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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写宇宙学cosmology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写宇宙学cosmology代写方面经验极为丰富，各种代写宇宙学cosmology相关的作业也就用不着说。

• 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

## 物理代写|宇宙学代写cosmology代考|Learning to count

At $T \gtrsim 100 \mathrm{GeV}$, all particles of the Standard Model were relativistic (see Table $3.2$ ). To determine the $g_{*}$ associated with these particles, we have to determine how many internal degrees of freedom each particle species has.

Let us start with the gauge bosons, the force carriers of the Standard Model. Photons – the mediators of the electromagnetic force-have $g_{\gamma}=2$ corresponding to two polarizations transverse to the direction of propagation. This is a general feature, any massless particle with spin has exactly two polarization states. A massive particle can have additional longitudinal polarizations, i.e. polarizations along the direction of propagation. In total, a massive particle of spin $s$ has $g=2 s+1$ polarization states. For the massive spin- 1 gauge bosons associated with the weak nuclear force, we therefore have $g_{W \pm, Z}=3$ and hence a total of $3 \times 3=9$ internal degrees of freedom. Gluons – the mediators of the strong nuclear force – are massless and therefore contribute $g_{g}=2$ internal degrees of freedom, like the photons. There are 8 of them, corresponding to the 8 generators of the group $S U(3)$, so we get $8 \times 2=16$.

Next, we consider the fermions – the matter particles of the Standard Model. The charged leptons $\left(e^{\pm}, \mu^{\pm}, \tau^{\pm}\right)$are massive spin- $-\frac{1}{2}$ particles and therefore contribute two spin states. Including a factor of 2 for the antiparticles, we have $3 \times 2 \times 2=12$. Similarly, each quark contributes two spin states. There are 6 flavours of quark $(t$, $b, c, s, d, u)$ and each comes in 3 different colors. Including a factor of 2 for the antiparticles, we then have $6 \times 2 \times 3 \times 2=72$. Lastly, we must talk about neutrinos. Although neutrinos are massive spin- $\frac{1}{2}$ particles, they only contribute 1 internal degree of freedom. The explanation is somewhat involved and will be given in the box below.

## 物理代写|宇宙学代写cosmology代考|Conservation of entropy

In statistical mechanics, a precise definition of entropy can be given in terms of the microstates of the system. Here, we will instead determine the entropy of the primordial plasma from the first law of thermodynamics.

The first law states that the change in the entropy ( $S)$ of a system is related to changes in its internal energy $(U)$ and volume $(V)$ as
$$T \mathrm{~d} S=\mathrm{d} U+P \mathrm{~d} V,$$
where we have assumed that any chemical potentials are small. Defining the entropy density as $s \equiv S / V$, we can write
$$\begin{gathered} T \mathrm{~d}(s V)=\mathrm{d}(\rho V)+P \mathrm{~d} V \ T s \mathrm{~d} V+T V \mathrm{~d} s=\rho \mathrm{d} V+V \mathrm{~d} \rho+P \mathrm{~d} V . \end{gathered}$$
Since $s$ and $\rho$ depend only on the temperature $T$, and not on the volume $V$, this implies
$$(T s-\rho-P) \mathrm{d} V+V\left(T \frac{d s}{d T}-\frac{d \rho}{d T}\right) \mathrm{d} T=0 .$$
In order for this to be satisfied for arbitrary variations $\mathrm{d} V$ and $\mathrm{d} T$, the two brackets have to vanish separately: The vanishing of the first bracket implies that the entropy density can be written as
$$s=\frac{\rho+P}{T},$$
while the vanishing of the second bracket enforces that
$$\frac{d s}{d T}=\frac{1}{T} \frac{d \rho}{d T} .$$

# 宇宙学代考

## 物理代写|宇宙学代写cosmology代考|Learning to count

$6 \times 2 \times 3 \times 2=72$. 最后，我们必须谈谈中微子。尽管中微子是巨大的自旋 $\frac{1}{2}$ 粒子，它们只贡献 1 个内部自由度。解释有些复杂，将在下面的框中给出。

## 物理代写|宇宙学代写cosmology代考|Conservation of entropy

$$T \mathrm{~d} S=\mathrm{d} U+P \mathrm{~d} V,$$

$$T \mathrm{~d}(s V)=\mathrm{d}(\rho V)+P \mathrm{~d} V T s \mathrm{~d} V+T V \mathrm{~d} s=\rho \mathrm{d} V+V \mathrm{~d} \rho+P \mathrm{~d} V .$$

$$(T s-\rho-P) \mathrm{d} V+V\left(T \frac{d s}{d T}-\frac{d \rho}{d T}\right) \mathrm{d} T=0 .$$

$$s=\frac{\rho+P}{T}$$

$$\frac{d s}{d T}=\frac{1}{T} \frac{d \rho}{d T}$$

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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。

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