assignmentutor™您的专属作业导师

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|The expanding universe

Just as the early navigators of the great oceans required sophisticated tools to help them find their way, we will need modern technology to help us work through the ramifications of an expanding universe. In this chapter, we introduce the metric and the distribution function, the first of which underlies general relativity and the second, statistical mechanics. We will use this language to derive some of the basic features of the smooth, expanding universe: the redshifting of light, the notion of distance needed to understand the arguments for dark energy, the evolution of the energy density with scale factor, and the epoch of equality $a_{\mathrm{eq}}$ shown in Fig. 1.3. We then go on to perform a cosmic inventory, identifying those constituents of the universe that dominate the energy budget at various epochs.
Implicit in this discussion will be the notion that the universe is smooth, more precisely: spatially homogeneous. That is, the densities of the various constituents such as matter and radiation do not vary in space. To make things even simpler, we will work under the assumption-which is observed to be correct and the reason for which is understood-that all the constituents have equilibrium distributions, as defined and explored in Sect. 2.3.

These simple assumptions form the basic framework within which cosmologists operate and around which they perturb, so that a good grasp of this “zeroth-order universe” is essential. In subsequent chapters, we will see that the deviations from smoothness and the equilibrium distributions are the source of much of the richness we observe in the universe.
From this chapter onward, we use units in which
$$\hbar=c=k_{\mathrm{B}}=1 .$$
Many research papers employ these units, so it is important to get accustomed to them. Please work through Exercise $2.1$ if you are uncomfortable with the idea of setting the speed of light, or Planck’s and Boltzmann’s constants to 1 .

## 物理代写|宇宙学代写cosmology代考|The metric

Rigorously defined, the metric returns the actual physical distance between two infinitesimally close points in spacetime defined in some arbitrary coordinate system. It will be an essential tool in our quest to make quantitative predictions in an expanding universe. In fact, long before Einstein, physicists such as Newton and Maxwell used a spacetime metric. However, their use of a metric was implicit, since they did not distinguish between space and the coordinates that describe it. Going back to Fig. $1.1$ from Ch. 1, we see that even if one knows the components of a separation vector between two points, say two grid points in that figure, the physical distance associated with this vector requires additional information; in this case, the value of the scale factor $a(t)$ at that time.

We are familiar with the metric for the Cartesian coordinate system $(x, y)$ which says that the square of the physical distance between two points separated by $d x$ and $d y$ in a $2 \mathrm{D}$ plane is $(d x)^2+(d y)^2$. However, if we use polar coordinates $(r, \theta)$ instead, the square of the physical distance no longer is the sum of the square of the two coordinate differences. Rather, if the differences $d r$ and $d \theta$ are small, the square of the distance between two points is $(d r)^2+r^2(d \theta)^2$. This distance is invariant: an observer using Cartesian coordinates to calculate it would get the same result as one using polar coordinates. Thus another way of stating what a metric does is this: it turns observer-dependent coordinates into invariants. Mathematically, in the 2D plane, the invariant distance squared is $d l^2=\sum_{i, j=1,2} g_{i j} d x^i d x^j$. The metric $g_{i j}$ in this $2 \mathrm{D}$ example is a $2 \times 2$ symmetric matrix. In Cartesian coordinates $\left(x_1=x, x_2=y\right)$ the metric is simply the identity matrix
$$g_{i j} \stackrel{\text { Cartesian }}{=}\left(\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right),$$
while in polar coordinates $\left(x_1=r, x_2=\theta\right)$ it instead becomes
$$g_{i j} \stackrel{\text { polar }}{=}\left(\begin{array}{cc} 1 & 0 \ 0 & r^2 \end{array}\right) .$$
Note that $g_{i j}$ can also depend on location (in this case through $r$ ). Both forms of the metric describe the same space: a 2D plane.

The concept of a metric really comes into its own when considering more general, curved spaces. Consider the surface of the Earth, which we can roughly approximate as a sphere. There are various ways to assign coordinates to a point on the Earth’s surface.

# 宇宙学代考

## 物理代写|宇宙学代写cosmology代考|膨胀的宇宙

$$\hbar=c=k_{\mathrm{B}}=1 .$$

## 物理代写|宇宙学代写cosmology代考| 度规

$$g_{i j} \stackrel{\text { Cartesian }}{=}\left(\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right),$$
，而在极坐标中$\left(x_1=r, x_2=\theta\right)$则变成
$$g_{i j} \stackrel{\text { polar }}{=}\left(\begin{array}{cc} 1 & 0 \ 0 & r^2 \end{array}\right) .$$

## 有限元方法代写

assignmentutor™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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师