The Universe Is Expanding — Question 7

Question 7

The same present Hubble rate allows different cosmic ages

Compare ideal power-law universes $a(t)=(t/t_0)^n$ for n=1/2 and n=2/3. Both have the same current Hubble time $H_0^{-1}=14$ Gyr.

1. Derive H(t) and determine the two ages.

2. For each model, find the cosmic time when a=1/4 as a fraction of the current age.

Question 7 - Solution

The same present Hubble rate allows different cosmic ages

(a) Method and calculation

$$H=\dot a/a=n/t\Rightarrow\boxed{t_0=nH_0^{-1}}.$$$$\boxed{t_{0,1/2}=7\ {\mathrm{Gyr}},\quad t_{0,2/3}=28/3\ {\mathrm{Gyr}}}.$$

(b) Interpretation and further calculation

$$t/t_0=a^{1/n}.$$$$\boxed{(t/t_0)_{1/2}=1/16,\quad(t/t_0)_{2/3}=1/8}.$$

Physical check

These are pure-component ideal histories. A realistic universe changes its dominant energy components with time.