# 物理代写|广义相对论代写General relativity代考|Dust Model

## 物理代写|广义相对论代写General relativity代考|Dust Model

For dust-filled universe, $p=0$. Eq. (11.53) yields
$$\frac{8 \pi G \rho}{3}=\frac{A}{a^3}$$
$[A=$ integration constant $]$
Equation (11.51) implies
$$\dot{a}^2=-k+\frac{A}{a}$$
Now we consider three different cases for different values of curvature parameter $k$.
(i) Einstein-de Sitter model: $(k=0)$
$k=0$ corresponds to the flat universe with zero curvature. The model with $k=0$ is often called the Einstein-de Sitter flat model.
Equation (11.63) yields
$$\dot{a}^2=\frac{A}{a} \Longrightarrow a=a_0\left(\frac{3 H_0 t}{2}\right)^{2 / 3}$$
Here $A=a_0^3 H_0^2$ and $a_0$ and $H_0$ are the present values of scale factor and Hubble constant, respectively,
$$a_0=a\left(t_0\right), \quad\left(\frac{\dot{a}}{a}\right)_{t=t_0}=H_0 .$$
Now, we can find the present epoch, which is given by
$$t=t_0=\frac{2}{3} H_0^{-1}$$
This present epoch $t_0$ is the age of the universe.

## 物理代写|广义相对论代写General relativity代考|Cosmology with

In 1915, Einstein formulated the general theory of relativity. At that time he, as well as people, believed that the universe was static. Since at that time expansion of the universe had not been discovered, so Einstein wanted to get static solutions of the universe from his field equation. Einstein field equations for $\mathrm{R}-\mathrm{W}$ metric indicate that the scale factor $a(t)$ can only be constant if
$$\begin{gathered} \rho=-3 p=\frac{3 k}{8 \pi G a^2} . \ \text { [field equations } \Rightarrow \ddot{a}=\frac{-4 \pi G}{3}(\rho+3 p) a ; 2 \frac{\ddot{a}}{a}+\frac{\dot{a}^2+k}{a^2}=-8 \pi G p ; \ \text { and put } a(t)=a=\text { constant] } \end{gathered}$$
It is well known that energy density $\rho>0$, so Eq. (11.87) indicates that the pressure $p$ must be negative. Definitely this is not a realistic solution. Also if one considers $p=0$, then, $\rho=0$. This is also not possible. In order to avoid this unrealistic situation, in 1917, Einstein introduced a constant term $\Lambda$ in his field equation. In the modified field equation, the adding term is known as the cosmological constant. The modified field equation is written as
$$R_{a b}-\frac{1}{2} g_{a b} R-\Lambda g_{a b}=8 \pi G T_{a b}$$
Then the field equations for $\mathrm{R}-\mathrm{W}$ metric can be written explicitly as
\begin{aligned} 3\left(\dot{a}^2+k\right) & =8 \pi G \rho a^2+\Lambda a^2, \ 2 a \ddot{a}+\dot{a}^2+k & =-8 \pi G p a^2+\Lambda a^2 . \end{aligned}
[Note that $\left.[\Lambda]=L^{-2}\right]$

# 广义相对论代考

## 物理代写|广义相对论代写General relativity代考|Dust Model

$$\frac{8 \pi G \rho}{3}=\frac{A}{a^3}$$
$[A=$ 积分常数 $]$

$$\dot{a}^2=-k+\frac{A}{a}$$

(i) 爱因斯坦-德西特模型: $(k=0)$
$k=0$ 对应于零曲率的平坦宇宙。该模型与 $k=0$ 通常称为 Einstein-de Sitter 平面模型。 方程 (11.63) 产生
$$\dot{a}^2=\frac{A}{a} \Longrightarrow a=a_0\left(\frac{3 H_0 t}{2}\right)^{2 / 3}$$

$$a_0=a\left(t_0\right), \quad\left(\frac{\dot{a}}{a}\right)_{t=t_0}=H_0 .$$

$$t=t_0=\frac{2}{3} H_0^{-1}$$

## 物理代写|广义相对论代写General relativity代考|Cosmology with

1915年，爱因斯坦提出了广义相对论。那时他和人们一样相信宇宙是静止的。由于那时还 没有发现宇宙膨胀，所以爱因斯坦想从他的场方程中得到宇宙的静态解。爱因斯坦场方程 $\mathrm{R}-$ Wmetric 表示比例因子 $a(t)$ 只有当
$$\rho=-3 p=\frac{3 k}{8 \pi G a^2} . \quad\left[\text { field equations } \Rightarrow \ddot{a}=\frac{-4 \pi G}{3}(\rho+3 p) a ; 2 \frac{\ddot{a}}{a}+\frac{\dot{a}^2+k}{a^2}=-8 \pi G p ;\right.$$

$$R_{a b}-\frac{1}{2} g_{a b} R-\Lambda g_{a b}=8 \pi G T_{a b}$$

$$3\left(\dot{a}^2+k\right)=8 \pi G \rho a^2+\Lambda a^2, 2 a \ddot{a}+\dot{a}^2+k=-8 \pi G p a^2+\Lambda a^2 .$$
[注意 $\left.[\Lambda]=L^{-2}\right]$

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