General Relativity: An Introduction for Physicists · Problem 14.12

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

Physics · galaxy-number-counts-flat-universe

Problem

A certain population of galaxies undergoes a short ultra-luminous phase at redshift

z = z_*

that lasts for a proper time interval

\Delta t

. After this phase, such galaxies are neither created or destroyed. If

z_* \ll 1

, show that for a spatially flat universe the total number of such galaxies in the sky that are in this phase is

N = \frac{4\pi c^3 n_0 \Delta t}{H_0^2}\left[z_* + \tfrac{1}{2}(1 - q_0)z_*^2 + \cdots\right],

where

n_0

is the present-day proper number density of these galaxies.