The Universe Is Expanding — Question 8

Question 8

Comoving coordinates remove the uniform part of the expansion

Two galaxies have fixed comoving separation chi=40 Mpc, with proper separation D=a chi. At one epoch a=.50 and $\dot a=.040$ Gyr$^{-1}$.

1. Calculate D, H and the rate of change of proper separation.

2. If a third object has peculiar velocity u relative to this flow, derive the additional term in its physical velocity for time-dependent chi.

Question 8 - Solution

Comoving coordinates remove the uniform part of the expansion

(a) Method and calculation

$$\boxed{D=.50(40)=20\ {\mathrm{Mpc}},\quad H=\dot a/a=.080\ {\mathrm{Gyr}}^{-1}}.$$$$\boxed{\dot D=\dot a\chi=1.60\ {\mathrm{Mpc/Gyr}}=HD}.$$

(b) Interpretation and further calculation

$$\dot D=\frac{d(a\chi)}{dt}=\dot a\chi+a\dot\chi.$$$$\boxed{\dot D=HD+u,\quad u=a\dot\chi}.$$

Physical check

Expansion does not imply that every object has fixed comoving position; gravitationally bound systems and peculiar motions break that simplification.