Biasing of galaxies in cosmological hydrodynamic simulations

4.4 Biasing of galaxies in cosmological hydrodynamic simulations

Popular models of the biasing based on the peak or the dark halos are successful in capturing some essential features of biasing. None of the existing models of bias, however, seems to be sophisticated enough for the coming precision cosmology era. The development of a more detailed theoretical model of bias is needed. A straightforward next step is to resort to numerical simulations which take account of galaxy formation even if phenomenological at this point. We show an example of such approaches from Yoshikawa et al. [103

] who apply cosmological smoothed particle hydrodynamic (SPH) simulations in the LCDM model with particular attention to the comparison of the biasing of dark halos and simulated galaxies (see also [78 ]).

Figure 9: Distribution of gas particles (upper right panel), dark matter particles (upper left panel), galaxies (lower right panel), and dark halos (lower left panel) in the volume of a $3 75h −1 × 75h− 1 × 30h −1 Mpc$ model at $z = 0$ . (Figure taken from [103

].)

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 $z = 0$ . In addition, they identify the surviving high-density substructures in dark halos, DM cores (see [103 ] for further details).

Figure 9 illustrates the distribution of dark matter particles, gas particles, dark halos, and galaxies at $z = 0$ where galaxies are more strongly clustered than dark halos. Figure 10 depicts a close-up snapshot of the most massive cluster at $z = 0$ with a mass $M ≃ 8 × 1014M ⊙$ . The circles in the lower panels indicate the positions of galaxies identified in our simulation.

Figure 10: Snapshots of the most massive cluster ( $14 M ≃ 8 × 10 M ⊙$ ) in the simulation at $z = 0$ . Upper left panel: dark matter; upper right panel: gas; lower left panel: DM cores; lower right panel: cold gas. The circles in the lower panels indicate the positions of galaxies identified according to our criteria. The comoving size of the box is $6.25h −1 Mpc$ per side. (Figure taken from [103

].)

Figure 11 shows the joint distribution of $δh$ and $δg$ with the mass density field $δm$ at redshift $z = 0$ , $1$ , and $2$ smoothed over $Rs = 12h −1 Mpc$ . The conditional mean relation $¯δi(δm)$ computed directly from the simulation is plotted in solid lines, while dashed lines indicate theoretical predictions of halo biasing by Taruya and Suto [87 ]. For a given smoothing scale, the simulated halos exhibit positive biasing for relatively small $δm$ in agreement with the predictions. On the other hand, they tend to be underpopulated for large $δm$ , 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.

Figure 11: Joint probability distributions of overdensity fields for dark halos and galaxies with dark matter overdensity smoothed over $−1 Rs = 12h Mpc$ at redshift $z = 0$ , $1$ , and $2$ . Solid lines indicate the conditional mean $¯ δi(δm )$ for each object. Dashed lines in each panel depict the theoretical prediction of conditional mean by Taruya and Suto [87

]. (Figure taken from [103

].)

We turn next to a more conventional biasing parameter defined through the two-point statistics:

where $ξii(r)$ and $ξmm (r)$ are two-point correlation functions of objects $i$ and of dark matter, respectively. While the above biasing parameter is ill-defined where either $ξ (r) ii$ or $ξ (r) mm$ becomes negative, it is not the case at clustering scales of interest ( $− 1 < 10h Mpc$ ).

Figure 12: Two-point correlation functions of dark matter, galaxies, and dark halos from cosmological hydrodynamical simulations. (Figure taken from [103

].)

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 $bξ(r )$ for those objects (lower panels) at $z = 0$ , $1$ , and $2$ . In the lower panels, we also plot the parameter $bvar,i ≡ σi∕σm$ , which are defined in terms of the one-point statistics (variance), for comparison on smoothing scales $Rs = 4h −1 Mpc$ , $−1 Rs = 8h Mpc$ , and $− 1 Rs = 12h Mpc$ at $r = Rs$ 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, $lmax = 0.05$ and $lmax = bh∕2$ . Correlation functions of DM cores identified with $lmax = 0.05$ are similar to those of galaxies. On the other hand, those identified with $l = b ∕2 max h$ 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 $lmax$ 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 $z = 0$ and $2$ . By contrast, the amplitude of the dark matter correlation functions evolve rapidly by a factor of $∼ 10$ from $z = 2$ to $z = 0$ . The biasing parameter $bξ,g$ is larger at a higher redshift, for example, $bξ,g ≃ 2– 2.5$ at $z = 2$ . The biasing parameter $bξ,h$ for dark halos is systematically lower than that of galaxies and DM cores again due to the volume exclusion effect. At $z = 0$ , galaxies and DM cores are slightly anti-biased relative to dark matter at $r ≃ 1h−1 Mpc$ . In lower panels, we also plot the one-point biasing parameter $bvar,i ≡ σi∕σm$ at $r = Rs$ for comparison. In general we find that $bξ,i$ is very close to $bvar,i$ at $z ∼ 0$ , but systematically lower than $bvar,i$ at higher redshifts.

For each galaxy identified at $z = 0$ , we define its formation redshift $zf$ by the epoch when half of its cooled gas particles satisfy our criteria of galaxy formation. Roughly speaking, $z f$ corresponds to the median formation redshift of stars in the present-day galaxies. We divide all simulated galaxies at $z = 0$ into two populations (the young population with $zf < 1.7$ and the old population with $zf > 1.7$ ) so as to approximate the observed number ratio of $3∕1$ for late-type and early-type galaxies.

Figure 13: Two-point correlation functions for the old and young populations of galaxies at $z = 0$ as well as that of the dark matter distribution. The profiles of bias parameters $b (r) ξ$ for both of the two populations are also shown in the lower panel. (Figure taken from [103

].)

The difference of the clustering amplitude can be also quantified by their two-point correlation functions at $z = 0$ 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 $∘ ---------- brξel,g ≡ ξold∕ξyoung$ ranges 1.5 and 2 for $1h −1 Mpc < r < 20h −1 Mpc$ , where $ξyoung$ and $ξold$ are the two-point correlation functions of the young and old populations.