Event Horizon in a closed, matter (dust) dominated universe

[Heisenberg1993]

Hi!

It is stated in V. Mukhanov's book "Physical foundations of Cosmology" the following (page 44, after equation 2.25): "In contrast, for the dust dominated universe, where η_max=2π, the event horizon exists only during the contraction phase when η>π." could someone please explain why is this true? the equation for the event horizon in a closed, dust dominated universe is (again from Mukhanov): d_e(t)=a_m(1-cosη)(η_max-η),(a_m is a constant) which clearly has non-vanishing values for η≤π

Thanks!

 

[George Jones]

For a closed, dust dominated universe, $0 < \eta < 2\pi$. During the expansion phase, $0 < \eta < \pi$. Combining this with
$$\chi_e \left( \eta \right) = \eta_\mathrm{max} - \eta = 2\pi - \eta$$
gives that, during the expansion phase,
$$\begin{align}
0 > -\eta > -\pi \\
2\pi> 2\pi - \eta > \pi \\
2\pi> \chi_e \left( \eta \right) > \pi
\end{align}$$
But this "solution" is unphysical, because, in a closed universe, $\chi$ is never larger that $\pi$.