Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 9B / a

Solution

Spatial homogeneity says that the velocity of

B

relative to

A

depends only on their separation. Comparing both galaxies with

C

gives

v (r_{B} - r_{A}) = v (r_{B} - r_{C}) - v (r_{A} - r_{C}) .

Thus

v (x - y) = v (x) - v (y)

, so continuity of the macroscopic velocity field turns this Cauchy functional equation into a linear map:

v_{i} = H_{ij} r_{j} .

Isotropy requires

R H = H R

R

. The only such rank-two tensor is a scalar multiple of the identity, and hence

v = H r .

This is the Hubble law. A comoving galaxy has physical position

r (t) = a (t) x

, so

v = \overset{a}{˙} x = \frac{a ˙}{a} r, H_{0} = \frac{a ˙}{a}_{t = t_{0}} .

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