In this case, the authors look at the relation between the galaxy group mass and richness. Richness is defined as the number of member galaxies in a galaxy group which can be as high as hundreds for some clusters or as low as a few. As you might imagine, massive groups tend to have more members, so the relation has a positive correlation. Therefore, to estimate the mass of a galaxy group, all you need to do is count the members! But wait! Galaxy clusters and groups can be very unique, and two systems with 15 member galaxies may have very different masses. This means there will be scatter in the relationship. The authors investigate reducing the scatter in the relations by invoking a third bit of information; the difference in magnitude between the first and second brightest galaxy in the group. This is called the magnitude gap.To test whether the magnitude gap carries information about a group’s mass, the authors use a large N-body simulation with galaxies painted onto dark matter sub-halos as a test to identify any trend. In Figure 1 they show that in the simulation, the groups that have large magnitude gaps tend to also be of larger mass than those with smaller magnitude gaps at the same richness. This means the magnitude gap can be used to improve the calibration of the mass-richness relation. In fact, they find that the scatter drops from 33% to 27% taking the magnitude gap into account. This suggests some sort of dynamical connection between the magnitude gap and richness; however, even in the absence of such a connection, we still expect large gap systems to have fewer members simply due to random draws from the luminosity function of galaxies (the Schecter function). Less draws = greater gap. But this should not be the case here because if the magnitude gap were purely statistical, it could not contain any information about the mass that would not already be provided by richness.
Now that the authors have shown that the magnitude gap carries information about the group in the simulations, they do a further test on the Sloan Digital Sky Survey DR7. In the real universe we no longer know the group’s true mass, so the authors choose the galaxy group’s velocity dispersion as a mass proxy. After correcting the sample to avoid biases, the authors show (Figure 2) that at a fixed richness (N) the groups with a large magnitude gap (red) have higher velocity dispersions therefore more mass than those with small magnitude gaps (purple). The authors don’t provide an explanation for this connection between magnitude gap and group mass, but the results agree with a recent paper I summarized on the nature of systems with large magnitude gaps.