When the universe is dominated by radiation, then the equation of state can be taken as
$$p=m \rho=\frac{1}{3} \rho$$
where the equation of state parameter $m=\frac{1}{3}$.
Equations (11.54) and (11.56) yield
$$\frac{\ddot{a}}{a}=-\frac{1+3 m}{6} A a^{-3(1+m)} .$$
$[A=$ integration constant $]$
Multiplying both sides by $2 \dot{a}$, we get
$$2 \dot{a} \ddot{a}=-\frac{1+3 m}{3} A \dot{a} a^{(-2-3 m)} .$$
After integrating, we obtain
$$\dot{a}^2=\frac{1}{3} A a^{-(1+3 m)}+D$$

where,
$$D=a_0^2 H_0^2-\frac{1}{3} A a_0^{-(1+3 m)}$$
Hence, putting $m=\frac{1}{3}$, we have
$$\dot{a}^2=\frac{A}{3 a^2}+D$$
Solving this we get
$$a^2(t)=\frac{1}{D}\left[D^2 t^2+\sqrt{\frac{4 A D^2}{3}} t\right] .$$
This indicates that the universe follows big-bang singularity and is expanding in nature.
This solution is not physically interesting as the present universe is far from radiation dominated, however, the behavior near $t=0$ is interesting.

## 物理代写|广义相对论代写General relativity代考|Cosmological Inflation

In 1981, Alan Guth proposed an important concept, known as cosmological inflation, which is a theory of the exponential expansion of space in the early universe. The inflationary scenario occurred around $10^{-36}$ second after the big bang. Following the inflationary period, there was a phase in which the universe continued to expand much faster (exponentially), than the rate given by standard cosmology. It is argued that in the inflationary phase, the vacuum energy density of the scalar field $\left(V_0\right.$ ) dominates energy density ( $\rho$ ) grows, i.e., we can use $\rho \approx V_0$. From Eq. (11.52), we have
\begin{aligned} \frac{3\left(\dot{a}^2+k\right)}{a^2} & =8 \pi G \rho, \ \Rightarrow \dot{a}^2 & =\frac{8 \pi G a^2 V_0}{3}-k . \end{aligned}
In the inflationary phase, the square of the scale factor $a^2(t)$ dominates curvature term, hence
$$\dot{a}^2=\frac{8 \pi G V_0}{3} a^2 \text {. }$$
Solving this we get,
$$a(t)=a_0 \exp (H t)$$
where, $H=\sqrt{\frac{8 \pi G V_0}{3}}$ and $a_0$ is the value of the scale factor when inflation began. Hence the universe expands exponentially due to inflation and it is often called a period of accelerated expansion as the distance between two fixed points in the universe is increasing exponentially. This new model is interesting as it can resolve some fundamental problems in cosmology like horizon and flatness problems. It is known that the universe is continually expanding or closed depending on whether the density parameter $\Omega(t)\left(=\frac{\rho(t)}{\rho_c}\right.$, where $\rho_c=\frac{3 H^2}{8 \pi G}$ is the critical density), $\Omega(t) \leq 1$ or $\Omega(t)>1$.

# 广义相对论代考

$$p=m \rho=\frac{1}{3} \rho$$

$$\frac{\ddot{a}}{a}=-\frac{1+3 m}{6} A a^{-3(1+m)} \text {. }$$
$[A=$ 积分常数 $]$

$$2 \dot{a} \ddot{a}=-\frac{1+3 m}{3} A \dot{a} a^{(-2-3 m)}$$

$$\dot{a}^2=\frac{1}{3} A a^{-(1+3 m)}+D$$

$$D=a_0^2 H_0^2-\frac{1}{3} A a_0^{-(1+3 m)}$$

$$\dot{a}^2=\frac{A}{3 a^2}+D$$

$$a^2(t)=\frac{1}{D}\left[D^2 t^2+\sqrt{\frac{4 A D^2}{3}} t\right]$$

## 物理代写|广义相对论代写General relativity代考|Cosmological Inflation

1981年，艾伦古斯提出了一个重要的概念，称为宇宙膨胀，这是一种早期宇宙空间指数膨 胀的理论。通胀情景发生在 $10^{-36}$ 大爆炸后的第二个。在暴胀期之后，有一个阶段宇宙继续 膨胀得比标准宇宙学给出的速度快得多 (指数) 。有人认为，在膨胀阶段，标量场的真空 能量密度 $\left(V_0\right)$ 支配能量密度 $(\rho)$ 增长，即我们可以使用 $\rho \approx V_0$. 从等式。(11.52)，我们有
$$\frac{3\left(\dot{a}^2+k\right)}{a^2}=8 \pi G \rho, \Rightarrow \dot{a}^2=\frac{8 \pi G a^2 V_0}{3}-k .$$

$$\dot{a}^2=\frac{8 \pi G V_0}{3} a^2$$

$$a(t)=a_0 \exp (H t)$$

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