General Relativity: An Introduction for Physicists · Problem 14.17

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

Physics · energy-conservation-expanding-universe

Problem

Consider a spherical comoving volume of the cosmological fluid whose surface is defined by

\chi = \text{constant}

. As the universe expands show that, for the infinitesimal time interval

t \to t + dt

, the conservation of energy requires that

c^2\rho V = c^2(\rho + d\rho)(V + dV) + p\,dV,

where

\rho c^2

and

p

are the energy density and pressure of the fluid respectively. Hence show that

\frac{d\rho}{dR} = -3(1+w)\frac{\rho}{R},

where

w = p/(\rho c^2)

and

R(t)

is the scale factor of the universe. Show that this equation has the solution

\rho \propto R^{-3(1+w)}