## 物理代写|宇宙学代写cosmology代考|PHYS4418

2022年12月28日

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
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## 物理代写|宇宙学代写cosmology代考|The concept of cross-section

The cross-section is primarily a statistical concept that allows us to quantitatively evaluate the propensity of two particles to interact with one another. The higher the cross-section, the more likely the interaction. Let us illustrate this notion with Thomson scattering, mentioned earlier (see reaction [1.16]), during which a photon collides with an electron. Without prejudice to the generality of the discussion, let us take the latter at rest and assume that all the photons with which it may interact have the same energy and follow the same incident direction, in this case, the descending vertical. Everything happens then as if the electron was undergoing a “photon rain”. By definition, the cross-section is the area it offers to this rain.

Similarly to an open umbrella that receives further drops for a larger horizontal extent, the electron interacts with further photons for a larger cross-section $\sigma_{\mathrm{T}}$. If it could have an umbrella, the electron would collect per unit of time all the “drops” contained in the vertical cylinder leaning on the surface $\sigma_{\mathrm{T}}$ and of height $c$, the speed of the photons. If the density of the latter is $n_\gamma$, the number of Thomson scatters that the electron undergoes each second amounts to:
$$\Gamma_e=\sigma_{\mathrm{T}} c n_\gamma$$
The previous term can also be understood as a probability of interaction per unit time. If the electron is part of a population of $n_{\mathrm{e}}$ particles per unit volume, then the number of Thomson scatterings per unit volume and time is equal to the product $\sigma_{\mathrm{T}} c n_\gamma n_{\mathrm{e}}$. By reverting the reasoning, the probability per unit of time that a photon scatters on one of the electrons is given by $\Gamma_\gamma=\sigma_{\mathrm{T}} c n_{\mathrm{e}}$.

Quantum field theory (QFT) makes it possible to calculate the cross-section $\sigma_{\mathrm{T}}$ which, in the case where the energy $\hbar \omega$ of the photon is small compared to the mass energy $M_e c^2$ of the electron, is reduced to: $\sigma_{\mathrm{T}}=\frac{8 \pi}{3} r_e^2$ where $r_e=\frac{\alpha_{\mathrm{em}}}{2 \pi}\left{\lambda_{\mathrm{C}} \equiv \frac{h}{M_e c}\right}$

## 物理代写|宇宙学代写cosmology代考|Thermal equilibrium breaking

Neutrinos are neutral and very light particles. Their mass was even assumed to be zero until their oscillations were discovered. After being emitted, an electron neutrino propagates, in effect, while changing flavor and will later manifest itself as a muon or tau neutrino. This mechanism requires a non-zero mass. Neutrinos interact with each other and with other species through forces whose intensity is low. At the energies of interest here, namely of the MeV order, the effective theory of Enrico Fermi suffices. It involves the contact interaction of two fermionic currents that couple at the same point in space-time. The strength of the weak force is characterized by the Fermi constant $G_{\mathrm{F}}=1.166 \times 10^{-5} \mathrm{GeV}^{-2}$ which, for dimensional reasons, is the inverse of a squared energy. As an example, the Fermi Lagrangian, whose expression is specified in equation [1.112], is used to describe the reactions that lead to transmutation between protons and neutrons. This will be the subject of Part 1.5.1 and its study will allow us to better understand the primordial nucleosynthesis of light elements.

The interaction cross-section $\sigma_\nu$ of a neutrino with another species contains as a factor the square $G_{\mathrm{F}}^2$ of the Fermi constant ${ }^7$. Now, $\sigma_\nu$ is a surface, and as such a squared length. In the system of units where $c=k_{\mathrm{B}}=\hbar=1$, a length is analogous to the inverse of an energy by virtue of relation [1.55]. Similarly, the cross-section behaves such as the inverse of a squared energy. We thus need to multiply $G_{\mathrm{F}}^2$, which is expressed in $\mathrm{GeV}^{-4}$, by the square of an energy so that $\sigma_\nu$ have the right dimensions. In order for $\sigma_\nu$ to be also a Lorentz invariant, the square of the total available energy $\sqrt{s}$ in the center of mass frame naturally seems obvious. Up to a numerical factor, the cross-section $\sigma_\nu$ is therefore equal to the product $G_{\mathrm{F}}^2 s$.

The exact calculation of $\sigma_\nu$ goes beyond the scope of this course. Since the analysis we are conducting here is a first approach, we shall simply replace $\sqrt{s}$ by the thermal energy $k_{\mathrm{B}} T$ of the fermions, which are involved in the neutrino thermalization. A fraction of a second after the Big Bang and when the temperature of the ylem is a few MeV, neutrinos interact with each other and with the plasma of electrons and positrons. All these particles are relativistic. We shall substitute the factor $k_{\mathrm{B}} T \equiv T$ for the energy $\sqrt{s}$. The numberr of collisions a neutrino undeergoes per unit time is then given by the product:
$$\Gamma_\nu=\left\langle\sigma_\nu v_{\text {rel }}\right\rangle\left(n_\nu+n_{\mathrm{e}^{-}}+n_{\mathrm{e}^{+}}\right)$$

## 物理代写|宇宙学代写cosmology代考|The concept of cross-section

$$\Gamma_e=\sigma_{\mathrm{T}} c n_\gamma$$

## 物理代写|宇宙学代写cosmology代考|Thermal equilibrium breaking

$G_{\mathrm{F}}=1.166 \times 10^{-5} \mathrm{GeV}^{-2}$ 由于尺寸原因，它是平方能量的倒数。例如，费米拉格朗日方程（其表达式在 方程 [1.112] 中指定) 用于描述导致质子和中子之间转 变的反应。这将是第 $1.5 .1$ 部分的主题，对它的研究将 使我们更好地理解轻元素的原始核合成。

$$\Gamma_\nu=\left\langle\sigma_\nu v_{\mathrm{rel}}\right\rangle\left(n_\nu+n_{\mathrm{e}^{-}}+n_{\mathrm{e}^{+}}\right)$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。