General Relativity: An Introduction for Physicists · Problem 14.8

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

Physics · look-back-time-proper-distance

Problem

In the FRW geometry, show that the look-back time to a nearby object at proper distance

d

t_0 - t = \frac{d}{c} - \frac{H_0 d^2}{2c^2} + \cdots.

Hence show that the redshift to the object is

z = \frac{H_0 d}{c} + \frac{1 + q_0}{2}\frac{H_0^2 d^2}{c^2} + \cdots.