# 物理代写|宇宙学代写cosmology代考|PHYC90009

## 物理代写|宇宙学代写cosmology代考|Scalar–vector–tensor decomposition

In the previous chapter, we have seen that the transformation to Fourier space simplified the perturbed Boltzmann equations considerably, by decoupling the different Fourier modes $\boldsymbol{k}$. The Einstein equations are a tensor equality and correspondingly comprise a set of equations that are in general coupled. However, there exists a decomposition of these equations that again allows us to decouple different modes. In fact, we have already implicitly relied on this result when writing the perturbed metric in the simple form of Eq. (3.49). We thus need to begin with this decomposition.

Let us consider an FLRW spacetime that is perturbed by a small amount. That is, we write the metric as
\begin{aligned} &g_{00}(t, \boldsymbol{x})=-1+h_{00}(t, \boldsymbol{x}), \ &g_{0 i}(t, \boldsymbol{x})=a(t) h_{0 i}(t, \boldsymbol{x})=a(t) h_{i 0}(t, \boldsymbol{x}), \ &g_{i j}(t, \boldsymbol{x})=a^2(t)\left[\delta_{i j}+h_{i j}(t, \boldsymbol{x})\right] \end{aligned}
where $h_{00}, h_{0 i}, h_{i j}$ are metric perturbations that are functions of space and time, and all of whose components are assumed to be small in magnitude. In this chapter, we will mostly use the physical time $t$. Keep in mind that $t$ and the conformal time $\eta$ are always related through $d t=a d \eta$, so that going back and forth is a simple variable transformation that does not touch the perturbations and is unrelated to the coordinate choice we will discuss below.

## 物理代写|宇宙学代写cosmology代考|From gauge to gauge

The effect of coordinate transformations on spacetime scalars, vectors, and tensors is described in Box 2.2. In the context of perturbation theory in relativity, a choice of coordinates is often referred to as gauge, and we will frequently use that term also. The ability to move back and forth between different gauges is useful when dealing with cosmological perturbations. Often, the equations simplify considerably in one gauge (one example is the spa-tially flat slicing when calculating perturbations generated during inflation, Sect. 7.4.3), while quantities that we actually measure observationally are more naturally calculated in another. So, different gauges have their advantages for different parts of the “cosmological perturbations” problem.

Let us start out with a scalar field $\phi(x)$, where in this section $x$ stands for a spacetime location $(t, x)$. We will encounter such a field when studying inflation in the next chapter. We are interested in small perturbations around the homogeneous universe, so we separate $\phi$ into background and perturbation:
$$\phi(x)=\bar{\phi}(t)+\delta \phi(t, x),$$
where the background field can only depend on $t$ since the background universe is homogeneous. We now want to derive how Eq. (6.7) changes when we transform coordinates to $x \rightarrow \hat{x}(x)$. In keeping with our interest in perturbations, it is sufficient to consider small coordinate transformations as well; otherwise, the transformed field would in general have large (unphysical) perturbations. Hence, we perform a Taylor series of $\hat{x}(x)$ in $x$, and keep only the zeroth-order piece, which is a shift in coordinates. That is, we write
$$\begin{gathered} t \rightarrow \hat{t}=t+\zeta(t, \boldsymbol{x}), \ x^i \rightarrow \hat{x}^i=x^i+\xi^{,}(t, \boldsymbol{x}), \end{gathered}$$
where the time shift is $\zeta$ while the spatial coordinate shift is written as the gradient of another scalar function $\xi$, since we are considering only scalar perturbations for now (we will get back to this point below).

## 物理代写|宇宙学代写cosmology代考|标量-向量-张量分解

\begin{aligned} &g_{00}(t, \boldsymbol{x})=-1+h_{00}(t, \boldsymbol{x}), \ &g_{0 i}(t, \boldsymbol{x})=a(t) h_{0 i}(t, \boldsymbol{x})=a(t) h_{i 0}(t, \boldsymbol{x}), \ &g_{i j}(t, \boldsymbol{x})=a^2(t)\left[\delta_{i j}+h_{i j}(t, \boldsymbol{x})\right] \end{aligned}
，其中$h_{00}, h_{0 i}, h_{i j}$是空间和时间的函数的度规摄动，并且它的所有分量都假设在量级上很小。在本章中，我们将主要使用物理时间$t$。请记住$t$和保形时间$\eta$总是通过$d t=a d \eta$联系在一起的，因此来回是一个简单的变量变换，不涉及摄动，也与我们将在下面讨论的坐标选择无关。

## 物理代写|宇宙学代写cosmology代考|从量规到量规

$$\phi(x)=\bar{\phi}(t)+\delta \phi(t, x),$$
，其中背景场只能依赖于$t$，因为背景宇宙是均匀的。现在我们想推导出当我们将坐标转换为$x \rightarrow \hat{x}(x)$时Eq.(6.7)的变化。为了保持我们对摄动的兴趣，考虑小的坐标变换也足够了;否则，变换后的场通常会有较大的(非物理的)扰动。因此，我们在$x$中执行$\hat{x}(x)$的泰勒级数，只保留零阶项，这是坐标上的偏移。也就是说，我们写
$$\begin{gathered} t \rightarrow \hat{t}=t+\zeta(t, \boldsymbol{x}), \ x^i \rightarrow \hat{x}^i=x^i+\xi^{,}(t, \boldsymbol{x}), \end{gathered}$$
，其中时移是$\zeta$，而空间坐标移被写成另一个标量函数$\xi$的梯度，因为我们现在只考虑标量扰动(我们将在下面回到这一点)

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