The Universe Is Expanding — Question 1

Question 1

A spectral line measures the relative cosmic scale factor

A spectral line emitted at 500 nm is observed at 650 nm. Assume the shift is entirely cosmological and normalize the present scale factor to one.

1. Find redshift and the emission scale factor.

2. Find the observed-to-emitted photon energy ratio and the emission proper separation of two comoving objects now 100 Mpc apart.

Question 1 - Solution

A spectral line measures the relative cosmic scale factor

(a) Method and calculation

$$\boxed{z=\lambda_o/\lambda_e-1=650/500-1=.30}.$$$$\boxed{a_e=1/(1+z)=10/13=.769231}.$$

(b) Interpretation and further calculation

$$\boxed{E_o/E_e=\lambda_e/\lambda_o=10/13}.$$$$\boxed{D_e=a_eD_0=76.9231\ {\mathrm{Mpc}}}.$$

Physical check

The separation comparison is at common cosmic times, not a light-travel distance. Bound systems do not generally follow this scaling.