General Relativity: An Introduction for Physicists · Problem 14.6

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

Physics · look-back-time-redshift-expansion

Problem

In the limit z≪1z \ll 1, show that the look-back time for a galaxy with redshift zz is t0−t=H0−1z−H0−1(1+12q0)z2+⋯ .t_0 - t = H_0^{-1}z - H_0^{-1}\left(1 + \tfrac{1}{2}q_0\right)z^2 + \cdots. Show also that, in this limit, the variation of the Hubble parameter with redshift is given by H(z)=H0[1+(1+q0)z−⋯ ].H(z) = H_0[1 + (1 + q_0)z - \cdots].

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