At fixed time and angles, proper radial distance is

D = a (t) \int_{0}^{Δ r} \frac{d r}{1 - k r ^{2}} .

Thus

D = ⎩ ⎨ ⎧ a Δ r, \frac{a}{k} sin^{- 1} (k Δ r), \frac{a}{- k} sinh^{- 1} (- k Δ r), k = 0, k > 0, k < 0.

These are respectively flat Euclidean, positively curved spherical, and negatively curved hyperbolic spatial geometries.

ii

Solution

d τ = d t / a (t)

. The radial coordinate and the angular-radius function are

k 0 k > 0 k < 0 χ (r) r k^{- 1/2} sin^{- 1} (k r) (- k)^{- 1/2} sinh^{- 1} (- k r) f (χ) χ k^{- 1/2} sin (k χ) (- k)^{- 1/2} sinh (- k χ) .

Substitution yields the stated conformal metric. A radial null ray satisfies

- c^{2} d τ^{2} + d χ^{2} = 0,

d χ / d τ = \pm c

and

χ = χ_{0} \pm c (τ - τ_{0})

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