Galaxies in their simulations are identified as clumps of cold and dense gas particles which satisfy the Jeans condition and have the SPH density more than 100 times the mean baryon density at each redshift. Dark halos are identified with a standard friend-of-friend algorithm; the linking length is 0.164 times the mean separation of dark matter particles, for instance, at . In addition, they identify the surviving high-density substructures in dark halos, DM cores (see  for further details).
Figure 9 illustrates the distribution of dark matter particles, gas particles, dark halos, and galaxies at where galaxies are more strongly clustered than dark halos. Figure 10 depicts a close-up snapshot of the most massive cluster at with a mass . The circles in the lower panels indicate the positions of galaxies identified in our simulation.
Figure 11 shows the joint distribution of and with the mass density field at redshift , , and smoothed over . The conditional mean relation computed directly from the simulation is plotted in solid lines, while dashed lines indicate theoretical predictions of halo biasing by Taruya and Suto . For a given smoothing scale, the simulated halos exhibit positive biasing for relatively small in agreement with the predictions. On the other hand, they tend to be underpopulated for large , or anti-biased. This is mainly due to the exclusion effect of dark halos due to their finite volume size which is not taken into account in the theoretical model. Since our simulated galaxies have smaller spatial extent than the halos, the exclusion effect is not so serious. This is clearly illustrated in the lower panels in Figure 11, and indeed they show much better agreement with the theoretical model.
We turn next to a more conventional biasing parameter defined through the two-point statistics:
Figure 12 shows two-point correlation functions of dark matter, galaxies, dark halos, and DM cores (upper and middle panels), and the profiles of biasing parameters for those objects (lower panels) at , , and . In the lower panels, we also plot the parameter , which are defined in terms of the one-point statistics (variance), for comparison on smoothing scales , , and at for each kind of objects by different symbols. In the upper panels, we show the correlation functions of DM cores identified with two different maximum linking lengths, and . Correlation functions of DM cores identified with are similar to those of galaxies. On the other hand, those identified with exhibit much weaker correlation, and are rather similar to those of dark halos. This is due to the fact that the present algorithm of group identification with larger tends to pick up lower mass halos which are poorly resolved in our numerical resolution.
The correlation functions of galaxies are almost unchanged with redshift, and the correlation functions of dark halos only slightly evolve between and . By contrast, the amplitude of the dark matter correlation functions evolve rapidly by a factor of from to . The biasing parameter is larger at a higher redshift, for example, at . The biasing parameter for dark halos is systematically lower than that of galaxies and DM cores again due to the volume exclusion effect. At , galaxies and DM cores are slightly anti-biased relative to dark matter at . In lower panels, we also plot the one-point biasing parameter at for comparison. In general we find that is very close to at , but systematically lower than at higher redshifts.
For each galaxy identified at , we define its formation redshift by the epoch when half of its cooled gas particles satisfy our criteria of galaxy formation. Roughly speaking, corresponds to the median formation redshift of stars in the present-day galaxies. We divide all simulated galaxies at into two populations (the young population with and the old population with ) so as to approximate the observed number ratio of for late-type and early-type galaxies.
The difference of the clustering amplitude can be also quantified by their two-point correlation functions at as plotted in Figure 13. The old population indeed clusters more strongly than the mass, and the young population is anti-biased. The relative bias between the two populations ranges 1.5 and 2 for , where and are the two-point correlation functions of the young and old populations.
© Max Planck Society and the author(s)