# 物理代写|宇宙学代写cosmology代考|ASTR3002

## 物理代写|宇宙学代写cosmology代考|Two components of the Einstein equations

We can now derive the evolution equations for $\Phi$ and $\Psi$, our scalar perturbations to the Friedmann-Lemaître-Robertson-Walker metric. We have several different options here, because the Einstein equations
$$G^\mu{ }_v=8 \pi G T_v^\mu$$
have 10 components and we need only two. All of the other eight components will either be zero at first-order or redundant. ${ }^3$

The first component we will use is the time-time component. Thus we need to evaluate
\begin{aligned} G_0^0 &=g^{00}\left[R_{00}-\frac{1}{2} g_{00} R\right] \ &=(-1+2 \Psi) R_{00}-\frac{R}{2} . \end{aligned}
Here one of the indices has been raised by multiplying $G_{00}$ by $g^{00}$ (recall that the $g^{0 i}$ vanish). This turns out to simplify the energy-momentum tensor (see Sect. $3.4$ and Exercise 3.12) which supplies the right-hand side. Also note that the second line follows from the first since $g^{00} g_{00}=1$. We have computed the time-time component of the Ricci tensor (Eq. (6.26)) and the perturbed Ricci scalar (Eq. (6.30)), so the first-order part of the timetime component of the Einstein tensor is
\begin{aligned} \delta G_0^0=&-6 \Psi \frac{\ddot{a}}{a}+\frac{k^2}{a^2} \Psi+3 \Phi_{, 00}-3 H\left(\Psi_{, 0}-2 \Phi_{, 0}\right) \ &+6 \Psi\left(H^2+\frac{\ddot{a}}{a}\right)-\frac{k^2}{a^2} \Psi-3 \Phi_{, 00} \ &+3 H\left(\Psi_{, 0}-4 \Phi_{, 0}\right)-2 \frac{k^2 \Phi}{a^2} \end{aligned}

## 物理代写|宇宙学代写cosmology代考|Tensor perturbations

Until now, we have derived equations applying to the scalar perturbations of the homogeneous FLRW universe. This focus is reasonable: as we have seen, scalar perturbations to the metric are sourced by density fluctuations and vice versa. For the most part, the density fluctuations that form the structure of the universe are our primary interest. Moreover, thanks to the decomposition theorem it is perfectly fine to study scalar perturbations in isolation.

Nonetheless, we have seen in Sect. $6.1$ that there are other types of gravitational perturbations, in particular tensor perturbations. In the next chapter we will see that the leading theory for the origin of scalar perturbations-inflation-also predicts tensor perturbations. Independently of cosmology, though, gravitational waves have emerged as a powerful probe of diverse astrophysical phenomena in the aftermath of their first detection by the LIGO collaboration. The wavelengths that LIGO is sensitive to are of order hundreds of kilometers, while we will be considering wavelengths of thousands of Mpc. However, the fundamental equation that governs their production and propagation is identical and we are now all set to derive that equation.

The most promising way to search for cosmological gravitational waves is through the distortions they induce in the CMB, especially on large scales. Sprinkled throughout the book, therefore, are exercises relating to tensor perturbations. The third type, vector perturbations, are also covered in the exercises. They are less interesting, since they are not sourced in appreciable amounts in most cosmological scenarios and, in any case, decay rapidly after they are produced. The tools needed to study vector and tensor modes are precisely those we crafted when studying scalar perturbations.

Tensor perturbations can be characterized by a metric perturbation (see Eq. (6.1)) with $h_{00}=-1, h_{0 i}=0$, and
$$\delta g_{i j}(t, \boldsymbol{x})=a^2(t) h_{i j}^{\mathrm{TT}}(t, \boldsymbol{x}), \quad h_{i j}^{\mathrm{TT}}=\left(\begin{array}{ccc} h_{+} & h_{\times} & 0 \ h_{\times} & -h_{+} & 0 \ 0 & 0 & 0 \end{array}\right)$$
That is, the perturbations to the metric are described by two functions, $h_{+}$and $h_{\times}$, assumed small. For definiteness, we have chosen the perturbations to be in the $x-y$ plane.

## 物理代写|宇宙学代写cosmology代考|爱因斯坦方程的两个分量

$$G^\mu{ }_v=8 \pi G T_v^\mu$$

\begin{aligned} G_0^0 &=g^{00}\left[R_{00}-\frac{1}{2} g_{00} R\right] \ &=(-1+2 \Psi) R_{00}-\frac{R}{2} . \end{aligned}

\begin{aligned} \delta G_0^0=&-6 \Psi \frac{\ddot{a}}{a}+\frac{k^2}{a^2} \Psi+3 \Phi_{, 00}-3 H\left(\Psi_{, 0}-2 \Phi_{, 0}\right) \ &+6 \Psi\left(H^2+\frac{\ddot{a}}{a}\right)-\frac{k^2}{a^2} \Psi-3 \Phi_{, 00} \ &+3 H\left(\Psi_{, 0}-4 \Phi_{, 0}\right)-2 \frac{k^2 \Phi}{a^2} \end{aligned}

## 物理代写|宇宙学代写cosmology代考|张量扰动

$$\delta g_{i j}(t, \boldsymbol{x})=a^2(t) h_{i j}^{\mathrm{TT}}(t, \boldsymbol{x}), \quad h_{i j}^{\mathrm{TT}}=\left(\begin{array}{ccc} h_{+} & h_{\times} & 0 \ h_{\times} & -h_{+} & 0 \ 0 & 0 & 0 \end{array}\right)$$

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