\boldsymbol{p}{3}-\boldsymbol{p}{4}\right) \delta_{\mathrm{D}}^{(1)}\left(E_{1}+E_{2}-E_{3}-E_{4}\right)|\mathcal{M}|^{2} \ & \times\left{f_{3} f_{4}\left[1 \pm f_{1}\right]\left[1 \pm f_{2}\right]-f_{1} f_{2}\left[1 \pm f_{3}\right]\left[1 \pm f_{4}\right]\right} \end{aligned}
Here, $E_{i}$ stands for $E_{i}\left(p_{i}\right)$ and $f_{i}$ for $f_{i}\left(p_{i}, t\right)$. We have thus obtained an integrodifferential equation for the phase-space distributions. Further, in principle at least, it must be supplemented with similar equations for the other species. In practice, these formidable obstacles can be overcome for many practical cosmological applications. The first, most important, realization is that scattering processes typically enforce kinetic equilibrium. That is, scattering takes place so rapidly that the distributions of the various species take on the generic Bose-Einstein/Fermi-Dirac forms (Eq. (2.65) and Eq. (2.66)) with equal temperature $T$ for each species. This form condenses all of the freedom in the distribution into the functions of time $T$ and $\mu$. If annihilations were also in equilibrium, the sum of the chemical potentials $\mu_{i}$ in any reaction would have to balance. For example, the reaction $e^{+}+e^{-} \leftrightarrow \gamma+\gamma$ would lead to $\mu_{e^{+}}+\mu_{e^{-}}=2 \mu_{\gamma}$. In the out-of-equilibrium cases we will study, the system will not be in chemical equilibrium and we will have to solve a differential equation for $\mu$. The great simplifying feature of kinetic equilibrium, though, is that this differential equation will be a single ordinary differential equation, as opposed to the very complicated form of Eq. (4.1).

## 物理代写|宇宙学代写cosmology代考|The collisionless Boltzmann equation for radiation

$\backslash 1$ eft 的分隔符缺失或无法识别

$$\frac{d f}{d t}=\frac{\partial f}{\partial t}+\frac{\hat{p}^{i}}{a} \frac{\partial f}{\partial x^{i}}-\left[H+\dot{\Phi}+\frac{1}{a} \hat{p}^{i} \frac{\partial \Psi}{\partial x^{i}}\right] p \frac{\partial f}{\partial p}$$

\eft 的分隔符缺失或无法识别

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