Testing the models for the variance

The volume-averaged two-point correlation functions have been obtained from the mass distribution and from the several halo samples in the simulations. For that, the procedure described in section (2.2) has been followed.

The analysis has been performed for two different cases. In the first, the high-order moments are calculated at the same time when the dark haloes are identified. In the second, haloes are identified at some high redshift while the calculations of the high-order moments are performed for their descendants at a later time. In all cases, the redshift at which halo identification is made is denoted by z1, while the redshift at which the

high-order moments are calculated is denoted by z0.

Figures (2.1)–(2.3) show the variance from the VIRGO ΛCDM simulation, together with the predictions from the MW and its ellipsoidal collapse extension. Both the MW model and the SMT ellipsoidal collapse extension work remarkably good in all cases. In these two figures the prediction of the MW model with ξ2 given by the perturbation

theory (see Bernardeau 1994) is also plotted. The fact that this prediction also matches the simulation results suggests that the moments obtained from the VIRGO simulations are not affected significantly by the finite-volume effect and confirms that the MW model is a good approximation to the second-order moment of haloes that are not much smaller thanM? [defined by σ(M?) = 1.68]. Similarly, Figures (2.4)–(2.6) show the variance from the GIF ΛCDM simulation, together with the predictions from the MW and its ellipsoidal collapse extension.

With their high mass resolutions, GIF simulations allow one to test the theoretical models for haloes with mass M _¿ M?. Since the GIF simulations have relatively small simulation boxes, the moments are expected to be affected by the finite-volume effect (Colombi et al. 1994). Nevertheless the finite-volume effect on the variance is expected to be negligible. In any case, this effect in each simulation is expected to be similar for both the mass distribution and the halo distribution. Thus, to test the bias model given in equation (2.32) by a numerical simulation we should use the value of ξ₂ obtained directly from the simulation, because it is the simulated power spectrum (not the theoretical spectrum) that is responsible for the clustering in the simulation. Figure (2.7) shows the results obtained for the GIF simulations for haloes identified and analyzed at the present epoch. As one can see, there is good agreement between model predictions and simulation

results. For haloes with masses much smaller than M?, the SMT model gives a better fit than the MW model.

Figure 2.1: Variance ξ₂ of dark haloes with different mass ranges obtained from the counts- in-cells analysis (symbols), from applying the bias model from MW (solid line) and its SMT extension (dashed-line). The moments for the mass distribution are shown by dotted lines. Thin lines correspond to quantities obtained using the variance of the mass given by perturba- tion theory (Bernardeau 1994), whereas thick lines correspond to quantities obtained using the variance of the mass directly from the simulations. Results are shown for the VIRGO ΛCDM simulations. The haloes have been identified and analyzed as indicated in the plot. The value of M∗ is also written for more information. Each box corresponds to a different range of masses of haloes. The quantities in parenthesis correspond to the number of haloes in each sample.

2.4. TEST BY N-BODY SIMULATIONS

Figure 2.2: Variance ξ₂ of haloes in the VIRGO ΛCDM simulations for haloes identified at

z= 1 and analyzed at the present time. The lines and symbols, as well as the notation have the same meaning as in figure2.1.

Figure 2.3: Variance ξ₂ of haloes in the VIRGO ΛCDM simulations for haloes identified at

z= 3 and analyzed at the present time. Lines, symbols and notation have the same meaning as in figure 2.1.

2.4. TEST BY N-BODY SIMULATIONS

Figure 2.4: Varianceξ₂ of dark haloes with different mass ranges obtained from the counts-in- cells analysis (symbols) and from applying the bias model from MW (solid line) and its SMT extension (dashed-line). The variance of the mass obtained directly from the simulations is shown by the dotted line. Results are shown for the GIF ΛCDM model and for haloes identified and analyzed at the present time.

Figure 2.5: Variance ξ₂ of dark haloes with different mass ranges obtained from counts-in- cells analysis (symbols) and from applying the bias model from MW (solid line) and its SMT extension (dashed-line). The variance of the mass obtained directly from the simulations is shown by the dotted line. Results are shown for the GIF ΛCDM model and for haloes identified atz= 1 and analyzed at the present time.

2.4. TEST BY N-BODY SIMULATIONS

Figure 2.6: Variance ξ₂ of dark haloes with different mass ranges obtained from counts-in- cells analysis (symbols) and from applying the bias model from MW (solid line) and its SMT extension (dashed-line). The variance of the mass obtained directly from the simulations is shown by the dotted line. Results are shown for the ΛCDM model and for haloes identified at

Figure 2.7: Varianceξ₂ obtained from the counts-in-cells analysis (symbols) and from applying the bias model from MW (solid line) and its ellipsoidal collapse extension (dashed-line) of haloes less massive than M∗. Each row in the panel corresponds to a different range of halo masses, as indicated in the boxes.

2.4. TEST BY N-BODY SIMULATIONS