General Relativity: An Introduction for Physicists · Problem 15.13

Cosmological models — Hobson, Efstathiou & Lasenby

Physics · turning-point-condition-scale-factor

Problem

Show that the condition for the

a(t)

-curve to have a turning point is

f(a) \equiv \Omega_{\Lambda,0}\, a^3 + (1 - \Omega_{\mathrm{m},0} - \Omega_{\Lambda,0})\, a + \Omega_{\mathrm{m},0} = 0.

In the case

\Omega_{\Lambda,0} > 0

, show by evaluating the derivatives

f'(a)

and

f''(a)

that the condition for

f(a)

to have a single positive root at

a = a_*

f(a_*) = f'(a_*) = 0

. Show further that this root occurs at

a_* = \left(\frac{\Omega_{\mathrm{m},0}}{2\Omega_{\Lambda,0}}\right)^{1/3}.

Hence show that the values

\Omega_{\mathrm{m},0}

and

\Omega_{\Lambda,0}

, along any dividing line in this plane that separates those models with a turning point in the

a(t)

-curve from those without, must satisfy

4(1 - \Omega_{\mathrm{m},0} - \Omega_{\Lambda,0})^3 + 27\Omega_{\mathrm{m},0}^2\,\Omega_{\Lambda,0} = 0.