The Universe Is Expanding — Question 10

Question 10

Lookback time follows an integral rather than redshift divided by a universal rate

Assume the ideal expansion law $H(z)=H_0(1+z)^{3/2}$. Use $dt/dz=-1/[(1+z)H(z)]$ and $H_0^{-1}=15$ Gyr.

1. Integrate to find lookback time to redshift z.

2. Evaluate at z=3 and compare with the model present age $2/(3H_0)$.

Question 10 - Solution

Lookback time follows an integral rather than redshift divided by a universal rate

(a) Method and calculation

$$t_L(z)=H_0^{-1}\int_0^z(1+z^\prime)^{-5/2}dz^\prime.$$$$\boxed{t_L=\frac{2}{3H_0}[1-(1+z)^{-3/2}]}.$$

(b) Interpretation and further calculation

$$\boxed{t_L(3)=10(1-1/8)=8.75\ {\mathrm{Gyr}}}.$$$$\boxed{t_0=10\ {\mathrm{Gyr}},\quad t_e=t_0-t_L=1.25\ {\mathrm{Gyr}}}.$$

Physical check

The result is specific to the supplied matter-only power law. The linear approximation z/H0 would badly overestimate this large-redshift lookback time.