# 如何重建宇宙演化历史（未完成）

Alan Sandage 曾经说过 1 Cosmology 的任务就是要寻找两个参数：哈勃常数 $H_0$ 和减速因子 $q_0$。这两个分别给出了宇宙膨胀的「速度」和「加速度」。如今我们的技术已经有了很大的进步，同样我们也早已不满足仅仅寻找这两个参数。

The present discussion is only a prelude to the coming decade. If work now in progress is successful, better values for both $H_0$ and $q_0$ (and perhaps even $Λ$ ) should be found, and the 30-year dream of choosing between world models on the basis of kinematics alone might possibly be realized.

$H^2(z) = \frac{8\pi G}{3} \sum\rho_i(z) = H_0^2 \sum \Omega_i(z)$

• Standard Candles
1. Luminosity distance of well known objects like SN Ⅰa
2. DAG Diﬀerential Ages of Galaxies
• Redshift Drift
• Standard Rulers
1. BAO Baryon Acoustic Oscillation
2. Sound wave of early universe

### Luminosity Distance

Luminosity distance $d_L$ 受到宇宙膨胀历史的影响，

$d_L = (1+z) \int _ z^0 \frac{\mathrm d\tilde z}{H(\tilde z)}$

### DAG

Hubble 方程 $H(z) = -\frac{1}{z+1}\frac{\mathrm dz}{\mathrm dt}$ 所以只要能够确定 $\mathrm dz/\mathrm dt$ 即可。

### Redshift Drift

$z(t_0+\Delta t_0) = \frac{a(t_0 + \Delta t_0)}{a(t_s + \Delta t_0)} - 1$

$\dot z= \mathrm dz / \mathrm dt_0 =\lim_{\Delta t_0 \rightarrow 0} \frac{\Delta z}{\Delta t_0} = \lim_{\Delta t_0\rightarrow 0} \frac{\dot a(t_0) - \dot a(t_s)}{a(t_s)} = (1+z)H_0 - H(z)$.

$\dot v = \frac{c H_0}{1+z} \left(1+z - \frac{H(z)}{H_0}\right)$

### BAO

BAO, baryon acoustic oscillation，是一种非常理想的标准尺。首先，我们可以建立对 BAO 在除了极早期外的整个宇宙历史中的描述。

## 尾注

1. A. Sandage, Physics Today, February 23, 34 (1970). ↩︎

By OctoMiao

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