11. If Omega is less than 1 at the present time, then (see problem 2) it will have been much closer to unity at high redshift. Hence Omega_matter~1 will have been a good approximation in this case.

If the cosmological constant is non-zero, then remember that the different contributions to the energy density of the Universe drop off at different rates as a increases (see the Friedmann equation incorporating the Lambda term from lecture 2). This means that for a critical density universe, radiation dominates at very early times, then matter, and finally the Lambda term. So at early epochs after z~1000 (i.e. once the matter dominates over radiation) we again expect Omega_matter~1.

At low redshift, Omega_matter drops (and observationally it appears to be ~0.3 at present) and this has the effect of reducing the rate of growth of perturbations. One might therefore expect to see that there is less merging taking place at low z.