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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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物理代写|宇宙学代写cosmology代考|Einstein equations

In the previous chapter, we have dealt with gravity only in terms of the metric, which gives us a notion of distances and straight lines (geodesics) in general spacetimes. These results were built on the principle of general covariance alone. We now turn to the second aspect of general relativity, which relates the metric to the constituents of the universe. This second part is contained in the Einstein equations, which relate the Einstein tensor describing the geometry to the energy-momentum tensor of matter. ${ }^{1}$ This set of equations can be summarized as the following celebrated tensor equality (Fig. 3.1):
$$G_{\mu \nu}+\Lambda g_{\mu \nu}=8 \pi G T_{\mu \nu} .$$

Here $G_{\mu \nu}$ is the Einstein tensor defined through
$$G_{\mu \nu} \equiv R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R .$$
$R_{\mu \nu}$ is the Ricci tensor, which depends only on the metric and its derivatives; $R$, the Ricci scalar, is the contraction of the Ricci tensor $\left(R \equiv g^{\mu v} R_{\mu v}\right.$ ). Further, $\Lambda$ is the famous cosmological constant, $G$ is Newton’s constant, and $T_{\mu \nu}$ is the energy-momentum tensor, whose expression in the background universe we have already encountered in Sect. 2.3. Thus, the left-hand side of Eq. (3.1) is a function of the metric, the right a function of the constituents of the universe: the Einstein equations relate the two.

The simplicity of Eq. (3.1) belies the rich physics encoded in the Einstein equations. They govern the evolution of the smooth universe as well as the growth of structure within it. On small scales, Newtonian gravity is included, as we will see in Sect. 3.3, as are black holes which we will not deal with in this book. We will later encounter a different purely general-relativistic effect contained in Eq. (3.1) though: gravitational waves.

物理代写|宇宙学代写cosmology代考|Boltzmann equation

After having treated gravity in the homogeneous universe, let us now turn to the equations governing matter and radiation. In cosmology, we are not interested in the fate of individual particles, but in their behavior in a statistical sense. Hence let us consider a collection of particles occupying some region of space, as we did in Sect. 2.3. In classical physics, these particles are completely described by the set $\left{\boldsymbol{x}{i}, \boldsymbol{p}{i}\right}$ of their positions $\boldsymbol{x}{i}$ and momenta $\boldsymbol{p}{i}$. We can then define the distribution function, as in Sect. 2.3, by relating it to the number of particles in a small phase-space element around $(\boldsymbol{x}, \boldsymbol{p})$ :
$$N(\boldsymbol{x}, \boldsymbol{p}, t)=f(\boldsymbol{x}, \boldsymbol{p}, t)(\Delta x)^{3} \frac{(\Delta p)^{3}}{(2 \pi)^{3}} .$$
In the limit of a large number of particles within the volume element considered, $f(\boldsymbol{x}, \boldsymbol{p}, t)$ approaches a continuous function describing the state of the collection of particles, and we no longer need to keep track of individual particles. We already saw that the appropriate integration measure (in natural units) is given by $d^{3} x d^{3} p /(2 \pi)^{3}$. Note that we do not need to include the energy as a separate variable, since, at any point in phase space, $E$ is completely determined by $(\boldsymbol{x}, \boldsymbol{p})$.

Now we would like to derive an equation governing this distribution function. This equation should uniquely follow from the equations of motion obeyed by the individual particles. Let us begin by neglecting any particle-particle interactions. Then, the only forces acting on the particles are long-range forces, which we can describe through a force field (more precisely, acceleration field) $\boldsymbol{a}(\boldsymbol{x}, \boldsymbol{p}, t)$. This could for example be gravity, in which case $a=-\nabla \Psi(x, t)$, where the gravitational potential $\Psi$ (defined in Eq. (3.49) below) is independent of the particle momenta, or it could be the Lorentz force due to electromagnetic fields. Then, using the definition of the momentum $p$, the equations of motion for nonrelativistic particles are
$$\dot{x}=\frac{p}{m} ; \quad \dot{p}=m a(x, p, t) .$$

物理代写|宇宙学代写cosmology代考|Boltzmann equation in an expanding universe

So far, we have studied the Boltzmann equation in the Minkowski-space context (cf. Eq. (3.16)), as appropriate for lab experiments on Earth. Let us now derive the generalization to an expanding spacetime. As we know from Sect. 2.1.2, the equations of motion Eq. (3.16) get generalized to the geodesic equation, and the three-momentum $\boldsymbol{p}$ is correspondingly promoted to a four-vector
$$P^{\mu} \equiv \frac{d x^{\mu}}{d \lambda}$$
where $\lambda$ again parametrizes the particle’s path, as in Eq. (2.20) (and again we will not need to specify $\lambda$ explicitly). However, the distribution function for a given collection of particles remains a function defined on a six-dimensional phase space: first, we keep track of time separately as before, and second, the four-momentum of each particle obeys the massshell constraint
$$P^{2} \equiv g_{\mu \nu} P^{\mu} P^{\nu}=-m^{2},$$

where $m$ is the particle rest mass (which could be zero, e.g. for photons). Defining the norm of the three-momentum $p$, by generalizing Eq. (2.32) to
$$p^{2} \equiv g_{i j} P^{i} P^{j},$$
Eq. (3.27) becomes, for the FLRW metric,
$$E^{2} \equiv\left(P^{0}\right)^{2}=p^{2}+m^{2}$$
Thus, we have eliminated $P^{0}$ in favor of $\boldsymbol{p}$, and we can write a relativistic Boltzmann equation for a distribution function $f(\boldsymbol{x}, \boldsymbol{p}, t)$ as before. It is convenient to separate the dependence on $p$ into a dependence on its magnitude $p \equiv \sqrt{p^{2}}$ and its unit vector $\hat{p}^{i}=\hat{p}{i}$, which satisfies $\delta{i j} \hat{p}^{i} \hat{p}^{j}=1$ by definition. We expect that $\hat{p}^{i}$ is proportional to the comoving momentum $P^{i}$; call the proportionality constant $C$ :
$$P^{i} \equiv C \hat{p}^{i} .$$

物理代写|宇宙学代写cosmology代考|Einstein equations

$$G_{\mu \nu}+\Lambda g_{\mu \nu}=8 \pi G T_{\mu \nu} .$$
$$G_{\mu \nu} \equiv R_{\mu \nu}-\frac{1}{2} g_{\mu \nu} R$$ 顿常数，并且 $T_{\mu \nu}$ 是能量动量张量，它在背景宇宙中的表达我们已经在教派中遇到过。2.3. 因此，等式的左侧。(3.1) 是度量的函数，右边 是宇宙成分的函数：爱因斯坦方程将两者联系起来。

物理代写|宇宙学代写cosmology代考|Boltzmann equation

2.3，通过将其与周围的小相空间元素中的粒子数相关联 $(\boldsymbol{x}, \boldsymbol{p})$ :
$$N(\boldsymbol{x}, \boldsymbol{p}, t)=f(\boldsymbol{x}, \boldsymbol{p}, t)(\Delta x)^{3} \frac{(\Delta p)^{3}}{(2 \pi)^{3}}$$

$$\dot{x}=\frac{p}{m} ; \quad \dot{p}=m a(x, p, t) .$$

物理代写|宇宙学代写cosmology代考|Boltzmann equation in an expanding universe

$$P^{\mu} \equiv \frac{d x^{\mu}}{d \lambda}$$

$$P^{2} \equiv g_{\mu \nu} P^{\mu} P^{\nu}=-m^{2},$$

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

$$E^{2} \equiv\left(P^{0}\right)^{2}=p^{2}+m^{2}$$

$$P^{i} \equiv C \hat{p}^{i} .$$

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MATLAB代写

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