Tests for the Shape of the Universe

I have already briefly mentioned these various tests. The problem with all of the tests proposed and applied up until is now is that questions around how the properties of galaxies evolve as the Universe ages add a great deal of uncertainty to the results. In addition, most of the tests try to see what the Universe looked like when it was much smaller, i.e., at large redshifts (z > 1) ===> how the Universe looks at large distances.

DEVIATION FROM HUBBLE FLOW AT LARGE REDSHIFTS

The problem is that it is difficult to determine distances to objects at large redshift (distances).

ANGULAR SIZE OF GALAXIES AT LARGE REDSHIFT

The angular size of nearby galaxies decreases in a simple way, However, for galaxies at large distances because the Universe the overall shape of the Universe starts to play a role and the angular size does not always decrease as the distance increases. For example, even for a flat universe, k = 0, we have that Note that for small redshift, z (and therefore distance from cz = H(now) distance) that Angular Size ~ 1/distance (as required).

DENSITY TEST

Due to the topology of the Universe, the variation in the volume of the Universe as the distance changes as we observe more distant parts of the Universe. Recall that this means we see more or less of the Universe depending upon its shape at large distances. This will affect the number density of galaxies observed at large distances.

This test was initially performed by Loh and Spillar. What they did was to look in different directions on the sky and count the number of galaxies in various redshift bins. For example, one could count the number of galaxies between z = 0.5 and 0.6, z = 0.6 and 0.7, z = 0.7 and 0.8, z = 0.8 and 0.9, and so on. Since, the redshift z indicates distances in the Universe what they were doing was measuring the number of galaxies in various shells (volumes) of the Universe. The manner in which the density of galaxies changed at large z would then indicate how the shape of the Universe was changing at large z.

The density of galaxies falls off strongly at large z (even for the flat model) because, Loh and Spillar did not plot their data directly, they plotted observed number density / proper density. The proper density takes account of the expansion of the number and thus increases as z increases (the Universe is smaller ---> density of galaxies is larger).