The Universe Is Expanding — Question 3

Question 3

Peculiar velocity biases a nearby Hubble estimate

A galaxy has distance 20 Mpc and observed recession velocity 1700 km/s. Its radial peculiar velocity is +300 km/s, where positive means away from the observer. Use the low-redshift additive approximation.

1. Find the Hubble-flow velocity and corrected H0.

2. Find the uncorrected estimate and fractional bias.

Question 3 - Solution

Peculiar velocity biases a nearby Hubble estimate

(a) Method and calculation

$$v_{\mathrm{obs}}=H_0D+v_{\mathrm{pec}}\Rightarrow\boxed{v_H=1700-300=1400\ {\mathrm{km/s}}}.$$$$\boxed{H_0=1400/20=70\ {\mathrm{km\,s^{-1}\,Mpc^{-1}}}}.$$

(b) Interpretation and further calculation

$$\boxed{H_{\mathrm{naive}}=1700/20=85\ {\mathrm{km\,s^{-1}\,Mpc^{-1}}}}.$$$$\boxed{(H_{\mathrm{naive}}-H_0)/H_0=15/70=.214286}.$$

Physical check

At fixed peculiar speed, the fractional bias decreases with distance within the regime where the linear Hubble law applies.