The Universe Is Expanding — Question 9

Question 9

Recession faster than light is not a local special-relativistic speed

At one cosmic time take H=70 km s$^{-1}$ Mpc$^{-1}$ and c=$3.00\times10^5$ km/s. Define proper Hubble-flow recession by v=HD.

1. Calculate the Hubble radius c/H and v/c at D=6000 Mpc.

2. Explain why this does not imply a local observer measures a galaxy passing faster than light, or that c/H always equals the particle horizon.

Question 9 - Solution

Recession faster than light is not a local special-relativistic speed

(a) Method and calculation

$$\boxed{R_H=c/H=4285.71\ {\mathrm{Mpc}}}.$$$$\boxed{v/c=D/R_H=1.40}.$$

(b) Interpretation and further calculation

$$v=\frac{dD}{dt}\bigg|_{\chi}\quad\text{compares distant worldlines on a cosmic-time slice}.$$$$\boxed{D_{\mathrm{particle}}=a(t)\int_0^t\frac{c\,dt^\prime}{a(t^\prime)}}.$$

Physical check

The horizon depends on the whole expansion history. The local speed of light remains c in a local inertial frame.