These results are preliminary, in the sense that I did not have a complete sample of halos to work with, but above 10^14, the sample is complete. Incidentally this required much brute-forcing in the form of restarting hte analysis over 25 times, in order to get enough halos in place to put a talk together. I need help fixing the pipeline.
First, the relation between the velocity dispersion of the dark matter particles in the po simulation, and the FoF halo masses:
Red points are halos for which subhalo data exists, blue are the ones I haven't got done yet (they overlay the red and saturate). One very clear thing here, the friends of friends algorithm is linking together some objects that are actually two separate halos, and thereby causing a large scatter upward. As you can also see, this is artifacty. Mike thinks this is because I didn't let halopick know that the velocities were peculiar velocities (i.e. do not include the hubble flow*) so it was subtracting off a large number before computing the velocity dispersion. In principle of course this should not matter (mathematically) but probably does numerically. Still, it's a funny manifestation and I am skeptical of the diagnosis. This needs to be re-run and checked.
Next, I'd like to show the mass function and dark matter velocity dispersion function for these halos:
It's impressive to have so many tens of 1e15 solar mass halos in the mass function. I would like to check the mass function against the Battycharia fit, just as a sanity check on these new simulations.
Again here you can see the impact of the relatively long linking length of b=0.2 in the velocity dispersion, where there are several tens of objects at the high end, whereas in the mass function, there are only a few in top several bins. Also, I think the relatively high scatter in the mass-vd relation is causing the vd function to roll more slowly, rather than being a fairly obvious powerlaw with a roll off, as in the mass function. This might make it slightly less useful as a cosmological probe, but maybe not. This particular diagnostic is not an observable anyway.
Next I'd like to show my preliminary work on connecting the subhalo population with various HOD populating techniques. If we identify the subhalos with galaxies, this next plot shows the HOD, that is, the mean occupation of halos as a function of their mass:
Note that the subhalos used to make this plot are strictly the "leaf" subhalos, that is, subhalos with no further substructure of their own. The top plot just gives you an idea of how many halos are in each bin (e.g. you should probably ignore the top three bins that have fewer than 10 halos in them). The four colored power laws in the bottom panel are for different subhalo mass thresholds. We see that these indeed look like a power law. One thing on the to-do list might be to fit a power law slope to these simulation. You should be skeptical of the 10^7 halos because the smallest "subhalos" in this sample only have a few particles. The 10^10 is robust, but as you can see, we start running out out of substructure at this mass. This is mainly because at the 10^11 level, most subhalos will have their own substructure, and therefore do not appear in the "leaf" set. Another thing on the to-do list: Get occupation from the olapt files instead of the find files. Olapt halos are bound subhalos that can contain the leaf halos, so for example, we expect many more of the larger objects than in the leaf sample. I don't know if it's interesting, but it might be cute to see if the slope of the HOD depends strongly on whether we use leaves only, or olapt instead. We should think about what it is we learn about when we compare these. Also, I suspect that this will strongly depend on how well resolved the simulation is.
Next I'd like to talk about the statistics in a particular bin of the plot above. You'll note that I've placed error bars on the plot, these represent the 25th and 75th percentile of the ranked occupations in the bin. Of course the masses do not evenly sample the bin because of the steeply sloping mass function, so the distribution of occupations is fairly skew. This next plot shows the distribution for a few narrower non-overlapping bins in halo mass:
This is for a particular subhalo mass threshold, and the host halo mass bins used to make this plot are 1-2e14, 3-4e14, 5-6e14, 7-8e14, 9-10e14. This plot shows that all the mass bins have roughly the same width in occupation. The assumption that is commonly used in the generation of mock catalogs is that the distribution is poisson about the mean. It would be nice to explicitly check this. It could be, for example, that the validity of this assumption could depend on the subhalo threshold that is used, since we know that tidal disruption timescales depend both on the infall times and the infall masses. The next two plots show more explicitly the dependance of this distribution on both host mass, and subhalo cutoff.
These red and green plots show occupation histograms for two different host halo masses (noted in the plot titles for each panel). The four panels correspond to four different subhalo mass thresholds. I see no convincing evidence that the occupation distribution depends in a significant way on the subhalo threshold, but this needs to be checked quantitatively. To do this, I need to figure out how to convolve the the poisson distribution with the underlying mass function to get the predicted distribution of subhalo occupations for a given mass bin, if they are really sampling from Poisson. This is another item on the to-do list. It will be interesting to see if these results change when using olapt halos instead of leaf halos. Incidentally, you might be fooled into thinking that the fourth panel in the red shows some qualitative difference in the distribution, but I think this is a side effect fo the fact that the lower half of the mass bin may be too small to host 10^10 solar mass leaves. So I don't think that change of shape is significant.
*this has changed in the file format since Feb of this year
