哈勃距离(哈勃视界,Hubble distance,Hubble horizon):
$$d_H=\frac{c}{H_0}.$$
共动距离(Comoving distance,\(\chi\)):
$$d_C=c\int_{t}^{t_0}\frac{\mathrm{d} t’}{a(t’)}\\=d_H\int_{a}^{1}\frac{\mathrm{d} a’}{a’^2E(t’)}=d_H\int_0^z\frac{\mathrm{d}z’}{E(z’)}.$$
固有距离(Proper distance):
$$d(t)=a(t)d_C.$$
横向共动距离(Transverse comoving distance, or Comoving coordinate distance, \(r\)):
$$d_{M}(z)=\left\{\begin{array}{ll}
\frac{d_{H}}{\sqrt{\Omega_{k}}} \sinh \left(\frac{\sqrt{\Omega_{k}} d_{C}(z)}{d_{H}}\right) & \Omega_{k}>0 \\
d_{C}(z) & \Omega_{k}=0 \\
\frac{d_{H}}{\sqrt{\left|\Omega_{k}\right|}} \sin \left(\frac{\sqrt{\left|\Omega_{k}\right|} d_{C}(z)}{d_{H}}\right) & \Omega_{k}<0
\end{array}\right..$$
角直径距离(Angular diameter distance):
$$d _ { A } ( z ) = \frac { d _ { M } ( z ) } { 1 + z }.$$
光度距离(Luminosity distance):
$$d _ { L } ( z ) = ( 1 + z ) d _ { M } ( z ).$$
回溯时间(Lookback time):
$$\Delta t=t_0-t=\int_t^{t_0}\mathrm{d}t’=\int _ { 0 } ^ { z } \frac { d z ^ { \prime } } { ( 1 + z ^ { \prime } ) H ( z ^ { \prime } ) }.$$
光传播距离(Light-travel distance, or lookback distance):
$$ d _ { T } ( z ) = c\Delta t = d _ { H } \int _ { 0 } ^ { z } \frac { d z ^ { \prime } } { ( 1 + z ^ { \prime } ) E ( z ^ { \prime } ) } .$$
共形时间(Conformal time,光走过的共动距离除以光速):
$$\eta=\int_0^t\frac{\mathrm{d}t’}{a(t’)}.$$
共动粒子视界(共动视界,comoving particle horizon):
$$d_P=c\int_{t_e}^{t}\frac{\mathrm{d} t’}{a(t’)}=c\int_{t_e}^{a}\mathrm{dln}a\left(\frac{1}{aH}\right).$$
其中\(t_e\)为暴涨结束
事件视界(Event horizon):
$$d_e=c\int_{t}^{t_{max}}\frac{\mathrm{d} t’}{a(t’)}.$$
暴涨,H~常数,a(eta)=-(1/Heta), a=0对应eta>-oo
辐射主导,a oc eta^2
物质主导,a oc eta
共动哈勃视界c/(aH)与共动粒子视界的区别:
共动视界内是可观测宇宙,之外代表从未有过联系
哈勃视界外代表目前不可以联系,但是不代表以后或者未来不可以联系
目前共动哈勃视界小于共动粒子视界。若某物体离我们的距离在两个视界之间,说明我们现在可以观测到这个物体(以前的样子),但是以后看不到这个物体(现在的样子)