## 物理代写|宇宙学代写cosmology代考|PHYS3003

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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After their kinetic decoupling, when the temperature of the ylem is now below $T_{\mathrm{kd}}$, neutrinos constitute a fossil population and are no longer in thermodynamic equilibrium with the rest of matter. They are not even in equilibrium with themselves anymore. It is therefore no longer possible to define a temperature in the thermodynamic sense. However, we shall show that the distribution of fossil neutrinos within the phase space remains constant and that it is then possible to define an effective temperature which, analogously to $\Theta_s$, plays the role of a scaling factor.
Since neutrinos are neither created nor destroyed after their kinetic decoupling, their density per covolume, that is, their number within a volume that follows the expansion of the universe, remains constant. Let us denote by $\mathrm{d}^3 \vec{x}0 \equiv d x_0 d y_0 d z_0$ an element of volume of the physical space taken today, at the time when the scale factor is $a_0=1$. During the expansion, this volume was smaller and was equal to $\mathrm{d}^3 \vec{x}-a^3 \mathrm{~d}^3 \vec{x}_0$. Since their kinetic decoupling, the number $\mathrm{d}^3 N\nu$ of neutrinos contained in the covolume $\mathrm{d}^3 \vec{x}$ has remained constant. If $n_\nu$ denotes the numerical density, then $\mathrm{d}^3 N_\nu=n_\nu \mathrm{d}^3 \vec{x} \equiv n_\nu a^3 \mathrm{~d}^3 \vec{x}0$. The product $n\nu a^3$, which measures the codensity, remains constant after the thermal freezing of the neutrinos.

The volume element $\mathrm{d}^3 \vec{p} \equiv d p_x d p_y d p_z$ also undergoes the expansion of the universe but, unlike $\mathrm{d}^3 \vec{x}$, it decreases as $a^{-3}$. Let us start with an intuitive, if not rigorous, argument. The momentum $p$ of a neutrino is associated by quantum mechanics to the length $\lambda_{\text {Broglie }}$ of the corresponding neutrinic wave. Now, $\lambda_{\text {Broglie }}$ undergoes the expansion of space such that it increases with time as the scale factor $a$. The momemtum $p$ decreases by the same amount, as $p \equiv \hbar / \lambda_{\text {Broglie }} \propto a^{-1}$. Each of the components $d p_x, d p_y$ and $d p_z$ varying as $a^{-1}$, the differential element $\mathrm{d}^3 \vec{p}$ decreases indeed just as $a^{-3}$.

During the expansion of the universe, the elementary cells of phase space have a volume $\mathrm{d}^3 \vec{x} \mathrm{~d}^3 \vec{p}$ that remains constant. If $\mathrm{d}^3 \vec{x}$ increases as $a^3$, the element $\mathrm{d}^3 \vec{p}$ of the momentum space evolves as $a^{-3}$, one offsetting the other. The distribution of neutrinos within phase space is described by the function $\mathcal{F}\nu$, which measures the number $\mathrm{d}^6 N\nu$ of particles in the cell located at point $(\vec{x}, \vec{p})$, so that:
$$\mathrm{d}^6 N_\nu=\mathcal{F}_\nu(t, \vec{x}, \vec{p}) \mathrm{d}^3 \vec{x} \mathrm{~d}^3 \vec{p}$$

## 物理代写|宇宙学代写cosmology代考|The Cowsik and McClelland limit

Around a temperature of $0.5 \mathrm{MeV}$, the electrons annihilate with the positrons. The heat release caused by these reactions benefits only the photons because neutrinos no longer interact with the rest of the plasma. The ylem has become transparent to them. They are thus no longer likely to receive energy. The temperature of the photons $T$ then increases with respect to that of the neutrinos $T_\nu$. The corresponding heating is calculated by writing that the entropy of the radiative mixture, constituting of electrons, positrons and photons, remains constant in time. The annihilation of electrons is in fact an internal phenomenon of this radiative gas. The entropy it possesses inside a covolume of typical size $a \propto T_\nu^{-1} \equiv \Theta_{\mathrm{s}}^{-1}$ is:
$$S_{\text {rad }}=h_{\text {eff }}(\mathrm{rad}) \times \mathcal{S}\gamma(T) \times T\nu^{-3}=\text { constant }$$
where $\mathcal{S}\gamma$ denotes the volume entropy of photons alone. The effective number of entropic degrees of freedom of the radiative mixture is described by the factor: $$h{\mathrm{eff}}(\mathrm{rad})=h_{\mathrm{eff}}(\gamma)+h_{\mathrm{eff}}\left(e^{-}\right)+h_{\mathrm{eff}}\left(e^{+}\right)=1+2 h_{\text {eff }}^e(T)$$
Before the annihilation of electrons, for a temperature higher than their mass $M_e=0.511 \mathrm{MeV}$, the coefficient $h_{\text {eff }}(\mathrm{rad})$ is $11 / 4$. Once the photons are alone, this coefficient is equal to 1 . The electron-positron annihilation thus leads to the heating:
$$\frac{T}{T_\nu}=\left{\frac{11}{4}\right}^{1 / 3} \simeq 1.401$$
By taking a CMB temperature $T_\gamma^0$ equal to $2.726$ (Fixsen 2009), we find that the current neutrino temperature is $T_\nu^0=T_\gamma^0 / 1.401=1.946 \mathrm{~K}$.

## 宇宙学代考

$$\mathrm{d}^6 N_\nu=\mathcal{F}_\nu(t, \vec{x}, \vec{p}) \mathrm{d}^3 \vec{x} \mathrm{~d}^3 \vec{p}$$

## 物理代写|宇宙学代写cosmology代考|The Cowsik and McClelland limit

$S_{\mathrm{rad}}=h_{\text {eff }}(\mathrm{rad}) \times \mathcal{S} \gamma(T) \times T \nu^{-3}=$ constant

$$h \operatorname{eff}(\mathrm{rad})=h_{\mathrm{eff}}(\gamma)+h_{\mathrm{eff}}\left(e^{-}\right)+h_{\mathrm{eff}}\left(e^{+}\right)=1+2 h$$

$\lfloor$ frac ${T}{T \backslash \ln }==$ left ${\backslash$ frac ${11}{4} \backslash$ right $} \wedge{1 / 3} \backslash$ simeq $1.401$

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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。