General Relativity: An Introduction for Physicists · Problem 14.7

The Friedmann–Robertson–Walker geometry — Hobson, Efstathiou & Lasenby

Physics · luminosity-angular-diameter-distances

Problem

In a spatially flat FRW geometry, show that the luminosity and angular diameter distances to an object of redshift zz are given, in the limit z≪1z \ll 1, by dL=cH0[z+12(1−q0)z2+⋯ ],d_{\mathrm{L}} = \frac{c}{H_0}\left[z + \tfrac{1}{2}(1 - q_0)z^2 + \cdots\right], dA=cH0[z−12(3+q0)z2+⋯ ].d_{\mathrm{A}} = \frac{c}{H_0}\left[z - \tfrac{1}{2}(3 + q_0)z^2 + \cdots\right]. Hence show that the angular diameter of a standard object can increase as zz increases. Do these results still hold in a spatially curved FRW geometry?

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