## 物理代写|宇宙学代写cosmology代考|PHYS3080

statistics-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|Introductory remarks

The description of the thermodynamic properties of ylem is considerably simplified when one realizes that it behaves like a perfect gas. The kinetic energies of its particles are indeed much higher than their potential energies associated with long-range interactions. In order to establish this property, let us take, for example, the universe shortly before primordial nucleosynthesis, when its temperature $T \sim 10^{10} \mathrm{~K}$ and its age $t \sim 1 \mathrm{~s}$. The ylem is then mainly composed of photons as well as of an ultrarelativistic gas of electrons and positrons, their antiparticles. Baryons exist in the form of traces. Photons are electrically neutral and do not interact with each other. Their energy is purely kinetic. Electrons and positrons carry electric charges $q=-e$ and $q=+e$, respectively, with $e=1.602 \times 10^{-19} \mathrm{C}$.

Depending on the sign of $q$, the interaction is attractive or repulsive. Electrons and positrons have the same density, equal to:
$$n_{\mathrm{e}^{-}}=n_{\mathrm{e}^{+}}=\frac{3 \zeta(3)}{2 \pi^2}\left(\frac{k_{\mathrm{B}} T}{\hbar c}\right)^3 \quad \text { with } \quad \zeta(3)=1.20206$$
and their average distance $\bar{r}$ is of the order of $\left(n_{\mathrm{e}^{-}}+n_{\mathrm{e}^{+}}\right)^{-1 / 3} \simeq 1.4 \hbar c / k_{\mathrm{B}} T$. The average electrostatic energy $E_{\mathrm{e}}$ of an electron or a positron corresponds, in absolute value, to that of two charges $q$ located at the distance $\bar{r}$ from each other, such that:
$$\left|E_{\mathrm{e}}\right|=|q V| \simeq \frac{e^2}{4 \pi \epsilon_0 \bar{r}} \simeq \frac{e^2}{4 \pi \epsilon_0 \hbar c} \frac{k_{\mathrm{B}} T}{1.4} \simeq \frac{\alpha_{\mathrm{em}}}{1.4} k_{\mathrm{B}} T$$
where $\alpha_{\mathrm{em}}=7.297 \times 10^{-3}$ is the fine structure constant of electromagnetism. This electrostatic energy is to be compared to the average kinetic energy $E_{\mathrm{k}}$ of electrons and positrons which, for each particle, is given by $3.15 k_{\mathrm{B}} T$. We then obtain a ratio $\left|E_{\mathrm{e}}\right| / E_{\mathrm{k}}$ of the order of $1.66 \times 10^{-3}$, which fully justifies that the ylem and its components behave like a perfect gas. We shall extend this assumption to the totality of the ylem.

## 物理代写|宇宙学代写cosmology代考|Numerical density

The number density of a generic population consisting of particles $A$ is given by the general expression:
$$n_A=\int \frac{d^3 \vec{p}}{h^3} \frac{g_A}{\exp \left(E / k_{\mathrm{B}} T\right)-\epsilon}=\frac{g_A}{2 \pi^2}\left(\frac{k_{\mathrm{B}} T}{\hbar c}\right)^3 \int_0^{\infty} x^2 d x\left(e^y-\epsilon\right)^{-1}$$
where the parameters $x=p c / k_{\mathrm{B}} T$ and $y=E / k_{\mathrm{B}} T$ are related by the equality $y^2=u^2+x^2$, the ratio $M c^2 / k_{\mathrm{B}} T$ between mass and temperature being denoted by $u$.

In order to explore the different limits of expression [1.26], we shall use the relation:
$$\int_0^{\infty} d x\left(\frac{x^n}{e^x-\epsilon}\right)=\Gamma(n+1) \zeta(n+1) \begin{cases}1 & \text { if } \epsilon=1 \ 1-\frac{1}{2^n} & \text { if } \epsilon=-1\end{cases}$$
where $\Gamma(s)$ denotes the Euler gamma function. For $s$ integer, $\Gamma(s+1)$ is equal to $s$ !, the product of the $s$ first numbers. The Riemann function $\zeta(s)$ is defined by the series:
$$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s}$$
Some particular values can be computed using Fourier series such as, for instance, $\zeta(2)=\pi^2 / 6$ or $\zeta(4)=\pi^4 / 90$. Others require a direct calculation such as $\zeta(3)$, an irrational equal to $1.20206$.

• In the low temperature limit, the gas behaves in a non-relativistic manner. Within this regime, the ratio $u=M / T=M c^2 / k_{\mathrm{B}} T$ is very large compared to 1 and the density simplifies to:
$$n_A=g_A T^3\left(\frac{u}{2 \pi}\right)^{3 / 2} e^{-u}$$
It becomes negligible compared to the densities of the relativistic components, being exponentially suppressed hy the factor $e^{-u}$.

# 宇宙学代考

## 物理代写|宇宙学代写cosmology代考|Introductory remarks

$$n_{\mathrm{e}^{-}}=n_{\mathrm{e}^{+}}=\frac{3 \zeta(3)}{2 \pi^2}\left(\frac{k_{\mathrm{B}} T}{\hbar c}\right)^3 \quad \text { with } \quad \zeta(3)=1.20206$$

$$\left|E_{\mathrm{e}}\right|=|q V| \simeq \frac{e^2}{4 \pi \epsilon_0 \bar{r}} \simeq \frac{e^2}{4 \pi \epsilon_0 \hbar c} \frac{k_{\mathrm{B}} T}{1.4} \simeq \frac{\alpha_{\mathrm{em}}}{1.4} k_{\mathrm{B}} T$$

## 物理代写|宇宙学代写cosmology代考|Numerical density

$$n_A=\int \frac{d^3 \vec{p}}{h^3} \frac{g_A}{\exp \left(E / k_{\mathrm{B}} T\right)-\epsilon}=\frac{g_A}{2 \pi^2}\left(\frac{k_{\mathrm{B}} T}{\hbar c}\right)^3 \int_0^{\infty} x^2 d x\left(e^y-\epsilon\right)^{-1}$$

$$\int_0^{\infty} d x\left(\frac{x^n}{e^x-\epsilon}\right)=\Gamma(n+1) \zeta(n+1)\left{1 \quad \text { if } \epsilon=11-\frac{1}{2^n} \quad \text { if } \epsilon=-1\right.$$

$$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s}$$

• 在低温极限下，气体以非相对论方式表现。在这个制度下，比率 $u=M / T=M c^2 / k_{\mathrm{B}} T$ 与 1 相 比非常大，密度简化为:
$$n_A=g_A T^3\left(\frac{u}{2 \pi}\right)^{3 / 2} e^{-u}$$
与相对论分量的密度相比，它变得可以忽略不计，被因子指数抑制 $e^{-u}$.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|宇宙学代写cosmology代考|PHYC90009

statistics-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|A Thermal History of the Universe and Primordial Nucleosynthesis

In the Friedmann-Lemaitre standard model, the universe is homogeneous and has identical characteristics everywhere. The theory however does not explain the reason for this homogeneity, which is well verified by observations. On closer inspection, it is even surprising that the CMB appears so isotropic. Whatever the direction of observation, the photons emitted by the last scattering surface all show the same temperature of $2.73 \mathrm{~K}$. However, they come from a multitude of regions that, in the strict framework of the theory, were not in causal contact at the time they emitted the CMB. Their number can be estimated at about a thousand.

Alan Guth proposed to solve this problem (Guth 1981) by assuming that the universe should undergo, shortly after the Planck era, a phase of exponential expansion called cosmic inflation, during which the scale factor $a(t)$ is multiplied by a factor $\sim e^{60}$. To generate this phenomenal acceleration, it is necessary to assume that the universe undergoes a phase transition during which its energy density $\rho$ is dominated by the potential energy of a scalar field. At the time, high-energy theoretical physicists were working on a theory for unifying the weak, strong and electromagnetic interactions. The models they arrived at, called grand unification models, naturally present all the qualities required to generate an inflationary period. The grand unified symmetry spontaneously breaks down around a cosmic time of about $10^{-35} \mathrm{~s}$ and the resulting inflation allows a single causally connected domain to become as large as the entire observable universe today. The original scenario has undergone a few modifications (Albrecht et al. 1982; Linde 1983) and now requires a dedicated scalar field called the inflaton. Its quantum fluctuations generate density perturbations that will much later transform into galaxies. Gravitational waves are also radiated during this period. They are a signature that future interferometers, such as LISA (Laser Interferometer Space Antenna), will try to identify. When inflation ends, around a cosmic age of $10^{-33}$ to $10^{-32} \mathrm{~s}$, the inflaton disintegrates and gives birth to the ylem during a reheating phase. The evolution of the universe is then described by the Friedmann-Lemaitre model.

The elementary particles sensitive to the strong interaction are called hadrons, from the Greek hadros which means strong. They are constructions of quarks bound together by gluons. The hadron family is divided into two categories. Mesons are pairs formed by a quark and an antiquark and include among them the neutral pion $\pi^0$ and the charged pion $\pi^{p m}$. Baryons are assemblies composed of three quarks, such as the proton $p$ and neutron $n$, which are themselves the constituents of atomic nuclei. Gluons carry the strong interaction and are vector bosons of zero mass with two helicity states. Their electromagnetic analogue is the photon. The standard model of high energy physics, in this case quantum chromodynamics, requires the existence of eight gluons, as well as six quarks grouped in three families of doublets. The lightest of them is composed of $u$ quarks (up) and $d$ quarks (down). It enables the proton and the neutron to be built.

In the primordial universe, the temperature is so high that quarks and gluons cease to be bound. Mesons and baryons are completely dissociated into a plasma of free quarks and gluons, called quark gluon plasma (QGP). Heavy ion collisions, achieved at CERN and studied through experimental collaborations such as ALICE, attempt to recreate QGP in the laboratory in order to study it. Yet, QGP exists freely in a natural way during the Big Bang. Nonetheless, around a cosmic age of $10 \mu$ s, a phase transition occurs in which quarks and gluons condense into hadrons. Protons, neutrons and pions then appear.

Neutrinos are very light, neutral elementary particles that experience only the weak force. They interact with each other as well as with electrons, muons and tauons ${ }^3$, with which they are associated in the family of leptons, a term derived from the Greek leptos and meaning weak. Leptons are grouped into three doublets, each consisting of a neutrino and its charged partner. In the primordial universe, neutrinos frequently interact with their environment. They are thermally coupled to the primordial plasma and share its temperature. But under the combined influence of dilution and cooling of the ylem, both resulting from the overall expansion, neutrinos have fewer and fewer collisions with other particles. Their thermodynamic coupling fades around a cosmic age of $1 \mathrm{~s}$. They then undergo a thermal freeze-out, also called kinetic decoupling, which makes them a fossilized population of particles that no longer interact with each other or with the rest of the plasma. Thermal decoupling of neutrinos will be the subject of section 1.4.

# 宇宙学代考

## 物理代写|宇宙学代写cosmology代考|A Thermal History of the Universe and Primordial Nucleosynthesis

Alan Guth 提出解决这个问题 (Guth 1981) 的方法是假设宇宙在普朗克时代之后不久会经历一个称为宇宙 憉胀的指数膨胀阶段，在此期间比例因子 $a(t)$ 乘以一个因数 $\sim e^{60}$. 为了产生这种惊人的加速度，有必要 假设宇宙经历了一个相变，在此期间它的能量密度 $\rho$ 由标量场的势能支配。当时，高能理论物理学家正在 研究统一弱相互作用、强相互作用和电磁相互作用的理论。他们得出的模型被称为大统一模型，自然而然 地呈现了产生通货膨胀时期所需的所有品质。大统一对称性在大约 10 秒的宇宙时间附近自发地破缺 $10^{-35}$ s由此产生的膨胀使得一个因果联系的领域变得和今天整个可观察的宇宙一样大。最初的场景经过 了一些修改（Albrecht 等人 1982 年；Linde 1983 年)，现在需要一个称为暴胀子的专用标量场。它的 量子涨落会产生密度扰动，这些扰动会在很久以后转变为星系。引力波也在此期间辐射。它们是末来干涉 仪 (例如 LISA (激光干涉仪空间天线) ) 将尝试识别的特征。当暴胀结束时，大约是一个宇宙时代 $10^{-33}$ 到 $10^{-32} \mathrm{~s}$ ，暴胀子在再加热阶段分解并产生 ylem。然后，弗里德曼-勒梅特模型描述了宇宙的演 化。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|宇宙学代写cosmology代考|ASTR3002

statistics-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|A useful refresher of the Friedmann–Lemaître model

The expansion of the universe and the Big Bang from which it originated are well described in the framework of the theory of general relativity. The cosmic fluid generates a curvature of space-time via its energy and momentum, which manifests itself by means of a dilation of space over time. The Friedmann-Lemaitre model is based on the cosmological principle according to which the universe is homogeneous and isotropic, a starting hypothesis that large-scale observations corroborate. The corresponding geometry is described by the Robertson-Walker metric:
$$c^2 d \tau^2=c^2 d t^2-a^2(t) R_0^2\left{\frac{d r^2}{1-k r^2}+r^2 d \theta^2+r^2 \cos ^2 \theta d \phi^2\right}$$

The cosmic time $t$ is none other than the proper time $\tau$ of an observer at rest and yet in free fall with respect to the universe. The radial variable $r$ is identified with the radius when the index of spatial curvature $k$ is zero ${ }^1$. The angular variables $\theta$ (latitude) and $\phi$ (longitude) have their usual meaning. In the case of a spherical metric, the index $k$ is $+1$ and $R_0$ is interpreted as the current value of the radius of curvature of the universe. Otherwise, $R_0$ is simply a length standard. The scale factor $a(t)$ is nowadays normalized to $a\left(t_0\right) \equiv a_0=1$. Its temporal variation follows the Friedmann-Lemaître equations, deriving from general relativity:
$$\left{H \equiv \frac{\dot{a}}{a}\right}^2=\frac{8 \pi G}{3 c^2} \rho-\frac{k c^2}{a^2 R_0^2}$$
and:
$$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3 c^2} \rho-\frac{4 \pi G}{c^2} P$$
where the variables $r h o$ and $P$ are, respectively, the energy density and the pressure of the cosmic fluid filling the universe.

The first relation describes the evolution of the expansion rate $H$, defined as the logarithmic derivative of the scale factor $a(t)$. Its current value, called the Hubble constant and denoted by $H_0$, is still heartily debated among cosmologists, although the range of possible values is gradually narrowing. Adam Riess and his team obtain $H_0=74.03 \pm 1.42 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$ from the variable $\delta$-cepheid stars of the Large Magellanic Cloud (Riess et al. 2019), while the final result of the Planck mission is $H_0=67.4 \pm 0.5 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$ (Aghanim et al. 2018). The Planck collaboration also finds good agreement between its CMB observations and the assumption of a spatially flat space. In this case, the index of curvature $k$ is zero and the energy density of the cosmic fluid is given by the critical value $\rho_{\mathrm{c}}^0$, known as the closure density, such that:
$$H_0^2=\frac{8 \pi G}{3 c^2} \rho_{\mathrm{c}}^0$$

## 物理代写|宇宙学代写cosmology代考|The major events of the Big Bang

The universe was born $13.8$ billion years ago (Aghanim et al. 2018) from a state where the temperature and density of the primordial plasma are virtually infinite. Physics does not like divergencies. These indicate, in general, that the theory is no longer suitable and wanders outside its domain of validity. Such is the case here. General relativity indeed proposes a geometrical vision of gravitation and relies on the notion of a classical space-time, where each point or event is associated with well-defined coordinates. There is no question of quantum mechanics in the Friedmann-Lemaître model. Yet, when the universe has just been born, for cosmic times $t$ tending toward 0 , quantum fluctuations prove to be very important. The Heisenberg uncertainty principle allows a quantity of energy $\Delta E \sim \hbar / \Delta t$ to appear spontaneously during the lapse of time $\Delta t$, which we can assimilate here to the cosmic time $t$. The reduced Planck constant is denoted by $\hbar \equiv h / 2 \pi$. The energy $\Delta E$ is all the stronger that $\Delta t \sim t$ is small. We can then define two characteristic distances. At cosmic time $t$, the distance $r_{\mathrm{c}}$ traveled by light since the Big Bang is equal to the product $c t$. It indicates the size of each causally connected domain at this time. Moreover, the energy fluctuations $\Delta E$ that appear at the same time are associated with the mass $\Delta M=\Delta E / c^2$ and the Schwarzschild radius:
$$r_{\mathrm{S}}=\frac{2 G \Delta M}{c^2} \simeq \frac{2 G}{c^4} \frac{\hbar}{t}$$

This defines the region where gravity is so strong that space closes in on itself. For $t$ very small, the causal distance $r_{\mathrm{c}}$ becomes smaller than the Schwarzschild radius $r_{\mathrm{S}}$ of quantum fluctuations. These fluctuations then dominate the geometrical behavior of space and it is no longer possible to neglect them. The Friedmann-Lemaitre model ceases to be valid. The introduction of a quantum theory of gravitation is therefore necessary as $r_{\mathrm{c}} \leq r_{\mathrm{S}}$, which corresponds to an age $t$ lower than the critical value:
$$t_{\mathrm{Pl}} \equiv \sqrt{\frac{2 G \hbar}{c^5}} \simeq 7,624 \times 10^{-44} \mathrm{~s}$$
called Planck time. The Friedmann-Lemaitre model can only really be used after the Planck era, when the cosmic time $t \geq t_{\mathrm{Pl}}$.

# 宇宙学代考

## 物理代写|宇宙学代写cosmology代考|A useful refresher of the Friedmann–Lemaître model

$c^{\wedge} 2 d \backslash \operatorname{tau}^{\wedge} 2=c^{\wedge} 2 d t^{\wedge} 2-a^{\wedge} 2(t) R_{-} 0^{\wedge} 2 \backslash$ eft $\left{\backslash f r a c\left{d r^{\wedge} 2\right}\left{1-k r^{\wedge} 2\right}+r^{\wedge} 2 d \backslash\right.$ theta^$a^{\wedge} 2+r^{\wedge} 2 \backslash \cos ^{\wedge} 2 \backslash$ theta $d$ phi^^ $2 \backslash r i g h t$.

\eft $\left{\mathrm{H} \text { lequiv } \backslash \text { frac }{\backslash d o t{a}}{a} \backslash r_{i g h t}\right}^{\wedge} 2=\backslash$ frac ${8 \backslash$ pi $G}\left{3 \mathrm{c}^{\wedge} 2\right} \backslash \operatorname{rho}-\backslash f r a c\left{k \mathrm{c}^{\wedge} 2\right}\left{\mathrm{a}^{\wedge} 2 \mathrm{R}{-} 0^{\wedge} 2\right}$ 和： $$\frac{\ddot{a}}{a}=-\frac{4 \pi G}{3 c^2} \rho-\frac{4 \pi G}{c^2} P$$ 其中变量 $r h o$ 和 $P$ 分别是充满宇宙的宇宙流体的能量密度和压力。 第一个关系描述了膨胀率的演变 $H$ ，定义为比例因子的对数导数 $a(t)$. 它的当前值，称为哈勃常数，表示 为 $H_0$ ，尽管可能值的范围正在逐渐缩小，但宇宙学家仍在热烈争论。Adam Riess 和他的团队获得 $H_0=74.03 \pm 1.42 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$ 从变量 $\delta$-大麦哲伦云的造父变星 (Riess 等人，2019 年)，而普 朗克任务的最终结果是 $H_0=67.4 \pm 0.5 \mathrm{~km} \mathrm{~s}^{-1} \mathrm{Mpc}^{-1}$ (Aghanim 等人，2018 年)。普朗克合作 还发现其 CMB 观测结果与空间平坦空间的假设非常吻合。在这种情况下，曲率指数 $k$ 为零，宇宙流体的 能量密度由临界值给出 $\rho{\mathrm{c}}^0$ ，称为闭包密度，使得:
$$H_0^2=\frac{8 \pi G}{3 c^2} \rho_{\mathrm{c}}^0$$

## 物理代写|宇宙学代写cosmology代考|The major events of the Big Bang

$$r_{\mathrm{S}}=\frac{2 G \Delta M}{c^2} \simeq \frac{2 G}{c^4} \frac{\hbar}{t}$$

$$t_{\mathrm{Pl}} \equiv \sqrt{\frac{2 G \hbar}{c^5}} \simeq 7,624 \times 10^{-44} \mathrm{~s}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|宇宙学代写cosmology代考|ASTR3002

statistics-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|The fundamental equations of cosmology

Almost all of cosmology consists of a series of applications of two fundamental equations of physics: the Einstein equations describing gravity; and the Boltzmann equation of statistical mechanics describing matter and radiation. In this chapter, we have provided a concise summary of these equations and applied them to the smooth and, in the case of the Boltzmann equation, perturbed universe.
The full Einstein equations are
$$G_{\mu \nu} \equiv R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R=8 \pi G T_{\mu \nu},$$
where we have included the cosmological constant (or other form of dark energy) on the right-hand side. Applied to the FLRW metric and assuming a Euclidean universe, we derived the Friedmann equation for the scale factor $a(t)$ :
$$\frac{H^2(t)}{H_0^2}=\frac{\rho(t)}{\rho_{\mathrm{cr}}}=\sum_{s=\mathrm{r}, \mathrm{m}, v, \mathrm{DE}} \Omega_s[a(t)]^{-3\left(1+w_s\right)} .$$
Later chapters will be wholly devoted to studying perturbations around the homogeneous universe. Including these, we write the perturbed metric as
\begin{aligned} &g_{00}(\boldsymbol{x}, t)=-1-2 \Psi(\boldsymbol{x}, t), \ &g_{0 i}(\boldsymbol{x}, t)=0, \ &g_{i j}(\boldsymbol{x}, t)=a^2(t) \delta_{i j}[1+2 \Phi(\boldsymbol{x}, t)], \end{aligned}
and work to linear order in $\Psi, \Phi$ throughout. Deferring the derivation of the Einstein equations in the perturbed universe to Ch. 6 , we solved the geodesic equation in the perturbed universe in this chapter. The comoving momentum becomes
$$P^\mu=\left[E(1-\Psi), p^i \frac{1-\Phi}{a}\right],$$
where $E=\sqrt{p^2+m^2}$ is the proper energy and $p$ is the physical momentum. The geodesic equation yields
$$\frac{d p^i}{d t}=-(H+\dot{\Phi}) p^i-\frac{E}{a} \Psi_{, i}-\frac{1}{a} \frac{p^i}{E} p^k \Phi_{, k}+\frac{p^2}{a E} \Phi_{, i},$$
a compact relation which contains such diverse physics as Newtonian dynamics and gravitational lensing and which we will make use of many times throughout this book.

## 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|宇宙学的基本方程

$$G_{\mu \nu} \equiv R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R=8 \pi G T_{\mu \nu},$$
，其中我们在右边包含了宇宙常数(或其他形式的暗能量)。应用于FLRW度规并假设一个欧几里德宇宙，我们导出了比例因子$a(t)$:
$$\frac{H^2(t)}{H_0^2}=\frac{\rho(t)}{\rho_{\mathrm{cr}}}=\sum_{s=\mathrm{r}, \mathrm{m}, v, \mathrm{DE}} \Omega_s[a(t)]^{-3\left(1+w_s\right)} .$$

\begin{aligned} &g_{00}(\boldsymbol{x}, t)=-1-2 \Psi(\boldsymbol{x}, t), \ &g_{0 i}(\boldsymbol{x}, t)=0, \ &g_{i j}(\boldsymbol{x}, t)=a^2(t) \delta_{i j}[1+2 \Phi(\boldsymbol{x}, t)], \end{aligned}
，并在$\Psi, \Phi$中始终按线性顺序工作。本章将摄动宇宙中爱因斯坦方程的推导推至第六章，求解摄动宇宙中的测地线方程。移动动量变为
$$P^\mu=\left[E(1-\Psi), p^i \frac{1-\Phi}{a}\right],$$

$$\frac{d p^i}{d t}=-(H+\dot{\Phi}) p^i-\frac{E}{a} \Psi_{, i}-\frac{1}{a} \frac{p^i}{E} p^k \Phi_{, k}+\frac{p^2}{a E} \Phi_{, i},$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|宇宙学代写cosmology代考|PHYS3080

statistics-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|The geodesic equation

In order to derive the Boltzmann equation, we need to know how particles move within the perturbed spacetime. Again, this is determined by the geodesic equation which we considered in Sect. 2.1.2, and which we now extend to include the spacetime perturbations $\Phi, \Psi$. In particular, our goal is to calculate $d x^i / d t, d p / d t$, and $d \hat{p}^i / d t$ to insert into Eq. (3.33). The mass-shell constraint for a particle with mass $m$ is now given by
$$g_{\mu \nu} P^\mu P^v=-(1+2 \Psi)\left(P^0\right)^2+p^2=-m^2,$$
where again
$$p^2 \equiv g_{i j} P^i P^j \text {. }$$
We will continue to define the energy as $E(p) \equiv \sqrt{p^2+m^2}$. In the massless case, we obviously have $E=p$. We can now eliminate the time component of $P^\mu$ through
$$P^0=\frac{E}{\sqrt{1+2 \Psi}}=E(1-\Psi) .$$

This last equality holds since we are doing first-order perturbation theory in the small quantity $\Psi$. Similarly, we can use Eq. (3.58) to derive $P^i$. This yields the four-momentum of a massive particle in a perturbed FLRW spacetime (which includes the massless case):
$$P^\mu=\left[E(1-\Psi), p^i \frac{1-\Phi}{a}\right] .$$
Here, we have defined $p^i$ through
$$p^i=p \hat{p}^i \quad \text { where } \quad \hat{p}^i=\hat{p}i$$ is a unit vector satisfying $\delta{i j} \hat{p}^i \hat{p}^j=1$ as before. Eq. (3.60) allows us to eliminate $P^0$ and $P^i$ in favor of $E(p), p$, the magnitude of the momentum, and $\hat{p}^i$ whenever they occur. Moreover, plugging these into Eq. (3.20) yields the expressions for the energy-momentum tensor in terms of the distribution function in the presence of metric perturbations (see Exercise $3.12$ ) which we will need later.

## 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|测地方程

$$g_{\mu \nu} P^\mu P^v=-(1+2 \Psi)\left(P^0\right)^2+p^2=-m^2,$$

$$p^2 \equiv g_{i j} P^i P^j \text {. }$$

$$P^0=\frac{E}{\sqrt{1+2 \Psi}}=E(1-\Psi) .$$ 消除$P^\mu$的时间成分

$$P^\mu=\left[E(1-\Psi), p^i \frac{1-\Phi}{a}\right] .$$

$$p^i=p \hat{p}^i \quad \text { where } \quad \hat{p}^i=\hat{p}i$$是一个满足$\delta{i j} \hat{p}^i \hat{p}^j=1$的单位向量。Eq.(3.60)允许我们剔除$P^0$和$P^i$，取而代之的是$E(p), p$，动量的大小，以及它们发生时的$\hat{p}^i$。此外，将它们代入式(3.20)，得到存在度规摄动(见练习$3.12$)时能量动量张量的分布函数表达式，这是我们以后需要的

## 物理代写|宇宙学代写cosmology代考|辐射的无碰撞玻尔兹曼方程

\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}

$$\frac{d f}{d t}=\frac{\partial f}{\partial t}+\frac{\hat{p}^i}{a} \frac{\partial f}{\partial x^i}-\left[H+\dot{\Phi}+\frac{1}{a} \hat{p}^i \frac{\partial \Psi}{\partial x^i}\right] p \frac{\partial f}{\partial p} .$$
Eq。(3.74)将直接引导我们得到控制CMB各向异性的方程

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|宇宙学代写cosmology代考|PHYC90009

statistics-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|Collision terms

The effect of direct particle interactions is, in the Boltzmann realm, referred to as “collisions.” Collisions include scattering as well as pair creation, annihilation, and particle decay. A common type of process is a reaction where particles of type 1 and 2 interact to form particles of type 3 and 4 :
$$(1)p+(2)_q \longleftrightarrow(3){p^{\prime}}+(4)_{q^{\prime}},$$
where the subscripts indicate momenta. Note that this includes scattering of electrons and photons for example, if we choose $(1)=(3)=\left(e^{-}\right)$and $(2)=(4)=(\gamma)$; or annihilation, if we choose $(1)=\left(e^{-}\right),(2)=\left(e^{+}\right)$and $(3)=(4)=(\gamma)$. Moreover, all microscopic physical processes conserve momentum and energy:
$$\boldsymbol{p}+\boldsymbol{q}=\boldsymbol{p}^{\prime}+\boldsymbol{q}^{\prime} ; \quad E_1(\boldsymbol{p})+E_2(\boldsymbol{q})=E_3\left(\boldsymbol{p}^{\prime}\right)+E_4\left(\boldsymbol{q}^{\prime}\right),$$
where $E_s(p)=\sqrt{p^2+m_s^2}$ denotes the energy-momentum relation for particle $s$ [Eq. (3.29)]. Each type of particle has its respective distribution function $f_s(x, p, t), s=1,2,3,4$. Often in cosmology, different states (e.g. spin) have the same distribution function. So, instead of following them with separate functions, we will assign appropriate statistical weights $g_s$.
How does the reaction Eq. (3.44) affect the evolution of the distribution functions $f_s$ of the particles involved? First, we are dealing with a local interaction in space and time, so all the distribution functions are evaluated at $(\boldsymbol{x}, t)$, and we only need to determine the momentum arguments. For $f_1(\boldsymbol{x}, \boldsymbol{p}, t)$, for example, Eq. (3.44) means that we have to subtract the particles of type 1 that get scattered away from momentum $p$ by the forward reaction, and add the particles of type 1 that get scattered to momentum $p$ by the reverse reaction (Fig. 3.3). Therefore we must sum over all other momenta $\left(\boldsymbol{q}, \boldsymbol{q}^{\prime}, \boldsymbol{p}^{\prime}\right)$ which affect $f_1(\boldsymbol{p})$.

## 物理代写|宇宙学代写cosmology代考|Perturbed spacetime

To begin, we must specify the form of the metric, accounting for perturbations around the smooth universe described by Eq. (2.12). Whereas the smooth universe is characterized by a single function, $a(t)$, which depends only on time and not on space, the perturbed universe requires two more functions, $\Psi$ and $\Phi$, both of which depend on space and time. In terms of these, the metric can be written as
\begin{aligned} &g_{00}(\boldsymbol{x}, t)=-1-2 \Psi(\boldsymbol{x}, t) \ &g_{0 i}(\boldsymbol{x}, t)=0 \ &g_{i j}(\boldsymbol{x}, t)=a^2(t) \delta_{i j}[1+2 \Phi(\boldsymbol{x}, t)] \end{aligned}
In the absence of $\Psi$ and $\Phi$, Eq. (3.49) is simply the FLRW metric of the zeroth-order homogeneous, Euclidean cosmology. Conversely, in the absence of expansion $(a(t)=1)$ this metric describes a weak gravitational field. The perturbations to the metric are $\Psi$, which corresponds to the Newtonian potential and governs the motion of slow-moving (nonrelativistic) bodies; and $\Phi$, the perturbation to the spatial curvature which, from Eq. (3.49), can also be interpreted as a local perturbation to the scale factor: $a(t) \rightarrow a(\boldsymbol{x}, t)=a(t) \sqrt{1+2 \Phi(\boldsymbol{x}, t)}$. In general, there is a tight relation between $\Phi$ and $\Psi$, as we will see in later chapters.

The typical magnitude of metric perturbations $\Psi, \Phi$ in our universe is less than $10^{-4}$. For this reason, it is an excellent approximation to work at linear order in these quantities. This means that we neglect all terms that are quadratic or of higher order in them. We will work under this approximation, which greatly simplifies the calculations, throughout the entire book.

There are two technical points about the metric in Eq. (3.49) that you do not need to worry about for most of this book, but which nonetheless are important to be aware of. We will cover these issues in Ch. 6, when we study gravity in the inhomogeneous universe in more detail. First, one can break up perturbations into those behaving as scalars, vectors, and tensors under a transformation from one 3D spatial coordinate system to another. Eq. (3.49) contains only scalar perturbations. On the other hand, tensor perturbations correspond to gravitational waves, which we know to exist. To take these into account, $g_{\mu v}$ requires other functions besides $\Psi$ and $\Phi$. For now we focus solely on the scalar perturbations; these are by far the most important ones for the origin and evolution of structure in the universe.

## 物理代写|宇宙学代写cosmology代考|碰撞条款

$$(1)p+(2)q \longleftrightarrow(3){p^{\prime}}+(4){q^{\prime}},$$
，其中下标表示动量。注意，这包括电子和光子的散射，例如，如果我们选择$(1)=(3)=\left(e^{-}\right)$和$(2)=(4)=(\gamma)$;或者湮灭，如果我们选择$(1)=\left(e^{-}\right),(2)=\left(e^{+}\right)$和$(3)=(4)=(\gamma)$。此外，所有微观物理过程都保存动量和能量:
$$\boldsymbol{p}+\boldsymbol{q}=\boldsymbol{p}^{\prime}+\boldsymbol{q}^{\prime} ; \quad E_1(\boldsymbol{p})+E_2(\boldsymbol{q})=E_3\left(\boldsymbol{p}^{\prime}\right)+E_4\left(\boldsymbol{q}^{\prime}\right),$$

## 物理代写|宇宙学代写cosmology代考|摄动时空

\begin{aligned} &g_{00}(\boldsymbol{x}, t)=-1-2 \Psi(\boldsymbol{x}, t) \ &g_{0 i}(\boldsymbol{x}, t)=0 \ &g_{i j}(\boldsymbol{x}, t)=a^2(t) \delta_{i j}[1+2 \Phi(\boldsymbol{x}, t)] \end{aligned}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|宇宙学代写cosmology代考|PHYS3080

statistics-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|Structure in the universe

The existence of structure in the universe was known long before the detection of CMB anisotropies: various efforts to map out the distribution of galaxies in the local universe clearly showed that they are not distributed homogeneously. The number of galaxies and volume covered by such surveys has grown exponentially. Two surveys in particular broke new ground: the Sloan Digital Sky Survey (SDSS; Fig. 1.8) and the Two Degree Field Galaxy Redshift Survey (2dF), which between them compiled the redshifts of, and hence the distances to, over a million galaxies. Projects over the ensuing decades have and will provide deeper and more detailed maps than these ground-breaking surveys, by orders of magnitude.

The galaxies in Fig. $1.8$ are clearly not distributed randomly: the universe has structure on large scales. To understand this structure, we must develop the tools to study perturbations around the smooth background. We will see that this is straightforward in theory, as long as the perturbations remain small. To compare theory with observations, we must thus try to avoid regimes that cannot be described by small perturbations. As an extreme example, we can never hope to understand cosmology by carefully examining rock formations on Earth. The intermediate steps-collapse of matter into a galaxy; star formation; planet formation; geology; etc. – are much too complicated to allow comparison between linear theory and observations. In fact, perturbations to the matter on small scales (less than about $10 \mathrm{Mpc}$ ) have become large in the late universe; that is, the fractional density fluctuations on these scales are not small, but comparable to or larger than unity. We say that these scales have grown nonlinear. On the other hand, large-scale perturbations are still small (quasi-linear). So they have been processed much less than the small-scale structure. Similarly, anisotropies in the CMB are small because they originated at early times and the photons that we observe from the CMB do not clump on their way to us. Because of this, the best ways to learn about the evolution of structure and to compare theory with observations are to look at anisotropies in the $\mathrm{CMB}$ and at large-scale structure (LSS), i.e. how galaxies and matter are distributed on large scales. However, we will learn in Chs. 12-13 that valuable cosmological information can also be extracted from smaller, nonlinear scales provided we choose our observables wisely.

## 物理代写|宇宙学代写cosmology代考|Standard Model of particle physics

The Standard Model of particle physics describes the known fundamental particles in nature and how they interact. The particles can be divided into two classes: spin-1/2 fermions and integer-spin bosons.
Fermions are the constituents of matter: the quarks, out of which baryons are built, and the leptons such as electrons and neutrinos. There are three generations with two quarks each for a total of six quarks, denoted $u, d ; s, c ; b, t$. Each generation of quarks is associated with a pair of leptons. For example, the $u, d$ pair is associated with the electron and its neutrino: $e^{-}, v_e$. The other lepton pairs are $\mu^{-}, v_\mu$ and $\tau^{-}, v_\tau$. The vast majority of matter in the universe is made up of the first generation, with the exception of neutrinos, which are mixed between the different generations. Unlike leptons, quarks do not exist on their own, but they form bound states under the strong interaction. Baryons, the most important ones being the proton and neutron, are made out of three quarks. Mesons are composed of a quark-antiquark pair.
Bosons contain the spin-1 (vector) force carriers, the most famous of which is the photon which mediates the electromagnetic force. There are eight gluons (massless, like the photon) that mediate the strong force. The weak force, responsible for example for neutron decay, is mediated by three massive bosons: the $Z, W^{+}$and $W^{-}$. These force mediators are complemented with the spin-0 (scalar) Higgs boson. The Higgs couples to all massive fermions as well as the $W$ and $Z$ bosons. This coupling gives mass to the particles through the Higgs’ homogeneous background field value.

The Standard Model has remained largely intact since its inception, gaining more and more experimental verification every year. However, neutrino masses are now a confirmed piece of physics beyond the Standard Model. Moreover, the evidence cosmologists have uncovered-that there is a need for dark matter, dark energy, and new physics leading to inflation-clearly shows that the Standard Model is not the final word in particle physics.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|宇宙学代写cosmology代考|PHYC90009

statistics-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|Big Bang nucleosynthesis

Armed with an understanding of the evolution of the scale factor and the densities of the constituents in the universe, we can extrapolate backwards to explore phenomena at early times. When the universe was much hotter and denser, and the temperature was of order $1 \mathrm{MeV} / k_{\mathrm{B}}$, there were no neutral atoms or even bound nuclei. The vast amounts of highenergy radiation in such a hot environment ensured that any atom or nucleus produced would be immediately destroyed by a high-energy photon. As the universe cooled well below typical nuclear binding energies, light elements began to form in a process known as Big Bang Nucleosynthesis ( $B B N$ ). Knowing the conditions of the early universe and the relevant nuclear cross-sections, we can calculate the expected primordial abundances of all the elements (Ch. 4).

Fig. 1.6 shows the BBN predictions for the abundances of helium and deuterium as a function of the mean baryon density, essentially the density of ordinary matter (Sect. 2.4) in the universe, in units of the critical density. The predicted abundances, in particular that of deuterium, which we will explore in detail in Ch. 4, depend on the density of protons and neutrons at the time of nucleosynthesis. The combined proton plus neutron density is equal to the baryon density since both protons and neutrons have baryon number one and these are the only baryons around at the time.

The horizontal lines in Fig. $1.6$ show the current measurements of the light element abundances. The deuterium abundance is measured in the intergalactic medium at high redshifts by looking for a subtle absorption feature in the spectrum of distant quasars (see Burles and Tytler, 1998; Cooke et al., 2018 and Exercise 1.3). These measurements of the abundances, combined with BBN calculations, give us a way of measuring the baryon density in the universe, constraining ordinary matter to contribute at most $5 \%$ of the critical density (note that the $x$-axis in Fig. $1.6$ is the baryon density divided by the critical density, but multiplied by $h^2 \simeq 0.5$ ). Since the total matter density today is significantly larger than this-as we will see throughout the book-nucleosynthesis provides a compelling argument for matter that is comprised of neither protons or neutrons. This new type of matter has been dubbed dark matter because it apparently does not emit light. One of the central questions in physics now is: “What is the Dark Matter?”

## 物理代写|宇宙学代写cosmology代考|The cosmic microwave background

Another phenomenon that falls out of energetics and a qualitative understanding of the evolution of the universe is the origin of the CMB. When the temperature of the radiation was of order $10^4 \mathrm{~K}$ (corresponding to energies of order an $\mathrm{eV}$ ), free electrons and protons combined to form neutral hydrogen. Before then, any hydrogen produced was quickly ionized by energetic photons. After that epoch, at $z \simeq 1100$, the photons that comprise the CMB ceased interacting with any particles and traveled freely through space. When we observe them today, we are thus looking at messengers from an early moment in the universe’s history. They are therefore the most powerful probes of the early universe. We will spend an inordinate amount of time in this book working through the details of what happened to the photons before they last scattered off of free electrons, and also developing the mathematics of the free-streaming process since then. Among many other aspects, we will understand how the CMB constrains the baryon density independently, and in agreement with $\mathrm{BBN}$ as shown in Fig. 1.6, providing a ringing confirmation of the concordance model.

For now, we are only concerned with the crucial fact that the interactions of photons with electrons before last scattering ensured that the photons were in equilibrium. That is, they should have a black-body spectrum. The specific intensity of a gas of photons with a black-body spectrum is
$$I_v=\frac{4 \pi \hbar v^3 / c^2}{\exp \left[2 \pi \hbar v / k_{\mathrm{R}} T\right]-1} .$$
Fig. $1.7$ shows the remarkable agreement between this prediction (see Exercise 1.4) of Big Bang cosmology and the observations by the FIRAS instrument aboard the COBE satellite. In fact, the CMB provides the best black-body spectrum ever measured. We have been told ${ }^2$ that detection of the $3 \mathrm{~K}$ background by Penzias and Wilson in the mid-1960s was sufficient evidence to decide the controversy in favor of the Big Bang over the Steady State universe, an alternative scenario without any expansion. Penzias and Wilson, though, measured the radiation at just one wavelength. If even their one-wavelength result was enough to tip the scales, the current data depicted in Fig. $1.7$ should send skeptics from the pages of physics journals to the far reaches of radical internet chat groups.

The most important fact we learned from our first 25 years of surveying the CMB was that the early universe was very smooth. No anisotropies were detected in the CMB. This period, while undoubtedly frustrating for observers searching for anisotropies, solidified the view of a smooth Big Bang. The satellite mission COBE discovered anisotropies in the $\mathrm{CMB}$ in 1992, indicating that the early universe was not completely smooth. There were small perturbations in the cosmic plasma, with fractional temperature fluctuations of order $10^{-5}$. By now, these small fluctuations have been mapped with exquisite precision, and the state of the art is to look for even more subtle effects such as CMB polarization and the effect of the intervening matter distribution through gravitational lensing. To understand all of these effects, we must clearly go beyond the smooth background universe and look at deviations from smoothness, or inhomogeneities. Inhomogeneities in the universe are often simply called structure.

## 物理代写|宇宙学代写cosmology代考|大爆炸核合成

1.6显示了BBN对氦和氘丰度的预测作为平均重子密度的函数，本质上是宇宙中普通物质的密度(第2.4节)，以临界密度为单位。预测的丰度，特别是氘的丰度，我们将在第4章中详细探讨，取决于核合成时质子和中子的密度。质子和中子的密度之和等于重子密度，因为质子和中子都有1号重子，而这些是当时周围唯一的重子 图$1.6$中的水平线显示了当前对轻元素丰度的测量。通过在遥远类星体的光谱中寻找细微的吸收特征(见Burles和Tytler, 1998;Cooke等人，2018和练习1.3)。这些丰度的测量，结合BBN的计算，为我们提供了一种测量宇宙重子密度的方法，约束普通物质最多贡献$5 \%$的临界密度(注意，图$1.6$中的$x$轴是重子密度除以临界密度，但乘以$h^2 \simeq 0.5$)。由于现在物质的总密度远远大于这个——正如我们将在全书中看到的那样——核合成为既不是由质子也不是由中子组成的物质提供了一个令人信服的论据。这种新型物质被称为暗物质，因为它显然不发光。现在物理学的核心问题之一是:“暗物质是什么?”

## 物理代写|宇宙学代写cosmology代考|宇宙微波背景

$$I_v=\frac{4 \pi \hbar v^3 / c^2}{\exp \left[2 \pi \hbar v / k_{\mathrm{R}} T\right]-1} .$$
$1.7$显示了大爆炸宇宙学的这一预测(见练习1.4)与COBE卫星上的FIRAS仪器的观测结果之间的显著一致。事实上，CMB提供了迄今为止测量过的最好的黑体光谱。我们被告知${ }^2$，彭齐亚斯和威尔逊在20世纪60年代中期对$3 \mathrm{~K}$背景的探测，足以证明这场争论支持大爆炸而不是稳定状态宇宙，稳定状态宇宙是没有任何膨胀的另一种情况。不过，彭齐亚斯和威尔逊只测量了一种波长的辐射。如果他们的单波长结果就足以扭转这一趋势，那么图$1.7$中所描述的当前数据应该会把怀疑论者从物理期刊的页面送到激进的互联网聊天群中去

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|宇宙学代写cosmology代考|ASTR3002

statistics-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|A nutshell history of the universe

We have solid evidence that the universe is expanding. This means that early in its history the distance between us and distant galaxies was smaller than it is today. It is convenient to describe this effect by introducing the scale factor $a$, whose present value is set to 1 by convention. At earlier times, $a$ was smaller than it is today. We can imagine placing a grid in space as in Fig. 1.1 which expands uniformly as time evolves. Points on the grid, which correspond to observers at rest, maintain their coordinates, so the comoving distance between two points-which just measures the difference between coordinates, and can be obtained by counting grid cells as indicated in Fig. 1.1-remains constant. However, the physical distance is proportional to the scale factor, and the physical distance does evolve with time.
A directly related effect is that the physical wavelength of light emitted from a distant object is stretched out proportionally to the scale factor, so that the observed wavelength is larger than the one at which the light was emitted. It is convenient to define this stretching factor as the redshift $z$ :
$$1+z \equiv \frac{\lambda_{\mathrm{obs}}}{\lambda_{\text {emit }}}=\frac{a_{\mathrm{obs}}}{a_{\text {emit }}}=\frac{1}{a_{\text {emit }}} .$$
In addition to the scale factor and its evolution, the smooth universe is characterized by one other parameter, its geometry. There are three possibilities: Euclidean, open, or closed universes. These different possibilities are best understood by considering two freely traveling particles which start their journeys moving parallel to each other. In a Euclidean universe, often also called a “flat universe,” the particles behave as Euclid himself expected them to: their trajectories remain parallel as long as they travel freely. If the universe is closed, the initially parallel particles gradually converge, just as in the case of the 2 -sphere all lines of constant longitude meet at the North and South Poles. The analogy of a closed universe to the surface of a sphere runs even deeper: both are spaces of constant positive curvature, the former in three spatial dimensions and the latter in two. Finally, in an open universe, the initially parallel paths diverge, as would two marbles rolling off a saddle.

## 物理代写|宇宙学代写cosmology代考|The Hubble diagram

If the universe is expanding as depicted in Fig. $1.1$, then galaxies should be moving away from each other. We should therefore see galaxies receding from us. Hubble (1929) first found that distant galaxies are in fact all apparently receding from us, i.e. redshifted. He also noticed the trend that the velocity increases with distance. This is exactly what we expect in an expanding universe, for the physical distance between two galaxies is $d=a x$ where $x$ is the comoving distance. ${ }^1$ In the absence of any comoving motion, $\dot{x} \equiv d x / d t=0$ (no peculiar velocity), the relative velocity $v$ is therefore equal to
$$v=\frac{d}{d t}(a x)=\dot{a} x=H_0 d \quad(v \ll c),$$
where overdots indicate derivatives with respect to time $t$. Therefore, the apparent velocity should increase linearly with distance (at least at low redshift) with a slope given by $H_0$, the Hubble constant. Eq. (1.8) is known as the Hubble-Lemaitre law. The value of the constant is simply determined by measuring the slope of the line in the Hubble diagram shown in Fig. 1.5.

In the next chapter, we will generalize the distance-redshift relation to larger distances, where Eq. (1.8) breaks down. Instead of recession velocities, this more rigorous derivation will be based on the stretching of the wavelength of light encoded in Eq. (1.1). For now, let us just point out that the distance-redshift relation depends on the energy content of the universe. Data from a variety of sources point to a current best-fit scenario that is Euclidean and contains about $70 \%$ of the energy in the form of a cosmological constant, or some other form of dark energy. This now forms the concordance cosmology that will be our working model throughout.

## 物理代写|宇宙学代写cosmology代考|宇宙的概略历史

$$1+z \equiv \frac{\lambda_{\mathrm{obs}}}{\lambda_{\text {emit }}}=\frac{a_{\mathrm{obs}}}{a_{\text {emit }}}=\frac{1}{a_{\text {emit }}} .$$

## 物理代写|宇宙学代写cosmology代考|哈勃图

$$v=\frac{d}{d t}(a x)=\dot{a} x=H_0 d \quad(v \ll c),$$
，其中overdots表示对时间$t$的导数。因此，视速度应该随距离线性增加(至少在低红移时)，斜率为$H_0$，哈勃常数。式(1.8)被称为哈勃-勒梅特定律。这个常数的值可以通过测量哈勃图中直线的斜率来确定，如图1.5所示

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 物理代写|宇宙学代写cosmology代考|ASTR3002

statistics-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|Robertson-Walker metric

In this problem, you will be guided through an alternative (more rigorous) derivation of the Robertson-Walker metric.

1. Explain why the most general metric for a homogeneous and isotropic universe is
$$\mathrm{d} s^{2}=-\mathrm{d} t^{2}+a^{2}(t) \gamma_{i j}(\mathbf{x}) \mathrm{d} x^{i} \mathrm{~d} x^{j},$$
where we have set $c=1$. In particular, explain why $g_{00}=-1$ and $g_{0 i}=0$.
2. Assume isotropy of the universe about a fixed point $r=0$. Show that the most general spatial metric takes the form
$$\mathrm{d} \ell^{2} \equiv \gamma_{i j} \mathrm{~d} x^{i} \mathrm{~d} x^{j}=e^{2 \alpha(r)} \mathrm{d} r^{2}+r^{2} \mathrm{~d} \Omega^{2}$$
where $\mathrm{d} \Omega^{2} \equiv \mathrm{d} \theta^{2}+\sin ^{2} \theta \mathrm{d} \phi^{2}$. Show that the scalar curvature associated with this metric is
$$R_{(3)}=\frac{2}{r^{2}}\left[1-\frac{d}{d r}\left(r e^{-2 \alpha(r)}\right)\right] .$$
Warning: this part is a bit tedious.

## 物理代写|宇宙学代写cosmology代考|Geodesics from a simple Lagrangian

In Appendix A, we derive the geodesics equation from the relativistic action of a point particle. In this problem, you will discover a simpler way to obtain the same result.

1. Consider the Lagrangian
$$\mathcal{L} \equiv-g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}$$
where $\dot{x}^{\mu} \equiv d x^{\mu} / d \lambda$, for a general parameter $\lambda$. Show that the EulerLagrange equation
$$\frac{d}{d \lambda}\left(\frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu}}\right)=\frac{\partial \mathcal{L}}{\partial x^{\mu}}$$
2. If $\mathcal{L}$ has no explicit dependence on $\lambda$, then $\partial \mathcal{L} / \partial \lambda=0$. Show that this implies that the “Hamiltonian” is a constant along the geodesics:
$$\mathcal{H} \equiv \mathcal{L}-\frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu}} \dot{x}^{\mu}=g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu} \text {. }$$
For a massive particle, we set $\lambda$ equal to the proper time $\tau$, and the constraint becomes $g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}=-1$. A nice feature of the Lagrangian method described in this problem is that is also applies to massless particles, in which case we must have $g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}=0$.

## 物理代写|宇宙学代写cosmology代考|Robertson-Walker metric

1. 解释为什么均匀和各向同性宇宙的最通用度量是
$$\mathrm{d} s^{2}=-\mathrm{d} t^{2}+a^{2}(t) \gamma_{i j}(\mathbf{x}) \mathrm{d} x^{i} \mathrm{~d} x^{j},$$
我们设置的地方 $c=1$. 特别说明原因 $g_{00}=-1$ 和 $g_{0 i}=0$.
2. 假设宇宙关于一个固定点的各向同性 $r=0$. 证明最一般的空间度量采用以下形式
$$\mathrm{d} \ell^{2} \equiv \gamma_{i j} \mathrm{~d} x^{i} \mathrm{~d} x^{j}=e^{2 \alpha(r)} \mathrm{d} r^{2}+r^{2} \mathrm{~d} \Omega^{2}$$
在哪里 $\mathrm{d} \Omega^{2} \equiv \mathrm{d} \theta^{2}+\sin ^{2} \theta \mathrm{d} \phi^{2}$. 表明与该度量相关的标量曲率是
$$R_{(3)}=\frac{2}{r^{2}}\left[1-\frac{d}{d r}\left(r e^{-2 \alpha(r)}\right)\right] .$$
警告：这部分有点乏味。

## 物理代写|宇宙学代写cosmology代考|Geodesics from a simple Lagrangian

1. 考虑拉格朗日
$$\mathcal{L} \equiv-g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}$$
在哪里 $\dot{x}^{\mu} \equiv d x^{\mu} / d \lambda$, 对于一般参数 $\lambda$. 证明 EulerLagrange 方程
$$\frac{d}{d \lambda}\left(\frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu}}\right)=\frac{\partial \mathcal{L}}{\partial x^{\mu}}$$
导致测地线方程。
2. 如果 $\mathcal{L}$ 没有明确的依赖 $\lambda$ ， 然后 $\partial \mathcal{L} / \partial \lambda=0$. 证明这意味着“哈密顿”是沿测地线的常数:
$$\mathcal{H} \equiv \mathcal{L}-\frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu}} \dot{x}^{\mu}=g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}$$
对于一个大质量粒子，我们设置 $\lambda$ 等于适当的时间 $\tau$ ，约束变为 $g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}=-1$. 这个问题中描述的拉格朗 日方法的一个很好的特点是它也适用于无质量粒子，在这种情况下，我们必须有 $g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}=0$.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。