### 物理代写|宇宙学代写cosmology代考|PHYS3080

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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

## 物理代写|宇宙学代写cosmology代考|The Expanding Universe

Our goal in this chapter is to derive, and then solve, the equations governing the evolution of the entire universe. This may seem like a daunting task. How can we hope to describe the long-term evolution of the cosmos when we have such a hard time just predicting the weather or the stability of the Solar System?

Fortunately, the coarse-grained properties of the universe are remarkably simple. While the distribution of galaxies is clumpy on small scales, it becomes more and more uniform on large scales. In particular, when averaged over sufficiently large distances (say larger than $100 \mathrm{Mpc}$ ), the universe looks isotropic (the same in all directions). Assuming that we don’t live at a special point in space-and that nobody else does either – the observed isotropy then implies that the universe is also homogeneous (the same at every point in space). This leads to a simple mathematical description of the universe because the spacetime geometry takes a very simple form.

Since a static universe filled with matter and energy is unstable, we expect the spacetime to be dynamical. Indeed, observations of the light from distant galaxies have shown that the universe is expanding. Running this expansion backwards in time, we predict that nearly 14 billion years ago our whole universe was in a hot dense state. The Big Bang theory describes what happened in this fireball, and how it evolved into the universe we see around us today. In Part I of this book, I will describe our modern understanding of this theory. In this chapter, we will study the geometry and dynamics of the homogeneous universe, while in the next chapter, we will discuss the many interesting events that occured in the hot Big Bang.
I will assume some familiarity with the basics of special relativity (at the level of manipulating spacetime tensors), but will introduce the necessary elements of general relativity (GR) as we go along. I will mostly state results in $\mathrm{GR}$ without derivation, which are then relatively easy to apply in the cosmological context. Although this plug-and-play approach loses some of the geometrical beauty of Einstein’s theory, it gets the job done and provides the fastest route to our explorations of cosmology. Further background on GR is given in Appendix A.

## 物理代写|宇宙学代写cosmology代考|Spacetime and Relativity

I will assume that you have been introduced to the concept of a metric before. Just to remind you, the metric is an object that turns coordinate distances into physical distances. For example, in three-dimensional Euclidean space, the physical distance between two points separated by the infinitesimal coordinate distances $\mathrm{d} x, \mathrm{~d} y$ and $\mathrm{d} z$ is
$$\mathrm{d} \ell^{2}=\mathrm{d} x^{2}+\mathrm{d} y^{2}+\mathrm{d} z^{2}=\sum_{i, j=1}^{3} \delta_{i j} \mathrm{~d} x^{i} \mathrm{~d} x^{j},$$
where I have introduced the notation $\left(x^{1}, x^{2}, x^{3}\right)=(x, y, z)$. In this example, the metric is simply the Kronecker delta $\delta_{i j}=\operatorname{diag}(1,1,1)$. However, you also know that if we were to use spherical polar coordinates, the square of the physical distance would no longer be the sum of the squares of the coordinate distances. Instead, we would get
$$\mathrm{d} \ell^{2}=\mathrm{d} r^{2}+r^{2} \mathrm{~d} \theta^{2}+r^{2} \sin ^{2} \theta \mathrm{d} \phi^{2} \equiv \sum_{i, j=1}^{3} g_{i j} \mathrm{~d} x^{i} \mathrm{~d} x^{j},$$
where $\left(x^{1}, x^{2}, x^{3}\right)=(r, \theta, \phi)$. In this case, the metric has taken a less trivial form, namely $g_{i j}=\operatorname{diag}\left(1, r^{2}, r^{2} \sin ^{2} \theta\right)$. Observers using different coordinate systems won’t necessarily agree on the coordinate distances between two points, but they will always agree on the physical distance, $\mathrm{d} \ell$. We say that $\mathrm{d} \ell$ is an invariant. Hence, the metric turns observer-dependent coordinates into invariants.

A fundamental object in relativity is the spacetime metric. It turns observerdependent spacetime coordinates $x^{\mu}=\left(c t, x^{i}\right)$ into the invariant line element ${ }^{1}$
$$\mathrm{d} s^{2}=\sum_{\mu, \nu=0}^{3} g_{\mu \nu} \mathrm{d} x^{\mu} \mathrm{d} x^{\nu} \equiv g_{\mu \nu} \mathrm{d} x^{\mu} \mathrm{d} x^{\nu} .$$
In special relativity, the spacetime is Minkowski space, $\mathbb{R}^{1,3}$, whose line element is
$$\mathrm{d} s^{2}=-c^{2} \mathrm{~d} t^{2}+\delta_{i j} \mathrm{~d} x^{i} \mathrm{~d} x^{j},$$
1 Throughout this book, I will use Einstein’s summation convention where repeated indices are summed over. Our metric signature will be $(-,+,+,+)$. In this chapter, I will keep the speed of light explicit, but in the rest of the book I will use natural units with $c \equiv 1$.

## 物理代写|宇宙学代写cosmology代考|Spacetime and Relativity

$$\mathrm{d} \ell^{2}=\mathrm{d} x^{2}+\mathrm{d} y^{2}+\mathrm{d} z^{2}=\sum_{i, j=1}^{3} \delta_{i j} \mathrm{~d} x^{i} \mathrm{~d} x^{j},$$

$\delta_{i j}=\operatorname{diag}(1,1,1)$. 但是，您也知道，如果我们使用球极坐标，物理距离的平方将不再是坐标距离的平方和。 相反，我们会得到
$$\mathrm{d} \ell^{2}=\mathrm{d} r^{2}+r^{2} \mathrm{~d} \theta^{2}+r^{2} \sin ^{2} \theta \mathrm{d} \phi^{2} \equiv \sum_{i, j=1}^{3} g_{i j} \mathrm{~d} x^{i} \mathrm{~d} x^{j}$$

$g_{i j}=\operatorname{diag}\left(1, r^{2}, r^{2} \sin ^{2} \theta\right)$. 使用不同坐标系的观察者不一定就两点之间的坐标距离达成一致，但他们总是会 就物理距离达成一致， $\mathrm{d} \ell$. 我们说 $\mathrm{d}$ 是一个不变量。因此，该度量将依赖于观察者的坐标转换为不变量。

$$\mathrm{d} s^{2}=\sum_{\mu, \nu=0}^{3} g_{\mu \nu} \mathrm{d} x^{\mu} \mathrm{d} x^{\nu} \equiv g_{\mu \nu} \mathrm{d} x^{\mu} \mathrm{d} x^{\nu} .$$

$$\mathrm{d} s^{2}=-c^{2} \mathrm{~d} t^{2}+\delta_{i j} \mathrm{~d} x^{i} \mathrm{~d} x^{j},$$
1 在本书中，我将使用爱因斯坦的求和约定，其中重复的索引被求和。我们的度量签名将是 $(-,+,+,+)$. 在本 章中，我将明确说明光速，但在本书的其余部分中，我将使用自然单位 $c \equiv 1$.

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